25,431 results
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2. Global estimates of fundamental solutions for higher-order Schrödinger equations
- Author
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Xiaohua Yao, JinMyong Kim, and Anton Arnold
- Subjects
Pointwise ,Polynomial ,42B20, 42B37, 35Q41, 35B65 ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Order (ring theory) ,Mathematics::Spectral Theory ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Operator (computer programming) ,Fundamental solution ,Applied mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics - Abstract
In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr\"odinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related Lp-Lq estimates on the Schr\"odinger solution. These estimates extend known results from the literature and are sharp. This result was latetly already generalized to a degenerate case (cf. [9]).
- Published
- 2011
3. Volume-Preserving Diffeomorphisms with the $$\mathcal {M}_0$$-Shadowing Properties
- Author
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Xu Zhang, Xinxing Wu, and Fu Sun
- Subjects
010101 applied mathematics ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,010102 general mathematics ,Diffeomorphism ,0101 mathematics ,Mathematics::Geometric Topology ,Mathematics::Symplectic Geometry ,01 natural sciences ,Symplectic geometry ,Mathematics ,Volume (compression) - Abstract
This paper studies the $$\mathcal {M}_0$$ -shadowing property for two types of volume-preserving diffeomorphisms defined on compact manifolds. For symplectic diffeomorphisms defined on symplectic manifolds, the $$C^1$$ -interior of the set of all symplectic diffeomorphisms with the $$\mathcal {M}_0$$ -shadowing property is described by the set of the Anosov diffeomorphisms. If a volume-preserving diffeomorphism in $$\mathrm{Diff}^1_{\mu }(M)$$ is a $$C^1$$ -stable $$\mathcal {M}_0$$ -shadowing diffeomorphism, then M admits a volume-hyperbolic dominated splitting.
- Published
- 2021
4. SATURATION FOR THE BUTTERFLY POSET
- Author
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Maria-Romina Ivan, Ivan, Maria [0000-0003-0817-3777], and Apollo - University of Cambridge Repository
- Subjects
General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,05D05 ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Saturation (chemistry) ,Partially ordered set ,Mathematics - Abstract
Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$., Comment: 13 pages
- Published
- 2023
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5. A curvature-free 𝐿𝑜𝑔(2𝑘-1) theorem
- Author
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Florent Balacheff and Louis Merlin
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,Curvature ,Mathematics::Geometric Topology ,01 natural sciences ,Volume entropy ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.
- Published
- 2023
6. Analogues of Entropy in Bi-Free Probability Theory: Microstates
- Author
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Paul Skoufranis and Ian Charlesworth
- Subjects
Free entropy ,General Mathematics ,FISHERS INFORMATION MEASURE ,01 natural sciences ,Upper and lower bounds ,46L54, 46L53, 47B80, 94A17 ,Ministate ,0103 physical sciences ,Subadditivity ,FOS: Mathematics ,Entropy (information theory) ,Statistical physics ,0101 mathematics ,Operator Algebras (math.OA) ,Central limit theorem ,Mathematics ,Science & Technology ,Mathematics::Operator Algebras ,010102 general mathematics ,Mathematics - Operator Algebras ,16. Peace & justice ,Free probability ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Physical Sciences ,010307 mathematical physics ,Random matrix - Abstract
In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbital version of bi-free entropy is examined which provides a tighter upper bound for the subadditivity of microstate bi-free entropy and provides an alternate characterization of bi-freeness in certain settings., Comment: (new version contains corrections, orbital bi-free entropy, and a new characterization of bi-freenness)
- Published
- 2023
7. A Laplace-Type Representation of the Generalized Spherical Functions Associated with the Root Systems of Type A
- Author
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P. Sawyer
- Subjects
Pure mathematics ,Laplace transform ,General Mathematics ,010102 general mathematics ,Harmonic (mathematics) ,010103 numerical & computational mathematics ,Type (model theory) ,Expression (computer science) ,01 natural sciences ,Operator (computer programming) ,Abel transform ,0101 mathematics ,Trigonometry ,Representation (mathematics) ,Mathematics - Abstract
In this paper, we extend the iterative expression for the generalized spherical functions associated with the root systems of type A previously obtained (Sawyer in Trans Am Math Soc 349(9):3569–3584, 1997; Sawyer in Q J Math Oxf Ser (2) 50(197):71–86, 1999) beyond regular elements. We also provide a similar expression in the corresponding flat case. From there, we derive a Laplace-type representation for the generalized spherical functions associated with the root systems of type A in the Dunkl setting as well as in the trigonometric Dunkl setting. This representation leads us to describe precisely the support of the generalized Abel transform. Thanks to a recent result of Gallardo and Rejeb (Support properties of the intertwining and the mean value operators in Dunkls analysis. Preprint [hal01331693], pp 1–10, 2016) and Rejeb (Harmonic and subharmonic functions associated with root systems. Mathematics, Universite Francois-Rabelais de Tours, Universite de Tunis El Manar, 2015), which allows us to give the support for the Dunkl intertwining operator.
