86 results
Search Results
2. Affine focal sets of codimension-2 submanifolds contained in hypersurfaces
- Author
-
Marcelo José Saia, Marcos Craizer, and Luis F. Sánchez
- Subjects
Pure mathematics ,General Mathematics ,020207 software engineering ,02 engineering and technology ,Codimension ,GEOMETRIA DIFERENCIAL CLÁSSICA ,01 natural sciences ,Darboux vector ,0104 chemical sciences ,010404 medicinal & biomolecular chemistry ,Hypersurface ,Hyperplane ,Affine focal set ,0202 electrical engineering, electronic engineering, information engineering ,Tangent space ,Affine sphere ,Affine transformation ,Mathematics - Abstract
In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.
- Published
- 2018
3. Flows of measures generated by vector fields
- Author
-
Emanuele Paolini and Eugene Stepanov
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Lipschitz continuity ,01 natural sciences ,Measure (mathematics) ,Integral curve ,Flow (mathematics) ,Ordinary differential equation ,0103 physical sciences ,Vector field ,010307 mathematical physics ,0101 mathematics ,Borel measure ,Smooth structure ,Mathematics - Abstract
The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.
- Published
- 2018
4. Four notions of conjugacy for abstract semigroups
- Author
-
João Araújo, Michael Kinyon, António Malheiro, and Janusz Konieczny
- Subjects
Pure mathematics ,Endomorphism ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Representation theory ,Automaton ,Conjugacy class ,Areas of mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Special classes of semigroups ,0101 mathematics ,Mathematics - Group Theory ,Group theory ,Mathematics - Abstract
The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations or transformation semigroups). Here we study them in maximum generality. The paper ends with a large list of open problems., Comment: The paper is now more focused on abstract semigroups and a fourth notion of conjugacy was introduced for its importance in representation theory and finite semigroups
- Published
- 2017
5. The Lax–Oleinik semi-group: a Hamiltonian point of view
- Author
-
Patrick Bernard, Université Paris sciences et lettres (PSL), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), European Project: 307062,EC:FP7:ERC,ERC-2012-StG_20111012,SAW(2012), Université Paris Dauphine-PSL, École normale supérieure - Paris (ENS-PSL), and Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris)
- Subjects
Pure mathematics ,Kolmogorov–Arnold–Moser theorem ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,Fixed point ,Invariant (physics) ,01 natural sciences ,Convexity ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Compact space ,symbols ,Configuration space ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Mathematics - Abstract
International audience; The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on R d and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications. 1. The method of characteristics, existence and uniqueness of regular solutions We consider a C 2 Hamiltonian H(t, q, p) : R × R d × R d * → R and study the associated Hamiltonian system ˙ q(t) = ∂ p H(t, q(t), p(t)), ˙ p(t) = −∂ q H(t, q(t), p(t)), (HS) * This paper is a late addition to the papers surveying active areas in partial differential equations , published in issue 141.2, which were based on a series of mini-courses held in the International Centre for Mathematical Sciences (ICMS) in Edinburgh during 2010. and Hamilton–Jacobi equation ∂ t u + H(t, q, ∂ q u(t, q)) = 0. (HJ) We denote by X H (x) = X H (q, p) the Hamiltonian vector field X H = J dH, where J is the matrix J = 0 I −I 0. The Hamiltonian system can be written in condensed terms ˙ x(t) = X H (t, x(t)). We shall always assume that the solutions extend to R. We denote by ϕ t τ = (Q t τ , P t τ): R d
- Published
- 2012
6. On the polynomial vector fields on
- Author
-
Jaume Llibre and Yulin Zhao
- Subjects
Combinatorics ,Polynomial ,Polynomial vector fields ,Degree (graph theory) ,General Mathematics ,Homogeneous polynomial ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Vector field ,Mathematics - Abstract
Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.
- Published
- 2011
7. Singular solutions for the Uehling–Uhlenbeck equation
- Author
-
Juan J. L. Velázquez, Stéphane Mischler, and Miguel Escobedo
- Subjects
Physical point ,Regular singular point ,Singular function ,Semigroup ,Singular solution ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like k −7/6 as k → 0. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function k −7/6 that we obtained in an earlier paper.
