50 results
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2. Mean values of derivatives of L-functions in function fields: III
- Author
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Julio Andrade
- Subjects
General Mathematics - Abstract
In this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.
- Published
- 2018
3. Affine focal sets of codimension-2 submanifolds contained in hypersurfaces
- Author
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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
4. Flows of measures generated by vector fields
- Author
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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
5. Four notions of conjugacy for abstract semigroups
- Author
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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
6. On two congruence conjectures of Z.-W. Sun involving Franel numbers
- Author
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Mao, Guo-Shuai and Liu, Yan
- Subjects
Mathematics - Number Theory ,General Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Number Theory (math.NT) ,Combinatorics (math.CO) - Abstract
In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{(-4)^k}\pmod{p^2},$$ and if $p\equiv1\pmod3$, then $$\sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{(-4)^k}\pmod{p^3},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ stands for the $n$th Franel number., 20 pages, revised some typos
- Published
- 2023
7. Is the Sacker–Sell type spectrum equal to the contractible set?
- Author
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Mengda Wu and Yonghui Xia
- Subjects
General Mathematics - Abstract
For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.
- Published
- 2023
8. On stability for generalized linear differential equations and applications to impulsive systems
- Author
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Gallegos, Claudio A. and Robledo, Gonzalo
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,34A06, 34D20 (Primary), 34A30, 34A37 (Secondary) - Abstract
In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterisation of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article., Comment: 25 pages
- Published
- 2023
9. A new inner approach for differential subordinations
- Author
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Adam Lecko
- Subjects
General Mathematics - Abstract
In this paper we introduce and examine the differential subordination of the form \[ p(z)+zp'(z)\varphi(p(z),zp'(z))\prec h(z),\quad z\in\mathbb{D}:=\{z\in\mathbb{C}:|z| where $h$ is a convex univalent function with $0\in h(\mathbb {D}).$ The proof of the main result is based on the original lemma for convex univalent functions and offers a new approach in the theory. In particular, the above differential subordination leads to generalizations of the well-known Briot-Bouquet differential subordination. Appropriate applications among others related to the differential subordination of harmonic mean are demonstrated. Related problems concerning differential equations are indicated.
- Published
- 2023
10. Girth, magnitude homology and phase transition of diagonality
- Author
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Yasuhiko Asao, Yasuaki Hiraoka, and Shu Kanazawa
- Subjects
General Mathematics ,Probability (math.PR) ,FOS: Mathematics ,Mathematics - Combinatorics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Combinatorics (math.CO) ,Mathematics - Probability - Abstract
This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth-Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behavior is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality., Comment: 21 pages, 5 figures, a reference added
- Published
- 2023
11. One-sided estimates via function
- Author
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María Lorente, F. J. Martin-Reyes, and Israel Pablo Rivera-Ríos
- Subjects
General Mathematics - Abstract
We recall that $w\in C_{p}^+$ if there exist $\varepsilon >0$ and $C>0$ such that for any $a< b< c$ with $c-b< b-a$ and any measurable set $E\subset (a,b)$ , the following holds \[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{(a,c)}\right)^{p}w This condition was introduced by Riveros and de la Torre [33] as a one-sided counterpart of the $C_{p}$ condition studied first by Muckenhoupt and Sawyer [30, 34]. In this paper we show that given $1< p< q if $w\in C_{q}^+$ then \[ \|M^+f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \] and conversely if such an inequality holds, then $w\in C_{p}^+$ . This result is the one-sided counterpart of Yabuta's main result in [37]. Combining this estimate with known pointwise estimates for $M^{\sharp,+}$ in the literature we recover and extend the result for maximal one-sided singular integrals due to Riveros and de la Torre [33] obtaining counterparts a number of operators.
- Published
- 2022
12. Borderline gradient continuity for fractional heat type operators
- Author
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Arya, Vedansh and Kumar, Dharmendra
- Subjects
Mathematics - Analysis of PDEs ,General Mathematics ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this paper, we establish gradient continuity for solutions to \[ (\partial_t - \operatorname{div}(A(x) \nabla u))^s =f,\ s \in (1/2, 1), \] when $f$ belongs to the scaling critical function space $L(\frac{n+2}{2s-1}, 1)$. Our main results Theorems 1.1 and 1.2 can be seen as a nonlocal generalization of a well-known result of Stein in the context of fractional heat type operators and sharpens some of the previous gradient continuity results which deals with $f$ in subcritical spaces. Our proof is based on an appropriate adaptation of compactness arguments, which has its roots in a fundamental work of Caffarelli in [13]., arXiv admin note: text overlap with arXiv:1905.02580, arXiv:1806.07652 by other authors
- Published
- 2022
13. On the topological complexity and zero-divisor cup-length of real Grassmannians
- Author
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Marko Radovanovic
- Subjects
General Mathematics - Abstract
Topological complexity naturally appears in the motion planning in robotics. In this paper we consider the problem of finding topological complexity of real Grassmann manifolds $G_k(\mathbb {R}^{n})$. We use cohomology methods to give estimates on the zero-divisor cup-length of $G_k(\mathbb {R}^{n})$ for various $2\leqslant k< n$, which in turn give us lower bounds on topological complexity. Our results correct and improve several results from Pavešić (Proc. Roy. Soc. Edinb. A151 (2021), 2013–2029).
