36 results
Search Results
2. 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
3. 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
4. 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
5. Corrigenda to 'Cohen-Macaulay bipartite graphs in arbitrary codimension'
- Author
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Rahim Zaare-Nahandi, Hassan Haghighi, and Siamak Yassemi
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Bipartite graph ,Codimension ,Mathematics - Abstract
A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.
- Published
- 2021
6. 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
7. 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
8. Convergence of spectral likelihood approximation based on q-Hermite polynomials for Bayesian inverse problems
- Author
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Zhiliang Deng and Xiaomei Yang
- Subjects
Hermite polynomials ,Applied Mathematics ,General Mathematics ,Convergence (routing) ,Bayesian probability ,Applied mathematics ,Inverse problem ,Mathematics - Abstract
In this paper, q-Gaussian distribution, q-analogy of Gaussian distribution, is introduced to characterize the prior information of unknown parameters for inverse problems. Based on q-Hermite polynomials, we propose a spectral likelihood approximation (SLA) algorithm of Bayesian inversion. Convergence results of the approximated posterior distribution in the sense of Kullback–Leibler divergence are obtained when the likelihood function is replaced with the SLA and the prior density function is truncated to its partial sum. In the end, two numerical examples are displayed, which verify our results.
- Published
- 2022
9. Members of thin Π₁⁰ classes and generic degrees
- Author
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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
10. On the density of multivariate polynomials with varying weights
- Author
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András Kroó and József Szabados
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Multivariate polynomials ,Mathematics - Abstract
In this paper we consider multivariate approximation by weighted polynomials of the form w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) , where p n p_n is a multivariate polynomial of degree at most n n , w w is a given nonnegative weight with nonempty zero set, and γ n ↑ ∞ \gamma _n\uparrow \infty . We study the question if every continuous function vanishing on the zero set of w w is a uniform limit of weighted polynomials w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) . It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that γ n = o ( n ) . \gamma _n=o(n).
- Published
- 2023
11. Rouché’s theorem and the geometry of rational functions
- Author
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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
12. Maps on positive definite cones of 𝐶*-algebras preserving the Wasserstein mean
- Author
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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
13. Rough singular integrals and maximal operator with radial-angular integrability
- Author
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Huoxiong Wu and Ronghui Liu
- Subjects
Applied Mathematics ,General Mathematics ,Singular integral ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the rough singular integral operator T Ω f ( x ) = p.v. ∫ R n f ( x − y ) Ω ( y ′ ) | y | n d y , \begin{equation*} T_\Omega f(x)=\text {p.v.}\int _{\mathbb {R}^n}f(x-y)\frac {\Omega (y’)}{|y|^n}dy, \end{equation*} and the corresponding maximal singular integral operator T Ω ∗ f ( x ) = sup ε > 0 | ∫ | y | ≥ ε f ( x − y ) Ω ( y ′ ) | y | n d y | , \begin{equation*} T^*_\Omega f(x)=\sup _{\varepsilon >0}\Big |\int _{|y|\geq \varepsilon }f(x-y)\frac {\Omega (y’)}{|y|^n}dy\Big |, \end{equation*} where the kernel Ω ∈ H 1 ( S n − 1 ) \Omega \in H^1(\mathrm {S}^{n-1}) has zero mean value and n ≥ 2 n\geq 2 . We prove that T Ω T_\Omega and T Ω ∗ T^*_\Omega are bounded on the mixed radial-angular spaces L | x | p L θ p ~ ( R n ) L_{|x|}^pL_{\theta }^{\tilde {p}}(\mathbb {R}^n) for some suitable indexes 1 > p , p ~ > ∞ 1>p, \tilde {p}>\infty . The corresponding vector-valued versions are also established.