- Published
- 2017
8. Homogenization of a neutronic critical diffusion problem with drift
- Author
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Yves Capdeboscq, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Nuclear reactor ,01 natural sciences ,Homogenization (chemistry) ,law.invention ,010101 applied mathematics ,Elliptic partial differential equation ,Nuclear reactor core ,Criticality ,law ,Neutron flux ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Diffusion (business) ,Eigenvalues and eigenvectors ,ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics - Abstract
In this paper we study the homogenization of an eigenvalue problem for a cooperative system of weakly coupled elliptic partial differential equations, called the neutronic multigroup diffusion model, in a periodic heterogeneous domain. Such a model is used for studying the criticality of nuclear reactor cores. In a recent work in collaboration with Grégoire Allaire, it is proved that, under a symmetry assumption, the first eigenvector of the multigroup system in the periodicity cell controls the oscillatory behaviour of the solutions, whereas the global trend is asymptotically given by a homogenized diffusion eigenvalue problem. It is shown here that when this symmetry condition is not fulfilled, the asymptotic behaviour of the neutron flux, corresponding to the first eigenvector of the multigroup system, is dramatically different. This result enables to consider new types of geometrical configurations in practical nuclear reactor core computations.
- Published
- 2016
9. The central sphere of an ALE space
- Author
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Nigel Hitchin
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Fixed point ,Space (mathematics) ,01 natural sciences ,Induced metric ,Action (physics) ,Set (abstract data type) ,Mathematics - Algebraic Geometry ,53C26 ,Differential Geometry (math.DG) ,0103 physical sciences ,Metric (mathematics) ,Algebraic surface ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
This paper focuses on the spherical fixed point set of a circle action on an ALE space endowed with Kronheimer's hyperkaehler metric. The induced metric on the sphere is described by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic., Dedicated to the memory of Michael Atiyah
- Published
- 2022
10. Global dynamics for the two-dimensional stochastic nonlinear wave equations
- Author
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Herbert Koch, Tadahiro Oh, Leonardo Tolomeo, and Massimiliano Gubinelli
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General Mathematics ,Mathematics::Analysis of PDEs ,damped nonlinear wave equation ,01 natural sciences ,renormalization ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Gibbs measure ,white noise ,Mathematics ,Forcing (recursion theory) ,35L71, 60H15 ,010102 general mathematics ,Probability (math.PR) ,Double exponential function ,Torus ,White noise ,Sobolev space ,stochastic nonlinear wave equation ,nonlinear wave equation ,Norm (mathematics) ,symbols ,Invariant measure ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure., 33 pages. To appear in Internat. Math. Res. Not. Minor typos corrected
- Published
- 2022
11. Local Uniformization of Abhyankar Valuations
- Author
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Steven Dale Cutkosky
- Subjects
Algebraic function field ,Pure mathematics ,13A18, 13H05, 14E15 ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Valuation ring ,Uniformization (probability theory) ,Separable space ,Ground field ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Residue field ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar., Comment: 28 pages. Final Final version
- Published
- 2022
12. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
- Author
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Elizabeth Gillaspy, Anna Duwenig, and Rachael Norton
- Subjects
General Mathematics ,01 natural sciences ,Section (fiber bundle) ,Combinatorics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,46L05, 22D25, 22A22 ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Twist ,Operator Algebras (math.OA) ,Mathematics::Representation Theory ,Quotient ,Mathematics ,Science & Technology ,Mathematics::Operator Algebras ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematics - Operator Algebras ,Cartan subalgebra ,C-ASTERISK-ALGEBRAS ,Physical Sciences ,010307 mathematical physics ,EQUIVALENCE - Abstract
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages
- Published
- 2022
13. On the existence of proper Nearly Kenmotsu manifolds
- Author
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Piotr Dacko, I. Küpeli Erken, Cengizhan Murathan, Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü., Erken, İrem Küpeli, Dacko, Piotr, Murathan, Cengizhan, ABE-8167-2020, and ABH-3658-2020
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Kähler manifold ,01 natural sciences ,Warped Product ,Kaehler Manifold ,Sasakian Space Form ,Almost contact metric manifold ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Real line ,Mathematics::Symplectic Geometry ,Mathematics, applied ,Mathematics ,Mathematics::Complex Variables ,010102 general mathematics ,Mathematical analysis ,Mathematics::Geometric Topology ,Manifold ,Kenmotsu manifold ,Differential Geometry (math.DG) ,Nearly Kenmotsu manifold ,Product (mathematics) ,010307 mathematical physics ,Mathematics::Differential Geometry ,53C25, 53C55, 53D15 - Abstract
This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly K\"ahler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.
