Stochastic reverse isoperimetric inequalities in the plane

Geometriae Dedicata - The Hitchin component $$\mathrm {Hit}_n(S)$$ of a closed surface S is a preferred component of the character variety $$\mathcal {X}_{\mathrm {PSL}_n(\mathbb {R})}(S)$$...     

The depth of a reflexive polytope

Given arbitrary integers d and r with $$d \ge 4$$ and $$1 \le r \le d + 1$$ , a reflexive polytope $${\mathscr {P}}\subset {\mathbb R}^d$$ of dimension d with $$\mathrm{depth}\,K[{\mathscr {P}}] = r$$ for which its dual polytope $${\mathscr {P}}^\vee $$ is normal will be constructed, where $$K[{\mathscr {P}}]$$ is the toric ring of $${\mathscr {P}}$$ .     

Thin equivalence relations in scaled pointclasses

For ordinals α beginning a Σ₁ gap in $\mathrm{L}(\mathbb {R})$, where $\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ is closed under number quantification, we give an inner model‐theoretic proof that every thin $\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ equivalence relation is $\Delta _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$ in a real parameter from the (optimal) hypothesis $\mathsf {AD}^{\mathrm{J}_{\alpha }(\mathbb {R})}$.     

Approximating Characteristics of the Nikol’skii–Besov Classes S1,θrBℝd$$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$

Ukrainian Mathematical Journal - We establish the exact-order estimates for the approximation of the classes $$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$ by entire functions of...     

A Wiener Lemma for the discrete Heisenberg group

Monatshefte für Mathematik - This article contains a Wiener Lemma for the convolution algebra $$\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})$$ and group $$C^*$$ -algebra...     

Automorphisms of threefolds of general type acting trivially in cohomology

Geometriae Dedicata - Let X be a minimal projective threefold of general type over $$\mathbb {C}$$ with only Gorenstein quotient singularities, and let $$\mathrm {Aut}_{\mathbb {Q}}(X)$$ be the...     

Capacity solution for a perturbed nonlinear coupled system

We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, the model problem we refer to is $$\begin{aligned} \left\{ \begin{array}{l} \Delta _p u+g(x,u)= \rho (u)|\nabla \varphi |^2 \quad \mathrm{in} \quad \Omega ,\\ {{\,\mathrm{div}\,}}(\rho (u)\nabla \varphi ) =0 \quad \mathrm{in} \quad \Omega ,\\ \varphi =\varphi _0 \quad \text{ on } \quad {\partial \Omega },\\ u=0 \quad \mathrm{on} \quad {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset \mathbb {R}^N$$, $$N\ge 2$$ and $$\Delta _p u=-{\text {div}}\left( |\nabla u|^{p-2} \nabla u\right) $$ is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic.     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

On certain degenerate Whittaker Models for cuspidal representations of $${\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}$$ GL k · n ( F q )

Mathematische Zeitschrift - Let $$\pi $$ be an irreducible cuspidal representation of $$\mathrm {GL}_{kn}(\mathbb {F}_q)$$ . Assume that $$\pi = \pi _{\theta }$$ , corresponds to a regular...     

Low Mach number limit of multidimensional steady flows on the airfoil problem

Given $$\alpha >0$$, we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$where $$W^{1,n}_{0,\mathrm{rad}}(B)$$ is the usual Sobolev spaces of radially symmetric functions on B in $${\mathbb {R}}^n$$ with $$n\ge 2$$. Without restricting to the class of functions $$W^{1,n}_{0,\mathrm{rad}}(B)$$, we should emphasize that the above inequalities fail in $$W^{1,n}_{0}(B)$$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.     

Equivariant quantum cohomology of the odd symplectic Grassmannian

Mathematische Zeitschrift - The odd symplectic Grassmannian $$\mathrm {IG}:=\mathrm {IG}(k, 2n+1)$$ parametrizes k dimensional subspaces of $${\mathbb {C}}^{2n+1}$$ which are isotropic with respect...     

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

Average of Dirichlet coefficients of cuspidal representations related to GL(2)

The Ramanujan Journal - Let $$\pi $$ be a cuspidal representation on $${{\,\mathrm{GL}\,}}(2,\mathbb {A}_{\mathbb {Q}}).$$ We give nontrivial lower and upper bounds for average of absolute values...     

Tangencies and polynomial optimization

Mathematical Programming - Given a polynomial function $$f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ and an unbounded closed semi-algebraic set $$S \subset {\mathbb {R}}^n,$$ we show that the...     

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space $$\Gamma ^{\infty }$$, in which $$\Gamma$$ is the Bohr compactification of the additive group $${\mathbb {R}}$$ of real numbers. In this way, we need to show that $$\Gamma ^\infty$$ is isomorphic to the spectrum of $$W({\mathbb {R}})$$, the norm closure of the invariant algebra generated by the maps $$\exp q(t)$$, where q(t) is a real polynomial on $${\mathbb {R}}$$.     

Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

Journal of Theoretical Probability - We study boundary non-crossing probabilities $$\begin{aligned} P_{f,u} := \mathrm {P}\big (\forall t\in {\mathbb {T}}\ X_t + f(t)\le u(t)\big ) \end{aligned}$$...     

Congruences with intervals and arbitrary sets

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).     

Maximal Operators in the Spherical Mean Setting

In this paper, we prove an existence result for \(\mathcal {L}^{\infty }\)-solutions for a class of semilinear delay evolution inclusions with measures and subjected to nonlocal initial conditions of the form

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)= \{Au(t)+f(t)\}\mathrm{d}t+\mathrm{d}h(t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle f(t)\in F(t,u_t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle u(t)=g(u)(t),&{}\quad t\in [\,-\tau ,0\,]. \end{array} \right. \end{aligned}$$

Here \(\tau \ge 0\), X is a Banach space, \(A:D(A)\subseteq X \rightarrow X \) is the infinitesimal generator of a \(C_0\)-semigroup, \(F:\mathbb {R}_+\times \mathcal {R}([\,-\tau ,0\,];X)\rightsquigarrow X\) is a u.s.c. multifunction with nonempty, convex and weakly compact values, \(h\in BV_{\mathrm{loc}}(\mathbb {R}_+;X)\) and the function \(g:\mathcal {R}_{b}(\mathbb {R}_+;X)\rightarrow \mathcal {R}([\,-\tau ,0\,];X)\) is nonexpansive.

     

Ideal class groups of imaginary quadratic fields

The Ramanujan Journal - Let d be a square-free positive integer and $$\mathrm{CL}(-d)$$ the ideal class group of the imaginary quadratic field $${\mathbb {Q}}(\sqrt{-d})$$ . In this paper, we show...     

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form