328 results
Search Results
2. Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur
- Author
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Kevin Zumbrun and Benjamin Texier
- Subjects
Conservation law ,Kullback–Leibler divergence ,Standard molar entropy ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Min entropy ,Shock strength ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS
- Published
- 2014
3. An elementary proof of the homotopy invariance of stabilized configuration spaces
- Author
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Malin, Connor
- Subjects
Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology - Abstract
In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual of $\Sigma^\infty_+ F(M,k)$ and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to ``generalized configuration spaces'' which have recently undergone study., Comment: Minor changes to structure of paper; to appear in "Proceedings of the American Mathematical Society"
- Published
- 2023
4. Spaces of countable free set number and PFA
- Author
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Dow, Alan and Juhasz, Istvan
- Subjects
Applied Mathematics ,General Mathematics ,General Topology (math.GN) ,FOS: Mathematics ,Mathematics::General Topology ,Mathematics - General Topology ,54A25, 54A35, 54D10, 03E04 - Abstract
The main result of this paper is that, under PFA, for every regular space X X with F ( X ) = ω F(X) = \omega we have | X | ≤ w ( X ) ω |X| \le w(X)^\omega ; in particular, w ( X ) ≤ c w(X) \le \mathfrak {c} implies | X | ≤ c |X| \le \mathfrak {c} . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . In fact, this space X X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X X intersect, which is much stronger than F ( X ) = ω F(X) = \omega . Moreover, any non-empty open set in X X also has size 2 c 2^\mathfrak {c} , and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points.
- Published
- 2023
5. Covering by homothets and illuminating convex bodies
- Author
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Alexey Glazyrin
- Subjects
Conjecture ,Applied Mathematics ,General Mathematics ,Discrete geometry ,Boundary (topology) ,Metric Geometry (math.MG) ,Upper and lower bounds ,Infimum and supremum ,Homothetic transformation ,Combinatorics ,Mathematics - Metric Geometry ,Hausdorff dimension ,FOS: Mathematics ,Mathematics::Metric Geometry ,Convex body ,Mathematics - Abstract
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
- Published
- 2021
6. On the Baum–Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture
- Author
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Adam Skalski and Yuki Arano
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Operator Algebras ,Quantum group ,Applied Mathematics ,General Mathematics ,Mathematics - Operator Algebras ,Crossed product ,Unimodular matrix ,Mathematics::K-Theory and Homology ,Primary 46L67, Secondary 46L80 ,FOS: Mathematics ,Baum–Connes conjecture ,Countable set ,Equivariant map ,Operator Algebras (math.OA) ,Quantum ,Mathematics - Abstract
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition., Comment: 15 pages, v2 corrects a few minor points. The final version of the paper will appear in the Proceedings of the American Mathematical Society
- Published
- 2021
7. The nilpotent cone for classical Lie superalgebras
- Author
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Daniel K. Nakano and L. Jenkins
- Subjects
Pure mathematics ,Nilpotent cone ,17B20, 17B10 ,Applied Mathematics ,General Mathematics ,Group Theory (math.GR) ,Representation theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.
- Published
- 2021
8. On the isochronous center of planar piecewise polynomial potential systems
- Author
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Changjian Liu and Shaoqing Wang
- Subjects
Applied Mathematics ,General Mathematics ,MathematicsofComputing_GENERAL - Abstract
This paper is devoted to the isochronicity of two kinds of planar piecewise polynomial potential systems separated by x x -axis and y y -axis, respectively. By means of the expansion of the period function near the infinity, we obtain in this paper that for these two cases, the piecewise potential system has an isochronous center at the origin if and only if the subsystems are linear. It is however not true in the analytic scenario; one can find in the paper by F. Mañosas and P. J. Torres [Proc. Amer. Math. Soc. 133 (2005), pp. 3027–3035] that the isochronicity of the center of the piecewise analytic potential system cannot imply the linearity of the two subsystems.
- Published
- 2022
9. Gossez’s skew linear map and its pathological maximally monotone multifunctions
- Author
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Simons, Stephen
- Subjects
Applied Mathematics ,Pure Mathematics ,Mathematical Sciences ,Skew linear operator ,maximal monotonicity ,duality map ,Pure mathematics - Abstract
In this note, we give a generalization of Gossez’s example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez’s original papers. We also discuss some new properties of Gossez’s skew linear operator and its adjoint.