- Published
- 2016
14. Common Fixed Point Results for Rational (α,β)φ-mω Contractions in Complete Quasi Metric Spaces
- Author
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Wasfi Shatanawi, Abdalla Tallafha, Tariq Qawasmeh, and Anwar Bataihah
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Pure mathematics ,General Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Distance mapping ,Fixed-point theorem ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Computer Science::Digital Libraries ,010101 applied mathematics ,Metric space ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,fixed point ,Computer Science (miscellaneous) ,Common fixed point ,contraction mappings ,Computer Science::Programming Languages ,quasi metric ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
The &omega, distance mapping is one of the important tools that can be used to get new contractions in fixed point theory. The aim of this paper is to use the concept of modified &omega, distance mapping to introduce the notion of rational ( &alpha, &beta, ) &phi, m &omega, contraction. We utilize our new notion to construct and formulate many fixed point results for a pair of two mappings defined on a nonempty set A. Our results modify many existing known results. In addition, we support our work by an example.
- Published
- 2019
15. On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation
- Author
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Giuseppe Maria Coclite and Lorenzo di Ruvo
- Subjects
Cauchy problem ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,uniqueness ,Existence ,stability ,01 natural sciences ,Rosenau–Korteweg-deVries–Kawahara type equation ,010101 applied mathematics ,Type equation ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Existence, uniqueness, stability, Rosenau–Korteweg-deVries–Kawahara type equation, Cauchy problem ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics ,Mathematical physics - Abstract
The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.
- Published
- 2022
16. Non-local solvable birth-death processes
- Author
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Giacomo Ascione, Enrica Pirozzi, Nikolai Leonenko, Ascione, G., Leonenko, N., and Pirozzi, E.
- Subjects
Statistics and Probability ,Class (set theory) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Structure (category theory) ,Eigenfunction ,Non local ,01 natural sciences ,60K15, 60G22, 33C45 ,Matrix decomposition ,Classical orthogonal polynomial of discrete variable ,010104 statistics & probability ,Orthogonal polynomials ,FOS: Mathematics ,Applied mathematics ,Quantitative Biology::Populations and Evolution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Invariant (mathematics) ,Representation (mathematics) ,Subordinator ,Mathematics - Probability ,Mathematics ,Bernstein function - Abstract
In this paper we study strong solutions of some non-local difference-differential equations linked to a class of birth-death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth-death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth-death processes., 26 pages
- Published
- 2022
17. Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula
- Author
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Mihalis Mourgoglou, Jonas Azzam, John Garnett, and Xavier Tolsa
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Monotonic function ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Measure (mathematics) ,Square (algebra) ,Mathematics - Abstract
Let $\Omega \subset {{\mathbb {R}}}^{n+1}$, $n\geq 2$, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly nonsymmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partial \Omega $ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $\Omega $ is $\varepsilon $-approximable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $\Omega $ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called “$S
- Published
- 2022
18. Annihilators of the ideal class group of a cyclic extension of an imaginary quadratic field
- Author
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Hugo Chapdelaine and Radan Kučera
- Subjects
Pure mathematics ,Degree (graph theory) ,Mathematics - Number Theory ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Ideal class group ,Elliptic unit ,Extension (predicate logic) ,01 natural sciences ,Primary 11R20, Secondary 11R27, 11R29 ,Prime (order theory) ,Annihilator ,0103 physical sciences ,FOS: Mathematics ,Quadratic field ,010307 mathematical physics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.
- Published
- 2017
19. The geometry of diagonal groups
- Author
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Peter J. Cameron, Cheryl E. Praeger, Csaba Schneider, R. A. Bailey, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, and University of St Andrews. Statistics
- Subjects
Mathematics(all) ,South china ,Primitive permutation group ,General Mathematics ,Diagonal group ,T-NDAS ,Library science ,Group Theory (math.GR) ,O'Nan-Scott Theorem ,01 natural sciences ,Hospitality ,FOS: Mathematics ,NCAD ,Mathematics - Combinatorics ,QA Mathematics ,0101 mathematics ,Diagonal semilattice ,QA ,Cartesian lattice ,Mathematics ,business.industry ,20B05 ,Applied Mathematics ,010102 general mathematics ,Latin square ,Semilattice ,Latin cube ,010101 applied mathematics ,Hamming graph ,Research council ,Diagonal graph ,Combinatorics (math.CO) ,business ,Mathematics - Group Theory ,Partition - Abstract
Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m
- Published
- 2022
20. On the transmission problems for the Oseen and Brinkman systems on Lipschitz domains in compact Riemannian manifolds
- Author
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Cornel Pintea, Mirela Kohr, Robert Gutt, and Wolfgang L. Wendland
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Perturbation (astronomy) ,Riemannian manifold ,Lipschitz continuity ,Differential operator ,01 natural sciences ,Potential theory ,010101 applied mathematics ,Sobolev space ,Compact space ,Lipschitz domain ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
The purpose of this work is to show the well-posedness in L2-Sobolev spaces of the Poisson-transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi-Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the Levi-Civita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds.
