Cardoso, Domingos M., Martins, Enide Andrade, Robbiano, Maria, Trevisan, Vilmar
Elsevier
In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of a regular graph or a path. Center for Research and Development in Mathematics and Applications FCT FEDER/POCI 2010 Fondecyt-IC Project 11090211 CNPq—Grants 309531/2009-8 CNPq—Grants 473815/2010-9
Moreno, A. Foulquié, Martínez-Finkelshtein, A., Sousa, V. L.
Springer
We consider the orthogonal polynomials on [-1, 1] with respect to the weight w(c)(x) = h(x)(1 - x)(alpha) (1+ x)beta Xi(c)(x), alpha, beta > -1, where h is real analytic and strictly positive on [-1, 1] and Xi(c) is a step-like function: Xi(c)(x) = 1 for x is an element of [-1, 0) and Xi(c) (x) = c(2), c > 0, for x is an element of [0, 1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in C, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n -> infinity. In particular, we prove for w(c) a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at x = 0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest. Junta de Andalucía-Spain- FQM-229 and P06- FQM-01735. Ministry of Science and Innovation of Spain - MTM2008-06689-C02-01 FCT -SFRH/BD/29731/2006
Plakhov, A. Yu.
MAIK Nauka/Interperiodica
published
Tchemisova, T. V., Olga, Kostyukova
Taylor & Francis
In the paper,we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided.
Abreu, Nair M.M., Cardoso, Domingos Moreira, Martins, Enide A., Robbiano, Maria, San Martin, B.
Consider the Laplacian and signless Laplacian spectrum of a graph G of order n, with k pairwise co-neighbor vertices. We prove that the number of shared neighbors is a Laplacian and a signless Laplacian eigenvalue of G with multiplicity at least k− 1. Additionally, considering a connected graph Gk with a vertex set defined by the k pairwise co-neighbor vertices of G, the Laplacian spectrum of Gk, obtained from G adding the edges of Gk, includes l + β for each nonzero Laplacian eigenvalue β of Gk. The Laplacian spectrum of G overlaps the Laplacian spectrum of Gk in at least n − k + 1 places.
Gomes, Helena, Gutman, Ivan, Martins, Enide Andrade, Robbiano, María, San Martin, B.
Mathematical Chemistry Monographs
A new spectral graph invariant sprR , called Randíc spread, is defined and investigated. This quantity is equal to the maximal difference between two eigenvalues of the Randi´c matrix, disregarding the spectral radius. Lower and upper bounds for sprR are deduced, some of which depending on the Randíc index of the underlying graph.
Gomes, Helena, Martins, Enide, Robbiano, María, San Martín, Bernardo
The Randi´c spread of a simple undirected graph G, sprR(G), is equal to the maximal difference between two eigenvalues of the Randi´c matrix, disregarding the spectral radius [Gomes et al., MATCH Commun. Math. Comput. Chem. 72 (2014) 249–266]. Using a rank-one perturbation on the Randi´c matrix of G it is obtained a new matrix whose matricial spread coincide with sprR(G). By means of this result, upper bounds for sprR(G) are obtained.
Cardoso, Domingos M., Martins, Enide A., Robbiano, Maria, Rojo, Oscar
A generalized H-join operation of a family of graphs G1, . . . , Gp, where H has order p, constrained by a family of vertex subsets Si ⊆V(Gi), i = 1, . . . , p, is introduced. When each vertex subset Si is (ki, τi)-regular, it is deduced that all non-main adjacency eigenvalues of Gi , different from ki−τi , remain as eigenvalues of the graph G obtained by this operation. If each Gi is ki-regular and all the vertex subsets are such that Si = V(Gi), the H-generalized join constrained by these vertex subsets coincides with the H-join operation. Furthermore, some applications on the spread of graphs are presented.
Gutman, Ivan, Martins, Enide A., Robbiano, María, Martín, Bernardo San
Let G be a simple undirected graph of order n with vertex set V(G) ={v1, v2, ..., vn}. Let di be the degree of the vertex vi. The Randić matrix R=(r_{i,j}) of G is the square matrix of order n whose (i, j)-entry is equal to 1/ didj if the vertices vi and vj are adjacent, and zero otherwise. The Randić energy is the sum of the absolute values of the eigenvalues of R. Let X, Y, and Z be matrices, such that X +Y=Z. Ky Fan established an inequality between the sum of singular values of X, Y, and Z. We apply this inequality to obtain bounds on Randić energy. We also present results pertaining to the energy of a symmetric partitioned matrix, as well as an application to the coalescence of graphs.
Abreu, Nair, Costa, Liliana, Martins, Enide Andrade
Every acyclic Birkhoff polytope is represented by a bicolored tree. In this paper we use the concept of T-component of a tree in order to cover it. In addition, the definitions of T-edge cover(respectively, T-vertex cover)subgraphs and of complementary coverage by vertices (edges) are introduced. Some consequences related to the dimension of the acyclic Birkhoff polytope are also obtained.
Abreu, Nair, Costa, Liliana, Dahl, Geir, Martins, Enide
For a fixed tree T with n vertices the corresponding acyclic Birkhoff polytope Ωn(T)consists of doubly stochastic matrices having support in positions specified by T. This is a face of the Birkhoff polytope Ωn(which consists of all n ×n doubly stochastic matrices). The skeleton of Ωn(T) is the graph where vertices and edges correspond to those of Ωn(T), and we investigate some properties of this graph. In particular, we characterize adjacency of pairs of vertices, compute the minimum degree of a vertex and show some properties of the maximum degree of a vertex in the skeleton. We also determine the maximum degree for certain classes of trees, including paths, stars and caterpillars.
Castro, L. P., Rodrigues, M. M., Saitoh, S.
We introduce a new method for solving general initial value problems by using the theory of reproducing kernels. The results are depending on the specific structure of each problem. Here, we give the general principle of the method and illustrate it with simple prototype examples. On the basis of the process, we have certain integral transforms, which are generated by each specific initial value problem, and need to be analysed. In view of this, we shall establish the basic relations among initial value problems for linear operator equations, eigenvalues and eigenfunctions in the related operator equations, integral transforms and associated reproducing kernels. Within this process, we will realize a general theory for operator equations and incorporate a time dependence in view to consider an associated regularization method.
Rodrigues, M. M., Vieira, N.
The aim of this paper is to construct a testing function space equipped with the topology generated by the L_{v,p}-multinorm of the differential operator Bx = -4x^2 d^2/dx^2- 1 + x^2 -ux, where u < 1/2, v>0, p in [1, \infinity[, and its k-iterates B^k_x where k = 0, 1,..., and B^0_x \phi=\phi. We also introduce the correspondent dual space for the index Whittaker transform on distributions. The existence, uniqueness, imbedding and inversion properties are investigated.
Castro, L. P., Kapanadze, D., Pesetskaya, E.
Wiley
We consider a steady-state heat conduction problem in 2D unbounded doubly periodic composite materials with temperature independent conductivities of their components. Imperfect contact conditions are assumed on the boundaries between the matrix and inclusions. By introducing complex potentials, the corresponding boundary value problem for the Laplace equation is transformed into a special R-linear boundary value problem for doubly periodic analytic functions. The method of functional equations is used for obtaining a solution. Thus, the R-linear boundary value problem is transformed into a system of functional equations which is analysed afterwards. A new improved algorithm for solving this system is proposed. It allows to compute the average property and reconstruct the temperature and the flux at an arbitrary point of the composite. Computational examples are presented.
Catalano, Domenico, Breda d'Azevedo, António, Karábas, Ján, Nedela, Roman
In the present paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by Magot and Zvonkin (2000), Singerman and Syddall (2003), and Girondo (2003). We identify a set of 10 operations in the above infinite family which produce vertex-transitive dessins out of regular ones. This set is complete in the following sense: if a vertex-transitive map arises from a regular dessin H applying an operation, then it can be obtained from a regular dessin on the same surface (possibly different from H) applying one of the 10 operations. The statement includes the classical case when the underlying surface is the sphere.
Catalano, Domenico, Breda d'Azevedo, António, Duarte, Rui
Pseudo-orientable maps were introduced by Wilson in 1976 to describe non-orientable regular maps for which it is possible to assign an orientation to each vertex in such a way that adjacent vertices have opposite orientations. This property extends naturally to non- orientable and orientable hypermaps. In this paper we classify the regular pseudo-oriented maps and hypermaps of characteristic χ > −3. With the help of GAP (The GAP group, 2014) and its library of small groups, we extend the classification down to characteristic χ = −16 (Tables 7–19 in the Appendix).
Almeida, Alexandre, Caetano, António
In this article we study atomic and molecular decompositions in 2-microlocal Besov and Triebel–Lizorkin spaces with variable integrability. We show that, in most cases, the convergence implied in such decompositions holds not only in the distributions sense, but also in the function spaces themselves. As an application, we give a simple proof for the denseness of the Schwartz class in such spaces. Some other properties, like Sobolev embeddings, are also obtained via atomic representations.
Azevedo, António Breda d', Fernandes, Maria Elisa
In this paper we classify the reflexible and chiral regular oriented maps with faces of valency, and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular oriented maps with faces. The limit depends on and for certain primes we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans (2014), has produced a nill asymptotic ratio.
Cameron, Peter J., Fernandes, Maria Elisa, Leemans, Dimitri, Mixer, Mark
If $\Gamma$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sn (other than Sn and the alternating group An), then the rank of is at most n/2+1, with nitely many exceptions (which are classi ed). It is conjectured that only the symmetric group has to be excluded.
Macedo, Pedro, Scotto, Manuel, Silva, Elvira
It is well-known that under fairly conditions linear regression becomes a powerful statistical tool. In practice, however, some of these conditions are usually not satisfied and regression models become ill-posed, implying that the application of traditional estimation methods may lead to non-unique or highly unstable solutions. Addressing this issue, in this paper a new class of maximum entropy estimators suitable for dealing with ill-posed models, namely for the estimation of regression models with small samples sizes affected by collinearity and outliers, is introduced. The performance of the new estimators is illustrated through several simulation studies.
We present an analytic and numerical analysis of several properties of a composite material with stiff imperfect interface conditions. Spaces of functions are identified where we can guarantee existence and uniqueness of solutions. In particular, formulas for the temperature distribution and flux are exhibited. Numerical calculations of the material characteristics such as temperature, flux and the effective conductivity are also performed and interpreted.
Gouveia, Sónia, Scotto, Manuel G., Pinna, G. D., Maestri, R., La Rovere, M. T., Ferreira, Paulo J.
Portland Press
Baroreflex sensitivity (BRS) is an important prognostic factor as a reduced BRS has been associated with an adverse cardiovascular outcome. The threshold for “reduced” BRS was established by the ATRAMI study at BRS <3 ms/mmHg in patients with a previous myocardial infarction and has shown to improve risk assessment in many other cardiac dysfunctions. The successful application of this cutoff to other populations suggests that it may reflect an inherent property of baroreflex functioning. Hence, our goal is to investigate whether it represents a “natural” partition of BRS values. Since reduced baroreflex responsiveness is also associated with aging, we also investigate whether a BRS estimate below 3 ms/mmHg can be the result of a process of physiologic senescence besides a sign of BRS dysfunction. This study involved 228 chronic heart failure (CHF) patients and 60 age-matched controls. Our novel method combines transfer function BRS estimation and automatic clustering of BRS probability distributions to define indicative levels of different BRS activities. The analysis produced a fit clustering (cophenetic coefficient 0.9 out of 1) and identified one group of homogeneous patients (well separated from the remaining by 3 ms/mmHg) with increased BRS based mortality risk (HR: 3.19 [1.73,5.89], p<0.001). The age dependent BRS cutoff, estimated by 5% quantile regression of log(BRS) with age (considering the age-matched controls), provides a similar mortality value (HR: 2.44 [1.37,4.43], p=0.003). In conclusion, the 3 ms/mmHg cutoff identifies two large clusters of homogeneous HF patients, thus supporting the hypothesis of being a natural cutoff in the HF population. Furthermore, age was found to have no statistical impact on risk assessment, thus suggesting that there is no need to establish age-based cutoffs as 3 ms/mmHg optimally identifies patients at high mortality risk.
Andrade, Enide, Cardoso, Domingos, Robbiano, Maria, Rodriguez, Jonnathan
The spread of an $n\times n$ complex matrix $B$ with eigenvalues $\beta _{1},\beta _{2},\ldots ,\beta _{n}$ is defined by \begin{equation*} s\left( B\right) =\max_{i,j}\left\vert \beta _{i}-\beta _{j}\right\vert , \end{equation*}% where the maximum is taken over all pairs of eigenvalues of $B$. Let $G$ be a graph on $n$ vertices. The concept of Laplacian spread of $G$ is defined by the difference between the largest and the second smallest Laplacian eigenvalue of $G$. In this work, by combining old techniques of interlacing eigenvalues and rank $1$ perturbation matrices new lower bounds on the Laplacian spread of graphs are deduced, some of them involving invariant parameters of graphs, as it is the case of the bandwidth, independence number and vertex connectivity.
Andrade, Enide, Gomes, Helena, Robbiano, Maria, Rodrigues, Jonnathan
The Laplacian spread of a graph $G$ is defined as the difference between the largest and the second smallest eigenvalue of the Laplacian matrix of $G$. In this work, an upper bound for this graph invariant, that depends on first Zagreb index, is given. Moreover, another upper bound is obtained and expressed as a function of the nonzero coefficients of the Laplacian characteristic polynomial of a graph.
Andelic, M., Andrade, E., Cardoso, D. M., Fonseca, C. M. da, Simic, S. K., Tosic, D. V.
In the set of all connected graphs with fixed order and size, the graphs with maximal index are nested split graphs, also called threshold graphs. It was recently (and independently) observed in [F.K.Bell, D. Cvetkovi´c, P. Rowlinson, S.K. Simi´c, Graphs for which the largest eigenvalue is minimal, II, Linear Algebra Appl. 429 (2008)] and [A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalue of bipartite graphs, Electron. J. Combin. 15 (2008), #144] that double nested graphs, also called bipartite chain graphs, play the same role within class of bipartite graphs. In this paper we study some structural and spectral features of double nested graphs. In studying the spectrum of double nested graphs we rather consider some weighted nonnegative matrices (of significantly less order) which preserve all positive eigenvalues of former ones. Moreover, their inverse matrices appear to be tridiagonal. Using this fact we provide several new bounds on the index (largest eigenvalue) of double nested graphs, and also deduce some bounds on eigenvector components for the index. We conclude the paper by examining the questions related to main versus non-main eigenvalues.
Correia, Alexandre C. M.
EDP Sciences
Cassini states correspond to equilibria of the spin axis of a body when its orbit is perturbed. They were initially described for satellites, but the spin axis of stars and planets undergoing strong dissipation can also evolve into some equilibria. For small satellites, the rotational angular momentum is usually much smaller than the total angular momentum, so classical methods for finding Cassini states rely on this approximation. Here we present a more general approach, which is valid for the secular quadrupolar non-restricted problem with spin. Our method is still valid when the precession rate and the mutual inclination of the orbits are not constant. Therefore, it can be used to study stars with close-in companions, or planets with heavy satellites, like the Earth-Moon system.
Delisle, J. -B., Correia, A. C. M., Laskar, J.
We study the stability of mean-motion resonances (MMR) between two planets during their migration in a protoplanetary disk. We use an analytical model of resonances and describe the effect of the disk by a migration timescale (T-m,T-i) and an eccentricity damping timescale (T-e,T-i) for each planet (i = 1; 2 for the inner and outer planets, respectively). We show that the resonant configuration is stable if T-e,T-1/T-e,T-2 > (e(1)/e(2))(2). This general result can be used to put constraints on specific models of disk-planet interactions. For instance, using classical prescriptions for type-I migration, we show that when the angular momentum deficit (AMD) of the inner orbit is greater than the outer's orbit AMD, resonant systems must have a locally inverted disk density profile to stay locked in resonance during the migration. This inversion is very atypical of type-I migration and our criterion can thus provide an evidence against classical type-I migration. That is indeed the case for the Jupiter-mass resonant systems HD 60532b, c (3: 1 MMR), GJ 876b, c (2: 1 MMR), and HD 45364b, c (3: 2 MMR). This result may be evidence of type-II migration (gap-opening planets), which is compatible with the high masses of these planets.
