- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 11:51:44
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Portugaliae Mathematica
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PM
0032-5155
1662-2758
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10.4171/PM
http://www.ems-ph.org/doi/10.4171/PM
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Zuerich, Switzerland
© European Mathematical Society (from 2008)
68
2011
1
Denseness of ergodicity for a class of volume-preserving flows
Mário
Bessa
Universidade do Porto, PORTO, PORTUGAL
Jorge
Rocha
Universidade do Porto, PORTO, PORTUGAL
Dominated splitting, partial hyperbolicity, volume-preserving flows, Lyapunov exponents, stable ergodicity
We consider the class of $C^1$ partially hyperbolic volume-preserving flows with one-dimensional central direction endowed with the $C^1$-Whitney topology. We prove that, within this class, any flow can be approximated by an ergodic $C^2$ volume-preserving flow and so, as a consequence, ergodicity is dense.
General
1
17
10.4171/PM/1878
http://www.ems-ph.org/doi/10.4171/PM/1878
Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback
Serge
Nicaise
Université de Valenciennes et du Hainaut Cambrésis, VALENCIENNES CEDEX 9, FRANCE
Salah-eddine
Rebiai
Université de Batna, BATNA, ALGERIA
Schrö}dinger equation, time delays, feedback stabilization
In this paper, we investigate the effect of time delays in boundary or internal feedback stabilization of the Schrödinger equation. In both cases, under suitable assumptions, we establish sufficient conditions on the delay term that guarantee the exponential stability of the solution. These results are obtained by using suitable energy functionals and some observability estimates.
Systems theory; control
Partial differential equations
General
19
39
10.4171/PM/1879
http://www.ems-ph.org/doi/10.4171/PM/1879
Global existence for two regularized MHD models in three space-dimension
Davide
Catania
Facoltà di Ingegneria, Università di Brescia, BRESCIA, ITALY
Paolo
Secchi
Facoltà di Ingegneria, Università di Brescia, BRESCIA, ITALY
Magnetohydrodynamics, MHD-$\alpha$ model, Bardina model, regularizing MHD, turbulence models, incompressible fluid
The global existence of solutions for the 3D incompressible Euler equations is a major open problem. For the 3D inviscid MHD system, the global existence is an open problem as well. Our main concern in this paper is to understand which kind of regularization, of the form of $\alpha$-regularization or partial viscous regularization, is capable to provide the global in time solvability for the 3D inviscid MHD system of equations. We consider two different regularized magnetohydrodynamic models for an incompressible fluid. In both cases, we provide a global existence result for the solution of the system.
Partial differential equations
Fluid mechanics
General
41
52
10.4171/PM/1880
http://www.ems-ph.org/doi/10.4171/PM/1880
Euclidean distance matrices, semidefinite programming and sensor network localization
Abdo
Alfakih
University of Windsor, WINDSOR, ONTARIO, CANADA
Philippe
Charron
Université de Montréal, Montréal, QC, CANADA
Veronica
Piccialli
Università di Roma Tor Vergata, ROMA, ITALY
Henry
Wolkowicz
University of Waterloo, WATERLOO, ONTARIO, CANADA
Euclidean distance matrix completions, sensor network localization, fundamental problem of distance geometry, semidefinite programming
The fundamental problem of distance geometry involves the characterization and study of sets of points based only on given values of some or all of the distances between pairs of points. This problem has a wide range of applications in various areas of mathematics, physics, chemistry, and engineering. Euclidean distance matrices play an important role in this context by providing elegant and powerful convex relaxations. They play an important role in problems such as graph realization and graph rigidity. Moreover, by relaxing the embedding dimension restriction, these matrices can be used to approximate the hard problems efficiently using semidefinite programming. Throughout this survey we emphasize the interplay between these concepts and problems. In addition, we illustrate this interplay in the context of the sensor network localization problem.