- Published
- 2015
21. Satisfiability in MultiValued Circuits
- Author
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Paweł M. Idziak and Jacek Krzaczkowski
- Subjects
FOS: Computer and information sciences ,Computational complexity theory ,General Computer Science ,68Q17, 08A05, 08A70 (Primary) 68Q05, 68T27, 03B25, 08B05, 08B10 (Secondary) ,Boolean circuit ,General Mathematics ,010102 general mathematics ,circuit satisfiability ,Distributive lattice ,0102 computer and information sciences ,Computational Complexity (cs.CC) ,01 natural sciences ,Satisfiability ,Algebra ,Computer Science - Computational Complexity ,Monotone polygon ,010201 computation theory & mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Lie algebra ,0101 mathematics ,Time complexity ,solving equations ,Equation solving ,Mathematics - Abstract
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time., 50 pages
- Published
- 2022
22. On $$\{P_1,P_2\}$$-Nekrasov Matrices
- Author
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Qilong Liu, Lei Gao, Chaoqian Li, and Yaotang Li
- Subjects
010101 applied mathematics ,Combinatorics ,Class (set theory) ,Generalization ,General Mathematics ,010102 general mathematics ,Physics::Atomic Physics ,0101 mathematics ,Permutation matrix ,01 natural sciences ,Mathematics - Abstract
The class of $$\{P_1,P_2\}$$ -Nekrasov matrices, defined in terms of permutation matrices $$P_1$$ and $$P_2$$ , is a generalization of the well-known class of Nekrasov matrices. In this paper, some computable error bounds for linear complementarity problems (LCPs) of $$\{P_1,P_2\}$$ -Nekrasov matrices are given, which depend only on the entries of the involved matrices and can be used to obtain the perturbation bounds of $$\{P_1,P_2\}$$ -Nekrasov matrices LCPs. Besides, some sufficient conditions ensuring that the subdirect sum of $$\{P_1,P_2\}$$ -Nekrasov matrices lies in the same class are also provided.
- Published
- 2021
23. Wild Cantor actions
- Author
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Ramón Barral Lijó, Hiraku Nozawa, Jesús A. Álvarez López, and Olga Lukina
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Closure (topology) ,Mathematics::General Topology ,Dynamical Systems (math.DS) ,16. Peace & justice ,Equicontinuity ,01 natural sciences ,Centralizer and normalizer ,Cantor set ,Group action ,Wreath product ,0103 physical sciences ,FOS: Mathematics ,Countable set ,2020: 37B05, 37E25, 20E08, 20E15, 20E18, 20E22, 22F05, 22F50 (Primary), 20F22, 57R30, 57R50 (Secondary) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper
- Published
- 2022
24. P-adic Integration on Bad Reduction Hyperelliptic Curves
- Author
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Eric Katz, Enis Kaya, and Algebra
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,Computation ,Mathematics::Number Theory ,010102 general mathematics ,Open set ,010103 numerical & computational mathematics ,Good reduction ,01 natural sciences ,Reduction (complexity) ,Mathematics - Algebraic Geometry ,Torsion (algebra) ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Abelian group ,10. No inequality ,Hyperelliptic curve ,Algebraic Geometry (math.AG) ,Mathematics ,Meromorphic function - Abstract
In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish., Comment: Comments Welcome! figure taken from arxiv::1606.09618; v2 minor revisions
- Published
- 2022
25. Sobolev estimates for solutions of the transport equation and ODE flows associated to non-Lipschitz drifts
- Author
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Elia Bruè and Quoc-Hung Nguyen
- Subjects
Integrable system ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Ode ,Lipschitz continuity ,01 natural sciences ,Euler equations ,Sobolev space ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,symbols ,Order (group theory) ,Vector field ,010307 mathematical physics ,0101 mathematics ,Convection–diffusion equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
It is known, after Jabin (J Differ Equ 260(5):4739–4757, 2016) and Alberti et al. (Ann PDE 5(1):9, 2019), that ODE flows and solutions of the transport equation associated to Sobolev vector fields do not propagate Sobolev regularity, even of fractional order. In this paper, we improve the result at Clop and Jylha (J Differ Equ 266(8):4544–4567, 2019) and show that some kind of propagation of Sobolev regularity happens as soon as the gradient of the drift is exponentially integrable. We provide sharp Sobolev estimates and new examples. As an application of our main theorem, we generalize a regularity result for the 2D Euler equation obtained by Bahouri and Chemin in Bahouri and Chemin (Arch Ration Mech Anal 127(2):159–181, 1994).