- Published
- 2019
10. The Lane-Emden equation with variable double-phase and multiple regime
- Author
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Vicenţiu D. Rădulescu and Claudianor O. Alves
- Subjects
Variable exponent ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Mathematical proof ,Supercritical fluid ,symbols.namesake ,Mathematics - Analysis of PDEs ,Criticality ,Feature (computer vision) ,Dirichlet boundary condition ,FOS: Mathematics ,symbols ,Lane–Emden equation ,Analysis of PDEs (math.AP) ,Variable (mathematics) ,Mathematics - Abstract
We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS
- Published
- 2020
11. On Lau’s conjecture II
- Author
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Khadime Salame
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Conjecture ,Weak topology ,Semigroup ,Applied Mathematics ,General Mathematics ,Mathematics - Abstract
In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer.
- Published
- 2020
12. On schizophrenic patterns in 𝑏-ary expansions of some irrational numbers
- Author
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László Tóth
- Subjects
Combinatorics ,Integer ,Square root ,Applied Mathematics ,General Mathematics ,Irrational number ,Schizophrenic number ,Function (mathematics) ,Decimal representation ,Extension (predicate logic) ,Special case ,Mathematics - Abstract
In this paper we study the b b -ary expansions of the square roots of the function defined by the recurrence f b ( n ) = b f b ( n − 1 ) + n f_b(n)=b f_b(n-1)+n with initial value f ( 0 ) = 0 f(0)=0 taken at odd positive integers n n , of which the special case b = 10 b=10 is often referred to as the “schizophrenic” or “mock-rational” numbers. Defined by Darling in 2004 2004 and studied in more detail by Brown in 2009 2009 , these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 b\geq 2 by formally defining the schizophrenic pattern present in the b b -ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.
- Published
- 2019
13. Precise large deviations for the first passage time of a random walk with negative drift
- Author
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Mariusz Maślanka and Dariusz Buraczewski
- Subjects
Applied Mathematics ,General Mathematics ,Statistics ,Large deviations theory ,Statistical physics ,First-hitting-time model ,Random walk ,Mathematics - Abstract
Let S n S_n be partial sums of an i.i.d. sequence { X i } \{X_i\} . We assume that E X 1 > 0 \mathbb {E} X_1 >0 and P [ X 1 > 0 ] > 0 \mathbb {P}[X_1>0]>0 . In this paper we study the first passage time τ u = inf { n : S n > u } . \begin{equation*} \tau _u = \inf \{n:\; S_n > u\}. \end{equation*} The classical Cramér’s estimate of the ruin probability says that P [ τ u > ∞ ] ∼ C e − α 0 u as u → ∞ , \begin{equation*} \mathbb {P}[\tau _u>\infty ] \sim C e^{-\alpha _0 u}\qquad \text {as } u\to \infty , \end{equation*} for some parameter α 0 \alpha _0 . The aim of the paper is to describe precise large deviations of the first crossing by S n S_n a linear boundary. More precisely for a fixed parameter ρ \rho we study asymptotic behavior of P [ τ u = ⌊ u / ρ ⌋ ] \mathbb {P}\big [\tau _u = \lfloor u/\rho \rfloor \big ] as u u tends to infinity.
- Published
- 2019
14. Gossez’s skew linear map and its pathological maximally monotone multifunctions
- Author
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Stephen Simons
- Subjects
Pure mathematics ,Generalization ,Applied Mathematics ,General Mathematics ,duality map ,Closure (topology) ,Regular polygon ,Skew ,Mathematics::General Topology ,Pure Mathematics ,Linear map ,Range (mathematics) ,Monotone polygon ,Skew linear operator ,Compactification (mathematics) ,maximal monotonicity ,Mathematics - Abstract
In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. While most of this paper uses elementary functional analysis, we correlate our results with those obtained by using the Stone-Cech compactification of the integers.