- Published
- 2021
14. Finite-time convergence of solutions of Hamilton-Jacobi equations
- Author
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Kaizhi Wang, Kai Zhao, and Jun Yan
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Dynamical Systems (math.DS) ,Hamilton–Jacobi equation ,Viscosity ,Mathematics - Analysis of PDEs ,Convergence (routing) ,FOS: Mathematics ,Mathematics - Dynamical Systems ,Viscosity solution ,Finite time ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper deals with the long-time behavior of viscosity solutions of evolutionary contact Hamilton-Jacobi equations w t + H ( x , w , w x ) = 0 , \begin{equation*} w_t+H(x,w,w_x)=0, \end{equation*} where H ( x , u , p ) H(x,u,p) is strictly decreasing in u u and satisfies Tonelli conditions in p p . We show that each viscosity solution of the ergodic contact Hamilton-Jacobi equation H ( x , u , u x ) = 0 H(x,u,u_x)=0 can be reached by many different viscosity solutions of the above evolutionary equation in a finite time.
- Published
- 2021
15. The global Kotake-Narasimhan theorem
- Author
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P. Rampazo and G. Hoepfner
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics - Abstract
In this paper we introduce the notion of global ultradifferentiable functions with respect to weight functions and include a discussion of its functional analytic theory and prove a characterization in terms of certain exponential decay of the Fourier-Broz-Iagolnitzer transform—a Paley-Wiener type theorem. As an application we investigate the regularity of global ultradifferentiable vectors proving a version of the Kotake-Narasimhan theorem in this setting.
- Published
- 2021
16. Regular evolution algebras are universally finite
- Author
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Antonio Viruel, Panagiote Ligouras, Alicia Tocino, and Cristina Costoya
- Subjects
Pure mathematics ,Finite group ,Functor ,Applied Mathematics ,General Mathematics ,Field (mathematics) ,Mathematics - Rings and Algebras ,Automorphism ,05C25, 17A36, 17D99 ,Rings and Algebras (math.RA) ,Simple (abstract algebra) ,Scheme (mathematics) ,Affine group ,FOS: Mathematics ,Algebraic number ,Mathematics - Abstract
In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc
- Published
- 2021
17. Two solutions to Kazdan-Warner’s problem on surfaces
- Author
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Li Ma
- Subjects
geography ,geography.geographical_feature_category ,Applied Mathematics ,General Mathematics ,Direct method ,Mathematical analysis ,Regular polygon ,Function (mathematics) ,Riemannian manifold ,symbols.namesake ,Variational method ,symbols ,Mountain pass ,Euler number ,Mathematics - Abstract
In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)
- Published
- 2021
18. Orthogonality relations on certain homogeneous spaces
- Author
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Chi-Wai Leung
- Subjects
Pure mathematics ,Orthogonality (programming) ,Homogeneous ,Applied Mathematics ,General Mathematics ,Mathematics - Abstract
Let G G be a locally compact group and let K K be its closed subgroup. Write G ^ K \widehat {G}_{K} for the set of irreducible representations with non-zero K K -invariant vectors. We call a pair ( G , K ) (G,K) admissible if for each irreducible representation ( π , V π ) (\pi , V_{\pi }) in G ^ K \widehat {G}_{K} , its K K -invariant subspace V π K V_{\pi }^{K} is of finite dimension. For each π \pi in G ^ K \widehat {G}_{K} , let π v i , ξ ¯ j \pi _{v_{i}, \overline {\xi }_{j}} ’s ( π v i , ξ ¯ j ( g K ) ≔ ⟨ v i , π ( g ) ξ j ⟩ ) (\pi _{v_{i}, \overline {\xi }_{j}}(gK)≔\langle v_{i}, \pi (g)\xi _{j}\rangle ) be the matrix coefficeints on G / K G/K induced by fixed orthonormal bases { v i } \{v_{i}\} and { ξ j } \{\xi _{j}\} for V π V_{\pi } and V π K V_{\pi }^{K} respectively. A probability measure μ \mu on G / K G/K is called a spectral measure if there is a subset Γ \Gamma of G ^ K \widehat {G}_{K} such that the set of all such matrix coefficients π v i , ξ ¯ j , π ∈ Γ , \pi _{v_{i}, \overline {\xi }_{j}},\ \pi \in \Gamma , constitutes an orthonormal basis for L 2 ( G / K , μ ) L^{2}(G/K, \mu ) with some suitable normalization of these matrix coordinate functions. In this paper, we shall give a characterization of a spectral measure for an admissible pair ( G , K ) (G,K) by using the Fourier transform on G / K G/K . Also, from this we show that there is a “local translation” (we call it locally regular representation in the sequel) of G G on L 2 ( G / K , μ ) L^{2}(G/K, \mu ) under a mild condition. This will give us some necessary conditions for the existence of spectral measures. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied.