- Published
- 2008
8. Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems
- Author
-
Messoud Efendiev, Alain Miranville, and Sergey Zelik
- Subjects
Projected dynamical system ,Dynamical systems theory ,General Mathematics ,Mathematical analysis ,Attractor ,Statistical physics ,Limit set ,Dynamical system ,Random dynamical system ,Mathematics ,Linear dynamical system ,Hamiltonian system - Abstract
We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.
- Published
- 2005
9. Constructing the symplectic Evans matrix using maximally analytic individual vectors
- Author
-
Gianne Derks and Thomas J. Bridges
- Subjects
Determinant ,Symplectic vector space ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Symplectic representation ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic matrix ,Mathematics ,Symplectic manifold ,Symplectic geometry - Abstract
For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.
- Published
- 2003
10. Lifting of recollements and gluing of partial silting sets
- Author
-
Alexandra Zvonareva and Manuel Saorín
- Subjects
Noetherian ,Pure mathematics ,General Mathematics ,01 natural sciences ,Lift (mathematics) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Category Theory (math.CT) ,16E35, 18E30 ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Equivalence (measure theory) ,Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Coproduct ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Rings and Algebras (math.RA) ,Bounded function ,Torsion (algebra) ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.
- Published
- 2021
11. Hyperasymptotics and the Stokes' phenomenon
- Author
-
A. B. Olde Daalhuis
- Subjects
symbols.namesake ,Integral representation ,General Mathematics ,Phenomenon ,Confluence ,Mathematical analysis ,symbols ,Hypergeometric function ,Series expansion ,Bessel function ,Mathematics - Abstract
SynopsisHyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.
- Published
- 1993
12. Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
- Author
-
Arkadiy Skopenkov and Diarmuid Crowley
- Subjects
Group (mathematics) ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,57R52, 57R67, 55R15 ,Mathematics::Geometric Topology ,01 natural sciences ,Connected sum ,010101 applied mathematics ,Combinatorics ,Mathematics - Geometric Topology ,Simply connected space ,FOS: Mathematics ,Torsion (algebra) ,Isotopy ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Invariant (mathematics) ,Signature (topology) ,Quotient ,Mathematics - Abstract
We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; \mathbb Z)$. Our main result is a readily calculable classification of embeddings $N\to\mathbb R^7$ up to isotopy, with an indeterminancy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\to\mathbb R^7$ acts on the set of embeddings $N\to\mathbb R^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1\ne0$, with an indeterminancy. Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008. For $N=S^1\times S^3$ we give a geometrically defined 1--1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_{12}$., Comment: 19 pages, exposition improved
- Published
- 2021
13. Bounded solutions for an ordinary differential system from the Ginzburg–Landau theory
- Author
-
Anne Beaulieu
- Subjects
Nonlinear system ,General Mathematics ,Ordinary differential equation ,Bounded function ,Linear system ,Applied mathematics ,Ginzburg–Landau theory ,Boundary value problem ,Upper and lower bounds ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg-Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg-Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem. AMS classification : 34B40: Ordinary Differential Equations, Boundary value problems on infinite intervals. 35J60: Nonlinear PDE of elliptic type. 35P15: Estimation of eigenvalues, upper and lower bound.