Blaya, Ricardo Abreu, Reyes, Juan Bory, Adán, Alí Guzmán, Kähler, Uwe
In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalize most of those previous results in this direction and additionally other consequent results are presented. In particular, the expression of the jump of the generalized Π-operator across the boundary of the domain is obtained as well as an estimate for the norm of the Π-operator is given. At the end an application of the generalized Π-operator to the solution of Beltrami equations is studied where we give conditions for a solution to realize a local and global homeomorphism.
Cerejeiras, Paula, Kähler, Uwe, Ku, Min
Taylor and Francis
We study discrete Hilbert boundary value problems in the case of the upper half lattice. The solutions are given in terms of the discrete Cauchy transforms for the upper and lower half space while the study of their solvability is based on the discrete Hardy decomposition for the half lattice. Furthermore, the solutions are proved to converge to those of the associated continuous Hilbert boundary value problems.
Krausshar, R. S., Rodrigues, M. M., Vieira, N.
In this paper we present an explicit construction for the fundamental solution of the heat operator, the Schrödinger operator and related first order parabolic Dirac operators on a class of some conformally flat non-orientable orbifolds. More concretely, we treat a class of projective cylinders and tori where we can study parabolic monogenic sections with values in different pin bundles. We present integral representation formulas together with some elementary tools of harmonic analysis that enable us to solve boundary value problems on these orbifolds.
Berti, Emanuele, Barausse, Enrico, Cardoso, Vitor, Gualtieri, Leonardo, Pani, Paolo, Sperhake, Ulrich, Stein, Leo C., Wex, Norbert, Yagi, Kent, Baker, Tessa, Burgess, C. P., Coelho, Flávio S., Doneva, Daniela, De Felice, Antonio, Ferreira, Pedro G., Freire, Paulo C. C., Healy, James, Herdeiro, Carlos, Horbatsch, Michael, Kleihaus, Burkhard, Klein, Antoine, Kokkotas, Kostas, Kunz, Jutta, Laguna, Pablo, Lang, Ryan N., Li, Tjonnie G. F., Littenberg, Tyson, Matas, Andrew, Mirshekari, Saeed, Okawa, Hirotada, Radu, Eugen, O'Shaughnessy, Richard, Sathyaprakash, Bangalore S., Van Den Broeck, Chris, Winther, Hans A., Witek, Helvi, Aghili, Mir Emad, Alsing, Justin, Bolen, Brett, Bombelli, Luca, Caudill, Sarah, Chen, Liang, Degollado, Juan Carlos, Fujita, Ryuichi, Gao, Caixia, Gerosa, Davide, Kamali, Saeed, Silva, Hector O., Rosa, João G., Sadeghian, Laleh, Sampaio, Marco, Sotani, Hajime, Zilhão, Miguel
IOP Publishing
One century after its formulation, Einstein's general relativity (GR) has made remarkable predictions and turned out to be compatible with all experimental tests. Most of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that GR should be modified when gravitational fields are strong and spacetime curvature is large. The best astrophysical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of GR. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einstein's theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime.
Debbouche, Amar, Torres, Delfim F. M.
Springer Verlag
We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.
Taverna, Giorgio S., Torres, Delfim F. M.
In this note, we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems, and the corresponding Euler-Lagrange and Hamilton equations are analyzed. The dependence of the equation of motion on the generalized kernel permits to obtain a wide range of different configurations of motion. Some examples are discussed and analyzed.Copyright (c) 2014 John Wiley & Sons, Ltd.
He, Fuli, Ku, Min, Kähler, Uwe
By the characterization of the matrix Hilbert transform in the Hermitian Clifford analysis, we introduce the matrix Szegö projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded sub-domain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. Furthermore, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a boundary value problem for matrix functions.
Micheli, G., Rosenthal, J., Vettori, P.
Necessary and sufficient conditions are given on matrices $A$, $B$ and $S$, having entries in some field $F$ and suitable dimensions, such that the linear span of the terms $A^iSB^j$ over $F$ is equal to the whole matrix space. This result is then used to determine the cardinality of subsets of $F[A]SF[B]$ when $F$ is a finite field.
Khosravian-Arab, Hassan, Almeida, Ricardo
We exhibit a numerical method to solve fractional variational problems, applying a decomposition formula based on Jacobi polynomials. Formulas for the fractional derivative and fractional integral of the Jacobi polynomials are proven. By some examples, we show the convergence of such procedure, comparing the exact solution with numerical approximations.
Almeida, Ricardo, Martins, Natália
We develop the new variational calculus introduced in 2011 by J. Cresson and I. Greff, where the classical derivative is substituted by a new complex operator called the scale derivative. In this paper we consider several nondifferentiable variational problems with free terminal point: with and without constraints, of first and higher-order type.
Tavares, Dina, Almeida, Ricardo, Torres, Delfim F. M.
We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integerorder) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.
Duarte, Rui, Guedes de Oliveira, António
In this article we study a construction, due to Pak and Stanley, with which every region RR of the Shi arrangement is (bijectively) labelled with a parking function λ(R). In particular, we construct an algorithm that returns R out of λ(R). This is done by relating λ to another bijection, that labels every region S of the braid arrangement with r(S), the unique central parking function f such that λ−1(f)⊆S. We also prove that λ maps the bounded regions of the Shi arrangement bijectively onto the prime parking functions. Finally, we introduce a variant (that we call “s-parking”) of the parking algorithm that is in the very origin of the term “parking function”. S-parking may be efficiently used in the context of our new algorithm, but we show that in some (well defined) cases it may even replace it.
KrauBhar, R. S., Rodrigues, M. M., Vieira, N.
Springer International Publishing
In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger-Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger-Günter problem on a class of conformally flat cylinders and tori.
Branquinho, A., Moreno, A. Foulquié, Paiva, A., Rebocho, M. N.
Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.
Branquinho, A., Moreno, A. Foulquié, Godoy, E., Area, I.
IOP Publising
The correspondence between dynamics of Delta-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. A method to solve inverse problem - integration of Delta-Toda equations - based on Padé approximates and continued fractions for the resolvent function is proposed. The main ingredient are orthogonal polynomials which satisfy an Appell condition, with respect to the forward difference operator Delta. Two examples related with Jacobi and Laguerre orthogonal polynomials and Delta-Toda equations are given.
Area, I., Branquinho, A., Moreno, A. Foulquié, Godoy, E.
In this paper we introduce the Δ-Volterra lattice which is interpreted in terms of symmetric orthogonal polynomials. It is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like (1+x2)1−tμ(x)(1+x2)1−tμ(x) where μ is a given positive Borel measure. Moreover, the Δ-Volterra lattice is related to the Δ-Toda lattice from Miura or Bäcklund transformations. The main ingredients are orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ and the characterization of the point spectrum of a Jacobian operator that satisfies a Δ-Volterra equation (Lax type theorem). We also provide an explicit example of solutions of Δ-Volterra and Δ-Toda lattices, and connect this example with the results presented in the paper.
Kostyukova, O. I., Tchemisova, T. V.
We consider a convex problem of Semi-Infinite Programming (SIP) with multidimensional index set. In study of this problem we apply the approach suggested in [20] for convex SIP problems with one-dimensional index sets and based on the notions of immobile indices and their immobility orders. For the problem under consideration we formulate optimality conditions that are explicit and have the form of criterion. We compare this criterion with other known optimality conditions for SIP and show its efficiency in the convex case.
Kostyukova, O. I., Tchemisova, T. V., Yermalinskaya, S. A.
We state a new implicit optimality criterion for convex semi-infinite programming (SIP) problems. This criterion does not require any constraint qualification and is based on concepts of immobile index and immobility order. Given a convex SIP problem with a continuum of constraints, we use an information about its immobile indices to construct a nonlinear programming (NLP) problem of a special form. We prove that a feasible point of the original infinite SIP problem is optimal if and only if it is optimal in the corresponding finite NLP problem. This fact allows us to obtain new efficient optimality conditions for convex SIP problems using known results of the optimality theory of NLP. To construct the NLP problem, we use the DIO algorithm. A comparison of the optimality conditions obtained in the paper with known results is provided.
We consider a convex semi-infinite programming (SIP) problem whose objective and constraint functions are convex w.r.t. a finite-dimensional variable x and whose constraint function also depends on a so-called index variable that ranges over a compact set inR. In our previous paper [O.I.Kostyukova,T.V. Tchemisova, and S.A.Yermalinskaya, On the algorithm of determination of immobile indices for convex SIP problems, IJAMAS Int. J. Math. Stat. 13(J08) (2008), pp. 13–33], we have proved an implicit optimality criterion that is based on concepts of immobile index and immobility order. This criterion permitted us to replace the optimality conditions for a feasible solution x0 in the convex SIP problem by similar conditions for x0 in certain finite nonlinear programming problems under the assumption that the active index set is finite in the original semi-infinite problem. In the present paper, we generalize the implicit optimality criterion for the case of an infinite active index set and obtain newfirst- and second-order sufficient optimality conditions for convex semi-infinite problems. The comparison with some other known optimality conditions is provided.
Tchemisova, T. V.
Optimality conditions for nonlinear problems with equality and inequality constraints are considered. In the case when no constraint qualification (or regularity) is assumed, the Lagrange multiplier corresponding to the objective function can vanish in first order necessary optimality conditions given by Fritz John and the corresponding extremum is called abnormal. In the paper we consider second order sufficient optimality conditions that guarantee the rigidity of abnormal extrema (i.e. their isolatedness in the admissible sets).
Plakhov, A. Yu., Tchemisova, T. V.
Pressure force exerted on a rough disk spinning in a flow of noninteracting particles is determined by considering that a flow of point particles impinges on a body spinning around a fixed point. The rough disk is identical with the sequence of sets and thus the sets can be viewed as successive approximations of the rough disk. A proper choice of sequence of sets shows that the characteristic of billiard scattering is independent of n, and the billiard scattering on the rough set is defined. The pressure force exerted on the disk is independent of its angular velocity and that the characteristic of the interaction that is the moment of the pressure force slows down the rotation of the rough disk. The transverse force aligned with the instantaneous velocity of the front point of the body results in Magnus effect.
Martins, I., Guedes, T., Rama, P., Ramos, J., Tchemisova, T.
Inderscience
A food bank is a non-pro t, social solidarity organization that typically distributes the donated food among a wide variety of local non-pro t, social solidarity institutions which in turn feed the lowincome people. The problem presented by the Portuguese Federation of Food Banks is to determine, for a speci c food bank, the quantities of the donated products that must be assigned to each local social solidarity institution in order to satisfy the needs of the supported people as much as possible, without favouring any institution. We propose a linear programming model followed by a rounding heuristic to obtain a solution to the problem described. Computational results are reported.
We consider two closely related optimization problems: a problem of convex Semi- Infinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions.
Baleanu, D., Bhrawy, A. H., Torres, D. F. M., Salahshour, S.
Hindawi Publishing Corporation
Resumo indisponível.
Andelic, M., Cardoso, Domingos M.
An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted.
Sciriha, I, Cardoso, Domingos M.
Charles Babbage Research Centre
A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.
Nina, H., Soto, R. L., Cardoso, Domingos M.
Recently, Cardon and Tuckfield (2011) [1] have described the Jordan canonical form for a class of zero-one matrices, in terms of its associated directed graph. In this paper, we generalize this result to describe the Jordan canonical form of a weighted adjacency matrix A in terms of its weighted directed graph.
Cardoso, Domingos M., Freitas, M. A. A. de, Martins, E. A., Robbiano, M.
Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.
Andelic, M., Cardoso, Domingos M., Simic, S. K .
Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.
Barbedo, I., Cardoso, Domingos M., Cvetkovic, D., Rama, P., Simic, S. K.
European Mathematical Society Publishing House
In spectral graph theory a graph with least eigenvalue 2 is exceptional if it is connected, has least eigenvalue greater than or equal to 2, and it is not a generalized line graph. A ðk; tÞ-regular set S of a graph is a vertex subset, inducing a k-regular subgraph such that every vertex not in S has t neighbors in S. We present a recursive construction of all regular exceptional graphs as successive extensions by regular sets.
Andrade, E., Cardoso, Domingos M., Pastén, G., Rojo, O.
Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m_L±(G)(1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that mL±(G)(1) is bounded below by p(G)−q(G). Let r(G) be the number of internal vertices of G. If r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G). If r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs.
Hartmann, Stefan
Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case.
Costa, Marco, Monteiro, Magda
Springer Berlin Heidelberg
The surface water quality monitoring is an important concern of public organizations due to its relevance to the public health. Statistical methods are taken as consistent and essential tools in the monitoring procedures in order to prevent and identify environmental problems. This work presents the study case of the hydrological basin of the river Vouga, in Portugal. The main goal is discriminate the water monitoring sites using the monthly dissolved oxygen concentration dataset between January 2002 and May 2013. This is achieved through the extraction of trend and seasonal components in a linear mixed-effect state space model. The parameters estimation is performed with both maximum likelihood method and distribution-free estimators in a two-step procedure. The application of the Kalman smoother algorithm allows to obtain predictions of the structural components as trend and seasonality. The water monitoring sites are discriminated through the structural components by a hierarchical agglomerative clustering procedure. This procedure identified different homogenous groups relatively to the trend and seasonality components and some characteristics of the hydrological basin are presented in order to support the results.
Papageorgiou, N. S., Santos, S. R. Andrade, Staicu, V.
We consider a nonlinear Dirichlet problem with a Carathéodory reaction which has arbitrary growth from below. We show that the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal. In the semilinear case (i.e., p =2), with the reaction f(z, .)being C1and with subcritical growth, we show that there is a second nodal solution, for a total of four nontrivial smooth solutions. Finally,when the reaction has concave terms and is subcritical and for the nonlinear problem (i.e., 1
ria.ua.pt
Aizicovici, S., Papageorgiou, N. S., Staicu, V.
Texas State University, Department of Mathematics
We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator plus an indefinite potential and a reaction having the competing effects of concave and convex terms. For the superlinear (concave) term we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem describing the set of positive solutions as the parameter varies.
American Mathematical Society
In this paper, we study a nonlinear elliptic equation driven by the sum of a p-Laplacian and a Laplacian ((p, 2)-equation), with a Carathéodory (p − 1)-(sub-)linear reaction. Using variational methods combined with Morse theory, we prove two multiplicity theorems providing precise sign information for all the solutions (constant sign and nodal solutions). In the process, we prove two auxiliary results of independent interest.
World Scientific
We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behaviour of the set of positive solutions as the parameter varies.
Mota, Maria João, Cardoso, Mirtha, Carvalho, Andreia, Marques, Alda, Sá-Couto, Pedro, Demain, Sara
IOS press
OBJECTIVES: This study investigated the self-reported prevalence and impact of low back pain (LBP) during pregnancy in primiparous and multiparous women, and their treatment-seeking rationales and experiences, including their use of physiotherapy. METHODS: A sample of 105 post-partum women was recruited. All participants answered a questionnaire; women who experienced LBP during pregnancy (n=71) continue in the study and later they were also interviewed. Content analysis, descriptive and inferential statistics were used to analyse the data. RESULTS: Reports of LBP were common (n=71; 67.6%) and slightly more frequent in primiparous (n=40; 56.3%) than multiparous (n=31; 43.7%) women. Multiparous women with LBP were significantly older (p< 0.001) and reported more sleep disturbances (p=0.026) than primiparous women with LBP. LBP prevented women performing their daily activities (n=41; 57.7%) and worsened with the advance of pregnancy (n=55; 77.5%), yet 93.0% (n=66) of these women received no treatment. CONCLUSION: LBP is a prevalent and important clinical condition affecting the daily life of many pregnant women. Nevertheless, few women seek any treatment and physiotherapy is rarely considered. Given the significant impact on quality of life, health professionals need to be proactive in asking women about LBP.