- Published
- 2020
26. Mathematical based control method and performance analysis of a novel hydromechatronics driving system Micro-Independent Metering
- Author
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Philip Godfrey, Karem Abuowda, Siamak Noroozi, and Mihai Dupac
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Poppet valve ,Fluid mechanics ,01 natural sciences ,Volumetric flow rate ,Flow control valve ,010101 applied mathematics ,Controllability ,Hydraulic cylinder ,Control theory ,Metering mode ,0101 mathematics ,Body orifice ,Mathematics - Abstract
This paper aims to investigate the performance of a hydraulic actuator controlled by the novel system micro-independent metering (MIM). This analysis has been performed by comparing the models of two systems which are the traditional independent metering, that depends on poppet valve, and the new hydro-mechatronics system micro-independent metering, that relies on a stepped rotary flow control valve. In general, independent metering is a hydraulic control system which guarantees a separation between the meter-in and the meter-out of the hydraulic actuator. A Valvistor valve, a special type of Poppet valves, was developed to be embedded into the independent metering (IM) system. This valve has controllability and stability shortcomings which prevent the system from spreading in the industrial applications. The Valvistor valve performance is highly affected by the fluid disturbances because the fluid is considered as a part of its control elements. A stepped rotary flow control valve has been developed to control hydraulic flow rate. The valve composed of a rotary orifice attached to a stepper motor. Using this valve instead of the traditional poppet type has led to a new configuration, that is termed by micro-independent metering. This form improves the hydraulic cylinder velocity performance by rejecting the fluid disturbances effect on the control circuit.
- Published
- 2022
27. Extrapolation of compactness on weighted spaces: Bilinear operators
- Author
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Stefanos Lappas, Tuomas Hytönen, Tuomas Hytönen / Principal Investigator, and Department of Mathematics and Statistics
- Subjects
Pure mathematics ,General Mathematics ,COMMUTATORS ,Mathematics::Classical Analysis and ODEs ,Extrapolation ,Bilinear interpolation ,NORM INEQUALITIES ,47B38 (Primary), 42B20, 42B35, 46B70, 47H60 ,Space (mathematics) ,Multilinear Muckenhoupt weights ,01 natural sciences ,Rubio de Francia extrapolation ,Compact operators ,111 Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Lp space ,Mathematics ,Calderon-Zygmund operators ,Fractional integral operators ,010102 general mathematics ,Muckenhoupt weights ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Range (mathematics) ,Compact space ,Mathematics - Classical Analysis and ODEs ,Bounded function ,Fourier multipliers ,INTEGRAL-OPERATORS - Abstract
In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages
- Published
- 2022
28. Filtrations in Module Categories, Derived Categories, and Prime Spectra
- Author
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Ryo Takahashi and Hiroki Matsui
- Subjects
Pure mathematics ,Noetherian ring ,Mathematics::Commutative Algebra ,General Mathematics ,13C60, 13D09, 13D45 ,010102 general mathematics ,Cohomological dimension ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Prime (order theory) ,Commutative diagram ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mod ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Commutative property ,Mathematics - Representation Theory ,Mathematics - Abstract
Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN
- Published
- 2022
29. The quantum Hikita conjecture
- Author
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Nicholas Proudfoot, Michael McBreen, and Joel Kamnitzer
- Subjects
Pure mathematics ,Conjecture ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Torus ,01 natural sciences ,Cohomology ring ,Mathematics - Algebraic Geometry ,Singularity ,0103 physical sciences ,FOS: Mathematics ,14N35, 16E40, 14F10 ,0101 mathematics ,Representation Theory (math.RT) ,Affine variety ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Symplectic geometry ,Quantum cohomology ,Resolution (algebra) ,Mathematics - Abstract
The Hikita conjecture relates the coordinate ring of a conical symplectic singularity to the cohomology ring of a symplectic resolution of the dual conical symplectic singularity. We formulate a quantum version of this conjecture, which relates the quantized coordinate ring of the first variety to the quantum cohomology of a symplectic resolution of the dual variety. We prove this conjecture for hypertoric varieties and for the Springer resolution. Our paper includes an appendix, written by Ben Webster, which studies highest weights for quantizations of symplectic resolutions with isolated torus actions., Comment: Version 2 contains an abstract by Ben Webster
- Published
- 2018
- Full Text
- View/download PDF
30. Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients
- Author
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Le Trong Thanh Bui and Quoc-Hung Nguyen
- Subjects
010101 applied mathematics ,General Mathematics ,Norm (mathematics) ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Applied mathematics ,0101 mathematics ,01 natural sciences ,Parabolic partial differential equation ,Mathematics - Abstract
In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.