- Published
- 2020
14. Local minimizers in absence of ground states for the critical NLS energy on metric graphs
- Author
-
Nicola Soave, Gianmaria Verzini, and Dario Pierotti
- Subjects
Pure mathematics ,General Mathematics ,Structure (category theory) ,FOS: Physical sciences ,non-linear Schrödinger equation ,01 natural sciences ,Schrödinger equation ,symbols.namesake ,Mathematics - Analysis of PDEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,non-compact metric graphs ,0101 mathematics ,Mathematical Physics ,Mathematics ,Energy functional ,010102 general mathematics ,L^2-critical exponent ,Mathematical Physics (math-ph) ,010101 applied mathematics ,Constraint (information theory) ,Normalized solutions ,Mathematics - Classical Analysis and ODEs ,Metric (mathematics) ,symbols ,Negative energy ,Ground state ,Energy (signal processing) ,Analysis of PDEs (math.AP) - Abstract
We consider the mass-critical nonlinear Schr\"odinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in arXiv:1605.07666, where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved., Comment: Accepted for publication on Proceedings of the Royal Society of Edinburgh Section A: Mathematics
- Published
- 2020
15. All finite transitive graphs admit a self-adjoint free semigroupoid algebra
- Author
-
Adam Dor-On and Christopher Linden
- Subjects
Transitive relation ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,47L55, 47L15, 05C20 ,0102 computer and information sciences ,Graph algebra ,Directed graph ,Disjoint sets ,01 natural sciences ,Algebra ,Semigroupoid ,010201 computation theory & mathematics ,FOS: Mathematics ,0101 mathematics ,Algebra over a field ,Operator Algebras (math.OA) ,Self-adjoint operator ,Mathematics - Abstract
In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is $B(\mathcal{H})$. This is accomplished through a new construction that reduces this problem to in-degree $2$-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras., Comment: Added missing reference. 16 pages 2 figures
- Published
- 2020
16. Nonsingular bilinear maps revisited
- Author
-
Carlos Dominguez and Kee Yuen Lam
- Subjects
General Mathematics ,010102 general mathematics ,Bilinear interpolation ,Vector bundle ,Mathematics - Rings and Algebras ,01 natural sciences ,law.invention ,010101 applied mathematics ,Algebra ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics::Algebraic Geometry ,Invertible matrix ,Perspective (geometry) ,Rings and Algebras (math.RA) ,law ,FOS: Mathematics ,Immersion (mathematics) ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Bilinear map ,Mathematics - Abstract
More than 47 years have passed without any new example of nonsingular bilinear maps appearing in literature. The purpose of this paper is to construct a new family of nonsingular bilinear maps., Comment: This version has a new title and a joint authorship
- Published
- 2020
17. Bilinear forms on potential spaces in the unit circle
- Author
-
Carme Cascante and Joaquin M. Ortega
- Subjects
Pure mathematics ,Functional analysis ,Equacions en derivades parcials ,General Mathematics ,Espais de Sobolev ,Potential theory (Mathematics) ,010102 general mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Bilinear form ,Partial differential equations ,01 natural sciences ,Teoria del potencial (Matemàtica) ,Unit circle ,Anàlisi funcional ,Sobolev spaces ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Mathematics - Abstract
In this paper we characterize the boundedness on the product of Sobolev spaces Hs(𝕋) × Hs(𝕋) on the unit circle 𝕋, of the bilinear form Λb with symbol b ∈ Hs(𝕋) given by $${{\Lambda}_b} (\varphi, \psi): = \int_{\open T} {\left( {{{( - {\Delta })}^s} + I} \right)(\varphi \psi )(\eta )b(\eta ) {\rm d}\sigma (\eta ).}$$
- Published
- 2019
18. Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras
- Author
-
Frederico A. M. Ribeiro, Csaba Schneider, and Stephen P. Glasby
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Duality (optimization) ,Group Theory (math.GR) ,010103 numerical & computational mathematics ,Computer Science::Digital Libraries ,01 natural sciences ,Prime (order theory) ,Simple (abstract algebra) ,FOS: Mathematics ,Exponent ,20D15, 20C20, 20E15, 20F28, 17A30, 17A36 ,0101 mathematics ,Algebra over a field ,Quotient group ,Mathematics - Group Theory ,Commutative property ,Mathematics - Abstract
Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group of exponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$, and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbb F}_p$-algebra $L$ such that the action of ${\rm Aut}(L)$ is irreducible on $L$; we call such an algebra IAC. This paper establishes a duality $G\leftrightarrow L$ between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the $m$-th symmetric power of the natural module of ${\rm SL}(2,{\mathbb F})$., Comment: 26 pages, 2 figures; revised and to appear in Proceedings of The Royal Society A
- Published
- 2019
19. Interlacing polynomials and the veronese construction for rational formal power series
- Author
-
Philip B. Zhang
- Subjects
Discrete mathematics ,Property (philosophy) ,Formal power series ,General Mathematics ,010102 general mathematics ,Interlacing ,0102 computer and information sciences ,01 natural sciences ,Identity (mathematics) ,Integer ,05A15, 13A02, 26C10, 52B20, 52B45 ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Descent (mathematics) ,Mathematics - Abstract
Fixing a positive integer $r$ and $0 \le k \le r-1$, define $f^{\langle r,k \rangle}$ for every formal power series $f$ as $ f(x) = f^{\langle r,0 \rangle}(x^r)+xf^{\langle r,1 \rangle}(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle}(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := \left( (1+x+\cdots+x^{r-1})^{n} h(x) \right)^{\langle r,k \rangle}$ has only nonpositive zeros for any $r \ge \deg h(x) -k$ and any positive integer $n$. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension $n$, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ge n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $\left( h^{\langle r,r-i \rangle}(x)\right)_{1\le i \le r}$ is interlacing, so is $\left( U^{n}_{r,r-i}\, h(x) \right)_{1\le i \le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations., Comment: 18 pages
- Published
- 2019
20. Hausdorff operators on holomorphic Hardy spaces and applications
- Author
-
Thai Thuan Quang, Luong Dang Ky, and Ha Duy Hung
- Subjects
Mathematics::Functional Analysis ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Hausdorff space ,Holomorphic function ,Hardy space ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Mathematics - Classical Analysis and ODEs ,Bounded function ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Complex Variables (math.CV) ,47B38, 42B30, 46E15 ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to characterize the nonnegative functions $\varphi$ defined on $(0,\infty)$ for which the Hausdorff operator $$\mathscr H_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt$$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given., Comment: Proc. Roy. Soc. Edinburgh Sect. A (to appear)
- Published
- 2019
21. Induction for locally compact quantum groups revisited
- Author
-
Paweł Kasprzak, Piotr M. Sołtan, Mehrdad Kalantar, and Adam Skalski
- Subjects
Containment (computer programming) ,Pure mathematics ,Mathematics::Operator Algebras ,General Mathematics ,Locally compact quantum group ,010102 general mathematics ,Mathematics - Operator Algebras ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Operator Algebras (math.OA) ,Quantum ,Mathematics - Abstract
In this paper we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment., Comment: 20 pages
- Published
- 2019
22. Existence of solutions for critical Choquard equations via the concentration-compactness method
- Author
-
Fashun Gao, Minbo Yang, Edcarlos D. Silva, and Jiazheng Zhou
- Subjects
010101 applied mathematics ,Lemma (mathematics) ,High energy ,Nonlinear system ,Pure mathematics ,Compact space ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematics ,Sobolev inequality - Abstract
In this paper, we consider the nonlinear Choquard equation $$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$ where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and $G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potential V(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.
- Published
- 2019
23. Approximations, ghosts and derived equivalences
- Author
-
Yiping Chen and Wei Hu
- Subjects
Pure mathematics ,Endomorphism ,General Mathematics ,Modulo ,18E30, 20K40 ,010102 general mathematics ,Mathematics - Rings and Algebras ,Algebraic geometry ,01 natural sciences ,Noncommutative geometry ,Representation theory ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,0103 physical sciences ,Mutation (genetic algorithm) ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Quotient ,Mathematics - Abstract
Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.