International Press
We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is ”concave” (i.e., (p − 1)− sublinear) near zero and ”convex” (i.e., (p − 1)− superlinear) near ±1. Using variational methods combined with truncation and comparison techniques, we show that for all small values of the parameter > 0, the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal). In the Hilbert space case (p = 2), using Morse theory, we produce a sixth nontrivial smooth solution but we do not determine its sign.
Rami, Mustapha Ait, Napp, Diego
IEEE
The aim of this paper is to provide an efficient control design technique for discrete-time positive periodic systems. In particular, stability, positivity and periodic invariance of such systems are studied. Moreover, the concept of periodic invariance with respect to a collection of boxes is introduced and investigated with connection to stability. It is shown how such concept can be used for deriving a stabilizing state-feedback control that maintains the positivity of the closed-loop system and respects states and control signals constraints. In addition, all the proposed results can be efficiently solved in terms of linear programming.
Tavares, João Paulo de Almeida, Silva, Alcione Leite da, Sá-Couto, Pedro, Boltz, Marie, Capezuti, Elizabeth
Portugal is impacted by the rapid growth of the aging population, which has significant implications for its health care system. However, nurses have received little education focusing on the unique and complex care needs of older adults. This gap in the nurses′ education has an enormous impact in their knowledge and attitudes and affects the quality of nursing care provided to older adults. A cross-sectional study was conducted among 1068 Portuguese nurses in five hospitals (northern and central region) with the following purposes: (i) explore the knowledge and attitudes of nurses about four common geriatric syndromes (pressure ulcer, incontinence, restraint use and sleep disturbance) in Portuguese hospitals; and (ii) evaluate the influence of demographic, professional and nurses' perception about hospital educational support, geriatric knowledge, and burden of caring for older adults upon geriatric nursing knowledge and attitudes. The mean knowledge and attitudes scores were 0.41 ± 0.15 and 0.40 ± 0.21, respectively (the maximum score was 1). Knowledge of nurses in Portuguese hospitals about the four geriatric syndromes (pressure ulcers, sleep disturbance, urinary incontinence and restraint use) was found inadequate. The nurses' attitudes towards caring for hospitalized older adults were generally negative. Nurses who work in academic hospitals demonstrated significantly more knowledge than nurses in hospital centers. The attitudes of nurses were significantly associated with the hospital and unit type, region, hospital educational support, staff knowledge, and perceived burden of caring for older adults. The study findings support the need for improving nurses' knowledge and attitudes towards hospitalized older adults and implementing evidence-based guidelines in their practice.
Ferreira, Brigida Costa, Marques, Rui Vale, Khouri, Leila, Santos, Tânia, Sá-Couto, Pedro, Lopes, Maria do Carmo
BioMed Central
Purpose: To evaluate the differences between three methods of classification of recurrences in patients with head and neck tumours treated with Radiation Therapy (RT). Materials and methods: 367 patients with head and neck tumours were included in the study. Tumour recurrences were delineated in the CT images taken during patient follow-up and deformable registration was used to transfer this volume into the planning CT. The methods used to classify recurrences were: methodCTV quantified the intersection volume between the recurrence and the Clinical Target Volume (CTV); methodTV quantified the intersection between the Treated Volume and the recurrence (for method CTV and TV, recurrences were classified in-field if more than 95% of their volume were inside the volume of interest, marginal if the intersection was between 20-95% and outfield otherwise); and methodCOM was based on the position of the Centre Of Mass of the recurrence. A dose assessment in the recurrence volume was also made. Results: The 2-year Kaplan-Meier locoregional recurrence incidence was 10%. Tumour recurrences occurred in 22 patients in a mean time of 16.5 ± 9.4 months resulting in 28 recurrence volumes. The percentage of in-field recurrences for methods CTV, TV and COM was 7%, 43% and 50%, respectively. Agreement between the three methods in characterizing individually in-field and marginal recurrences was found only in six cases. Methods CTV and COM agreed in 14. The percentage of outfield recurrences was 29% using all methods. For local recurrences (in-field or marginal to gross disease) the average difference between the prescribed dose and D 98% in the recurrence volume was -5.2 ± 3.5% (range: -10.1%-0.9%). Conclusions: The classification of in-field and marginal recurrences is very dependent on the method used to characterize recurrences. Using methods TV and COM the largest percentage of tumour recurrences occurred in-field in tissues irradiated with high doses. Keywords: Head and neck tumours, Radiation therapy, Characterization of tumour recurrences, Geometric methods, Dosimetric assessment
Loureiro, Cláudia Chaves, Sá-Couto, Pedro, Todo-Bom, Ana, Bousquet, Jean
Background: Unbiased cluster analysis using clinical parameters has identified asthma pheno- types. Adding inflammatory biomarkers to this analysis provided a better insight into the disease mechanisms. This approach has not yet been applied to asthmatic Portuguese patients. Aim: To identify phenotypes of asthma using cluster analysis in a Portuguese asthmatic popu- lation treated in secondary medical care. Methods: Consecutive patients with asthma were recruited from the outpatient clinic. Patients were optimally treated according to GINA guidelines and enrolled in the study. Procedures were performed according to a standard evaluation of asthma. Phenotypes were identified by cluster analysis using Ward’s clustering method. Results: Of the 72 patients enrolled, 57 had full data and were included for cluster analysis. Distribution was set in 5 clusters described as follows: cluster (C) 1, early onset mild aller- gic asthma; C2, moderate allergic asthma, with long evolution, female prevalence and mixed inflammation; C3, allergic brittle asthma in young females with early disease onset and no evidence of inflammation; C4, severe asthma in obese females with late disease onset, highly symptomatic despite low Th2 inflammation; C5, severe asthma with chronic airflow obstruction, late disease onset and eosinophilic inflammation. Conclusions: In our study population, the identified clusters were mainly coincident with other larger-scale cluster analysis. Variables such as age at disease onset, obesity, lung function, FeNO (Th2 biomarker) and disease severity were important for cluster distinction.
Alvarelhão, Joaquim, Queirós, Alexandra, Sá-Couto, Pedro, Rocha, Nelson Pacheco
The linking process of information to ICF is a common task in different strategies used in rehabilitation practise but is a time consuming process mainly due to reliability issues. This work aims to developed additional rules to those already published in order to improve reliability of the linking process to ICF. The results are encouraging and these work could help to develop in formation technologies tools for facilitate this process.
Silva, Anabela Gonçalves, Queirós, Alexandra, Sá-Couto, Pedro, Rocha, Nelson Pacheco
American Physical Therapy Association
Background. Measurement of function usually involves the use of both performance-based and self-report instruments. However, the relationship between both types of measures is not yet completely understood, in particular for older adults attending primary care. Objective. The main objective of the study was to investigate the association between the World Health Organization Disability Assessment Schedule 2.0 (WHODAS 2.0) and the Short Physical Performance Battery (SPPB) for older adults at primary care. A secondary objective was to determine the influence of sociodemographic and health-related variables on this relationship. Design. This was a cross-sectional study. Methods. A total of 504 participants aged 60 years and older from 18 different primary care centers underwent a one-session assessment including: sociodemographic variables, comorbidities, performance, self-reported disability, pain, depressive symptoms, and physical activity. Performance was assessed using the SPPB, and self-reported disability was assessed using the WHODAS 2.0. Results. The correlation between WHODAS 2.0 and SPPB scores was strong (r=.65). Regression analysis showed that the SPPB total score explained 41.7% of the variance in WHODAS 2.0 scores (adjusted R2=41.6%). A second model including the SPPB subtests (balance, gait, and sit-to-stand), depressive symptoms, number of pain sites, pain intensity, and level of physical activity explained 61.7% of the variance in WHODAS 2.0 scores (adjusted R2=60.4%). No model improvement was found when considering the 6 WHODAS 2.0 individual domains. Limitations. The cross-sectional nature of the study does not allow inferences on causal relationships. Conclusions. This study's findings confirm that self-report and performance-based measures relate to different aspects of functioning. Further study is needed to determine if primary care interventions targeting lower extremity performance and depressive symptoms improve self-reported disability.
Madeira, Alexandre, Martins, Manuel A., Barbosa, Luis S., Hennicker, Rolf
Hybrid logics, which add to the modal description of transition structures the ability to refer to specific states, offer a generic framework to approach the specification and design of reconfigurable systems, i.e., systems with reconfiguration mechanisms governing the dynamic evolution of their execution configurations in response to both external stimuli or internal performance measures. A formal representation of such systems is through transition structures whose states correspond to the different configurations they may adopt. Therefore, each node is endowed with, for example, an algebra, or a first-order structure, to precisely characterise the semantics of the services provided in the corresponding configuration. This paper characterises equivalence and refinement for these sorts of models in a way which is independent of (or parametric on) whatever logic (propositional, equational, fuzzy, etc) is found appropriate to describe the local configurations. A Hennessy–Milner like theorem is proved for hybridised logics.
Bolmont, Emeline, Raymond, Sean N., Leconte, Jeremy, Hersant, Franck, Correia, Alexandre C. M.
A large proportion of observed planetary systems contain several planets in a compact orbital configuration, and often harbor at least one close-in object. These systems are then most likely tidally evolving. We investigate how the effects of planet-planet interactions influence the tidal evolution of planets. We introduce for that purpose a new open-source addition to the Mercury N-body code, Mercury-T, which takes into account tides, general relativity and the effect of rotation-induced flattening in order to simulate the dynamical and tidal evolution of multi-planet systems. It uses a standard equilibrium tidal model, the constant time lag model. Besides, the evolution of the radius of several host bodies has been implemented (brown dwarfs, M-dwarfs of mass 0.1 M-circle dot, Sun-like stars, Jupiter). We validate the new code by comparing its output for one-planet systems to the secular equations results. We find that this code does respect the conservation of total angular momentum. We applied this new tool to the planetary system Kepler-62. We find that tides influence the stability of the system in some cases. We also show that while the four inner planets of the systems are likely to have slow rotation rates and small obliquities, the fifth planet could have a fast rotation rate and a high obliquity. This means that the two habitable zone planets of this system, Kepler-62e ad f are likely to have very different climate features, and this of course would influence their potential at hosting surface liquid water.
Santos, N. C., Martins, J. H. C., Boue, G., Correia, A. C. M., Oshagh, M., Figueira, P., Santerne, A., Sousa, S. G., Melo, C., Montalto, M., Boisse, I., Ehrenreich, D., Lovis, C., Pepe, F., Udry, S., Munoz, A. Garcia
EDP Sciences, ESO
Aims. In this paper we explore the possibility that the recently detected reflected light signal of 51 Peg b could be caused by a ring system around the planet. Methods. We use a simple model to compare the observed signal with the expected signal from a short-period giant planet with rings. We also use simple dynamical arguments to understand the possible geometry of such a system. Results. We provide evidence that, to a good approximation, the observations are compatible with the signal expected from a ringed planet, assuming that the rings are non-coplanar with the orbital plane. However, based on dynamical arguments, we also show that this configuration is unlikely. In the case of coplanar rings we then demonstrate that the incident flux on the ring surface is about 2% the value received by the planet, a value that renders the ring explanation unlikely. Conclusions. The results suggest that the signal observed cannot in principle be explained by a planet+ring system. We discuss, however, the possibility of using reflected light spectra to detect and characterize the presence of rings around short-period planets. Finally, we show that ring systems could have already been detected by photometric transit campaigns, but their signal could have been easily misinterpreted by the expected light curve of an eclipsing binary.
Leleu, Adrien, Robutel, Philippe, Correia, Alexandre C. M.
Several celestial bodies in co-orbital configurations exist in the solar system. However, co-orbital exoplanets have not yet been discovered. This lack may result from a degeneracy between the signal induced by co-orbital planets and other orbital configurations. Here we determine a criterion for the detectability of quasi-circular co-orbital planets and develop a demodulation method to bring out their signature from the observational data. We show that the precision required to identify a pair of co-orbital planets depends only on the libration amplitude and on the planet's mass ratio. We apply our method to synthetic radial velocity data, and show that for tadpole orbits we are able to determine the inclination of the system to the line of sight. Our method is also valid for planets detected through the transit and astrometry techniques.
Correia, Alexandre C. M., Leleu, Adrien, Rambaux, Nicolas, Robutel, Philippe
We investigate the resonant rotation of circumbinary bodies in planar quasi-circular orbits. Denoting n(b) and n the orbital mean motion of the inner binary and of the circumbinary body, respectively, we show that spin-orbit resonances exist at the frequencies n +/- kv/2, where v = n(b) - n, and k is an integer. Moreover, when the libration at natural frequency has the same magnitude as v, the resonances overlap and the rotation becomes chaotic. We apply these results to the small satellites in the Pluto-Charon system, and conclude that their rotations are likely chaotic. However, the rotation can also be stable and not synchronous for small axial asymmetries.
Cunha, Diana, Correia, Alexandre C. M., Laskar, Jacques
Cambridge University Press
Planets with masses between 0.1 and 10M(circle plus) are believed to host dense atmospheres. These atmospheres can play an important role on the planet's spin evolution, since thermal atmospheric tides, driven by the host star, may counterbalance gravitational tides. In this work, we study the long-term spin evolution of Earth-sized exoplanets. We generalize previous works by including the effect of eccentric orbits and obliquity. We show that under the effect of tides and core-mantle friction, the obliquity of the planets evolves either to 0 degrees or 180 degrees. The rotation of these planets is also expected to evolve into a very restricted number of equilibrium configurations. In general, none of these equilibria is synchronous with the orbital mean motion. The role of thermal atmospheric tides becomes more important for Earth-sized planets in the habitable zones of their systems; so they cannot be neglected when we search for their potential habitability.
Cunha, Pedro V. P., Herdeiro, Carlos A. R., Radu, Eugen, Runarsson, Helgi F.
American Physical Society
Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHSH). KBHSH interpolate continuously between Kerr BHs and boson stars (BSs), so we start by investigating the lensing of light due to BSs. Moving from the weak to the strong gravity region, BSs-which by themselves have no shadows-are classified, according to the lensing produced, as (i) noncompact, which yield not multiple images, (ii) compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere, and (iii) ultracompact, which possess light rings, yielding an infinite number of images with (we conjecture) a self-similar structure. The shadows of KBHSH, for Kerr-like horizons and noncompact BS-like hair, are analogous to, but distinguishable from, those of comparable Kerr BHs. But for non-Kerr-like horizons and ultracompact BS-like hair, the shadows of KBHSH are drastically different: novel shapes arise, sizes are considerably smaller, and multiple shadows of a single BH become possible. Thus, KBHSH provide quantitatively and qualitatively new templates for ongoing (and future) very large baseline interferometry observations of BH shadows, such as those of the Event Horizon Telescope.
Herdeiro, Carlos A. R., Radu, Eugen, Runarsson, Helgi
The maximal Arnowitt-Deser-Misner (ADM) mass for (mini) boson stars (BSs)-gravitating solitons of Einstein's gravity minimally coupled to a free, complex, mass mu, Klein-Gordon field-is M-ADM(max) similar to M-Pl(2)/mu. Adding quartic self-interactions to the scalar field theory, described by the Lagrangian L-1 = lambda vertical bar Psi vertical bar(4), the maximal ADM mass becomes M-ADM(max) similar to root lambda M-Pl(3)/mu(2). Thus, for mini-BSs, astrophysically interesting masses require ultralight scalar fields, whereas self-interacting BSs can reach such values for bosonic particles with Standard Model range masses. We investigate how these same self-interactions affect Kerr black holes with scalar hair (KBHsSH) [C. A. R. Herdeiro and E. Radu, Kerr Black Holes with Scalar Hair, Phys. Rev. Lett. 112, 221101 (2014).], which can be regarded as (spinning) BSs in stationary equilibrium with a central horizon. Remarkably, whereas the ADM mass scales in the same way as for BSs, the horizon mass M-H does not increases with the coupling., and, for fixed mu, it is maximized at the "Hod point," corresponding to the extremal Kerr black hole obtained in the vanishing hair limit. This mass is always M-H(max) similar to M-Pl(2)/mu. Thus, introducing these self-interactions, the black hole spacetimes may become considerably "hairier" but the trapped regions cannot become "heavier." We present evidence that this observation also holds in a model with L-1 = beta vertical bar Psi vertical bar(6) - lambda vertical bar Psi vertical bar(4); if it extends to general scalar field models, KBHsSH with astrophysically interesting horizon masses require ultralight scalar fields. Their existence, therefore, would be a smoking gun for such (beyond the Standard Model) particles.