- Published
- 2022
31. Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols
- Author
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Hung Le
- Subjects
Class (set theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Bessel potential ,Order (ring theory) ,Lipschitz continuity ,01 natural sciences ,010101 applied mathematics ,Alpha (programming language) ,Mathematics - Analysis of PDEs ,76B15, 76B03, 35S30, 35A20 ,FOS: Mathematics ,0101 mathematics ,Bifurcation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order $\alpha$ for $\alpha > 1$. Based on the properties of the convolution operator, we apply a global bifurcation technique to show the existence of a highest, even, $2\pi$-periodic traveling-wave solution. The regularity of this wave is proved to be exactly Lipschitz., Comment: 22 pages. arXiv admin note: text overlap with arXiv:1810.00248 by other authors
- Published
- 2022
32. Non-homogeneous thermoelastic Timoshenko systems
- Author
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To Fu Ma, J.E. Muñoz Rivera, M. A. Jorge Silva, and Margareth S. Alves
- Subjects
Timoshenko beam theory ,General Mathematics ,010102 general mathematics ,Constitutive equation ,Mathematical analysis ,Dissipation ,01 natural sciences ,Exponential stability ,010101 applied mathematics ,Thermoelastic damping ,Polynomial stability ,Timoshenko systems ,Shear stress ,Bending moment ,SISTEMAS DINÂMICOS ,Observability ,0101 mathematics ,Non-homogeneous coefficients ,Thermoelasticity ,Mathematics - Abstract
The well-established Timoshenko system is characterized by a particular relation between shear stress and bending moment from its constitutive equations. Accordingly, a (thermal) dissipation added on the bending moment produces exponential stability if and only if the so called “equal wave speeds” condition is satisfied. This remarkable property extends to the case of non-homogeneous coefficients. In this paper, we consider a non-homogeneous thermoelastic system with dissipation restricted to the shear stress. To this new problem, by means of a delicate control observability analysis, we prove that a local version of the equal wave speeds condition is sufficient for the exponential stability of the system. Otherwise, we study the polynomial stability of the system with decay rate depending on the regularity of initial data.
- Published
- 2017
33. On a Waring's problem for Hermitian lattices
- Author
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Jingbo Liu
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,Upper and lower bounds ,Ring of integers ,Hermitian matrix ,Waring's problem ,Combinatorics ,Integer ,Rank (graph theory) ,Quadratic field ,Orthonormal basis ,0101 mathematics ,Mathematics - Abstract
Assume E is an imaginary quadratic field and O is its ring of integers. For each positive integer m, let I m be the free Hermitian lattice of rank m over O having an orthonormal basis. For each positive integer n, let S O ( n ) be the set of all Hermitian lattices of rank n over O that can be represented by some I m . Denote by g O ( n ) the smallest positive integer g such that each Hermitian lattice in S O ( n ) can be represented by I g . In this paper, we shall provide an explicit upper bound for g O ( n ) for all imaginary quadratic fields E and all positive integers n.
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- 2022
34. The VC-dimension of axis-parallel boxes on the Torus
- Author
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Clemens Müllner, Thomas Lachmann, and Pierre Gillibert
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Computer Science - Machine Learning ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Discrete Mathematics (cs.DM) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Torus ,0102 computer and information sciences ,01 natural sciences ,Machine Learning (cs.LG) ,Combinatorics ,VC dimension ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Computer Science - Discrete Mathematics ,Mathematics - Abstract
We show in this paper that the VC-dimension of the family of d-dimensional axis-parallel boxes and cubes on the d-dimensional torus are both asymptotically d log 2 d . This is especially surprising as in most other examples the VC-dimension usually grows linearly with d in similar settings.
- Published
- 2022
35. Large Deviations for the Single-Server Queue and the Reneging Paradox
- Author
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Ruoyu Wu, Paul Dupuis, Amarjit Budhiraja, and Rami Atar
- Subjects
Statement (computer science) ,Scale (ratio) ,General Mathematics ,010102 general mathematics ,Single server queue ,Management Science and Operations Research ,01 natural sciences ,Computer Science Applications ,010104 statistics & probability ,Laplace principle ,Applied mathematics ,Large deviations theory ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.
- Published
- 2022
36. Variational Analysis of Composite Models with Applications to Continuous Optimization
- Author
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M. Ebrahim Sarabi, Ashkan Mohammadi, and Boris S. Mordukhovich
- Subjects
Continuous optimization ,Mathematical optimization ,021103 operations research ,General Mathematics ,Parametric optimization ,010102 general mathematics ,Composite number ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Computer Science Applications ,Optimization and Control (math.OC) ,FOS: Mathematics ,0101 mathematics ,Variational analysis ,Mathematics - Optimization and Control ,Mathematics - Abstract
The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way, we develop extended calculus rules for first-order and second-order generalized differential constructions while paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second-order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers, strong metric subregularity of Karush-Kuhn-Tucker systems in parametric optimization, and so on.
- Published
- 2022
37. Asymptotic results for the best-choice problem with a random number of objects
- Author
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Masami Yasuda
- Subjects
Statistics and Probability ,Multivariate random variable ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Random function ,01 natural sciences ,Integral equation ,010104 statistics & probability ,Random variate ,Convergence of random variables ,Stochastic simulation ,Applied mathematics ,Optimal stopping ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Central limit theorem - Abstract
This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.
- Published
- 1984
38. Existence and concentration of ground state solutions for a class of fractional Schrödinger equations
- Author
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Zhuo Chen and Chao Ji
- Subjects
010101 applied mathematics ,Class (set theory) ,symbols.namesake ,General Mathematics ,010102 general mathematics ,symbols ,0101 mathematics ,Ground state ,01 natural sciences ,Mathematics ,Schrödinger equation ,Mathematical physics - Abstract
In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ), ϵ is a positive parameter, N > 2 α, ( − Δ ) α stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions.