- Published
- 2019
24. Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians
- Author
-
Sandra Kabisch and Jonathan J. Bevan
- Subjects
Global energy ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Stationary point ,010101 applied mathematics ,49N60, 74G40 ,symbols.namesake ,Mathematics - Analysis of PDEs ,Shear (geology) ,Jacobian matrix and determinant ,FOS: Mathematics ,symbols ,Uniqueness ,0101 mathematics ,Twist ,Nonlinear elasticity ,Analysis of PDEs (math.AP) ,Mathematics ,Energy functional - Abstract
In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian $\det \nabla u$ and the uniqueness and regularity of energy minimizers $u$ that are either twist maps or shear maps. We exhibit \emph{explicit} twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma}: \Omega\to \mathbb{R}^2$ in a model, two-dimensional case. The shear map minimizer has the properties that (i) $\det \nabla u_{\sigma}$ is strictly positive on one part of the domain $\Omega$, (ii) $\det \nabla u_{\sigma} = 0$ necessarily holds on the rest of $\Omega$, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma}$ is not continuous on the whole domain., Comment: 2 figures
- Published
- 2019
25. On critical exponents of ak-Hessian equation in the whole space
- Author
-
Yutian Lei and Yun Wang
- Subjects
Hessian matrix ,Hessian equation ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Sobolev inequality ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,symbols ,0101 mathematics ,Critical exponent ,Mathematical physics ,Mathematics - Abstract
In this paper, we study negative classical solutions and stable solutions of the followingk-Hessian equation$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$with radial structure, wheren⩾ 3, 1
1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to researchk-Hessian equations without radial structure. - Published
- 2019
26. Logarithmic upper bounds for weak solutions to a class of parabolic equations
- Author
-
Xiangsheng Xu
- Subjects
Class (set theory) ,Mathematics - Analysis of PDEs ,Logarithm ,General Mathematics ,Weak solution ,Mathematical analysis ,FOS: Mathematics ,Boundary value problem ,Parabolic partial differential equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation $\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in $\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate $$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$ provided that $q \gt 1+{N}/{2}$, where Ω is a bounded domain in ${\open R}^N$ with Lipschitz boundary, T > 0, $\partial _p\Omega _T$ is the parabolic boundary of $\Omega _T$, $\omega \in L^1(\Omega _T)$ with $\omega \ges 0$, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if $q=1+{N}/{2}$. In this paper, we show that the linear growth of the upper bound in $\Vert f \Vert_{q,\Omega _T}$ can be improved. To be precise, we establish $$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$
- Published
- 2019
27. A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects
- Author
-
Wen Yang and Aleks Jevnikar
- Subjects
Inequality ,Turbulence ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,010305 fluids & plasmas ,Vortex ,Mathematics - Analysis of PDEs ,Argument ,Mean field equation ,Phenomenon ,0103 physical sciences ,FOS: Mathematics ,35J61, 35J20, 35R01, 35B44 ,010306 general physics ,Analysis of PDEs (math.AP) ,Variable (mathematics) ,Mathematics ,Probability measure ,media_common - Abstract
We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.
- Published
- 2018
28. A problem of integer partitions and numerical semigroups
- Author
-
M. B. Branco, J. C. Rosales, Stuart White, University of Oxford, UK, and Professor Stuart White University of Oxford, UK
- Subjects
Integer partitions ,Combinatorics ,numerical semigroups ,General Mathematics ,Mathematics - Abstract
Let C be a set of positive integers. In this paper, we obtain an algorithm for computing all subsets A of positive integers which are minimals with the condition that if x1 + … + xn is a partition of an element in C, then at least a summand of this partition belongs to A. We use techniques of numerical semigroups to solve this problem because it is equivalent to give an algorithm that allows us to compute all the numerical semigroups which are maximals with the condition that has an empty intersection with the set C.
- Published
- 2018
29. Time-dependent attractors for non-autonomous non-local reaction–diffusion equations
- Author
-
Pedro Marín-Rubio, Marta Herrera-Cobos, and Tomás Caraballo
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Pullback attractor ,Non local ,01 natural sciences ,010101 applied mathematics ,Strong solutions ,Bounded function ,Reaction–diffusion system ,Attractor ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.
- Published
- 2018
30. On the finite-element approximation of ∞-harmonic functions
- Author
-
Tristan Pryer
- Subjects
Discretization ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Infinity ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Harmonic function ,Limit (mathematics) ,0101 mathematics ,Galerkin method ,Laplace operator ,media_common ,Mathematics - Abstract
In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.
- Published
- 2018
31. Integrable zero-Hopf singularities and three-dimensional centres
- Author
-
Isaac A. García
- Subjects
Pure mathematics ,Integrable system ,General Mathematics ,010102 general mathematics ,Quadratic function ,Parameter space ,01 natural sciences ,010101 applied mathematics ,Singularity ,Phase space ,Gravitational singularity ,0101 mathematics ,Affine variety ,Mathematics ,Poincaré map - Abstract
In this paper we show that the well-known Poincaré–Lyapunov non-degenerate analytic centre problem in the plane and its higher-dimensional version, expressed as the three-dimensional centre problem at the zero-Hopf singularity, have a lot of common properties. In both cases the existence of a neighbourhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré–Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincaré return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centres corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.