Herdeiro, Carlos, Radu, Eugen
We discuss electrostatics in Anti-de-Sitter (AdS) spacetime, in global coordinates. We observe that the multipolar expansion has two crucial differences to that in Minkowski spacetime. First, there are everywhere regular solutions, with finite energy, for every multipole moment exceptfor the monopole. Second, all multipole moments decay with the same inverse power of the areal radius, 1/r, as spatial infinity is approached. The first observation suggests there may be regular, self-gravitating, EinsteinMaxwell solitons in AdSspacetime. The second observation, renders a Lichnerowicz-type no-soliton theorem inapplicable. Consequently, we suggest Einstein-Maxwell solitons exist in AdS, and we support this claim by computing the first order metric perturbations sourced by test electric field multipoles, which are obtained analytically in closed form.
Herdeiro, Carlos A. R., Radu, Eugen
World Scientific Publishing
Kerr black holes (BHs) have their angular momentum, J, bounded by their mass, M: Jc <= GM(2). There are, however, known BH solutions violating this Kerr bound. We propose a very simple universal bound on the rotation, rather than on the angular momentum, of four-dimensional, stationary and axisymmetric, asymptotically flat BHs, given in terms of an appropriately defined horizon linear velocity, v(H). The v(H) bound is simply that v(H) cannot exceed the velocity of light. We verify the v(H) bound for known BH solutions, including some that violate the Kerr bound, and conjecture that only extremal Kerr BHs saturate the v(H) bound.
Herdeiro, Carlos, Kunz, Jutta, Radu, Eugen, Subagyo, Bintoro
We construct rotating boson stars and Myers-Perry black holes with scalar hair (MPBHsSH) as fully non-linear solutions of five dimensional Einstein gravity minimally coupled to a complex, massive scalar field. The MPBHsSH are, in general, regular on and outside the horizon, asymptotically flat, and possess angular momentum in a single rotation plane. They are supported by rotation and have no static limit. Such hairy BHs may be thought of as bound states of boson stars and singly spinning, vacuum MPBHs and inherit properties of both these building blocks. When the horizon area shrinks to zero, the solutions reduce to (in a single plane) rotating boson stars; but the extremal limit also yields a zero area horizon, as for singly spinning MPBHs. Similarly to the case of equal angular momenta, and in contrast to Kerr black holes with scalar hair, singly spinning MPBHsSH are disconnected from the vacuum black holes, due to a mass gap. We observe that for the general case, with two unequal angular momenta, the equilibrium condition for the existence of MPBHsSH is w = m(1)Omega(1)+m(2)Omega(2), where Omega(i) are the horizon angular velocities in the two independent rotation planes and w, m(i), i = 1, 2, are the scalar field's frequency and azimuthal harmonic indices.
Benone, Carolina L., Crispino, Luis C. B., Herdeiro, Carlos A. R., Radu, Eugen
We discuss stationary bound states, a.k.a. clouds, for a massless test scalar field around Kerr black holes (BHs) and spinning acoustic BH analogues. In view of the absence of a mass term, the trapping is achieved via enclosing the BH - scalar field system in a cavity and imposing Dirichlet or Neumann boundary conditions. We discuss the variation of these bounds states with the discrete parameters that label them, as well as their spatial distribution, complementing results in our previous work [C. L. Benone, L. C. B. Crispino, C. Herdeiro and E. Radu, Phys. Rev. D 91 (2015) 104038].
We consider the status of black hole (BH) solutions with nontrivial scalar fields but no gauge fields, in four-dimensional asymptotically flat spacetimes, reviewing both classical results and recent developments. We start by providing a simple illustration on the physical difference between BHs in electro-vacuum and scalar-vacuum. Next, we review no-scalar-hair theorems. In particular, we detail an influential theorem by Bekenstein and stress three key assumptions: (1) The type of scalar field equation; (2) the spacetime symmetry inheritance by the scalar field and (3) an energy condition. Then, we list regular (on and outside the horizon), asymptotically flat BH solutions with scalar hair, organizing them by the assumption which is violated in each case and distinguishing primary from secondary hair. We provide a table summary of the state-of-the-art.
Kerr black holes (BHs) with scalar hair are solutions of the Einstein-Klein-Gordon field equations describing regular (on and outside an event horizon), asymptotically flat BHs with scalar hair (Herdeiro and Radu 2014 Phys. Rev. Lett. 112 221101). These BHs interpolate continuously between the Kerr solution and rotating boson stars in D = 4 spacetime dimensions. Here we provide details on their construction, discussing properties of the ansatz, the field equations, the boundary conditions and the numerical strategy. Then, we present an overview of the parameter space of the solutions, and describe in detail the space-time structure of the BH's exterior geometry and of the scalar field for a sample of reference solutions. Phenomenological properties of potential astrophysical interest are also discussed, and the stability properties and possible generalizations are commented on. As supplementary material to this paper we make available numerical data files for the sample of reference solutions discussed, for public use (see stacks.iop.org/cqg/32/144001/mmedia).
Benone, Carolina L., Crispino, Luis C. B., Herdeiro, Carlos, Radu, Eugen
Under certain conditions sound waves in fluids experience an acoustic horizon with analogue properties to those of a black hole event horizon. In particular, a draining bathtub-like model can give rise to a rotating acoustic horizon and hence a rotating black hole (acoustic) analogue. We show that sound waves, when enclosed in a cylindrical cavity, can form stationary waves around such rotating acoustic holes. These acoustic perturbations display similar properties to the scalar clouds that have been studied around Kerr and Kerr-Newman black holes; thus they are dubbed acoustic clouds. We make the comparison between scalar clouds around Kerr black holes and acoustic clouds around the draining bathtub explicit by studying also the properties of scalar clouds around Kerr black holes enclosed in a cavity. Acoustic clouds suggest the possibility of testing, experimentally, the existence and properties of black hole clouds, using analog models.
Cardoso, Vitor, Gualtieri, Leonardo, Herdeiro, Carlos, Sperhake, Ulrich
Living Reviews
The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We review techniques for solving Einstein's equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology.
Abreu Blaya, Ricardo, Bory Reyes, Ricardo, Guzman Adan, Ali, Kähler, Uwe
Wiley-VCH Verlag
Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field H. It relies heavily on results on functions defined on domains in R4 or R3 with values in H. This theory is centred around the concept of ψ-hyperholomorphic functions related to a so-called structural set ψ of H4 or H3 respectively. The main goal of this paper is to develop the nucleus of the (φ,ψ)-hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy-Riemann operators associated to a pair φ,ψ of structural sets of H4. Following a matrix approach, a generalized Borel-Pompeiu formula and the corresponding Plemelj-Sokhotzki formulae are established.
Bartrum, Sam, Berera, Arjun, Rosa, João G.
We show that dissipative effects have a significant impact on the evolution of cosmological scalar fields, leading to friction, entropy production and field fluctuations. We explicitly compute the dissipation coefficient for different scalar fields within the standard model and some of its most widely considered extensions, in different parametric regimes. We describe the generic consequences of fluctuation-dissipation dynamics in the postinflationary universe, focusing in particular on friction and particle production, and analyze in detail two important effects. First, we show that dissipative friction delays the process of spontaneous symmetry breaking and may even damp the motion of a Higgs field sufficiently to induce a late period of warm inflation. Along with dissipative entropy production, this may parametrically dilute the abundance of dangerous thermal relics. Second, we show that dissipation can generate the observed baryon asymmetry without symmetry restoration, and we develop in detail a model of dissipative leptogenesis. We further show that this generically leads to characteristic baryon isocurvature perturbations that can be tested with cosmic microwave background observations. This work provides a fundamental framework to go beyond the leading thermal equilibrium semiclassical approximation in addressing fundamental problems in modern cosmology.
Rosa, João G.
We show that the magnetic dipole and gravitational radiation emitted by a pulsar can undergo superradiant scattering off a spinning black hole companion. We find that the relative amount of superradiant modes in the radiation depends on the pulsar's angular position relative to the black hole's equatorial plane. In particular, when the pulsar and black hole spins are aligned, superradiant modes are dominant at large angles, leading to an amplification of the pulsar's luminosity, whereas for small angles the radiation is dominantly composed of non-superradiant modes and the signal is attenuated. This results in a characteristic orbital modulation of the pulsar's luminosity, up to the percent level within our approximations, which may potentially yield a signature of superradiant scattering in astrophysical black holes and hence an important test of general relativity.
Morais, J., Cação, I.
Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this development. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quaternionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with $ \mathbb{R}^3$ and $ \mathbb{R}^4$). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second order homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation.
Costa, Raul, Morais, António, Sampaio, Marco, Santos, Rui
Motivated by the dark matter and the baryon asymmetry problems, we analyse a complex singlet extension of the Standard Model (SM) with a Z2 symmetry (which provides a dark matter candidate). After a detailed two-loop calculation of the renormalization group equations for the new scalar sector, we study the radiative stability of the model up to a high energy scale (with the constraint that the 126 GeV Higgs boson found at the LHC is in the spectrum) and find it requires the existence of a new scalar state mixing with the Higgs with a mass larger than 140 GeV. This bound is not very sensitive to the cut-off scale as long as the latter is larger than 10^10 GeV. We then include all experimental and observational constraints/measurements from collider data, dark matter direct detection experiments and from the Planck satellite and in addition force stability at least up to the GUT scale, to find that the lower bound is raised to about 170 GeV, while the dark matter particle must be heavier than about 50 GeV.
Espy Kehoe, AJ, Kehoe, TJJ, Colwell, JE, Dermott, SF
American Astronomical Society
We have performed detailed dynamical modeling of the structure of a faint dust band observed in coadded InfraRed Astronomical Satellite data at an ecliptic latitude of 17 degrees that convincingly demonstrates that it is the result of a relatively recent (significantly less than 1Ma) disruption of an asteroid and is still in the process of forming. We show here that young dust bands retain information on the size distribution and cross-sectional area of dust released in the original asteroid disruption, before it is lost to orbital and collisional decay. We find that the Emilkowalski cluster is the source of this partial band and that the dust released in the disruption would correspond to a regolith layer similar to 3 m deep on the similar to 10 km diameter source body's surface. The dust in this band is described by a cumulative size-distribution inverse power-law index with a lower bound of 2.1 (implying domination of cross-sectional area by small particles) for dust particles with diameters ranging from a few mu m up to a few cm. The coadded observations show that the thermal emission of the dust band structure is dominated by large (mm-cm size) particles. We find that dust particle ejection velocities need to be a few times the escape velocity of the Emilkowalski cluster source body to provide a good fit to the inclination dispersion of the observations. We discuss the implications that such a significant release of material during a disruption has for the temporal evolution of the structure, composition, and magnitude of the zodiacal cloud.
Luis Blazquez-Salcedo, Jose, Kunz, Jutta, Navarro-Lerida, Francisco, Radu, Eugen
Rotating black holes in Einstein-Maxwell-Chern-Simons theory possess remarkable features when the Chern-Simons coupling constant reaches a critical value. Representing single asymptotically flat black holes with horizons of spherical topology, they exhibit nonuniqueness. In particular, there even exist extremal and nonextremal black holes with the same sets of global charges. Both extremal and nonextremal black holes form sequences of radially excited solutions that can be labeled by the node number of the magnetic gauge potential function. The extremal Reissner-Nordstrm solution is no longer always located on the boundary of the domain of existence of these black holes, nor does it remain the single extremal solution with vanishing angular momentum. Instead a whole sequence of rotating extremal J = 0 solutions is present, whose mass converges towards the mass of the Reissner-Nordstrm solution. These radially excited extremal solutions are all associated with the same near horizon solution. Moreover, there are near horizon solutions that are not realized as global solutions.
Kichakova, Olga, Kunz, Jutta, Radu, Eugen, Shnir, Yasha
We investigate the thermodynamics of spherically symmetric black hole solutions in a four-dimensional Einstein-Yang-Mills-SU(2) theory with a negative cosmological constant. Special attention is paid to configurations with a unit magnetic charge. We find that a set of Reissner-Nordstrom-Anti-de Sitter black holes can become unstable to forming non-Abelian hair. However, the hairy black holes are never thermodynamically favoured over the full set of abelian monopole solutions. The thermodynamics of the generic configurations possessing a noninteger magnetic charge is also discussed. (C) 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license.
Brihaye, Yves, Manvelyan, Ruben, Radu, Eugen, Tchrakian, D. H.
We construct black holes with a Ricci-flat horizon in Einstein-Yang-Mills theory with a negative cosmological constant, which approach asymptotically an AdS(d) spacetime background (with d >= 4). These solutions are isotropic, i.e. all space directions in a hypersurface of constant radial and time coordinates are equivalent, and possess both electric and magnetic fields. We find that the basic properties of the non-Abelian solutions are similar to those of the dyonic isotropic branes in Einstein-Maxwell theory (which, however, exist in even spacetime dimensions only). These black branes possess a nonzero magnetic field strength on the flat boundary metric, which leads to a divergent mass of these solutions, as defined in the usual way. However, a different picture is found for odd spacetime dimensions, where a non-Abelian Chern-Simons term can be incorporated in the action. This allows for black brane solutions with a magnetic field which vanishes asymptotically. (C) 2015 The Authors. Published by Elsevier B.V.
Almeida, Ricardo, Bastos, Nuno R.O.
ILIRIAS Research Institute
The objective of this paper is to present an approximation formula for the Katugampola fractional integral, that allows us to solve fractional problems with dependence on this type of fractional operator. The formula only depends on first-order derivatives, and thus converts the fractional problem into a standard one. With some examples, we show the accuracy of the method, and then we present the utility of the method by solving a fractional integral equation.
Clain, Teresa Costa
Société Mathématique de France
Au xvie siècle furent publiés les premiers ouvrages sur l arithmé- tique marchande imprimés au Portugal tels que le Tratado da Pratica Darismetica de Gaspar Nicolas édité pour la première fois en 1519, la Pratica d Arismetica de Ruy Mendes de 1540 et le Tratado da arte de Arismetica de Bento Fernandes en 1555. Dans tous ces traités sont présents des modèles arithmétiques liés aux opérations nancières, sous la forme de règles propres issues du commerce portugais des épices et de sa diffusion dans toute l Europe. Nous donnerons une brève présentation de la Pratica d Arismetica de Ruy Mendes que nous replacerons dans le corpus des uvres d arithmétique marchande au Portugal, en indiquant leurs sources et leurs in uences relativement au contexte ibérique. In the sixteenth century began the publication of mercantile arithmetic treatises printed in Portugal, such as the Tratado da Pratica Darismetica of Gaspar Nicolas, rst published in 1519, the Pratica d Arismetica of Ruy Mendes, with only one edition in 1540 and the Tratado da arte de Arismetica published by Bento Fernandes in 1555. We can nd, in these treatises, arithmetical models linked to nancial transactions with speci c rules of Portuguese trade of spices and its distribution in Europe. We will present a brief introduction of the Pratica d Arismetica of Ruy Mendes and its place with respect to other works in mercantile arithmetic produced in Portugal. We also highlight sources and in uences with respect to the Iberian context. No século XVI inciou-se a publicação de obras de aritmética mercantil impressas em Portugal, tais como o "Tratado da Pratica Darismetica" de Gaspar Nicolas, publicado pela primeira vez em 1519, a "Pratica d'Arismetica" de Ruy Mendes de 1540 e o "Tratado da arte de Arismetica", de Bento Fernandes em 1555. Nos tratados encontramos uma modelação aritmética ligada às operações financeiras, na forma de regras próprias do comércio português das especiarias e da sua distribuição pela Europa. Faremos uma breve apresentação da "Pratica d'Arismetica" de Ruy Mendes e do seu enquadramento nas obras em aritmética mercantil produzidas em Portugal, referenciando ainda fontes e influências presentes tendo em conta o contexto ibérico.