- Published
- 2021
19. Contingency tables and the generalized Littlewood–Richardson coefficients
- Author
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Mark Colarusso, William Q. Erickson, and Jeb F. Willenbring
- Subjects
Combinatorics ,Polynomial (hyperelastic model) ,Contingency table ,Tensor product ,Irreducible polynomial ,Applied Mathematics ,General Mathematics ,General linear group ,Multiplicity (mathematics) ,Lambda ,Mathematics ,Symplectic geometry - Abstract
The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.
- Published
- 2021
20. Bohr’s inequality for non-commutative Hardy spaces
- Author
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Sneh Lata and Dinesh Singh
- Subjects
Pure mathematics ,Trace (linear algebra) ,Nuclear operator ,Applied Mathematics ,General Mathematics ,Order (ring theory) ,Hardy space ,Square matrix ,Bohr model ,symbols.namesake ,Von Neumann algebra ,symbols ,Commutative property ,Mathematics - Abstract
In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.
- Published
- 2021
21. New congruence properties for Ramanujan’s 𝜙 function
- Author
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Ernest X. W. Xia
- Subjects
symbols.namesake ,Pure mathematics ,Applied Mathematics ,General Mathematics ,S function ,symbols ,Congruence (manifolds) ,Theta function ,Congruence relation ,Ramanujan's sum ,Mathematics - Abstract
In 2012, Chan proved a number of congruences for the coefficients of Ramanujan’s ϕ \phi function. In this paper, we prove some new congruences modulo powers of 2 and 3 for Ramanujan’s ϕ \phi function by employing Newman’s identities and theta function identities.
- Published
- 2021
22. The Schreier space does not have the uniform 𝜆-property
- Author
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Kevin Beanland and Hùng Việt Chu
- Subjects
Mathematics::Group Theory ,Mathematics::Functional Analysis ,Pure mathematics ,Property (philosophy) ,46B99 ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Extreme point ,Uniform property ,Space (mathematics) ,Functional Analysis (math.FA) ,Mathematics - Abstract
The λ \lambda -property and the uniform λ \lambda -property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the λ \lambda -property and asked if it has the uniform λ \lambda -property. In this paper, we show that Schreier space does not have the uniform λ \lambda -property. Furthermore, we show that the dual of the Schreier space does not have the uniform λ \lambda -property.
- Published
- 2021
23. The operator norm on weighted discrete semigroup algebras ℓ¹(𝑆,𝜔)
- Author
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H. V. Dedania and J. G. Patel
- Subjects
Pure mathematics ,Semigroup ,Applied Mathematics ,General Mathematics ,Operator norm ,Mathematics - Abstract
Let ω \omega be a weight on a right cancellative semigroup S S . Let ‖ ⋅ ‖ ω \|\cdot \|_{\omega } be the weighted norm on the weighted discrete semigroup algebra ℓ 1 ( S , ω ) \ell ^1(S, \omega ) . In this paper, we prove that the weight ω \omega satisfies F-property if and only if the operator norm ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} of ‖ ⋅ ‖ ω \| \cdot \|_{\omega } is exactly equal to another weighted norm ‖ ⋅ ‖ ω ~ 1 \| \cdot \|_{\widetilde {\omega }_1} . Though its proof is elementary, the result is unexpectedly surprising. In particular, the operator norm ‖ ⋅ ‖ 1 o p \| \cdot \|_{1 op} is same as the ℓ 1 \ell ^1 - norm ‖ ⋅ ‖ 1 \| \cdot \|_1 on ℓ 1 ( S ) \ell ^1(S) . Moreover, various examples are discussed to understand the relation among ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} , ‖ ⋅ ‖ ω \| \cdot \|_{\omega } , and ℓ 1 ( S , ω ) \ell ^1(S, \omega ) .