- Published
- 2019
15. Generic linear perturbations
- Author
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Shunsuke Ichiki
- Subjects
Pure mathematics ,Yield (engineering) ,Transversality ,business.industry ,Applied Mathematics ,General Mathematics ,Geometric Topology (math.GT) ,Modular design ,Submanifold ,Stability (probability) ,Mathematics - Geometric Topology ,Transverse plane ,FOS: Mathematics ,Mathematics::Differential Geometry ,57R45, 58K25, 57R40 ,business ,Mathematics::Symplectic Geometry ,Subspace topology ,Mathematics ,Vector space - Abstract
In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear perturbations of a smooth mapping from a submanifold of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ yield a transverse mapping with respect to a given modular submanifold. Moreover, applications of this result are given., Comment: 11 pages, to appear in Proceedings of the American Mathematical Society
- Published
- 2018
16. Functions of triples of noncommuting self-adjoint operators under perturbations of class $\boldsymbol {S}_p$
- Author
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V. V. Peller
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics - Spectral Theory ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Spectral Theory (math.SP) ,Self-adjoint operator ,Mathematics - Abstract
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$., 14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961
- Published
- 2018
17. On Bohr sets of integer-valued traceless matrices
- Author
-
Alexander Fish
- Subjects
Applied Mathematics ,General Mathematics ,Torus ,Ergodic Ramsey theory ,Random walk ,Bohr model ,Combinatorics ,Matrix (mathematics) ,symbols.namesake ,Conjugacy class ,Integer ,symbols ,Analytic number theory ,Mathematics - Abstract
In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].
- Published
- 2017
18. On the Auslander-Reiten conjecture for Cohen-Macaulay local rings
- Author
-
Goto, Shiro and Takahashi, Ryo
- Subjects
Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,Mathematics::Representation Theory - Abstract
This paper studies vanishing of Ext modules over Cohen-Macaulay local rings. The main result of this paper implies that the Auslander-Reiten conjecture holds for maximal Cohen-Macaulay modules of rank one over Cohen-Macaulay normal local rings. It also recovers a theorem of Avramov-Buchweitz-Şega and Hanes-Huneke, which shows that the Tachikawa conjecture holds for Cohen-Macaulay generically Gorenstein local rings.
- Published
- 2017
19. On the Auslander–Reiten conjecture for Cohen–Macaulay local rings
- Author
-
Ryo Takahashi and Shiro Goto
- Subjects
Discrete mathematics ,Pure mathematics ,Conjecture ,Cohen–Macaulay ring ,Applied Mathematics ,General Mathematics ,Gorenstein ring ,Local ring ,Mathematics - Abstract
This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.
- Published
- 2017
20. Coefficients of McKay-Thompson series and distributions of the moonshine module
- Author
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Hannah Larson
- Subjects
Mathematics - Number Theory ,Series (mathematics) ,Distribution (number theory) ,Applied Mathematics ,General Mathematics ,Regular representation ,Asymptotic distribution ,Combinatorics ,Conjugacy class ,Character table ,Irreducible representation ,FOS: Mathematics ,Number Theory (math.NT) ,Group ring ,Mathematics - Abstract
In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module V ♮ = ⨁ n V n ♮ V^\natural = \bigoplus _n V_n^\natural . Their results show that as n n tends to infinity, V n ♮ V_n^\natural is dominated by direct sums of copies of the regular representation. That is, if we view V n ♮ V_n^\natural as a module over the group ring Z [ M ] \mathbb {Z}[\mathbb {M}] , the free part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster’s character table. We find analogous results for the other monster modules V ( − m ) V^{(-m)} and W ♮ W^\natural studied by Duncan, Griffin, and Ono.
- Published
- 2016
21. Explicit computations with the divided symmetrization operator
- Author
-
Tewodros Amdeberhan
- Subjects
Polynomial ,Permutohedron ,Applied Mathematics ,General Mathematics ,Algebra ,Symmetric function ,Operator (computer programming) ,Simple (abstract algebra) ,Linear form ,FOS: Mathematics ,Mathematics - Combinatorics ,Symmetrization ,Combinatorics (math.CO) ,Variety (universal algebra) ,Mathematics - Abstract
Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable and available to the research practitioner in these areas. Also, the methods of proof utilized here are simple and amenable to many more analogous computations. We conclude the paper with a list of such formulas., Comment: 13 pages, no figures
- Published
- 2015
22. Kurepa trees and the failure of the Galvin property
- Author
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Tom Benhamou, Shimon Garti, and Saharon Shelah
- Subjects
Mathematics::Logic ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics::General Topology ,Mathematics - Logic ,03E02, 03E35, 03E55 ,Logic (math.LO) - Abstract
We force the existence of a non-trivial κ \kappa -complete ultrafilter over κ \kappa which fails to satisfy the Galvin property. This answers a question asked by Benhamou and Gitik [Ann. Pure Appl. Logic 173 (2022), Paper No. 103107].