- Published
- 2022
39. A Sharp Multidimensional Hermite–Hadamard Inequality
- Author
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Simon Larson
- Subjects
Subharmonic function ,General Mathematics ,010102 general mathematics ,Regular polygon ,Function (mathematics) ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Dirichlet boundary condition ,Hermite–Hadamard inequality ,Bounded function ,symbols ,Nabla symbol ,0101 mathematics ,Mathematics - Abstract
Let $\Omega \subset {\mathbb{R}}^d $, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to{\mathbb{R}}$ be a non-negative subharmonic function. In this paper, we prove the inequality $$\begin{equation*} \frac{1}{|\Omega|}\int_{\Omega} f(x)\, \textrm{d}x \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\, \textrm{d}\sigma(x)\,. \end{equation*}$$Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset{\mathbb{R}}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then $$\begin{equation*} \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \end{equation*}$$Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. [2].
- Published
- 2022
40. Some significant remarks on multivalued Perov type contractions on cone metric spaces with a directed graph
- Author
-
Aleksandra Sretenovic, Nicola Fabiano, Ana Savić, Stojan Radenović, and Nikola Mirkov
- Subjects
Pure mathematics ,General Mathematics ,cone metric space ,010102 general mathematics ,multivalued mapping ,graphic contraction ,Directed graph ,common fixed point ,Fixed point ,Type (model theory) ,Mathematical proof ,directed graph ,01 natural sciences ,Cone (formal languages) ,c-sequence ,010101 applied mathematics ,Metric space ,QA1-939 ,0101 mathematics ,Contraction principle ,perov's type results ,Mathematics ,Complement (set theory) - Abstract
Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.
- Published
- 2022
41. An evolutionary Haar-Rado type theorem
- Author
-
Rudolf Rainer, Thomas Stanin, and Jarkko Siltakoski
- Subjects
osittaisdifferentiaaliyhtälöt ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,variaatiolaskenta ,Algebraic geometry ,Type (model theory) ,01 natural sciences ,Parabolic partial differential equation ,Omega ,Modulus of continuity ,Convexity ,010101 applied mathematics ,Combinatorics ,Number theory ,0101 mathematics ,Mathematics - Abstract
In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.
- Published
- 2022
42. Arbitrarily small nodal solutions for nonhomogeneous Robin problems
- Author
-
Nikolaos S. Papageorgiou and Shengda Zeng
- Subjects
010101 applied mathematics ,General Mathematics ,010102 general mathematics ,Applied mathematics ,0101 mathematics ,NODAL ,01 natural sciences ,Mathematics - Abstract
In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).
- Published
- 2022
- Full Text
- View/download PDF
43. Automorphisms and opposition in spherical buildings of exceptional type, I
- Author
-
James Parkinson and Hendrik Van Maldeghem
- Subjects
Pure mathematics ,automorphism ,Diagram (category theory) ,General Mathematics ,Root (chord) ,Group Theory (math.GR) ,0102 computer and information sciences ,Type (model theory) ,Unipotent ,01 natural sciences ,Mathematics::Group Theory ,Group of Lie type ,FOS: Mathematics ,Mathematics - Combinatorics ,domestic ,0101 mathematics ,Algebraic number ,Mathematics ,Simplex ,Exceptional spherical buildings ,010102 general mathematics ,Automorphism ,Mathematics and Statistics ,opposition diagram ,010201 computation theory & mathematics ,20E42, 51E24, 51B25, 20E45 ,Combinatorics (math.CO) ,Mathematics - Group Theory - Abstract
To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf {E}_6$ , $\mathsf {F}_4$ , and $\mathsf {G}_2$ . Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.
- Published
- 2022
44. Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials
- Author
-
Regilene Oliveira, Nicolae Vulpe, Dana Schlomiuk, and Alex C. Rezende
- Subjects
Pure mathematics ,General Mathematics ,INVARIANTES ,010102 general mathematics ,Diagram ,Ellipse ,Bifurcation diagram ,01 natural sciences ,010101 applied mathematics ,Quadratic equation ,Limit cycle ,Limit (mathematics) ,0101 mathematics ,Invariant (mathematics) ,Quadratic differential ,Mathematics - Abstract
Consider the class QS of all non-degenerate planar quadratic systems and its subclass QSE of all its systems possessing an invariant ellipse. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of $${{\mathbb {R}}}^{12}$$ through its coefficients. In this paper we provide necessary and sufficient conditions for a system in QS to have at least one invariant ellipse. We give the global “bifurcation” diagram of the family QS which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or it is not a limit cycle. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for any quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle.