Geometry
Convex and discrete geometry
Operations research, mathematical programming
General
53
102
10.4171/PM/1881
http://www.ems-ph.org/doi/10.4171/PM/1881
Special divisors of large dimension on curves with many points over finite fields
José Felipe
Voloch
University of Texas at Austin, AUSTIN, UNITED STATES
Algebraic curves over finite fields, special divisors, modular forms
We prove a non-existence result for special divisors of large dimension on curves over finite fields with many points. We also give a family of examples where such divisors exist under less stringent hypotheses.
Algebraic geometry
Number theory
General
103
107
10.4171/PM/1882
http://www.ems-ph.org/doi/10.4171/PM/1882
On the porosity of the free boundary in the $p(x)$-obstacle problem
Samia
Challal
York University, TORONTO, ONTARIO, CANADA
Abdeslem
Lyaghfouri
Research in Mathematical Sciences, TORONTO, ONTARIO, CANADA
Obstacle problem, $p(x)$-Laplacian, free boundary, porosity
In this paper we consider the obstacle problem for the $p(x)$-Laplace operator. Assuming that $p$ is locally Lipschitz continuous, we establish the growth rate of the solution near the free boundary from which we deduce its porosity.
Partial differential equations
General
109
123
10.4171/PM/1883
http://www.ems-ph.org/doi/10.4171/PM/1883
2
Fixed point theorems for 1-set weakly contractive and pseudocontractive operators on an unbounded domain
Afif
Ben Amar
Université de Gafsa, GAFSA, TUNISIA
Jesús
Garcia-Falset
Universitat de València, BURJASSOT (València), SPAIN
Weakly 1-set contractions, pseudocontractive maps, measures of weak noncompactness, fixed points
We establish new fixed point results for some nonlinear weakly condensing, 1-set weakly contractive, pseudo-contractive and nonexpansive operators defined on unbounded domains under different boundary conditions as well as other additional assumptions. An existence result of positive eigenvalues and eigenvectors for nonlinear weakly condensing operators and an application to generalized Hammerstein integral equations are given.
Operator theory
General
125
147
10.4171/PM/1884
http://www.ems-ph.org/doi/10.4171/PM/1884
Linear motions in a periodically forced Kepler problem
Rafael
Ortega
Universidad de Granada, GRANADA, SPAIN
Collision, Sundman’s integral, periodic solution, twist map
The periodically forced Kepler problem has at most one classical periodic solution and very simple dynamics. In this paper it is shown that when collisions are considered, many other periodic solutions appear as well as a dynamics of twist type.
Ordinary differential equations
Dynamical systems and ergodic theory
Mechanics of particles and systems
General
149
176
10.4171/PM/1885
http://www.ems-ph.org/doi/10.4171/PM/1885
On the endomorphism monoid of a profinite semigroup
Benjamin
Steinberg
City College of New York, NEW YORK, UNITED STATES
Profinite semigroups, endomorphism monoids, automorphism groups
Necessary and sufficient conditions are given for the endomorphism monoid of a profinite semigroup to be profinite. A similar result is established for the automorphism group.
Topological groups, Lie groups
Group theory and generalizations
General
177
183
10.4171/PM/1886
http://www.ems-ph.org/doi/10.4171/PM/1886
The Fibonacci version of the Brocard–Ramanujan Diophantine equation
Diego
Marques
Universidade de Brasília, BRASÍLIA-DF, BRAZIL
Diophantine equation, Fibonacci, Brocard–Ramanujan
In this note we prove that the Fibonacci version of the Brocard–Ramanujan Diophantine equation $n!+1=m^2$, that is, $F_n\dots F_1+1=F_m^2$ has no solution in positive integers $n$, $m$.