Chen, Jianqing, Rocha, Eugénio M.
In this paper, we prove the existence of four nontrivial solutions of a elliptic equation with a Hardy singularity and show that at least one of them is sign changing. Our results extend some previous works on the literature, as Tarantello(1993), Kang-Deng(2005) and Hirano-Shioji(2005).
Climent, J.-J., Napp, D., Perea, C., Pinto, Raquel
Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate $k/n$ and degree $delta $ , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate $k/n$ and degree $delta $ that reach such bound when $n geq k (({(lfloor ({delta }/{k}) rfloor + 2)(lfloor ({delta }/{k}) rfloor + 3)})/{2})$ if $k {nmid } delta $ , or $n geq k (({(({delta }/{k}) + 1)(({delta }/{k}) + 2)})/{2})$ if $k mid delta $ , by presenting a concrete constructive procedure.
Kleihaus, Burkhard, Kunz, Jutta, Radu, Eugen
We propose a general framework for the study of asymptotically flat black objects with k+1 equal magnitude angular momenta in d >= 5 spacetime dimensions (with 0 <= k <= [d-5/2]). In this approach, the dependence on all angular coordinates but one is factorized, which leads to a codimension-two problem. This framework can describe black holes with spherical horizon topology, the simplest solutions corresponding to a class of Myers-Perry black holes. A different set of solutions describes balanced black objects with Sn+1 x S2k+1 horizon topology. The simplest members of this family are the black rings (k = 0). The solutions with k > 0 are dubbed black ringoids. Based on the nonperturbative numerical results found for several values of (n, k), we propose a general picture for the properties and the phase diagram of these solutions and the associated black holes with spherical horizon topology: n = 1 black ringoids repeat the k = 0 pattern of black rings and Myers-Perry black holes in 5 dimensions, whereas n > 1 black ringoids follow the pattern of higher dimensional black rings associated with 'pinched' black holes and Myers-Perry black holes.
Tinetti, Giovanna, Drossart, Pierre, Eccleston, Paul, Hartogh, Paul, Isaak, Kate, Linder, Martin, Lovis, Christophe, Micela, Giusi, Ollivier, Marc, Puig, Ludovic, Ribas, Ignasi, Snellen, Ignas, Swinyard, Bruce, Allard, France, Barstow, Joanna, Cho, James, Coustenis, Athena, Cockell, Charles, Correia, Alexandre, Decin, Leen, de Kok, Remco, Deroo, Pieter, Encrenaz, Therese, Forget, Francois, Glasse, Alistair, Griffith, Caitlin, Guillot, Tristan, Koskinen, Tommi, Lammer, Helmut, Leconte, Jeremy, Maxted, Pierre, Mueller-Wodarg, Ingo, Nelson, Richard, North, Chris, Palle, Enric, Pagano, Isabella, Piccioni, Guseppe, Pinfield, David, Selsis, Franck, Sozzetti, Alessandro, Stixrude, Lars, Tennyson, Jonathan, Turrini, Diego, Zapatero-Osorio, Mariarosa, Beaulieu, Jean-Philippe, Grodent, Denis, Guedel, Manuel, Luz, David, Norgaard-Nielsen, Hans Ulrik, Ray, Tom, Rickman, Hans, Selig, Avri, Swain, Mark, Banaszkiewicz, Marek, Barlow, Mike, Bowles, Neil, Branduardi-Raymont, Graziella, du Foresto, Vincent Coude, Gerard, Jean-Claude, Gizon, Laurent, Hornstrup, Allan, Jarchow, Christopher, Kerschbaum, Franz, Kovacs, Geza, Lagage, Pierre-Olivier, Lim, Tanya, Lopez-Morales, Mercedes, Malaguti, Giuseppe, Pace, Emanuele, Pascale, Enzo, Vandenbussche, Bart, Wright, Gillian, Ramos Zapata, Gonzalo, Adriani, Alberto, Azzollini, Ruyman, Balado, Ana, Bryson, Ian, Burston, Raymond, Colome, Josep, Crook, Martin, Di Giorgio, Anna, Griffin, Matt, Hoogeveen, Ruud, Ottensamer, Roland, Irshad, Ranah, Middleton, Kevin, Morgante, Gianluca, Pinsard, Frederic, Rataj, Mirek, Reess, Jean-Michel, Savini, Giorgio, Schrader, Jan-Rutger, Stamper, Richard, Winter, Berend, Abe, L., Abreu, M., Achilleos, N., Ade, P., Adybekian, V., Affer, L., Agnor, C., Agundez, M., Alard, C., Alcala, J., Allende Prieto, C., Alonso Floriano, F. J., Altieri, F., Alvarez Iglesias, C. A., Amado, P., Andersen, A., Aylward, A., Baffa, C., Bakos, G., Ballerini, P., Banaszkiewicz, M., Barber, R. J., Barrado, D., Barton, E. J., Batista, V., Bellucci, G., Belmonte Aviles, J. A., Berry, D., Bezard, B., Biondi, D., Blecka, M., Boisse, I., Bonfond, B., Borde, P., Boerner, P., Bouy, H., Brown, L., Buchhave, L., Budaj, J., Bulgarelli, A., Burleigh, M., Cabral, A., Capria, M. T., Cassan, A., Cavarroc, C., Cecchi-Pestellini, C., Cerulli, R., Chadney, J., Chamberlain, S., Charnoz, S., Jessen, N. Christian, Ciaravella, A., Claret, A., Claudi, R., Coates, A., Cole, R., Collura, A., Cordier, D., Covino, E., Danielski, C., Damasso, M., Deeg, H. J., Delgado-Mena, E., Del Vecchio, C., Demangeon, O., De Sio, A., De Wit, J., Dobrijevic, M., Doel, P., Dominic, C., Dorfi, E., Eales, S., Eiroa, C., Espinoza Contreras, M., Esposito, M., Eymet, V., Fabrizio, N., Fernandez, M., Femena Castella, B., Figueira, P., Filacchione, G., Fletcher, L., Focardi, M., Fossey, S., Fouque, P., Frith, J., Galand, M., Gambicorti, L., Gaulme, P., Garcia Lopez, R. J., Garcia-Piquer, A., Gear, W., Gerard, J. -C., Gesa, L., Giani, E., Gianotti, F., Gillon, M., Giro, E., Giuranna, M., Gomez, H., Gomez-Leal, I., Gonzalez Hernandez, J., Gonzalez Merino, B., Graczyk, R., Grassi, D., Guardia, J., Guio, P., Gustin, J., Hargrave, P., Haigh, J., Hebrard, E., Heiter, U., Heredero, R. L., Herrero, E., Hersant, F., Heyrovsky, D., Hollis, M., Hubert, B., Hueso, R., Israelian, G., Iro, N., Irwin, P., Jacquemoud, S., Jones, G., Jones, H., Justtanont, K., Kehoe, T., Kerschbaum, F., Kerins, E., Kervella, P., Kipping, D., Koskinen, T., Krupp, N., Lahav, O., Laken, B., Lanza, N., Lellouch, E., Leto, G., Licandro Goldaracena, J., Lithgow-Bertelloni, C., Liu, S. J., Lo Cicero, U., Lodieu, N., Lognonne, P., Lopez-Puertas, M., Lopez-Valverde, M. A., Rasmussen, I. Lundgaard, Luntzer, A., Machado, P., MacTavish, C., Maggio, A., Maillard, J. -P., Magnes, W., Maldonado, J., Mall, U., Marquette, J. -B., Mauskopf, P., Massi, F., Maurin, A. -S., Medvedev, A., Michaut, C., Miles-Paez, P., Montalto, M., Montanes Rodriguez, P., Monteiro, M., Montes, D., Morais, H., Morales, J. C., Morales-Calderon, M., Morello, G., Moro Martin, A., Moses, J., Moya Bedon, A., Murgas Alcaino, F., Oliva, E., Orton, G., Palla, F., Pancrazzi, M., Pantin, E., Parmentier, V., Parviainen, H., Pena Ramirez, K. Y., Peralta, J., Perez-Hoyos, S., Petrov, R., Pezzuto, S., Pietrzak, R., Pilat-Lohinger, E., Piskunov, N., Prinja, R., Prisinzano, L., Polichtchouk, I., Poretti, E., Radioti, A., Ramos, A. A., Rank-Lueftinger, T., Read, P., Readorn, K., Rebolo Lopez, R., Rebordao, J., Rengel, M., Rezac, L., Rocchetto, M., Rodler, F., Sanchez Bejar, V. J., Lavega, A. Sanchez, Sanroma, E., Santos, N., Sanz Forcada, J., Scandariato, G., Schmider, F. -X., Scholz, A., Scuderi, S., Sethenadh, J., Shore, S., Showman, A., Sicardy, B., Sitek, P., Smith, A., Soret, L., Sousa, S., Stiepen, A., Stolarski, M., Strazzulla, G., Tabernero, H. M., Tanga, P., Tecsa, M., Temple, J., Terenzi, L., Tessenyi, M., Testi, L., Thompson, S., Thrastarson, H., Tingley, B. W., Trifoglio, M., Martin Torres, J., Tozzi, A., Turrini, D., Varley, R., Vakili, F., de Val-Borro, M., Valdivieso, M. L., Venot, O., Villaver, E., Vinatier, S., Viti, S., Waldmann, I., Waltham, D., Ward-Thompson, D., Waters, R., Watkins, C., Watson, D., Wawer, P., Wawrzaszk, A., White, G., Widemann, T., Winek, W., Wisniowski, T., Yelle, R., Yung, Y., Yurchenko, S. N.
The discovery of almost two thousand exoplanets has revealed an unexpectedly diverse planet population. We see gas giants in few-day orbits, whole multi-planet systems within the orbit of Mercury, and new populations of planets with masses between that of the Earth and Neptune-all unknown in the Solar System. Observations to date have shown that our Solar System is certainly not representative of the general population of planets in our Milky Way. The key science questions that urgently need addressing are therefore: What are exoplanets made of? Why are planets as they are? How do planetary systems work and what causes the exceptional diversity observed as compared to the Solar System? The EChO (Exoplanet Characterisation Observatory) space mission was conceived to take up the challenge to explain this diversity in terms of formation, evolution, internal structure and planet and atmospheric composition. This requires in-depth spectroscopic knowledge of the atmospheres of a large and well-defined planet sample for which precise physical, chemical and dynamical information can be obtained. In order to fulfil this ambitious scientific program, EChO was designed as a dedicated survey mission for transit and eclipse spectroscopy capable of observing a large, diverse and well-defined planet sample within its 4-year mission lifetime. The transit and eclipse spectroscopy method, whereby the signal from the star and planet are differentiated using knowledge of the planetary ephemerides, allows us to measure atmospheric signals from the planet at levels of at least 10(-4) relative to the star. This can only be achieved in conjunction with a carefully designed stable payload and satellite platform. It is also necessary to provide broad instantaneous wavelength coverage to detect as many molecular species as possible, to probe the thermal structure of the planetary atmospheres and to correct for the contaminating effects of the stellar photosphere. This requires wavelength coverage of at least 0.55 to 11 mu m with a goal of covering from 0.4 to 16 mu m. Only modest spectral resolving power is needed, with R similar to 300 for wavelengths less than 5 mu m and R similar to 30 for wavelengths greater than this. The transit spectroscopy technique means that no spatial resolution is required. A telescope collecting area of about 1 m(2) is sufficiently large to achieve the necessary spectro-photometric precision: for the Phase A study a 1.13 m(2) telescope, diffraction limited at 3 mu m has been adopted. Placing the satellite at L2 provides a cold and stable thermal environment as well as a large field of regard to allow efficient time-critical observation of targets randomly distributed over the sky. EChO has been conceived to achieve a single goal: exoplanet spectroscopy. The spectral coverage and signal-to-noise to be achieved by EChO, thanks to its high stability and dedicated design, would be a game changer by allowing atmospheric composition to be measured with unparalleled exactness: at least a factor 10 more precise and a factor 10 to 1000 more accurate than current observations. This would enable the detection of molecular abundances three orders of magnitude lower than currently possible and a fourfold increase from the handful of molecules detected to date. Combining these data with estimates of planetary bulk compositions from accurate measurements of their radii and masses would allow degeneracies associated with planetary interior modelling to be broken, giving unique insight into the interior structure and elemental abundances of these alien worlds. EChO would allow scientists to study exoplanets both as a population and as individuals. The mission can target super-Earths, Neptune-like, and Jupiter-like planets, in the very hot to temperate zones (planet temperatures of 300-3000 K) of F to M-type host stars. The EChO core science would be delivered by a three-tier survey. The EChO Chemical Census: This is a broad survey of a few-hundred exoplanets, which allows us to explore the spectroscopic and chemical diversity of the exoplanet population as a whole. The EChO Origin: This is a deep survey of a subsample of tens of exoplanets for which significantly higher signal to noise and spectral resolution spectra can be obtained to explain the origin of the exoplanet diversity (such as formation mechanisms, chemical processes, atmospheric escape). The EChO Rosetta Stones: This is an ultra-high accuracy survey targeting a subsample of select exoplanets. These will be the bright "benchmark" cases for which a large number of measurements would be taken to explore temporal variations, and to obtain two and three dimensional spatial information on the atmospheric conditions through eclipse-mapping techniques. If EChO were launched today, the exoplanets currently observed are sufficient to provide a large and diverse sample. The Chemical Census survey would consist of > 160 exoplanets with a range of planetary sizes, temperatures, orbital parameters and stellar host properties. Additionally, over the next 10 years, several new ground- and space-based transit photometric surveys and missions will come on-line (e.g. NGTS, CHEOPS, TESS, PLATO), which will specifically focus on finding bright, nearby systems. The current rapid rate of discovery would allow the target list to be further optimised in the years prior to EChO's launch and enable the atmospheric characterisation of hundreds of planets.
Aizicovici, S., Papageorgiou, N. S., Staicu, Vasile
Yokohama Publishers
We consider nonlinear periodic equations driven by the scalar p-Laplacian and with a Carath eodory reaction which does not satisfy a global growth condition. Using truncation-perurbation techniques, variational methods and Morse theory, we prove a "three solutions theorem", providing sign information for all the solutions. In the semilinear case (p = 2), we produce a second nodal solution, for a total of four nontrivial solutions. We also cover problems which are resonant at zero.
Plakhov, A., Gouveia, P. D. F.
A two-dimensional body moves through a rarefied medium; the collisions of the medium particles with the body are absolutely elastic. The body performs both translational and slow rotational motion. It is required to select the body, from a given class of bodies, such that the average force of resistance of the medium to its motion is maximal. Numerical and analytical results concerning this problem are presented. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1.5 times the resistance of K. The maximum is attained on a sequence of bodies with a very complicated boundary. The numerical study was made for somewhat more restricted classes of bodies. The obtained values of resistance are slightly lower, but the boundary of obtained bodies is much simpler, as compared with the analytical solutions. © 2007 IOP Publishing Ltd and London Mathematical Society.
Almeida, R., Torres, D.F.M.
Hindawi
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.
Branquinho, Amilcar, Fidalgo Prieto, Ulises, Foulquie Moreno, Ana
We give a general sufficient condition for the uniform convergence of sequences of type II Hermite-Padé approximants associated with Nikishin systems of functions.
Barrios Rolanía, Dolores, Branquinho, Amilcar, Foulquie Moreno, Ana
The relation between the solutions of the full Kostant–Toda lattice and the discrete Korteweg–de Vries equation is analyzed. A method for constructing solutions of these systems is given. As a consequence of the matricial interpretation of this method, the transform of Darboux is extended for general Hessenberg banded matrices.