- Published
- 2022
23. A remark on the global dynamics of competitive systems on ordered Banach spaces
- Author
-
Daniel Munther and King-Yeung Lam
- Subjects
Pure mathematics ,Exponential stability ,Applied Mathematics ,General Mathematics ,Stability theory ,Product (mathematics) ,Zero (complex analysis) ,Banach space ,Order (group theory) ,Banach manifold ,Complement (set theory) ,Mathematics - Abstract
A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on X + = X 1 + × X 2 + X^+ = X_1^+ \times X_2^+ , the product of two cones in respective Banach spaces, if ( u ∗ , 0 ) (u^*,0) and ( 0 , v ∗ ) (0,v^*) are the global attractors in X 1 + × { 0 } X_1^+ \times \{0\} and { 0 } × X 2 + \{0\}\times X_2^+ respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of ( u ∗ , 0 ) , ( 0 , v ∗ ) (u^*,0), (0,v^*) attracts all trajectories initiating in the order interval I = [ 0 , u ∗ ] × [ 0 , v ∗ ] I = [0,u^*] \times [0,v^*] . However, it was demonstrated by an example that in some cases neither ( u ∗ , 0 ) (u^*,0) nor ( 0 , v ∗ ) (0,v^*) is globally asymptotically stable if we broaden our scope to all of X + X^+ . In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) among all trajectories in X + X^+ . Namely, one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice.
- Published
- 2015
24. Coisotropic subalgebras of complex semisimple Lie bialgebras
- Author
-
Nicole Rae Kroeger
- Subjects
Pure mathematics ,Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Subalgebra ,Torus ,Fixed point ,symbols.namesake ,17B62 ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,symbols ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,Variety (universal algebra) ,Mathematics::Representation Theory ,Mathematics::Symplectic Geometry ,Mathematics - Representation Theory ,Lagrangian ,Mathematics - Abstract
In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.
- Published
- 2015
25. On the building dimension of closed cones and Almgren’s stratification principle
- Author
-
Andrea Marchese
- Subjects
Combinatorics ,Building dimension ,Conjecture ,Packing dimension ,Dimension (vector space) ,Applied Mathematics ,General Mathematics ,Geometry ,Stratification ,Stratification (mathematics) ,Mathematics - Abstract
In this paper we disprove a conjecture stated in (4) on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family C of closed cones and a set S such that every blow-up of S at every point x 2 S is contained in some element of C, is it true that the dimension of S is smaller than or equal to the largest dimension of a vector space contained is some element of C?
- Published
- 2015
26. Analytic isomorphisms of compressed local algebras
- Author
-
Juan Elias and Maria Evelina Rossi
- Subjects
Discrete mathematics ,Pure mathematics ,Class (set theory) ,Compressed Algebras ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Dimension (graph theory) ,Local ring ,"Hilbert function" ,Type (model theory) ,Conductor ,Socle ,Mathematics::Group Theory ,Embedding ,Mathematics - Abstract
In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Froberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see (6). This unexpected result has been extended to compressed level K-algebras of socle degree 3 in (4). In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s � 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.
- Published
- 2014
27. A separation between tropical matrix ranks
- Author
-
Shitov, Yaroslav
- Subjects
Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Physics::Atmospheric and Oceanic Physics - Abstract
We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a $01$-matrix with many ones and without large all-one submatrices, and we put our results together and construct an $n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov rank $n-o(n)$., Comment: 9 pages
- Published
- 2022
28. A unified approach towards the impossibility of finite time vanishing depth for incompressible free boundary flows
- Author
-
Zhiyuan Geng, Rafael Granero-Belinchón, and Universidad de Cantabria
- Subjects
Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,Fluid Dynamics (physics.flu-dyn) ,FOS: Mathematics ,FOS: Physical sciences ,Physics - Fluid Dynamics ,Analysis of PDEs (math.AP) - Abstract
In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows. The first author was supported by the Basque Government through the BERC 2022-2025 program and by the Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project PID2020-114189RB-I00 funded by Agencia Estatal de Investigaci´on (PID2020-14189RB-I00 / AEI / 10.13039/501100011033). The second author was supported by the project “Mathematical Analysis of Fluids and Applications” Grant PID2019-109348GA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and acronym “MAFyA”. This publication is part of the project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. The second author was also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation.