- Published
- 2022
45. On the Arithmetic Mean of the Square Roots of the First n Positive Integers
- Author
-
Mircea Merca
- Subjects
Root of unity modulo n ,Discrete mathematics ,Contraharmonic mean ,General Mathematics ,010102 general mathematics ,Landau–Ramanujan constant ,010103 numerical & computational mathematics ,01 natural sciences ,Education ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Square root ,Geometric–harmonic mean ,symbols ,Asymptotic formula ,0101 mathematics ,Geometric mean ,Mathematics ,Arithmetic mean - Abstract
SummaryIn the paper, the author presents an asymptotic formula for the arithmetic mean of the square roots of the first n positive integers using only tools from undergraduate calculus.
- Published
- 2017
46. Fejer type inequalities for higher order convex functions and quadrature formulae
- Author
-
M. Ribičić Penava, Josipa Barić, Josip Pečarić, and Lj. Kvesić
- Subjects
Mathematics::Functional Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hermite-Hadamard inequalitiesFejer inequalitiesHigher order convex functionsQuadrature formulae ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Quadrature (mathematics) ,Discrete Mathematics and Combinatorics ,Order (group theory) ,Applied mathematics ,Point (geometry) ,Integral formula ,0101 mathematics ,Convex function ,Mathematics - Abstract
The aim of this paper is to obtain Fejer type inequalities for higher-order convex functions and the general weighted integral formula involving w- harmonic sequences of functions. Also, Fejer type inequalities are obtained for the general three, four, and five-point quadrature formulae. Further, Fejer type inequalities for the corrected three corrected four and corrected five-point quadrature formulae are considered. In special cases, Fejer type estimates for Simpson, Maclaurin, corrected Simpson, and corrected Maclaurin quadrature rules are derived.
- Published
- 2022
47. A toy model for the Drinfeld–Lafforgue shtuka construction
- Author
-
Dennis Gaitsgory, David Kazhdan, Yakov Varshavsky, and Nick Rozenblyum
- Subjects
Pure mathematics ,Trace (linear algebra) ,Toy model ,Functor ,General Mathematics ,010102 general mathematics ,Excursion ,01 natural sciences ,Action (physics) ,0103 physical sciences ,Point of departure ,010307 mathematical physics ,0101 mathematics ,Categorical variable ,Mathematics - Abstract
The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.
- Published
- 2022
48. New extensions of Jacobson’s lemma and Cline’s formula
- Author
-
V. G. Miller and H. Zguitti
- Subjects
Lemma (mathematics) ,General Mathematics ,010102 general mathematics ,Linear operators ,Spectral properties ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,01 natural sciences ,law.invention ,Combinatorics ,Algebra ,Invertible matrix ,law ,Bounded function ,0101 mathematics ,Mathematics - Abstract
In an associative ring \(\mathcal {R}\), if elements a, b and c satisfy \(aba=aca\) then Corach et al. (Comm Algebra 41:520–531, 2013) proved that \(1-ac\) is (left/right) invertible if and only if \(1-ba\) is left/right invertible; which is an extension of the Jacobson’s lemma. Also, Lian and Zeng (Turk Math J 40:166–165, 2016) and Zeng and Zhong (J Math Anal Appl 427:830–840, 2015) proved that if the product ac is (generalized/pseudo) Drazin invertible, then so is ba extending the Cline’s formula to the case of the (generalized/pseudo) Drazin invertibility. In this paper, for elements a, b, c, d in an associative ring \(\mathcal {R}\) satisfying $$\begin{aligned} \left\{ \begin{array}{c}acd=dbd,\\ dba=aca,\end{array}\right. \end{aligned}$$ we study common spectral properties for \(1-ac\) (resp. ac) and \(1-bd\) (resp. bd). So, we extend Jacobson’s lemma for (left/right) invertibility and we generalize Cline’s formula to the case of the (generalized/pseudo) Drazin invertibility. In particular, as application, for bounded linear operators A, B, C, D satisfying \( {ACD}= {DBD}\) and \( {DBA}= {ACA}\), we show that AC is B-Weyl operator if and only if BD is B-Weyl operator.
- Published
- 2017
49. Supersingular O'Grady Varieties of Dimension Six
- Author
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Lie Fu, Zhiyuan Li, and Haitao Zou
- Subjects
Pure mathematics ,Conjecture ,Group (mathematics) ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,14J28, 14J42, 14G17, 14D22, 14M20, 14C15, 14C25 ,01 natural sciences ,Cohomology ,Moduli space ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Crepant resolution ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Algebraic Geometry (math.AG) ,Mathematics ,Symplectic geometry ,Tate conjecture - Abstract
O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces., Comment: Final version. To appear in I.M.R.N
- Published
- 2022
50. On the motivic oscillation index and bound of exponential sums modulo p via analytic isomorphisms
- Author
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Willem Veys and Kien Huu Nguyen
- Subjects
Polynomial ,Pure mathematics ,Conjecture ,Jet (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Resolution of singularities ,Algebraic number field ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,Isomorphism class ,0101 mathematics ,Local zeta-function ,Mathematics - Abstract
Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.
- Published
- 2022
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