Cardoso, D. M., Cerdeira, J. O.
Let p and t, p ≥ t, be positive integers. A t-composition of p is an ordered t-tuple of positive integers summing p. If T = (s_1, s_2, . . . , s_t) is a t-composition p and W is a (p − (t − 1)) × t matrix, then W(T) = w_(s_1)+ w_(s_2)2... + w_(s_k)k is called the weight of the t-composition T. We show that finding a minimum weight t-composition of p can be reduced to the determination of the shortest path in a certain digraph with O(tp) vertices. This study was motivated by a problem arising from the automobile industry, and the presented result is useful when dealing with huge location problems.
Cardoso, D. M., Lozin, V. V.
We show that the class of graphs with quadratic stability number is not hereditary. Then we prove that this class contains a unique maximal hereditary subclass and, finally, we characterize this subclass by two forbidden induced subgraphs.
Cardoso, D. M., Silva, M. E., Szymanski, J.
Considering the partitions of a set into nonempty subsets, we obtain an expression for the number of all partitions of a given type. The chromatic polynomial of a graph subdivision is generalized, considering two sets of colors, and a general explicit expression is obtained for this generalization. Using these results, we determine the generalized chromatic polynomial for the particular case of complete graph subdivision.
Cruz, Sérgio, Pinto, Catarina, Abrunheiro, Lígia
ISCA
O desenvolvimento das sociedades, a crescente educação e consciencialização dos indivíduos acerca dos impostos e medidas fiscais, conduziu a alterações no comportamento dos contribuintes no que diz respeito à perceção e cumprimento fiscal. Recolhendo e sintetizando a literatura existente, efetuámos uma revisão de literatura de forma a perceber quais as variáveis que influenciam os comportamentos dos contribuintes no que respeita ao seu (in)cumprimento e de que forma se manifestam na sociedade de hoje. A revisão efetuada permitiu-nos concluir que existem diversos fatores influentes no comportamento de cumprimento fiscal, podendo-se destacar os fatores económicos, os fatores comportamentais, sociais e psicológicos e os fatores políticos ou institucionais. Alguns estudos permitiram concluir que tanto a auditoria fiscal como a penalização fiscal são dissuasoras de evasão fiscal, embora haja opiniões divergentes no que respeita a qual das duas medidas seja mais eficaz no combate à evasão fiscal. No que diz respeito à perceção de equidade do sistema fiscal, verificou-se que esta é influenciada pelo conhecimento fiscal e por fatores sociais dos contribuintes, sendo que este último afeta também o comportamento de cumprimento dos contribuintes quando acreditam que o incumprimento é consistente com as expectativas e normas do grupo social em que se inserem. Verificou-se que na presença de um nível elevado de educação, há uma maior aceitação dos impostos sobre o rendimento e das taxas corretivas, porém, os contribuintes são movidos pelo seu interesse próprio. Por fim, concluiu-se que os fatores políticos ou institucionais devem ter em conta sobretudo o conhecimento e perceção que os contribuintes têm acerca da atuação da administração fiscal e a opinião destes acerca das alterações de forma a não fomentar o comportamento de incumprimento fiscal. The development of society, the increasing education and consciousness of individuals about taxes and fiscal measures leads to changes in taxpayer’s behavior regarding tax compliance and tax perception. In this paper, we present a literature review in order to understand which variables have impact on these behaviors, using a synthesis methodology that allows an interpretation of social actions from previous studies. The results show that either the tax audit as the penalty rate are deterrents of tax evasion, although there are divergent opinions about which one is most effective. Regarding equity perception of tax system, it have been observed that the level of knowledge and social factors affect that equity perception, when the taxpayers believe that noncompliance is consistent with in-group expectation and norms. On the other hand, it was found evidence that in the presence of a high level of education, there are a great acceptability of income taxes and corrective taxes. Nevertheless, taxpayers are often moved by self-interest. We concluded also that it is interesting to consider political factors. In particular, the analyses of taxpayers’ perception concerning on the activities of the tax administration and the fiscal changes to not induce a noncompliance behavior.
Castro, L. P., Rojas, E. M., Saitoh, S., Tuan, N. M., Tuan, P. D.
By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section.
Castro, L. P., Kapanadze, D.
The problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background. In fact, we may recognize significant research on this topic for more than one century. Despite this fact, up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate Sobolev space setting. In the present paper, we are considering the corresponding boundary value problems for the Helmholtz equation, with complex wave number, admitting combinations of Dirichlet and Neumann boundary conditions. The main ideas are based on a convenient combination of potential representation formulas associated with (weighted) Mellin pseudo-differential operators in appropriate Sobolev spaces, and a detailed Fredholm analysis. Thus, we prove that the problems have unique solutions (with continuous dependence on the data), which are represented by the single and double layer potentials, where the densities are solutions of derived pseudo-differential equations on the half-line.
Castro, L. P., Silva, A. S.
Duke University Press, Tusi Mathematical Research Group
Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators in both frameworks of standard and variable exponent Lebesgue spaces are considered in this paper. The main aim is to describe certain dependencies between the Fredholm property of some Wiener-Hopf operators acting between variable exponent Lebesgue spaces and the invertibility of Wiener-Hopf plus and minus Hankel operators on all the standard Lebesgue spaces. Different types of Fourier symbols will be used but special focus will be considered on the Wiener subclass of almost periodic matrix functions. In the first part of the paper we will give a survey of investigations on related results. This will be useful at the end of the paper to derive the above mentioned dependencies between the operators under study.
Vieira, Nelson
In this paper, we establish the fractional Cauchy-Kovalevskaya extension (FCK-extension) theorem for fractional monogenic functions defined on R^d. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We studied the connection between the FCK-extension of functions of the form x^\alpha P_l and the classical Gegenbauer polynomials. Finally we present two examples of FCK-extension.
Ferreira, Milton
In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$.
JGSP
In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of R$^n$, $n \in N,$ centered at the origin and with arbitrary radius $t \in R^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$-convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on $R^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.
Robaina-Alves, Margarita, Moutinho, Victor, Macedo, Pedro
This study aims to evaluate the resource and environment efficiency problem of European countries. We specify a new stochastic frontier model where Gross Domestic Product (GDP) is considered as the desirable output and Greenhouse Gases (GHG) emissions as the undesirable output. Capital, Labour, Fossil fuels and Renewable Energy consumption are regarded as inputs. GDP/GHG ratio is maximized given the values of the other four variables. The study is divided into two distinct periods: 2000-2004 and 2005-2011. This division is related to the implementation of the Kyoto Protocol in 2005, and will allow us to evaluate the difference between the levels of efficiency before and after the establishment of environmental targets. Since stochastic frontier models are typically ill-posed, a new maximum entropy approach to assess technical efficiency, which combines information from the data envelopment analysis and the structure of composed error from the stochastic frontier approach without requiring distributional assumptions, is presented in this work.
Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal f in the Wiener space can be expanded into a unique l2 series on a lattice at critical density 1, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion.
Almeida, R., Torres, D. F. M.
We present a method to solve fractional optimal control problems, where the dynamic control system depends on integer order and Caputo fractional derivatives. Our approach consists in approximating the initial fractional order problem with a new one that involves integer order derivatives only. The latter problem is then discretized, by application of finite differences, and solved numerically. We illustrate the effectiveness of the procedure with an example.
We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound formula for the error is provided. We exemplify our method by applying the proposed numerical procedure to the solution of a fractional differential equation and a fractional variational problem with dependence on the Hadamard-Marchaud fractional derivative.
Razminia, K., Razminia, A., Torres, D. F. M.
We obtain an analytical solution for the pressure-transient behavior of a vertically hydraulic fractured well in a heterogeneous reservoir. The heterogeneity of the reservoir is modeled by using the concept of fractal geometry. Such reservoirs are called fractal reservoirs. According to the theory of fractional calculus, a temporal fractional derivative is applied to incorporate the memory properties of the fractal reservoir. The effect of different parameters on the computed wellbore pressure is fully investigated by various synthetic examples.
Tavares, D., Almeida, R., Torres, D. F. M.
We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.
Benkhettou, N., Brito da Cruz, A. M. C., Torres, D. F. M.
We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.
Santos, S. P. S., Martins, N., Torres, D. F. M.
American Institute of Mathematical Sciences (AIMS)
We extend the DuBois-Reymond necessary optimality condition and Noether's first theorem to variational problems of Herglotz type with time delay. Our results provide, as corollaries, the DuBois-Reymond necessary optimality condition and the first Noether theorem for variational problems with time delay recently proved in [Numer. Algebra Control Optim. 2(2012), no. 3, 619-630]. Our main result is also a generalization of the first Noether-type theorem for the generalized variational principle of Herglotz proved in [Topol. Methods Nonlinear Anal. 20(2002), no. 2, 261-273].
Dryl, M., Torres, D. F. M.
We consider a general problem of the calculus of variations on time scales with a cost functional that is the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange delta-nabla differential equations are proved, which lead to important insights in the process of discretisation. Application of the obtained results to a firm that wants to program its production and investment policies to reach a given production rate and to maximise its future market competitiveness is discussed.
Rachah, A., Torres, D. F. M.
The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.
Jahanshahi, S., Babolian, E., Torres, D. F. M., Vahidi, A.
We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.
Denysiuk, R., Silva, C. J., Torres, D. F. M.
Mathematical modelling can help to explain the nature and dynamics of infection transmissions, as well as support a policy for implementing those strategies that are most likely to bring public health and economic benefits. The paper addresses the application of optimal control strategies in a tuberculosis model. The model consists of a system of ordinary differential equations, which considers reinfection and post-exposure interventions. We propose a multiobjective optimization approach to find optimal control strategies for the minimization of active infectious and persistent latent individuals, as well as the cost associated to the implementation of the control strategies. Optimal control strategies are investigated for different values of the model parameters. The obtained numerical results cover a whole range of the optimal control strategies, providing valuable information about the tuberculosis dynamics and showing the usefulness of the proposed approach.
Silva, C. J., Torres, D. F. M.
We propose a population model for TB-HIV/AIDS coinfection transmission dynamics, which considers antiretroviral therapy for HIV infection and treatments for latent and active tuberculosis. The HIV-only and TB-only sub-models are analyzed separately, as well as the TB-HIV/AIDS full model. The respective basic reproduction numbers are computed, equilibria and stability are studied. Optimal control theory is applied to the TB-HIV/AIDS model and optimal treatment strategies for co-infected individuals with HIV and TB are derived. Numerical simulations to the optimal control problem show that non intuitive measures can lead to the reduction of the number of individuals with active TB and AIDS.
Benharrat, M., Torres, D. F. M.
We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing us to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.
Brito da Cruz, A. M. C., Martins, N., Torres, D. F. M.
We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as versions of Holder’s, Cauchy–Schwarz’s, and Minkowski’s inequalities.
Cambridge Scientific Publishers
We study two wave diffraction problems modeled by the Helmholtz equation in a half-plane with a crack characterized by Dirichlet and impedance boundary conditions. The existence and uniqueness of solutions is proved by an appropriate combination of general operator theory, Fredholm theory, potential theory and boundary integral equation methods. This combination of methods leads also to integral representations of solutions. Moreover, in Sobolev spaces, a range of smoothness parameters is obtained in which the solutions of the problems are valid.
Paiva, Rui C., Ferreira, Milton, Mendes, Ana G., Eusébio, Augusto M. J.
SAGE Journals
This paper presents a research study addressing the development, implementation, evaluation and use of Interactive Modules for Online Training (MITO) of mathematics in higher education. This work was carried out in the context of the MITO project, which combined several features of the learning and management system Moodle, the computer-aided assessment for mathematics STACK, the mathematical software GeoGebra, several packages from the type-setting program LaTeX, and tutorial videos. A total of 1962 students participated in this study. Two groups of students taking a Calculus course were selected for a deeper analysis. In regard to usability and functionality, the results indicate that MITO scored well in almost all aspects, which is fundamental for their introduction into formal university courses. The analysis of the data reveals that the use of MITO educational contents by students mainly occurs about one week and a half prior the evaluations. Moreover, there is a strong correlation between the results of online assessments on MITO in a continuous assessment model and the final grade on the course.
Caetano, António, Haroske, Dorothee
Duke University Press
Let $\Gamma$ be a fractal $h$-set and ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ be a trace space of Besov type defined on $\Gamma$. While we dealt in [9] with growth envelopes of such spaces mainly and investigated the existence of traces in detail in [12], we now study continuous embeddings between different spaces of that type on $\Gamma$. We obtain necessary and sufficient conditions for such an embedding to hold, and can prove in some cases complete characterisations. It also includes the situation when the target space is of type $L_r(\Gamma)$ and, as a by-product, under mild assumptions on the $h$-set $\Gamma$ we obtain the exact conditions on $\sigma$, $p$ and $q$ for which the trace space ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ exists. We can also refine some embedding results for spaces of generalised smoothness on $\mathbb R^n$.
Almeida, Alexandre, Harjulehto, P., Hästö, P., Lukkari, T.
We prove optimal integrability results for solutions of the p(x)-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials map L1 to variable exponent weak Lebesgue spaces.
Hofmann, Dirk, Mynard, Frédéric, Seal, Gavin J.
Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.
Hofmann, Dirk, Nora, Pedro
In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on the other. Furthermore, we investigate the monoidal structure induced by Cartesian product on the relational side and show that in some cases the corresponding operation on the algebraic side represents bimorphisms.
Hofmann, Dirk, Seal, Gavin J.
University of Houston
In this note we present a characterisation of exponentiable approach spaces in terms of ultrafilter convergence.
Shahid Beheshti University
In this work, we describe an adjunction between the comma category of Set-based monads under the V-powerset monad and the category of associative lax extensions of Set-based monads to the category of V-relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.
Almeida, Paulo J.
Institut de Mathématiques de Bordeaux
Almeida, Paulo J., Pinto, Raquel, Napp, Diego
This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F . Such construction works for any given choice of characteristic of the field F and code parameters ( n , k ,δ) such that ( n − k ) | δ . We also discuss the size of F needed so that the proposed matrices are superregular.
Almeida, Juliana, Alonso, Hugo, Ribeiro, Pedro, Rocha, Paula
The aim of this paper is to present a method based on a 2D Hopfield Neural Network for online damage detection in beams subjected to external forces. The underlying idea of the method is that a significant change in the beam model parameters can be taken as a sign of damage occurrence in the structural system. In this way, damage detection can be associated to an identification problem. More concretely, a 2D Hopfield Neural Network uses information about the way the beam vibrates and the external forces that are applied to it to obtain time-evolving estimates of the beam parameters at the different beam points. The neural network organizes its input information based on the Euler-Bernoulli model for beam vibrations. Its performance is tested with vibration data generated by means of a different model, namely Timonshenko's, in order to produce more realistic simulation conditions.
Castro, L. P., Guerra, R. C., Tuan, N. M.
Georgian National Academy of Sciences; A. Razmadze Mathematical Institute
We present a detailed study of structural properties for certain algebraic operators generated by the Fourier transform and a reflection. First, we focus on the determination of the characteristic polynomials of such algebraic operators, which, e.g., exhibit structural differences when compared with those of the Fourier transform. Then, this leads us to the conditions that allow one to identify the spectrum, eigenfunctions, and the invertibility of this class of operators. A Parseval type identity is also obtained, as well as the solvability of integral equations generated by those operators. Moreover, new convolutions are generated and introduced for the operators under consideration.
Almeida, R., Pooseh, S., Torres, D.F.M.
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of the previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered. © 2011 Elsevier Ltd. All rights reserved.
Cruz, Artur M. C. Brito da, Martins, Natália, Torres, Delfim F. M.
We prove a necessary optimality condition of Euler-Lagrange type for quantum variational problems involving Hahn's derivatives of higher-order. (C) 2011 Elsevier Ltd. All rights reserved.
Xarez, J.J.
