Related papers: Partial difference equations over compact Abelian …

For functions $f: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, Bloch and Okounkov defined a power series $\langle f\rangle_q$ that is the "weighted average" of $f$. As Fourier series in $q=e^{2\pi i z}$, such $q$-brackets generate the…

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…

By the approximation method introduced in \cite{FYW}, the existence and uniqueness are proved for a class of distribution-dependent stochastic functional differential equations (DDSFDEs). Moreover, combining the Harnack and shift-Harnack…

In this paper we present a Calder\'{o}n-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $\gamma\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable…

We give some coefficient bounds and distortion theorems for a subclass of univalent functions in the unit disk, and defined using the S\^{a}l\^{a}gean differential operator. The results generalize and unify some well known results for…

Functional integrals are defined in terms of locally compact topological groups and their associated Banach-valued Haar integrals. This approach generalizes the functional integral scheme of Cartier and DeWitt-Morette. The definition allows…

Returning to a classical question in Harmonic Analysis we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers Z which is not in the norm-closure of the algebra B(Z)…

In this article, we consider the following fractional {Hardy-type} inequality: \begin{align} \label{Fractional Hardy_abst} \int_{\mathbb{R}^N} |w(x)||u(x)|^p \mathrm{d}x \leq C \int_{\mathbb{R}^N \times \mathbb{R}^N}…

A general study of non-abelian duality is presented. We first identify a possible obstruction to the conformal invariance of the dual theory for non-semisimple groups. We construct the exact non-abelian dual for any Wess-Zumino-Witten (WZW)…

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula…

A finite algebra $\bA=\alg{A;\cF}$ is \emph{dualizable} if there exists a discrete topological relational structure $\BA=\alg{A;\cG;\cT}$, compatible with $\cF$, such that the canonical evaluation map $e\_{\bB}\colon \bB\to \Hom(…

We study two discretisations of the nonlinear Fourier transform of AKNS-ZS type, ${\cal F}^E$ and ${\cal F}^D$. Transformation ${\cal F}^D$ is suitable for studying the distributions of the form $u = \sum_{n = 1}^N u_n \, \delta_{x_n}$,…

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We…

In this paper, we introduce a new subclass of harmonic functions $f=s+\overline{t}$ in the open unit disk $U =\left \{ z\in C:\left \vert z\right \vert <1\right \} $ satisfying ${\text{Re}}\left[ \gamma s^{\prime }(z)+\delta zs^{\prime…

A characterization of the minimal $\mathcal{W}$-algebras associated with the Deligne exceptional series at level $-h^\vee/6$ is obtained by using one-parameter family of modular linear differential equations of order $4$. In particular, the…

For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda…

We present an abstract multiscale analysis scheme for matrix functions $(H_{\varepsilon}(m,n))_{m,n\in \mathfrak{T}}$, where $\mathfrak{T}$ is an Abelian group equipped with a distance $|\cdot|$. This is an extension of the scheme developed…

A generalized modular relation of the form $F(z, w, \alpha)=F(z, iw,\beta)$, where $\alpha\beta=1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann $\Xi$-function. It is a two-variable…

In this article, we introduce a differentiability concept for fuzzy functions $\tilde{f}: F(\mathbb{R}) \to F(\mathbb{R})$, where $F(\mathbb{R})$ is the set of all fuzzy numbers. With the help of the proposed differentiability notion, we…

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg-Witten functional that in particular…