- Published
- 2017
32. Partial inverse problems for Sturm–Liouville operators on trees
- Author
-
Chung-Tsun Shieh and Natalia P. Bondarenko
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Inverse ,Sturm–Liouville theory ,Inverse problem ,Edge (geometry) ,01 natural sciences ,Tree (graph theory) ,Constructive ,010101 applied mathematics ,Graph (abstract data type) ,A priori and a posteriori ,0101 mathematics ,Mathematics - Abstract
In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.
- Published
- 2017
33. Geometric aspects of self-adjoint Sturm–Liouville problems
- Author
-
Yicao Wang
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Orbit (dynamics) ,Boundary (topology) ,Sturm–Liouville theory ,Boundary value problem ,Diffeomorphism ,Mathematics::Spectral Theory ,Principal type ,Self-adjoint operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.
- Published
- 2017
34. On the Hardy–Sobolev equation
- Author
-
Francesca Gladiali, E. N. Dancer, and Massimo Grossi
- Subjects
010101 applied mathematics ,Sobolev space ,Pure mathematics ,Singularity ,Bifurcation theory ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Bifurcation ,Mathematics ,Sobolev inequality - Abstract
In this paper we study the problemwhere Ω = ℝN or Ω = B1, N ⩾ 3, p > 1 and . Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.
- Published
- 2017
35. Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations
- Author
-
D. J. Needham and J. C. Meyer
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Hölder condition ,Lipschitz continuity ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,Stochastic partial differential equation ,Elliptic partial differential equation ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics ,Numerical partial differential equations - Abstract
We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.
- Published
- 2016
36. Whitney regularity of the image of the Chevalley mapping
- Author
-
Gérard P. Barbançon
- Subjects
Mathematics - Classical Analysis and ODEs ,business.industry ,General Mathematics ,Image (category theory) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Pattern recognition ,Artificial intelligence ,business ,Mathematics - Abstract
A compact set K ⊂ ℝn is Whitney 1-regular if the geodesic distance in K is equivalent to the Euclidean distance. Let P be the Chevalley map defined by an integrity basis of the algebra of polynomials invariant by a reflection group. This paper gives the Whitney 1-regularity of the image by P of any closed ball centred at the origin. The proof uses the works of Givental', Kostov and Arnol'd on the symmetric group. It needs a generalization of a property of the Vandermonde determinants to the Jacobian of the Chevalley mappings.
- Published
- 2016
37. Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal
- Author
-
Gergő Nemes
- Subjects
General Mathematics ,Representation (systemics) ,Classification of discontinuities ,Exponential function ,Exponential growth ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,33B15, 30E15, 65D20 ,Applied mathematics ,Gamma function ,Asymptotic expansion ,Reciprocal ,Mathematics - Abstract
In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a resurgence representation for the gamma function, exploiting the reformulation of the method of steepest descents by M. Berry and C. Howls (Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities, and asymptotics for the late coefficients. The main aim of this paper is to modify the resurgence formula of Boyd making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors., Comment: 22 pages, accepted for publication in Proceedings of the Royal Society of Edinburgh, Section A: Mathematical and Physical Sciences
- Published
- 2015
38. Global attractivity and extinction for Lotka–Volterra systems with infinite delay and feedback controls
- Author
-
Teresa Faria and Yoshiaki Muroya
- Subjects
education.field_of_study ,Extinction ,General Mathematics ,Population ,Boundary (topology) ,Dynamical Systems (math.DS) ,34K20, 34K25, 92D25, 93B52 ,Multiple species ,Lyapunov functional ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Mathematics - Dynamical Systems ,education ,Mathematics ,Diagonally dominant matrix - Abstract
The paper deals with a multiple species Lotka-Volterra model with infinite distributed delays and feedback controls, for which we assume a weak form of diagonal dominance of the instantaneous negative intra-specific terms over the infinite delay effect in both the population variables and controls. General sufficient conditions for the existence and attractivity of a saturated equilibrium are established. When the saturated equilibrium is on the boundary of $\R^n_+$, sharper criteria for the extinction of all or part of the populations are given. While the literature usually treats the case of competitive systems only, here no restrictions on the signs of the intra- and inter-specific delayed terms are imposed. Moreover, our technique does not require the construction of Lyapunov functionals., 27 pages
- Published
- 2015
39. Estimates on the non-real eigenvalues of regular indefinite Sturm–Liouville problems
- Author
-
Shaozhu Chen, Jiangang Qi, Friedrich Philipp, and Jussi Behrndt
- Subjects
Pure mathematics ,General Mathematics ,Sturm–Liouville theory ,Mathematics::Spectral Theory ,Differential expression ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Regular Sturm–Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this paper we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.