Mount Allison University
It is shown that, for a finitely-complete category C with coequalizers of kernel pairs, if every product-regular epi is also stably-regular then there exist the reflections (R)Grphs(C) → (R)Rel(C), from (reflexive) graphs into (reflexive) relations in C, and Cat(C) → Preord(C), from categories into preorders in C. Furthermore, such a sufficient condition ensures as well that these reflections do have stable units. This last property is equivalent to the existence of a monotone-light factorization system, provided there are sufficiently many effective descent morphisms with domain in the respective full subcategory. In this way, we have internalized the monotone-light factorization for small categories via preordered sets, associated with the reflection Cat → Preord, which is now just the special case C = Set.
Pooseh, S., Almeida, R., Torres, D. F. M.
We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method.
We consider fractional order optimal control problems in which the dynamic control system involves integer and fractional order derivatives and the terminal time is free. Necessary conditions for a state/control/terminal- time triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the obtained necessary optimality conditions become sufficient. Numer- ical methods to solve the problems are presented, and some computational simulations are discussed in detail.
Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M.
In 2009, for the first time in Cape Verde, an outbreak of dengue was reported and more than 20,000 people were infected. Only a few prophylactic measures were taken. The effects of vector control on disease spreading, such as insecticide(larvicide and adulticide) and mechanical control, as well as an hypothetical vaccine, are estimated through simulations with the Cape Verde data. © 2013 Copyright Taylor and Francis Group, LLC.
Rodrigues, H. S., Monteiro, M. T. T., Torres, D. F. M.
A model with six mutually exclusive compartments related to dengue is studied. Three vector control tools are considered: insecticides (larvicide and adulticide) and mechanical control. The basic reproduction number associated to the model is presented. The problem is studied using an optimal control approach. The human data are based on the dengue outbreak that occurred in Cape Verde. Control measures are simulated in different scenarios and their consequences analysed. © 2013 Taylor & Francis.
Razminia, A., Torres, D. F. M.
We consider a new fractional order chaotic system displaying an interesting behavior. A necessary condition for the system to remain chaotic is derived. It is found that chaos exists in the system with order less than three. Using the Routh-Hurwitz and the Matignon stability criteria, we analyze the novel chaotic fractional order system and propose a control methodology that is better than the nonlinear counterparts available in the literature, in the sense of simplicity of implementation and analysis. A scalar control input that excites only one of the states is proposed, and sufficient conditions for the controller gain to stabilize the unstable equilibrium points derived. Numerical simulations confirm the theoretical analysis. © 2013 Elsevier Ltd. All rights reserved.
Odzijewicz, T., Malinowska, A.B., Torres, D.F.M.
Polish Academy of Sciences
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem.
Dryl, M., Malinowska, A. B., Torres, D. F. M.
Both inflation and unemployment inflict social losses. When a tradeoff exists between the two, what would be the best combination of inflation and unemployment? A well known approach in economics to address this question is writing the social loss as a function of the rate of inflation p and the rate of unemployment u, with different weights, and then, using known relations between p, u, and the expected rate of inflation π, to rewrite the social loss function as a function of π. The answer is achieved by applying the calculus of variations in order to find an optimal path π that minimizes total social loss over a given time interval. Economists dealing with this question use a continuous or a discrete variational problem. Here we propose to use a time-scale model, unifying the results available in the literature. Moreover, the new formalism allows for obtaining new insights into the classical models when applied to real data of inflation and unemployment.
Sidi Ammi, M. R., Malinowska, A. B., Torres, D. F. M.
We study the optimal control of a steady-state dead oil isotherm problem. The problem is described by a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of mechanics of a continuous medium. Existence and regularity results of the optimal control are proved, as well as necessary optimality conditions.
Odzijewicz, T., Malinowska, A. B., Torres, D. F. M.
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether's theorem, a formulation of Dirichlet's principle and an uniqueness result are given. © 2013 EDP Sciences and Springer.
Debbouche, A., Torres, D. F. M.
We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus and Schauders fixed point theorem. Multi-delay controls and a fractional nonlocal condition are introduced. Furthermore, we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi-delay control system to be approximately controllable. An example to illustrate the abstract results is given. © 2013 Taylor & Francis.
Pooseh, S., Almeida, R., Torres, D.F.M.
Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grünwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.
We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions, considering reinfection and post-exposure interventions. They depend on the parameters of the model and reduce effectively the number of active infectious and persistent latent individuals. The time that the optimal controls are at the upper bound increase with the transmission coefficient. A general explicit expression for the basic reproduction number is obtained and its sensitivity with respect to the model parameters is discussed. Numerical results show the usefulness of the optimization strategies. © 2013 Elsevier Inc.
Frederico, G. S. F., Torres, D. F. M.
We prove Noether-type theorems for fractional isoperimetric variational problems with Riemann-Liouville derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples, in the fractional context of the calculus of variations, are discussed. © 2013 Polish Scientific Publishers.
We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo. © 2013 Versita Warsaw and Springer-Verlag Wien.
The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation. © 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society.
Sidi Ammi, M. R., Torres, D. F. M.
We obtain, by using the Leggett-Williams fixed point theorem, sufficient conditions that ensure the existence of at least three positive solutions to some p-Laplacian boundary value problems on time scales. © 2013 Moulay Rchid Sidi Ammi and Delfim F. M. Torres.
We study three types of generalized partial fractional order operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.
We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can be symmetric differentiable. © 2012 Elsevier Ltd. All rights reserved.
Lazo, M. J., Torres, D. F. M.
Derivatives and integrals of noninteger order were introduced more than three centuries ago but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler-Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois-Reymond lemma and showing how Euler-Lagrange equations involving only Caputo derivatives can be obtained. © 2012 Springer Science+Business Media New York.
Girejko, E., Torres, D. F. M.
The problem of the existence of solutions to nabla differential equations and nabla differential inclusions on time scales is considered. Under a special form of the set-valued constraint map, sufficient conditions for the existence of at least one solution, that stays in the constraint set, are derived. © 2011 Elsevier Ltd. All rights reserved.
We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the calculus of variations with general fractional operators. Necessary optimality conditions of Euler-Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann-Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered.
Ammi, M.R.S., Torres, D.F.M.
We are concerned with the optimal control problem of the well known nonlocal thermistor problem, i.e. in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature. Existence of an optimal control is proved. The optimality system consisting of the state system coupled with adjoint equations is derived, together with a characterisation of the optimal control. Uniqueness of solution to the optimality system, and therefore, the uniqueness of the optimal control, is established. The last part is devoted to numerical simulations.
Cresson, J., Malinowska, A. B., Torres, D. F. M.
We introduce differential, integral, and variational delta embeddings. We prove that the integral delta embedding of the Euler-Lagrange equations and the variational delta embedding coincide on an arbitrary time scale. In particular, a new coherent embedding for the discrete calculus of variations that is compatible with the least-action principle is obtained. © 2012 Elsevier Ltd. All rights reserved.
Martins, N., Torres, D. F. M.
We obtain necessary optimality conditions for higher-order infinite horizon problems of the calculus of variations via discrete quantum operators. © 2012 Elsevier Ltd. All rights reserved.
Malinowska, A. B., Torres, D. F. M.
A fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type. Editorial Note: The authors of this paper (A.B. Malinowska and D.F.M. Torres), together with T. Odzijewicz, have recently received a prestigeous award at the 5th International Symposium "Fractional Differentiation and Applications' 2012" in China, May 14-17, 2012. This is the "Gr̈unwald- Letnikov Award" for Best FDA Student Paper (theory), FDA #084: Green's Theorem for Generalized Fractional Derivatives. See details at http://em.hhu.edu.cn/fda12/Awards.html. © 2012 Diogenes Co., Sofia.
We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed. Copyright 2012 Tatiana Odzijewicz et al.
Ammi, M.R.S., El Kinani, E.H., Torres, D.F.M.
Using a fixed point theorem in a Banach algebra, we prove an existence result for a fractional functional differential equation in the Riemann- Liouville sense. Dependence of solutions with respect to initial data and an uniqueness result are also obtained.
We obtain Euler-Lagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and discrete settings: Our results seem new and interesting even in the particular cases when the time scale is the set of real numbers or the set of integers. © 2012 Springer Science+Business Media, LLC.
Girejko, E., Malinowska, A.B., Torres, D.F.M.
The calculus of variations on time scales is considered. We propose a new approach to the subject that consists of applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler-Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results. © 2012 Taylor and Francis Group, LLC.
Aldwoah, K. A., Malinowska, A. B., Torres, D. F. M.
Watam Press
We introduce the power difference calculus based on the operator D n; qf(t) =f(qt n)-f(t)qt n-t , where n is an odd positive integer and 0 < q < 1. Properties of the new operator and its inverse - the d n; q integral - are proved. As an application, we consider power quantum Lagrangian systems and corresponding n; q-Euler-Lagrange equations. © 2012 Watam Press.
We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. EulerLagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved. © 2011 Elsevier Ltd. All rights reserved.
Rodrigues, H. S., Monteiro, M. T. T., Torres, D. F. M., Zinober, A.
Dengue is one of the major international public health concerns. Although progress is underway, developing a vaccine against the disease is challenging. Thus, the main approach to fight the disease is vector control. A model for the transmission of dengue disease is presented. It consists of eight mutually exclusive compartments representing the human and vector dynamics. It also includes a control parameter (insecticide) in order to fight the mosquito. The model presents three possible equilibria: two disease-free equilibria (DFE) and another endemic equilibrium. It has been proved that a DFE is locally asymptotically stable, whenever a certain epidemiological threshold, known as the basic reproduction number, is less than one. We show that if we apply a minimum level of insecticide, it is possible to maintain the basic reproduction number below unity. A case study, using data of the outbreak that occurred in 2009 in Cape Verde, is presented. Copyright © 2012 Taylor and Francis Group, LLC.
The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for multiobjective fractional variational problems. © 2011 Elsevier Inc. All rights reserved.
Khosravian-Arab, H., Torres, D.F.M.
CSP; I&S Publishers
It is well known that for every f ε Cm there exists a polynomial pn such that pn (k) → f(k), k = 0,..m,Here we prove such a result for fractional (non-integer) derivatives. Moreover, a numerical method is proposed for fractional differential equations. The convergence rate and stability of the proposed method are obtained. Illustrative examples are discussed.
Pereira, Ricardo, Simões, Rita, Rocha, Paula
The new concept of 2D structured system is defined and characterizations of global reachability are obtained. This paper extends well known results for the 1D case, according to which a structured system (Aλ, Bλ) is (generically) reachable if and only if its graph is spanned by a cactus, or, equivalently, if and only if the pair (Aλ, Bλ) is full generically row rank and irreducible.
As the development of a dengue vaccine is ongoing, we simulate an hypothetical vaccine as an extra protection to the population. In a first phase, the vaccination process is studied as a new compartment in the model, and different ways of distributing the vaccines investigated: pediatric and random mass vaccines, with distinct levels of efficacy and durability. In a second step, the vaccination is seen as a control variable in the epidemiological process. In both cases, epidemic and endemic scenarios are included in order to analyze distinct outbreak realities. © 2013 Elsevier Inc.
We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler{Lagrange type and a sufficient optimality condi- tion for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided.
Dryl, M., Torres, D.F.M.
American Institute of Mathematical Sciences
We prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and its nabla derivative, as well as a nabla indefinite integral that depends on the unknown function.
We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations and optimal control with delays.
We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions. Numerical simulations are given using data from Angola.
Frederico, G. S. F., Odzijewicz, T., Torres, D. F. M.
We obtain a non-smooth extension of Noether's symmetry theorem for variational problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed Euler-Lagrange extremals are restricted to those that satisfy the DuBois-Reymond necessary optimality condition. The important case of delayed variational problems with higher order derivatives is considered as well. © 2013 Taylor & Francis.
We prove a necessary condition for a dynamic integrodifferential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. An example of a second order dynamic equation, which is not an Euler-Lagrange equation on an arbitrary time scale, is given. © 2014 Monika Dryl and Delfim F. M. Torres.
Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler-Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives. © 2014 Taylor & Francis.
We introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions. Approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a Hilbert space is studied. The results are obtained by using the fractional power of operators, multi-valued analysis, and Sadovskii's fixed point theorem. Main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable. As an example, a fractional partial nonlocal control functional differential inclusion is considered. © 2014 Elsevier Inc. All rights reserved.
Silva, F. C., Simões, R.
The concept of rank partition of a family of vectors v1,..., vm is a generalization of that has been useful for studying problems in Multilinear Algebra, namely, establishing conditions for non-vanishing decomposable symmetrized tensors and conditions for the equality of decomposable symmetrized tensors. A previous paper has described the rank partitions that can be obtained with arbitrarily small perturbations of the vectors v 1,..., v m. The purpose of the present article is to describe the pairs of row rank partitions and column rank partitions that can be obtained with arbitrarily small perturbations of a matrix. © 2011 Copyright Taylor and Francis Group, LLC.
Plakhov, Alexander, Cruz, João Pedro Antunes Ferreira da
An algorithm of searching a zero of an unknown function $\vphi : \, \R \to \R$ is considered: $\, x_{t} = x_{t-1} - \gamma_{t-1} y_t$,\, $t=1,\ 2,\ldots$, where $y_t = \varphi(x_{t-1}) + \xi_t$ is the value of $\vphi$ measured at $x_{t-1}$ and $\xi_t$ is the measurement error. The step sizes $\gam_t > 0$ are modified in the course of the algorithm according to the rule: $\, \gamma_t = \min\{u\, \gamma_{t-1},\, \mstep\}$ if $y_{t-1} y_t > 0$, and $\gamma_t = d\, \gamma_{t-1}$, otherwise, where $0 < d < 1 < u$,\, $\mstep > 0$. That is, at each iteration $\gam_t$ is multiplied either by $u$ or by $d$, provided that the resulting value does not exceed the predetermined value $\mstep$. The function $\vphi$ may have one or several zeros; the random values $\xi_t$ are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on $\vphi$, $\xi_t$, and $\mstep$, the conditions on $u$ and $d$ guaranteeing a.s. convergence of the sequence $\{ x_t \}$, as well as a.s. divergence, are determined. In particular, if $\P (\xi_1 > 0) = \P (\xi_1 < 0) = 1/2$ and $\P (\xi_1 = x) = 0$ for any $x \in \R$, one has convergence for $ud < 1$ and divergence for $ud > 1$. Due to the multiplicative updating rule for $\gam_t$, the sequence $\{ x_t \}$ converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximates one of the zeros of $\vphi$. By adjusting the parameters $u$ and $d$, one can reach arbitrarily high precision of the approximation; higher precision is obtained at the expense of lower convergence rate.
Aleksenko, A., Plakhov, A.
We consider a body in a parallel flow of non-interacting particles. The interaction of particles with the body is perfectly elastic. We introduce the notions of a body of zero resistance, a body that leaves no trace, and an invisible body, and prove that all such bodies do exist. © 2009 IOP Publishing Ltd and London Mathematical Society.
Cerveira, Adelaide, Agra, Agostinho, Bastos, Fernando, Gromicho, Joaquim
Springer US
Our paper considers a classic problem in the field of Truss Topology Design, the goal of which is to determine the stiffest truss, under a given load, with a bound on the total volume and discrete requirements in the cross-sectional areas of the bars. To solve this problem we propose a new two-stage Branch and Bound algorithm. In the first stage we perform a Branch and Bound algorithm on the nodes of the structure. This is based on the following dichotomy study: either a node is in the final structure or not. In the second stage, a Branch and Bound on the bar areas is conducted. The existence or otherwise of a node in this structure is ensured by adding constraints on the cross-sectional areas of its incident bars. In practice, for reasons of stability, free bars linked at free nodes should be avoided. Therefore, if a node exists in the structure, then there must be at least two incident bars on it, unless it is a supported node. Thus, a new constraint is added, which lower bounds the sum of the cross-sectional areas of bars incident to the node. Otherwise, if a free node does not belong to the final structure, then all the bar area variables corresponding to bars incident to this node may be set to zero. These constraints are added during the first stage and lead to a tight model. We report the computational experiments conducted to test the effectiveness of this two-stage approach, enhanced by the rule to prevent free bars, as compared to a classical Branch and Bound algorithm, where branching is only performed on the bar areas.