Number theory
General
185
189
10.4171/PM/1887
http://www.ems-ph.org/doi/10.4171/PM/1887
Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory
Melvyn
Nathanson
Lehman College, CUNY, BRONX, UNITED STATES
Bi-Lipschitz equivalence, metric geometry, g-adic representation, geometric group theory, additive number theory, combinatorial number theory
There is a standard “word length” metric canonically associated to any set of generators for a group. In particular, for any integers $a$ and $b$ greater than $1$, the additive group $\mathbb{Z}$ has generating sets $\{ a^i \}_{i=0}^{\infty}$ and $\{b^j\}_{j=0}^{\infty}$ with associated metrics $d_A$ and $d_B$, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers $m$ and $n$ such that $a^m = b^n$.
Number theory
Group theory and generalizations
Geometry
General
191
203
10.4171/PM/1888
http://www.ems-ph.org/doi/10.4171/PM/1888
Uniform decay rates of coupled anisotropic elastodynamic/Maxwell equations with nonlinear damping
Cleverson Roberto
da Luz
Universidade Federal de Santa Catarina, FLORIANOPOLIS - SC, BRAZIL
Gustavo Alberto
Perla Menzala
, PETROPOLIS, RJ, BRAZIL
Anisotropic Maxwell equations, anisotropic elastodynamic models, exterior domains, nonlinearly damped system, asymptotic behavior
This work is devoted to study the asymptotic behavior of the total energy associated with a coupled system of anisotropic hyperbolic models: the elastodynamic equations and Maxwell's system in the exterior of a bounded body in $\mathbb{R}^3$. Our main result says that in the presence of nonlinear damping, a unique solution of small initial data exists globally in time and the total energy as well as higher order energies decay at a uniform rate as $t \rightarrow + \infty$.
Number theory
General
205
238
10.4171/PM/1889
http://www.ems-ph.org/doi/10.4171/PM/1889
3
Agreeable solutions of variational problems
Alexander
Zaslavski
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Agreeable function, c-optimal function, good function, infinite horizon problem, integrand
In this paper we study solutions of infinite horizon variational problems associated with a certain class of integrands. We consider c-optimal solutions, which were introduced and used for models of solid-state physics and in the theory of thermodynamical equilibrium for materials and agreeable solutions introduced for models of economic dynamics. We show that if an integrand possesses an asymptotic turnpike property, then these two optimality notions are equivalent.
Calculus of variations and optimal control; optimization
General
239
257
10.4171/PM/1890
http://www.ems-ph.org/doi/10.4171/PM/1890
A universal enveloping algebra of Malcev superalgebras
Elisabete
Barreiro
Universidade de Coimbra, COIMBRA, PORTUGAL
Malcev superalgebras, universal enveloping algebra
In this paper an enveloping superalgebra is presented for Malcev superalgebra. An extension of the Poincaré–Birkhoff–Witt Theorem to this class of superalgebras is obtained.
Nonassociative rings and algebras
General
259
278
10.4171/PM/1891
http://www.ems-ph.org/doi/10.4171/PM/1891
On the stabilization and controllability for a third order linear equation
Patrícia Nunes
da Silva
Universidade do Estado do Rio de Janeiro, RIO DE JANEIRO - RJ, BRAZIL
Carlos Frederico
Vasconcellos
Universidade do Estado do Rio de Janeiro, RIO DE JANEIRO - RJ, BRAZIL
Exponential decay, stabilization, exact control
We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation: $$ iu_t+i\gamma u_x+ \alpha u_{xx} + i\beta u_{xxx} =0, $$ where $u=u(x,t)$ is a complex valued function defined in $(0,L)\times(0,+\infty)$ and $\alpha$, $\beta$ and $\gamma$ are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths $L$, named $\mathcal{X}$, in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters $\alpha$, $\beta$ and $\gamma$.
Partial differential equations
General
279
296
10.4171/PM/1892
http://www.ems-ph.org/doi/10.4171/PM/1892
Sums of seventh powers in the polynomial ring $\mathbb{F}_{2^{m}}[T]$
Mireille
Car
Université Aix-Marseille III, MARSEILLE CEDEX 20, FRANCE
Finite fields, polynomials, Waring’s problem
Let $F$ be a finite field with even characteristic and $q\geq16$ elements. We study representations of polynomials $P\in F[T]$ as sums $P = X_{1}^{7} +\cdots + X_{s}^{7}$.