- Published
- 2014
40. Bifurcation at isolated singular points of the Hadamard derivative
- Author
-
Charles Alexander Stuart
- Subjects
Discrete mathematics ,Nonlinear system ,Hadamard transform ,General Mathematics ,Banach space ,Neighbourhood (graph theory) ,Center (group theory) ,Differentiable function ,Lipschitz continuity ,Lambda ,Mathematics - Abstract
For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F (λ, u) = 0, where F : ℝ × X → Y with F (λ, 0) = 0 for all λ ∈ ℝ. The focus is on the situation where F (λ, ·) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Fréchet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on ℝN, where Fréchet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.
- Published
- 2014
41. Permanence criteria for Kolmogorov systems with delays
- Author
-
Zhanyuan Hou
- Subjects
Class (set theory) ,Distribution (mathematics) ,Simple (abstract algebra) ,General Mathematics ,Attractor ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,Algebraic number ,34D05, 34K12, 34K25, 34C11, 92D25 ,Mathematics - Abstract
In this paper, a class of Kolmogorov systems with delays are studied. Sufficient conditions are provided for a system to have a compact uniform attractor. Then Jansen's result (J. Math. Biol. Vol. 25 (1987) 411-422) for autonomous replicator and Lotka-Volterra systems has been extended to delayed nonautonomous Kolmogorov systems with periodic or autonomous Lotka-Volterra subsystems. Thus, simple algebraic conditions are obtained for partial permanence and permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays., To be published in Proceedings of Royal Society of Edinburgh
- Published
- 2014
42. Sharp boundary trace inequalities
- Author
-
Giles Auchmuty
- Subjects
Sobolev space ,Weight function ,Pure mathematics ,Trace (linear algebra) ,General Mathematics ,Bounded function ,Trace inequalities ,Boundary (topology) ,Function (mathematics) ,Domain (mathematical analysis) ,Mathematics - Abstract
This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ ℝN. The inequalities bound (semi-)norms of the boundary trace by certain norms of the function, its gradient on the region and by two specific constants κρ and κΩ associated with the domain and a weight function, respectively. These inequalities are sharp in that there exist functions for which equality holds. Explicit inequalities in some special cases when the region is a ball, or the region between two balls, are evaluated.
- Published
- 2014
43. Estimates for lower bounds of eigenvalues of the poly-Laplacian and the quadratic polynomial operator of the Laplacian
- Author
-
Qing-Ming Cheng, Guoxin Wei, He-Jun Sun, and Lingzhong Zeng
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Operator (physics) ,Quadratic function ,Mathematics::Spectral Theory ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we investigate the Dirichlet eigenvalue problems of the poly-Laplacian with any order and the quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first k eigenvalues.
- Published
- 2013
44. Uniqueness of positive solutions for a class of elliptic boundary value problems
- Author
-
Ratnasingham Shivaji and Alfonso Castro
- Subjects
Pure mathematics ,Class (set theory) ,Elliptic curve ,Partial differential equation ,Uniqueness theorem for Poisson's equation ,General Mathematics ,Mathematical analysis ,Uniqueness ,Boundary value problem ,Mathematics - Abstract
SynopsisUniqueness of non-negative solutions conjectured in an earlier paper by Shivaji is proved. Our methods are independent of those of that paper, where the problem was considered only in a ball. Further, our results apply to a wider class of nonlinearities.