Agra, Agostinho, Constantino, Miguel
We consider the mixed-integer set X = {(s, x, y) ∈ R × Z^n × Z^m : s + a1 x j ≥ bj , ∀ j ∈ N1, s + a2 y j ≥ dj, j ∈ N2} where N1 = {1,...,n}, N2 = {1,...,m} and a1, a2 ∈ Z_+\{0}. This set may arise in a relaxation of mixed-integer problems such as lot-sizing problems.We decompose X into a small number of subsets whose convex hull description is trivial. The convex hull of X is equal to the closure of the convex hull of the union of those polyhedra. Using a projection theorem from Balas (Discret Appl Math 89:3–44, 1998) we obtain a compact characterization of the facets of the convex hull of X. Then by studying the projection cone we characterize all the facet-defining inequalities of the convex hull of X in the space of the original variables. Each of those inequalities is either a mixed MIR inequality (Günlük and Pochet in Math Programm 90:429–457, 2001), or it is based on a directed cycle on a special bipartite graph. When a1 and a2 are relative prime, the convex hull of X is described by the mixed MIR inequalities.
Cruz, João Pedro Antunes Ferreira da, Plakhov
We consider the following stochastic approximation algorithm of searching for the zero point x∗ of a function ϕ: xt+1 = xt − γtyt, yt = ϕ(xt) + ξt, where yt are observations of ϕ and ξt is the random noise. The step sizes γt of the algorithm are random, the increment γt+1 − γt depending on γt and on yt yt−1 in a rather general form. Generally, it is meant that γt increases as ytyt−1 > 0, and decreases otherwise. It is proved that the algorithm converges to x∗ almost surely. This result generalizes similar results of Kesten (1958) and Plakhov and Almeida (1998), where γt+1 − γt is assumed to depend only on γt and sgn(ytyt−1) and not on the magnitude of ytyt−1. CEOC
Kostyukova, O., Tchemisova, T., Yermalinskaya, S.
We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x0 of the convex SIP problem by such conditions for x0 in some nonlinear programming (NLP) problem denoted by NLP(I(x0)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x0)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.
Plakhov, A., Roshchina, V.
The problem of invisibility for bodies with a mirror surface is studied in the framework of geometrical optics. A closely related problem concerning the existence of bodies that have zero aerodynamical resistance is also studied here. We construct bodies that are invisible/have zero resistance in two directions, and prove that bodies which are invisible/have zero resistance do not exist in all possible directions of incidence. © 2011 IOP Publishing Ltd & London Mathematical Society.
Bachurin, P., Khanin, K., Marklof, J., Plakhov, A.
We construct semi infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties. © 2011 AIMSciences.
Plakhov, A., Tchemisova, T., Gouveia, P.
We study the Magnus effect: deflection of the trajectory of a spinning body moving in a gas. It is well known that in rarefied gases, the inverse Magnus effect takes place, which means that the transversal component of the force acting on the body has opposite signs in sparse and relatively dense gases. The existing works derive the inverse effect from nonelastic interaction of gas particles with the body. We propose another (complementary) mechanism of creating the transversal force owing to multiple collisions of particles in cavities of the body surface. We limit ourselves to the two-dimensional case of a rough disc moving through a zero-temperature medium on the plane, where reflections of the particles from the body are elastic and mutual interaction of the particles is neglected. We represent the force acting on the disc and the moment of this force as functionals depending on 'shape of the roughness;, and determine the set of all admissible forces. The disc trajectory is determined for several simple cases. The study is made by means of billiard theory, Monge-Kantorovich optimal mass transport and by numerical methods. This journal is © 2010 The Royal Society.
Plakhov, A., Aleksenko, A.
Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric bodies satisfying the single impact assumption [Comte and Lachand-Robert, J. Anal. Math. 83 (2001) 313-335]. In particular, the minimal resistance in our class is smaller than in Newton's problem; the ratio goes to 1/2 as (length)/(width of the body) → 0, and to 1/4 as (length)/(width) → +∞. © EDP Sciences, SMAI, 2008.
Mishuris, G., Plakhov, A.
MAGNUS EFFECT consists in deflection of the trajectory of a rotating body moving in a gas. It is a direct consequence of the interaction between the body surface and the gas particles. In this paper, we study the so-called inverse Magnus effect which can be observed in rarefied gases. We restrict ourselves to the two-dimensional case, namely a spinning disc moving through a sparse zero-temperature medium. We consider general non-elastic interaction between the disc and the particles depending on the incidence angle. We give a classification of auxiliary parameters with respect to possible dynamical response. In the absence of other forces, three kinds of trajectories are possible: (i) a converging spiral, (ii) a curve converging to a straight line and (iii) a circumference, the case intermediate between the two first ones. A specific 2-D parameter space has been introduced to provide respective classification. © 2009 by IPPT PAN.
Plakhov, A.
Recently Comte and Lachand-Robert [SIAM J. Math. Anal., 34 (2002), pp. 101-120] stated a very interesting and actual problem of minimizing mean specific resistance of infinite surfaces in a parallel flow of noninteracting point particles. They also constructed surfaces having resistance 0.593 and proved that they are minimizers. Unfortunately, their proof is incorrect. In this comment we provide a counterexample showing that the least value of resistance is not attained and is less than 0.581 (but greater than or equal to 0.5). Therefore, the problem remains open. Copyright by SIAM.
Plakhov, A.Yu.
This paper contains results relating to billiards and their applications to various resistance optimization problems generalizing Newton's aerodynamic problem. The results can be divided into three groups. First, minimum resistance problems for bodies moving translationally in ahighly rarefied medium are considered. It is shown that generically the infimum of the resistance is zero, that is, there are almost 'perfectly streamlined' bodies. Second, arough body is defined and results on characterization of billiard scattering on non-convex and rough bodies are presented. Third, these results are used to reduce some problems on minimum and maximum resistance of moving and slowly rotating bodies to special problems on optimal mass transfer, which are then explicitly solved. In particular, the resistance of a3-dimensional convex body can be at most doubled or at most reduced by3.05% by grooving its surface. Bibliography: 27 titles. © 2009 Russian Academy of Sciences, (DoM) and London Mathematical Society, Turpion Ltd.
A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body's shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ⊂ ℝ2 we assign a measure νB on ∂(conv B)×[-π/2, π/2] generated by the billiard in ℝ2\B and characterize the set of measures {νB}. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge-Kantorovich problems. © Springer-Verlag 2008.
Aleksenko, A.I., Plakhov, A.Yu.
Resumo não disponível...
A body moving in a rarefied medium of point fixed particles is considered. The center of mass of the body moves with a constant speed, and, moreover, the body executes slow rotational motions. The resistance force of the medium to the body's motion is defined. The concept of a rough body is introduced, and it is proved that the ratio between the resistance of the rough body and that of the smooth body corresponding to it lies in the limits from 0.969 up to 2. © 2007 Springer Science+Business Media, Inc.
The Monge-Kantorovich problem on finding a measure realizing the transportation of mass from ℝ to ℝ at minimum cost is considered. The initial and resulting distributions of mass are assumed to be the same and the cost of the transportation of the unit mass from a point x to y is expressed by an odd function f(x - y) that is strictly concave on ℝ+. It is shown that under certain assumptions about the distribution of the mass the optimal measure belongs to a certain family of measures depending on countably many parameters. This family is explicitly described: it depends only on the distribution of the mass, but not on f. Under an additional constraint on the distribution of the mass the number of the parameters is finite and the problem reduces to the minimization of a function of several variables. Examples of various distributions of the mass are considered.
Consider a body Ω at rest in d-dimensional Euclidean space and a homogeneous flow of particles falling on it with unit velocity υ. The particles do not interact and they collide with the body perfectly elastically. Let ℛΩ(υ) be the resistance of the body to the flow. The problem of the body of minimum resistance, which goes back to Newton, consists in the minimization of the quantity (ℛΩ(υ) | υ) over a prescribed class of bodies. Assume that one does not know in advance the direction υ of the flow or that one measures the resistance repeatedly for various directions of υ. Of interest in these cases is the problem of the minimization of the mean value of the resistance ℛ̃(Ω) = ∫ Sd-1 (ℛΩ(υ) | (υ) dυ. This problem is considered (P̃d) in the class of bodies of volume 1 and (P̃dc) in the class of convex bodies of volume 1. The solution of the convex problem P̃dc is the d-dimensional ball. For the non-convex 2-dimensional problem P̃2 the minimum value ℛ̃(Ω ) is found with accuracy 0.61%. The proof of this estimate is carried out with the use of a result related to the Monge problem of mass transfer, which is also solved in this paper. This problem is as follows: find inf T∈script T sign ∫ Π f(ℓ, τ; T(ℓ τ)) dμ(ℓ, τ), where Π = [-π/2, π/2] × [0,1], dμ(ℓ, τ) = cos ℓ dℓ dτ, f(ℓ, τ ℓ′, τ′) = 1 + cos(ℓ + ℓ′), and script T sign is the set of one-to-one maps of Π onto itself preserving the measure μ. Another problem under study is the minimization of ℛ̄(Ω) = ∫ Sd-1 |ℛΩ(υ)| dυ. The solution of the convex problem P̄dc and the estimate for the non-convex 2-dimensional problem P̄2 obtained in this paper are the same as for the problems P̃dc and P̃2.
We consider Newton's problem of minimal resistance for unbounded bodies in Euclidean space ℝd, d ≥ 2. A homogeneous flow of noninteracting particles of velocity v falls onto an immovable body containing a half-space {x : (x, n) < 0} ⊂ ℝd, (v, n) < 0. No restriction is imposed on the number of (elastic) collisions of the particles with the body. For any Borel ser A ⊂ {v}⊥ of finite measure, consider the flow of cross-section A: the part of initial flow that consists of particles passing through A. We construct a sequence of bodies that minimize resistance to the flow of cross-section A, for arbitrary A. This sequence approximates the half-space; any particle collides with any body of the sequence at most twice. The infimum of resistance is always one half of corresponding resistance of the half-space. © 2004 Plenum Publishing Corporation.
Kostyukova, Olga I., Tchemisova, Tatiana V., Yermalinskaya, Svetlana A.
CESER Publications
We consider convex Semi-Infinite Programming (SIP) problems with a continuum of constraints. For these problems we introduce new concepts of immobility orders and immobile indices. These concepts are objective and important characteristics of the feasible sets of the convex SIP problems since they make it possible to formulate optimality conditions for these problems in terms of optimality conditions for some NLP problems (with a finite number of constraints). In the paper we describe a finite algorithm (DIO algorithm) of determination of immobile indices together with their immobility orders, study some important properties of this algorithm, and formulate the Implicit Optimality Criterion for convex SIP without any constraint qualification conditions (CQC). An example illustrating the application of the DIO algorithm is provided.
We present a constructive approach to solving convex programming problems in separable form and new constructive methods for simultaneous solving pairs of primal and dual geometric programming problems. These methods are based on the new principle of accumulation of approximate functions, which involves an approximation of the objective functions by a special function (which is minorant for the primal problem and majorant for the dual problem) constructed in a form that uses certain information about some preceding steps of the method. The choice of the optimal direction for current iteration for solution of primal or dual problem is performed basing on extreme properties of this function. The construction of the approximate functions is based on the convexity, concavity, and separability properties of objective functions and on duality results. The support of the problem which is the index set of independent variables is modified from one iteration to another. The method involves iterations of primal and dual type. During the iterations of primal, type we resolve the primal problem using information about the best dual approximation obtained up to this moment. During the iterations of dual type the dual problem is solved and the best primal approximation is used. The method interlaces primal and dual iterations providing the possibility of their interaction.
Plakhov, Alexander
Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object — notched angle — is a new one; a proof of its retroreflectivity is given.Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object — notched angle — is a new one; a proof of its retroreflectivity is given.
The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on T3 describing billiard scattering on the body. The main result is characterization of the set of measures generated by rough bodies. This result can be used to solve various problems of least aerodynamical resistance.
Napp Avelli, Diego, Perea, Carmen, Pinto, Raquel
Two-dimensional convolutional codes are considered, with codewords having compact support indexed in N^2 and taking values in F^n, where F is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented. © 2010 AIMS-SDU.
Kuijper, M., Pinto, R.
Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also well known that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = ℤ_{p^r} by introducing a so-called "p-encoder". We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over ℤ_{p^r} to produce a particular type of p-encoder ("minimal p-encoder") whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p^γ, where γ is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over ℤ_{p^r} admits a delay-free p-encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over ℤ_{p^r} admits a noncatastrophic p-encoder. © 2009 IEEE.
Barrios Rolanía, Dolores, Branquinho, Amilcar, Foulquié Moreno, Ana Pilar
The correspondence between a high-order non-symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for our integrable systems. The method of investigation is based on the evolutions of the matrical moments.
Branquinho, Amílcar, Cotrím, Luís, Foulquié Moreno, Ana Pilar
In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given.
Branquinho, Amílcar, Foulquie Moreno, Ana Pilar, Marcellán, Francisco, Rebocho, Maria das Neves
In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle. Some examples are given.
Branquinho, Amílcar, Fidalgo Prieto, Ulises, Moreno, Ana Pilar Foulquie
Angelesco systems of measures with Jacobi-type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szego-type functions. In the procedure, an approach from the Riemann Hilbert problem plays a fundamental role.
Branquinho, Amílcar, Barrios Rolania, Dolores, Foulquie Moreno, Ana Pilar
High-order non-symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems.
Branquinho, Amílcar, Bustamante, J, Foulquie Moreno, Ana Pilar, López Lagomasino, Guillermo
We improve the class of indices for which normality takes place in a Nikishin system and apply this in Hermite-Padé approximation of such systems of functions.
Xarez, JJ
Let K : B -> A be a functor such that the image of the objects in B is a cogenerating set of objects for A. Then, the right Kan extensions adjunction Set(K) (sic) Ran(K) induces necessarily an epireflection with stable units and a monotone-light factorization. This result follows from the one stating that an adjunction produces an epireflection in a canonical way, provided there exists a prefactorization system which factorizes all of its unit morphisms through epimorphisms. The stable units property, for the so obtained epireflections, is thereafter equivalently restated in such a manner it is easily recognizable in the examples. Furthermore, having stable units, there is a strong but quite simple sufficient condition for the existence of an associated monotone-light factorization, which has proved to be fruitful.
Pereira, Ricardo, Rocha, Paula
John Wiley and Sons
In this paper, we propose a definition of determinant for quaternionic, polynomial matrices inspired by the well-known Dieudonne determinant for the constant case. This notion allows to characterize the stability of linear dynamical systems with quaternionic coefficients, yielding results which generalize the ones obtained for the real and complex cases.
Pereira, Ricardo, Vettori, Paolo
The main goal of this paper is to characterize stability and bounded-input-bounded-output (BIBO)-stability of quaternionic dynamical systems. After defining the quaternion skew-field, algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated. Having established these results, the Smith and the Smith-McMillan forms of quaternionic matrices are introduced and studied. Finally, all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework.
Pereira, Ricardo, Rocha, Paula, Vettori, Paolo
In this paper we study behavioral systems whose trajectories are given as solutions of quaternionic difference equations. As happens in the commutative case, it turns out that quaternionic polynomial matrices play an important role in this context. Therefore we pay special attention to such matrices and derive new results concerning their Smith form. Based on these results, we obtain a characterization of system theoretic properties such as controllability and stability of a quaternionic behavior.
Monteiro, M., Scotto, M.G., Pereira, I.
In this article, we introduce a class of self-exciting threshold integer-valued autoregressive models driven by independent Poisson-distributed random variables. Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the least squares-type and likelihood-based ones. Their performance is compared through a simulation study. Copyright © 2012 Taylor and Francis Group, LLC.