Number theory
General
297
316
10.4171/PM/1893
http://www.ems-ph.org/doi/10.4171/PM/1893
Product approximations for solutions to a class of evolution equations in Hilbert space
Pierre-A.
Vuillermot
Université Henri Poincaré, VANDOEUVRE-LÈS-NANCY CEDEX, FRANCE
Walter
Wreszinski
Universidade de São Paulo, SÃO PAULO SP, BRAZIL
Schrödinger equations, weak solutions, weak operator limits
In this article we prove approximation formulae for a class of unitary evolution operators $U(t,s)_{s,t\in [0,T] }$ associated with linear non-autonomous evolution equations of Schr\"{o}dinger type defined in a Hilbert space $\mathcal{H}$. An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for $U(t,s)$ involve weak operator limits of products of suitable approximating functions taking values in $\mathcal{L(H)}$, the algebra of all linear bounded operators on $\mathcal{H}$. Our results may be relevant to the numerical analysis of $U(t,s)$ and we illustrate them by considering two typical examples, including one related to the theory of time-dependent singular perturbations of self-adjoint operators.
Operator theory
Numerical analysis
Quantum theory
General
317
343
10.4171/PM/1894
http://www.ems-ph.org/doi/10.4171/PM/1894
Null controllability of degenerate parabolic cascade systems
El Mustapha
Ait Ben Hassi
Université Cadi Ayyad, MARRAKECH, MOROCCO
Farid
Ammar Khodja
Université de Franche-Comté, BESANCON CEDEX, FRANCE
Abdelkarim
Hajjaj
Université Hassan 1, SETTAT, MOROCCO
Lahcen
Maniar
Université Cadi Ayyad, MARRAKECH, MOROCCO
Semigroups, Carleman estimates, degenerate parabolic equations, cascade, observability inequality, null controllability
In this paper, we study the null controllability of degenerate semilinear cascade parabolic systems with one control force. The key tool is the Carleman estimates developed recently for degenerate one dimension parabolic equations. We develop a Carleman estimate for these systems and then an observability inequality for the linear adjoint system. We conclude by linearization and fixed point arguments.
Partial differential equations
Systems theory; control
General
345
367
10.4171/PM/1895
http://www.ems-ph.org/doi/10.4171/PM/1895
4
Scarf lattice ideals
Hossein
Sabzrou
Institute for Research in Fundamental Sciences, TEHRAN, IRAN
Lattice ideals, minimal free resolution, Scarf complexes
This paper deals with the Scarf property of lattice ideals initiated by Peeva and Sturmfels[10], [11]. We will present a Scarf lattice ideal that is neither generic nor of codimension 2 and show that this property gives rise to several algebraic and combinatorial properties. In particular, we prove that for monomial curves, this property coincides with the notion of genericity, and that certain Scarf lattice ideals can have certain Scarf initial ideals.
Commutative rings and algebras
Convex and discrete geometry
General
369
380
10.4171/PM/1896
http://www.ems-ph.org/doi/10.4171/PM/1896
A note on n-gerbes and transgressions
Mauro
Spera
Università degli Studi di Verona, VERONA, ITALY
Gerbes, transgression, Euler class, geometric quantization, string structures
In this note we provide an explicit interpretation of a class of $(q-1)$-gerbes with multi-layered connections in terms of transgression of a fibrewise closed $q$-form on a fibration to a closed $(q+1)$-form on the base manifold, with the basic example of the Euler class of an oriented vector bundle in mind ($q \geq 0$). Picken's and Ferreira–Gothen’s $n$-gerbopoles are discussed from this point of view. Furthermore, string structures (à la Cocquereaux–Pilch and à la Spera–Wurzbacher) are briefly addressed and recast within the proposed framework.
