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Answering a question of P\'alfy and Pyber, we first prove the following extension of the k(GV)-Problem: Let G be a finite group and A\le Aut(G) such that (|G|,|A|)=1. Then the number of conjugacy classes of the semidirect product GA is at…

In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an…

In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a $p$-block of a finite group with abelian defect group $D$ is bounded by $|D|$…

Topological string theory near the conifold point of a Calabi-Yau threefold gives rise to factorially divergent power series which encode the all-genus enumerative information. These series lead to infinite towers of singularities in their…

Let $Y=(Y(t))_{t\geq0}$ be a zero-mean Gaussian stationary process with covariance function $\rho:\mathbb{R}\to\mathbb{R}$ satisfying $\rho(0)=1$. Let $f:\mathbb{R}\to\mathbb{R}$ be a square-integrable function with respect to the standard…

We provide a much shorter but even more powerful proof of an algebraic identity, which can be used to establish the direct and the converse inequality under Type IV superorthogonality. As an application, we obtain the optimal order of the…

In 1967, Arveson invented a non-commutative generalization of classical $H^{\infty},$ known as finite maximal subdiagonal subalgebras, for a finite von Neumann algebra $\mathcal M$ with a faithful normal tracial state $\tau$. In 2008,…

It is shown that a relativistic (i.e. a Poincar{\' e} invariant) theory of extended objects (called p-branes) is not necessarily invariant under reparametrizations of corresponding $p$-dimensional worldsheets (including worldlines for $p =…

Let $E=((e_{ij}))_{n\times n}$ be a fixed array of real numbers such that $e_{ij}=e_{ji}, e_{ii}=0$ for $1\le i,j \le n$. Let the permutation group be denoted by $S_n$ and the collection of involutions with no fixed points by $\Pi_n$, that…

Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is on a path…

Brou\'{e}'s Abelian Defect Conjecture predicts interesting derived equivalences between derived categories of modular representations of finite groups. We investigate a generalization of Brou\'{e}'s Conjecture to ring spectrum coefficients…

In a previous work we established a super Schur-Weyl-Brauer duality between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. This led to a proof of the first fundamental theorem of…

We study invariant and bi-invariant metrics on groups focusing on finite groups $G$. We show that non-equivalent (bi) invariant metrics on $G$ are in 1-1 correspondence with unitary symmetric (conjugate) partitions on $G$. To every metric…

Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{\varphi_{\lambda}(x) = A_{\lambda}x + v_{\lambda}\}_{\lambda\in\Lambda}$ and a probability vector $p=(p_{\lambda})_{\lambda}>0$. Assume the strong…

We study numerical invariants of 2-blocks with minimal nonabelian defect groups. These groups were classified by R\'edei. If the defect group is also metacyclic, then the block invariants are known. In the remaining cases there are only two…

For any prime $p$ and real number and $\alpha$, the $p$-adic Littlewood Conjecture due to de Mathan and Teuli\'e asserts that \[\inf_{|m|\ge1}|m|_p\cdot |m|\cdot |\left\langle\alpha m\right\rangle|=0.\] Above, $|m|$ is the usual absolute…

We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\sqrt{p-1}$. This confirms a conjecture of H\'{e}thelyi and K\"{u}lshammer for…

We compute and explicitly describe the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the full and pure braid groups of the sphere $\mathbb{S}^2$, the real projective plane $\mathbb{R}P^2$ and specially the torus $\mathbb{T}$ and the Klein…

We investigate the geometric behavior of $\tau(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $\tau(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key…

We introduce enumerative invariants $F_{g,n}$ $(g\geq0$, $n \geq 1)$ associated to a cyclic $A_\infty$ algebra and a splitting of its non-commutative Hodge filtration. These invariants are defined by explicitly computable Feynman sums, and…