- Published
- 2021
24. Blowup criterion of classical solutions for a parabolic-elliptic system in space dimension 3
- Author
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Yuxiang Li and Bin Li
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Space dimension ,Mathematics - Abstract
This paper is concerned with a parabolic-elliptic system, which was originally proposed to model the evolution of biological transport networks. Recent results show that the corresponding initial-boundary value problem possesses a global weak solution, which, in particular, is also classical in the one and two dimensional cases. In this work, we establish a Serrin-type blowup criterion for classical solutions in the three dimensional setting.
- Published
- 2021
25. Limiting profile for stationary solutions maximizing the total population of a diffusive logistic equation
- Author
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Jumpei Inoue
- Subjects
Applied Mathematics ,General Mathematics ,Applied mathematics ,Total population ,Limiting ,Logistic function ,Mathematics - Abstract
This paper focuses on the stationary problem of the diffusive logistic equation on a bounded interval. We consider the ratio of a population size of a species to a carrying capacity which denotes a spatial heterogeneity of an environment. In one-dimensional case, Wei-Ming Ni proposed a variational conjecture that the supremum of this ratio varying a diffusion coefficient and a carrying function is 3. Recently, Xueli Bai, Xiaoqing He, and Fang Li [Proc. Amer. Math. Soc. 144 (2016), pp. 2161–2170] settled the conjecture by finding a special sequence of diffusion coefficients and carrying functions. Our interest is to derive a profile of the solutions corresponding to this maximizing sequence. Among other things, we obtain the exact order of the maximum and the minimum of solutions of the sequence. The proof is based on separating the stationary problem into two ordinary differential equations and smoothly adjoining each solution.
- Published
- 2021
26. On the fragmentation phenomenon in the population optimization problem
- Author
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Jun Young Heo and Yeonho Kim
- Subjects
education.field_of_study ,Optimization problem ,Applied Mathematics ,General Mathematics ,Reaction–diffusion system ,Population ,Fragmentation (computing) ,Shape optimization ,Statistical physics ,education ,Mathematics - Abstract
In this paper, we study the population optimization problem in the logistic reaction-diffusion model. The issue of maximizing the total population in a heterogeneous environment has attracted the attention of many researchers. For the n n -dimensional box domain, it has recently been shown that resource fragmentation is better than concentration in order to maximize the total population when the diffusion rate is sufficiently small. As resource concentration is known to be beneficial for the survival of the species, this contrasting phenomenon is quite surprising. We proved that the fragmentation phenomenon occurs for any general bounded domains in R n \mathbb {R}^n if the diffusion rate is sufficiently small.
- Published
- 2021
27. Blow-up conditions of the incompressible Navier-Stokes equations in terms of sequentially defined Besov spaces
- Author
-
Hantaek Bae
- Subjects
Physics::Fluid Dynamics ,Mathematics::Functional Analysis ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Compressibility ,Navier–Stokes equations ,Mathematics - Abstract
In this paper, we introduce sequentially defined Besov spaces adapted to the scaling invariant property of the 3D incompressible Navier-Stokes equations. We first show that these spaces are Banach spaces. We then establish regularity conditions in these spaces.
- Published
- 2021
28. On Lie group representations and operator ranges
- Author
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J. Oliva-Maza
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Lie group ,Bitwise operation ,Mathematics - Abstract
In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R \mathcal {R} be an operator range of a Hilbert space H \mathcal {H} . Given the set Λ \Lambda of R \mathcal {R} -invariant operators, and given a Lie group representation ρ : G → GL ( H ) \rho :G\rightarrow \text {GL}(\mathcal {H}) , we discuss the induced semigroup homomorphism ρ ~ : ρ − 1 ( Λ ) → B ( R ) \widetilde {\rho }: \rho ^{-1}(\Lambda ) \rightarrow \mathcal {B(R)} for the operator range topology on R \mathcal {R} . In one direction, we work under the assumption ρ − 1 ( Λ ) = G \rho ^{-1} (\Lambda ) = G , so ρ ~ : G → B ( R ) \widetilde {\rho }:G\rightarrow \mathcal {B}(\mathcal {R}) is in fact a group representation. In this setting, we prove that ρ ~ \widetilde {\rho } is continuous (and smooth) if and only if the tangent map d ρ d\rho is R \mathcal {R} -invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H \mathcal {H} , the set ρ − 1 ( Λ ) \rho ^{-1}(\Lambda ) is neither a group for a large set of nonclosed operator ranges R \mathcal {R} nor closed for all nonclosed operator ranges R \mathcal {R} . Both results are proved by means of explicit counterexamples.