- Published
- 2022
29. Finite determinacy and approximation of flat maps
- Author
-
Patel, Aftab
- Subjects
Mathematics - Algebraic Geometry ,32S05, 32S10, 32B99, 32C07, 14P15, 14P20, 13H10, 13C14 ,Mathematics::Algebraic Geometry ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real or complex analytic spaces. It is shown that the flatness of analytic maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection and Cohen-Macaulay analytic germs can be arbitrarily closely approximated by algebraic and Nash maps respectively in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone., Comment: 13 pages. Substantial improvements in writing, existing theorems strengthened, several results added. arXiv admin note: text overlap with arXiv:1910.11498
- Published
- 2022
30. Endofunctors of singularity categories characterizing Gorenstein rings
- Author
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Ryo Takahashi and Takuma Aihara
- Subjects
Discrete mathematics ,Noetherian ring ,Pure mathematics ,Derived category ,Functor ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Gorenstein ring ,Local ring ,Cohen–Macaulay ring ,Mathematics::Category Theory ,Quotient module ,Krull dimension ,Mathematics - Abstract
In this paper, we prove that certain contravariant endofunctors of singularity categories characterize Gorenstein rings. Let Λ be a noetherian ring. Denote by Dsg(Λ) the singularity category of Λ, that is, the Verdier quotient of the bounded derived category D(Λ) of finitely generated (right) Λ-modules by the full subcategory consisting of bounded complexes of finitely generated projective Λ-modules. We are interested in the following question. Question 1. What contravariant endofunctor of Dsg(Λ) characterizes the Iwanaga– Gorenstein property of Λ? In this paper we shall consider this question in the case where Λ is commutative and Cohen–Macaulay. Let R be a commutative Cohen–Macaulay local ring of Krull dimension d. Denote by CM(R) the category of (maximal) Cohen–Macaulay R-modules and by CM(R) its stable category : the objects of CM(R) are the Cohen–Macaulay R-modules, and the hom-set HomCM(R)(M,N) is defined as HomR(M,N), the quotient module of HomR(M,N) by the submodule consisting homomorphisms factoring through finitely generated projective (or equivalently, free) R-modules. The natural full embedding functor CM(R) → D(R) induces an additive covariant functor η : CM(R) → Dsg(R). Furthermore, the assignment M 7→ ΩTrM , where Ω and Tr stand for the syzygy and transpose functors respectively (see [1, Chapter 2, §1] for details of the functors Ω and Tr), makes an additive contravariant functor
- Published
- 2015
31. A Cayley–Bacharach theorem and plane configurations
- Author
-
Jake Levinson and Brooke Ullery
- Subjects
Applied Mathematics ,General Mathematics - Abstract
In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.
- Published
- 2022
32. Gap theorems for ends of smooth metric measure spaces
- Author
-
Hua, Bobo and Wu, Jia-Yong
- Subjects
Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics::Differential Geometry ,53C20 - Abstract
In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n, g,e^{-f}dv)$ with the Bakry-\'Emery Ricci tensor $\mathrm{Ric}_f\ge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When $\mathrm{Ric}_f\ge 0$ and $f$ has some degeneration outside $B_o(R)$, we show that there exists an $\epsilon=\epsilon(n,\sup_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\le\epsilon$. When $\mathrm{Ric}_f\ge\frac 12$ and $f(x)\le\frac 14d^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion., Comment: Proc AMS, final version
- Published
- 2022
33. A rigidity theorem for parabolic 2-Hessian equations
- Author
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He, Yan, Pan, Cen, and Xiang, Ni
- Subjects
Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this paper, we consider the entire solution to the parabolic 2 2 -Hessian equation − u t σ 2 ( D 2 u ) = 1 -u_t\sigma _2(D^2 u)=1 in R n × ( − ∞ , 0 ] \mathbb {R}^n\times (-\infty ,0] . We prove a rigidity theorem for the parabolic 2 2 -Hessian equation in R n × ( − ∞ , 0 ] \mathbb {R}^n\times (-\infty ,0] by establishing Pogorelov type estimates for 2 2 -convex-monotone solutions.
- Published
- 2022
34. Mixed 𝐴₂-𝐴_{∞} estimates of the non-homogeneous vector square function with matrix weights
- Author
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Sergei Treil
- Subjects
Matrix (mathematics) ,Applied Mathematics ,General Mathematics ,Non homogeneous ,Mathematical analysis ,Mathematics - Abstract
This paper extends the results from a work of Hytönen, Petermichl, and Volberg about sharp A 2 A_2 - A ∞ A_\infty estimates with matrix weights to the non-homogeneous situation.