- Published
- 2022
14. Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres
- Author
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Marín Pérez, David, Saavedra, M., and Villadelprat Yagüe, Jordi
- Subjects
Period function ,Dulac time ,Asymptotic expansions ,General Mathematics ,Saddle-node unfolding - Abstract
In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \]parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.
- Published
- 2021
15. Lifting of recollements and gluing of partial silting sets
- Author
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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
16. Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
- Author
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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
17. The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation
- Author
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Teresa Luque, Juan Antonio Barceló, Magali Folch-Gabayet, María de la Cruz Vilela, and Salvador Pérez-Esteva
- Subjects
Physics ,Primary 35J05, Secondary 42B10, 46E35 ,Helmholtz equation ,General Mathematics ,Operator (physics) ,010102 general mathematics ,010103 numerical & computational mathematics ,Extension (predicate logic) ,Characterization (mathematics) ,01 natural sciences ,Sobolev space ,symbols.namesake ,Mathematics - Analysis of PDEs ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Wave function ,Laplace operator ,Analysis of PDEs (math.AP) ,Mathematical physics - Abstract
The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝd) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.
- Published
- 2020
18. 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
19. A monotonicity result under symmetry and Morse index constraints in the plane
- Author
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Francesca Gladiali
- Subjects
Physics ,Pure mathematics ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,Monotonic function ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Monotone polygon ,Bounded function ,FOS: Mathematics ,0101 mathematics ,Symmetry (geometry) ,Invariant (mathematics) ,Analysis of PDEs (math.AP) - Abstract
This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.
- Published
- 2020
20. Local minimizers in absence of ground states for the critical NLS energy on metric graphs
- Author
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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
21. All finite transitive graphs admit a self-adjoint free semigroupoid algebra
- Author
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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
22. Nonsingular bilinear maps revisited
- Author
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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
23. On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials
- Author
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Jianfu Yang and Jinge Yang
- Subjects
Physics ,General Mathematics ,010102 general mathematics ,Limiting ,Ellipse ,01 natural sciences ,Supercritical fluid ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,Nonlinear system ,Lagrange multiplier ,Excited state ,symbols ,0101 mathematics ,Nonlinear Schrödinger equation ,Mathematical physics - Abstract
In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.
- Published
- 2019
24. Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter
- Author
-
Hiroaki Kikuchi, Slim Ibrahim, and Takafumi Akahori
- Subjects
Physics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Instability ,Schrödinger equation ,Power (physics) ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,Nonlinear system ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,symbols ,0101 mathematics ,Ground state ,Critical exponent ,Scalar field ,Analysis of PDEs (math.AP) - Abstract
We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In [3], it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.
- Published
- 2019
25. 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
26. Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras
- Author
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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
27. 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
28. 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
29. On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations
- Author
-
Woocheol Choi, Jinmyoung Seok, and Younghun Hong
- Subjects
Physics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Supercritical fluid ,Schrödinger equation ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Bounded function ,symbols ,0101 mathematics ,Nonlinear Schrödinger equation ,Mathematical physics - Abstract
In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = \vert u \vert^{p-1}u\quad {\rm in}~{\open R}^n~(n \ges 2) $$ involving an H1/2-critical/supercritical power-type nonlinearity, that is, p ⩾ ((n + 1)/(n − 1)). We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is H1/2-critical/supercritical but it is H1-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either (i) if the nonlinearity is H1/2-critical/supercritical in the ultra-relativistic regime or (ii) if the nonlinearity is H1-critical/supercritical in all cases.
- Published
- 2019
30. 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
31. 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
32. 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
33. 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
34. 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
35. 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
36. Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations
- Author
-
Mimi Dai and Alexey Cheskidov
- Subjects
General Mathematics ,Mathematics::Analysis of PDEs ,FOS: Physical sciences ,35Q35, 37L30 ,01 natural sciences ,law.invention ,Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,law ,Intermittency ,Attractor ,FOS: Mathematics ,Fluid dynamics ,Wavenumber ,0101 mathematics ,Navier–Stokes equations ,Physics ,Turbulence ,010102 general mathematics ,Mathematical analysis ,Degrees of freedom ,Fluid Dynamics (physics.flu-dyn) ,Physics - Fluid Dynamics ,Nonlinear Sciences::Chaotic Dynamics ,010101 applied mathematics ,Bounded function ,Analysis of PDEs (math.AP) - Abstract
Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier-Stokes equations, is a mathematical analog of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier-Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme., Revised version
- Published
- 2019
37. 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
38. 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
39. 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
40. 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
41. 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
42. 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
43. 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
44. 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
45. 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
46. 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
47. 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
48. 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
49. 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
50. 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
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