- Published
- 2021
29. Matrix weighted Triebel-Lizorkin bounds: A simple stopping time proof
- Author
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Joshua Isralowitz
- Subjects
Mathematics::Functional Analysis ,Matrix (mathematics) ,Pure mathematics ,Mathematics - Classical Analysis and ODEs ,Simple (abstract algebra) ,Applied Mathematics ,General Mathematics ,Stopping time ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics::Classical Analysis and ODEs ,42B20 ,Mathematics - Abstract
In this paper we will give a simple stopping time proof in the $\mathbb{R}^d$ setting of the matrix weighted Triebel-Lizorkin bounds proved by F. Nazarov/S. Treil and A. Volberg, respectively. Furthermore, we provide explicit matrix A${}_p$ characteristic dependence and also discuss some interesting open questions., 13 pages, no figures, submitted
- Published
- 2021
30. A rank question for homogeneous polynomials
- Author
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Kevin Palencia and Jennifer Brooks
- Subjects
Combinatorics ,Homogeneous ,Applied Mathematics ,General Mathematics ,Rank (graph theory) ,Mathematics - Abstract
A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) on C n + 1 \mathbb {C}^{n+1} . In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z}\right \rVert ^2 under certain additional hypotheses. When r r has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of P = S Q P=SQ when Q Q is a homogeneous polynomial and S ( x ) = ∑ j = 0 n x j S(x) = \sum _{j=0}^n x_j . In this paper, we use techniques from commutative algebra to estimate the minimum rank of P = S Q P=SQ under the additional hypothesis that Q Q has maximum rank. The problem has already been solved for n + 1 ≤ 3 n+1 \leq 3 , and so we consider n + 1 ≥ 4 n+1\geq 4 . We obtain a minimum rank estimate that is sharp when n + 1 = 4 n+1=4 , and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for n + 1 > 4 n+1>4 that, while not sharp, is non-trivial.
- Published
- 2021
31. Minimal value set polynomials over fields of size 𝑝³
- Author
-
Herivelto Borges and Lucas Reis
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Value set ,Mathematics - Abstract
For any prime number p p , and integer k ⩾ 1 k\geqslant 1 , let F p k \mathbb {F}_{p^k} be the finite field of p k p^k elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials F ∈ F p k [ x ] F\in \mathbb {F}_{p^k}[x] for which the value set { F ( α ) : α ∈ F p k } \{F(\alpha ): \alpha \in \mathbb {F}_{p^k}\} has the minimum possible size ⌊ ( p k − 1 ) / deg F ⌋ + 1 \left \lfloor (p^k-1)/\deg F \right \rfloor +1 . For k ⩽ 2 k\leqslant 2 , the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for k = 3 k=3 .