- Published
- 2023
35. Weakly saturated hypergraphs and a conjecture of Tuza
- Author
-
Shapira, Asaf and Tyomkyn, Mykhaylo
- Subjects
Mathematics::Combinatorics ,Computer Science::Discrete Mathematics ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) - Abstract
Given a fixed hypergraph H H , let wsat ( n , H ) \operatorname {wsat}(n,H) denote the smallest number of edges in an n n -vertex hypergraph G G , with the property that one can sequentially add the edges missing from G G , so that whenever an edge is added, a new copy of H H is created. The study of wsat ( n , H ) \operatorname {wsat}(n,H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H H very little is known regarding wsat ( n , H ) \operatorname {wsat}(n,H) , Alon proved in 1985 that for every graph H H there is a limiting constant C H C_H so that wsat ( n , H ) = ( C H + o ( 1 ) ) n \operatorname {wsat}(n,H)=(C_H+o(1))n . Tuza conjectured in 1992 that Alon’s theorem can be (appropriately) extended to arbitrary r r -uniform hypergraphs. In this paper we prove this conjecture.
- Published
- 2023
36. Torus Lorenz links obtained by full twists along torus links
- Author
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de Paiva, Thiago
- Subjects
Mathematics - Geometric Topology ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Geometric Topology (math.GT) ,57k10, 57K32, 57k35 ,Physics::Classical Physics ,Mathematics::Symplectic Geometry ,Mathematics::Geometric Topology - Abstract
All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman., Comment: 8 pages, 1 figure. V2 contains minor changes. To appear in Proceedings of the AMS
- Published
- 2023
37. 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
38. Congruences concerning generalized central trinomial coefficients
- Author
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Jia-Yu Chen and Chen Wang
- Subjects
Mathematics - Number Theory ,Mathematics::Number Theory ,11A07, 11B75, 11B65, 05A10 ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Number Theory (math.NT) ,Combinatorics (math.CO) - Abstract
For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summation $\sum_{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400]., 13 pages. This is a preliminary manuscript
- Published
- 2022
39. Asymptotic free independence and entry permutations for Gaussian random matrices
- Author
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Popa, Mihai
- Subjects
Mathematics::Combinatorics ,Applied Mathematics ,General Mathematics ,Mathematics - Operator Algebras ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,46L54, 15B52, 05A05 ,Operator Algebras (math.OA) - Abstract
The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics., Comment: Second version, some errors corrected, less pages, more results
- Published
- 2022
40. Isometric rigidity of Wasserstein spaces: The graph metric case
- Author
-
Kiss, Gergely and Titkos, Tam��s
- Subjects
Mathematics - Metric Geometry ,Mathematics - Classical Analysis and ODEs ,54E40, 46E27 (Primary), 54E70, 05C12 (Secondary) ,Applied Mathematics ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics::Metric Geometry ,Metric Geometry (math.MG) - Abstract
The aim of this paper is to prove that the $p$-Wasserstein space $\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a countable graph metric space. As a consequence, we obtain that for every countable group $H$ and any $p\geq 1$ there exists a $p$-Wasserstein space whose isometry group is isomorphic to $H$., 14 pages with 3 figures
- Published
- 2022
41. The plethystic inverse of the odd Lie representations
- Author
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Sundaram, Sheila
- Subjects
05E10, 20C30 ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics - Representation Theory - Abstract
The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the multilinear component of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\sum_{n\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations. We determine the plethystic inverse of the alternating sum of the odd Lie characteristics., Comment: 12 pages; Section 3 reorganised per referee comments; added Lemma 3.2 and Proposition 3.9. To appear in Proceedings of the American Mathematical Society
- Published
- 2022
42. Central units of integral group rings of monomial groups
- Author
-
Gurmeet Bakshi and Gurleen Kaur
- Subjects
Applied Mathematics ,General Mathematics - Abstract
In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring Z G \mathbb {Z}G for a subgroup closed monomial group G G with the property that every cyclic subgroup of order not a divisor of 4 4 or 6 6 is subnormal in G G . If G G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) . Furthermore, for a generalized strongly monomial group G G , the rank of Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G G .