- Published
- 1984
45. On J-self-adjoint operators with stable C-symmetries
- Author
-
Seppo Hassi and Sergii Kuzhel
- Subjects
Class (set theory) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Spectral Theory ,Space (mathematics) ,01 natural sciences ,Kernel (algebra) ,Development (topology) ,0103 physical sciences ,Homogeneous space ,0101 mathematics ,010306 general physics ,Focus (optics) ,Self-adjoint operator ,Mathematics ,Symmetric operator - Abstract
The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.
- Published
- 2013
46. Construction of symplectic structures on 4-manifolds with a free circle action
- Author
-
Stefano Vidussi and Stefan Friedl
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,01 natural sciences ,Manifold ,Action (physics) ,Mathematics - Geometric Topology ,53D05 ,Cone (topology) ,Mathematics - Symplectic Geometry ,Product (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,0101 mathematics ,Orbit (control theory) ,Mathematics::Symplectic Geometry ,Symplectic geometry ,Mathematics - Abstract
Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fern\'andez-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-manifolds with a free circle action" (arXiv:0801.1313 [math.GT])., Comment: 11 pages
- Published
- 2012
47. Exit manifolds for lattice differential equations
- Author
-
J. Douglas Wright and Aaron Hoffman
- Subjects
Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Perturbation (astronomy) ,Weak interaction ,01 natural sciences ,010101 applied mathematics ,Superposition principle ,Linear differential equation ,Lattice (order) ,Stability theory ,Embedding ,0101 mathematics ,Mathematics - Abstract
We study the weak interaction between a pair of well-separated coherent structures in possibly non-local lattice differential equations. In particular, we prove that if a lattice differential equation in one space dimension has asymptotically stable (in the sense of a paper by Chow et al.) travelling-wave solutions whose profiles approach limiting equilibria exponentially fast, then the system admits solutions which are nearly the linear superposition of two such travelling waves moving in opposite directions away from one another. Moreover, such solutions are themselves asymptotically stable. This result is meant to complement analytic or numeric studies into interactions of such pulses over finite times which might result in the scenario treated here. Since the travelling waves are moving in opposite directions, these solutions are not shift-periodic and hence the framework of Chow et al. does not apply. We overcome this difficulty by embedding the original system in a larger one wherein the linear part can be written as a shift-periodic piece plus another piece which, although it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.
- Published
- 2011
48. Supports of weight modules over Witt algebras
- Author
-
Kaiming Zhao and Volodymyr Mazorchuk
- Subjects
General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,01 natural sciences ,Algebra ,Simple (abstract algebra) ,17B10, 17B20, 17B65, 17B66, 17B68 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras $W_n$, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a $\Z^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\oplus_{\alpha\in\Z^n}\mathfrak {g}_\alpha$ with respect to the abelian subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_\alpha,\mathfrak {g}_\beta]= \mathfrak {g}_{\alpha+\beta}$ for all $\alpha,\beta\in\Z^n$, $\alpha\neq \beta$ (this class contains the algebra $W_n$)., Comment: 17 pages
- Published
- 2011
49. On two-dimensional ferromagnetism
- Author
-
Pablo Pedregal and Baisheng Yan
- Subjects
Mathematical optimization ,Work (thermodynamics) ,Magnetization ,Field (physics) ,Ferromagnetism ,General Mathematics ,Applied mathematics ,Minification ,Focus (optics) ,Energy (signal processing) ,Domain (mathematical analysis) ,Mathematics - Abstract
We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.
- Published
- 2009
50. A singular Gierer—Meinhardt system with different source terms
- Author
-
Marius Ghergu and Vicentiu Radulescu
- Subjects
symbols.namesake ,General Mathematics ,Dirichlet's principle ,Dirichlet boundary condition ,Mathematical analysis ,symbols ,Mixed boundary condition ,Boundary value problem ,Uniqueness ,Mathematics - Abstract
We study the existence and non-existence of classical solutions to a general Gierer—Meinhardt system with Dirichlet boundary condition. The main feature of this paper is that we are concerned with a model in which both the activator and the inhibitor have different sources given by general nonlinearities. Under some additional hypotheses and in the case of pure powers in nonlinearities, regularity and uniqueness of the solution in one dimension is also presented.
- Published
- 2008
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.