- Published
- 2021
32. Adams’ inequality with logarithmic weights in ℝ⁴
- Author
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Lianfang Wang and Maochun Zhu
- Subjects
Pure mathematics ,Inequality ,Logarithm ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Mathematics ,media_common - Abstract
Trudinger-Moser inequality with logarithmic weight was first established by Calanchi and Ruf [J. Differential Equations 258 (2015), pp. 1967–1989]. The aim of this paper is to address the higher order version; more precisely, we show the following inequality sup u ∈ W 0 , r a d 2 , 2 ( B , ω ) , ‖ Δ u ‖ ω ≤ 1 ∫ B exp ( α | u | 2 1 − β ) d x > + ∞ \begin{equation*} \sup _{u \in W_{0,rad}^{2,2}(B,\omega ),{{\left \| {\Delta u} \right \|}_\omega } \le 1} \int _B {\exp \left ( {\alpha {{\left | u \right |}^{\frac {2}{{1 - \beta }}}}} \right )} dx > + \infty \end{equation*} holds if and only if \[ α ≤ α β = 4 [ 8 π 2 ( 1 − β ) ] 1 1 − β , \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}}, \] where B B denotes the unit ball in R 4 \mathbb {R}^{4} , β ∈ ( 0 , 1 ) \beta \in \left ( {0,1} \right ) , ω ( x ) = ( log 1 | x | ) β \omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta } or ( log e | x | ) β {\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta } , and W 0 , r a d 2 , 2 ( B , ω ) W_{0,rad}^{2,2}(B,\omega ) is the weighted Sobolev spaces. Our proof is based on a suitable change of variable that allows us to represent the laplacian of u u in terms of the second derivatives with respect to the new variable; this method was first used by Tarsi [Potential Anal. 37 (2012), pp. 353–385].
- Published
- 2021
33. Probabilistic pointwise convergence problem of Schrödinger equations on manifolds
- Author
-
Xiangqian Yan, Wei Yan, and Junfang Wang
- Subjects
Pointwise convergence ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Probabilistic logic ,symbols ,Applied mathematics ,Mathematics ,Schrödinger equation - Abstract
In this paper, we investigate the probabilistic pointwise convergence problem of Schrödinger equation on the manifolds. We prove probabilistic pointwise convergence of the solutions to Schrödinger equations with the initial data in L 2 ( T n ) L^{2}(\mathrm {\mathbf {T}}^{n}) , where T = [ 0 , 2 π ) \mathrm {\mathbf {T}}=[0,2\pi ) , which require much less regularity for the initial data than the rough data case. We also prove probabilistic pointwise convergence of the solutions to Schrödinger equation with Dirichlet boundary condition for a large set of random initial data in ∩ s > 1 2 H s ( Θ ) \cap _{s>\frac {1}{2}}H^{s}(\Theta ) , where Θ \Theta is three dimensional unit ball, which require much less regularity for the initial data than the rough data case.
- Published
- 2021
34. A new centroaffine characterization of the ellipsoids
- Author
-
Cheng Xing and Zejun Hu
- Subjects
Applied Mathematics ,General Mathematics ,Geometry ,Ellipsoid ,Mathematics ,Characterization (materials science) - Abstract
In this paper, we establish an integral inequality on centroaffine hyperovaloids in R n + 1 \mathbb {R}^{n+1} , in terms of the Ricci curvature in direction of the Tchebychev vector field and the norm of the covariant differentiation of the difference tensor with respect to the Levi-Civita connection of the centroaffine metric. This integral inequality is optimal, and its equality case provides a new centroaffine characterization of the ellipsoids.
- Published
- 2021
35. A Bailey type identity with applications related to integer representations
- Author
-
Mohamed El Bachraoui
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Identity (philosophy) ,media_common.quotation_subject ,Type (model theory) ,Mathematics ,Integer (computer science) ,media_common - Abstract
In this paper we shall deduce a Bailey type formula as a consequence of the residual identity of a q q -series transformation due to Gasper. Our formula leads to a variety of q q -series identities which are related to the arithmetic function counting integer representations of the form \[ n ( A n + B ) 2 + r ( C r + D ) 2 + E n r . \frac {n(An+B)}{2}+\frac {r(Cr+D)}{2} + Enr. \]
- Published
- 2021
36. Intersections and unions of a general family of function spaces
- Author
-
Guanlong Bao, Fangqin Ye, and Hasi Wulan
- Subjects
symbols.namesake ,Pure mathematics ,Function space ,Applied Mathematics ,General Mathematics ,Blaschke product ,symbols ,Space (mathematics) ,General family ,Mathematics - Abstract
In this paper, we investigate the strict inclusion relation associated with intersections and unions of a general family of function spaces. We answer partially a question left open in Korhonen and Rättyä [Comput. Methods Funct. Theory 5 (2005), pp. 459–469].
- Published
- 2021
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