- Published
- 2022
43. Members of thin Π₁⁰ classes and generic degrees
- Author
-
Guohua Wu, Bowen Yuan, Frank Stephan, and School of Physical and Mathematical Sciences
- Subjects
Mathematics [Science] ,Pure mathematics ,Turing Degrees ,Applied Mathematics ,General Mathematics ,Pi ,Genericity ,Mathematics - Abstract
A Π 1 0 \Pi ^{0}_{1} class P P is thin if every Π 1 0 \Pi ^{0}_{1} subclass Q Q of P P is the intersection of P P with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin Π 1 0 \Pi ^{0}_{1} classes, and proved that degrees containing no members of thin Π 1 0 \Pi ^{0}_{1} classes can be recursively enumerable, and can be minimal degree below 0 ′ \mathbf {0}’ . In this paper, we work on this topic in terms of genericity, and prove that all 2-generic degrees contain no members of thin Π 1 0 \Pi ^{0}_{1} classes. In contrast to this, we show that all 1-generic degrees below 0 ′ \mathbf {0}’ contain members of thin Π 1 0 \Pi ^{0}_{1} classes.
- Published
- 2022
44. On singularities of Ericksen-Leslie system in dimension three
- Author
-
Huang, Tao and Wang, Peiyong
- Subjects
Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second example is constructed for an initial data with small energy but nontrivial topology. A counter example of maximum principle to the system is constructed by utilizing the Poiseuille flow in dimension one., arXiv admin note: text overlap with arXiv:1504.01080
- Published
- 2023
45. Some improvements of the Katznelson-Tzafriri theorem on Hilbert space
- Author
-
David Seifert
- Subjects
Mathematics - Functional Analysis ,Large class ,Pure mathematics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Bounded function ,FOS: Mathematics ,Hilbert space ,symbols ,Abelian group ,Functional Analysis (math.FA) ,Mathematics - Abstract
This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations., 13 pages, to appear in Proceedings of the American Mathematical Society
- Published
- 2015
46. Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition
- Author
-
Kunikawa, Keita and Sakurai, Yohei
- Subjects
Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Spectral Theory - Abstract
In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation., 10 pages
- Published
- 2022
47. Convolution properties of Orlicz spaces on hypergroups
- Author
-
Ali Reza Bagheri Salec, Vishvesh Kumar, and Seyyed Mohammad Tabatabaie
- Subjects
Applied Mathematics ,General Mathematics - Abstract
In this paper, for a locally compact commutative hypergroup K K and for a pair ( Φ 1 , Φ 2 ) (\Phi _1, \Phi _2) of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of K , K, for the convolution f ∗ g f\ast g to exist a.e., where f f and g g are arbitrary elements of Orlicz spaces L Φ 1 ( K ) L^{\Phi _1}(K) and L Φ 2 ( K ) L^{\Phi _2}(K) , respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from L w 1 ( K ) L^1_w(K) into L w Φ ( K ) L^\Phi _w(K) for a weight w w on a locally compact hypergroup K K .
- Published
- 2022
48. Manifolds with two projective bundle structures
- Author
-
Gianluca Occhetta, Eleonora Romano, and Luis Solá Conde
- Subjects
Mathematics - Algebraic Geometry ,Primary 14M99, Secondary 14J45, 14M17 ,Mathematics::Algebraic Geometry ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana--Peternell conjecture for varieties of Picard number one admitting $\mathbb C^*$-actions of a certain kind., Comment: 15 pages
- Published
- 2022
49. Rouché’s theorem and the geometry of rational functions
- Author
-
Trevor Richards
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Rouché's theorem ,Rational function ,Mathematics - Abstract
In this paper, we use Rouché’s theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results and bounds regarding the geometry of the zeros, poles, and critical points of a rational function. Included is an improvement on a result by Alexander and Walsh regarding the “exclusion region” around a given zero or pole of a rational function in which no critical point may lie.
- Published
- 2022
50. Maps on positive definite cones of 𝐶*-algebras preserving the Wasserstein mean
- Author
-
Lajos Molnár
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Positive-definite matrix ,Mathematics - Abstract
The primary aim of this paper is to present the complete description of the isomorphisms between positive definite cones of C ∗ C^* -algebras with respect to the recently introduced Wasserstein mean and to show the nonexistence of nonconstant such morphisms into the positive reals in the case of von Neumann algebras without type I 2 _2 , I 1 _1 direct summands. A comment on the algebraic properties of the Wasserstein mean relating associativity is also made.
- Published
- 2022
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