Differential geometry
Algebraic topology
Manifolds and cell complexes
Quantum theory
381
387
10.4171/PM/1897
http://www.ems-ph.org/doi/10.4171/PM/1897
Global existence of small solutions to the Kerr–Debye model for the three-dimensional Cauchy problem
Mohamed
Kanso
Université Bordeaux 1, TALENCE CEDEX, FRANCE
Nonlinear Maxwell equations, Kerr model, Kerr–Debye model, Cauchy problem, global existence of solutions
We consider the Kerr–Debye model, describing the electromagnetic wave propagation in a nonlinear medium exhibiting a finite response time. This model is quasilinear hyperbolic and endowed with a dissipative entropy. We consider the Cauchy problem in the three-dimensional case and show that, if the initial data are sufficiently small, the solutions are global in time.
General
389
409
10.4171/PM/1898
http://www.ems-ph.org/doi/10.4171/PM/1898
Two results on the rank partition of a matroid
Andrew
Berget
University of California at Davis, DAVIS, UNITED STATES
Matroid, matroid polytope, rank partition, straightening law
The rank partition of a matroid M is the maximum dominance ordered partition $\rho$ such that the ground set of M can be partitioned into independent sets of sizes $\rho_1$, $\rho_2$, …. We prove two structural results on this partition, both motivated by representation theory of the general linear group. The first result characterizes the rank partition in terms of standard Young tableaux with a certain matroidal property. The second result says that the rank partition interacts nicely with certain polytopal decompositions of the matroid polytope of M. We also describe the representation theoretical motivation of these results.
Combinatorics
Geometry
General
411
423
10.4171/PM/1899
http://www.ems-ph.org/doi/10.4171/PM/1899
Cross varieties of aperiodic monoids with central idempotents
Edmond W. H.
Lee
Nova Southeastern University, FORT LAUDERDALE, UNITED STATES
Monoid, aperiodic monoid, central idempotent, variety, Cross variety
Let A denote the class of all aperiodic monoids with central idempotents. A description of all Cross subvarieties of A, based on identities that they satisfy and monoids that they cannot contain, is given. The two limit subvarieties of A, published by Marcel Jackson in 2005, turn out to be the only finitely generated, almost Cross subvarieties of A. It follows that it is decidable in quartic time if a finite monoid in A generates a Cross variety.
General
425
429
10.4171/PM/1900
http://www.ems-ph.org/doi/10.4171/PM/1900
Newton’s method and secant methods: A longstanding relationship from vectors to matrices
Marlliny
Monsalve
Universidad Central de Venezuela, CARACAS, VENEZUELA
Marcos
Raydan
Universidad Simón Bolívar, CARACAS, VENEZUELA
Newton's method, secant method, nonlinear matrix problems
Nonlinear matrix equations arise in different scientific topics, such as applied statistics, control theory, and financial mathematics, among others. As in many other scientific areas, Newton’s method has played an important role when solving these matrix problems. Under standard assumptions, the specialized Newton methods that have been developed for specific problems exhibit local and q-quadratic convergence and require a suitable initial guess. They also require, as usual, a significant amount of computational work per iteration, that in this case involve several matrix factorizations per iterations. As expected, whenever a Newton method can be developed, a secant method can also be developed. Indeed, more recently, secant methods for solving specific nonlinear matrix problems have been developed opening a new line of research. As in previous scenarios, these specialized secant methods exhibit local and q-superlinear convergence, also require a suitable initial guess, and avoid the use of derivatives in the formulation of the schemes. In this review we start by recalling the presence of Newton’s method and the secant methods, and also their classical relationship, in different and sometimes unexpected scenarios for vector problems. Then we present and describe the state of the art in the use of Newton's method and also the secant method in the space of matrices. A second objective is to present a unified approach for describing the features of these classical schemes, that in the space of matrices represent an interesting research area with special features to be explored.
Ordinary differential equations
Calculus of variations and optimal control; optimization
Game theory, economics, social and behavioral sciences
General
431
475
10.4171/PM/1901
http://www.ems-ph.org/doi/10.4171/PM/1901