Kantorovich–Wright integral and representation of quasi-Banach lattices

The purpose of this paper is two-fold: first, to outline a purely order-based integral of the type of the Kantorovich–Wright integral of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a K _σ-space (that is, a Dedekind σ-complete vector lattice) and, secondly, prove new theorems on the representation of Dedekind complete vector lattices and quasi-Banach lattices in the form of lattices of functions integrable or “weakly” integrable with respect to an appropriate vector measure. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method does not apply, the Kantorovich–Wright integral is more flexible than the Bartle–Dunford–Schwartz integral.     

Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral

In this paper we introduce a nonlinear integral equation such that the system of global solutions to this equation represents the class of a very narrow beam as T → ∞ (an analog of the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral (1918). The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown, and Ivic.     

Černý’s conjecture and group representation theory

Let us say that a Cayley graph Γ of a group G of order n is a Černy Cayley graph if every synchronizing automaton containing Γ as a subgraph with the same vertex set admits a synchronizing word of length at most (n−1)². In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of Černy Cayley graphs.     

Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

On representation theory and the cohomology rings of irreducible compact hyperkähler manifolds of complex dimension four

In this paper, we continue the study of the possible cohomology rings of compact complex four dimensional irreducible hyperkähler manifolds. In particular, we prove that in the case b ₂=7, b ₃=0 or 8. The latter was achieved by the Beauville construction.     

Deligne’s congruence and supersingular reduction of Drinfeld modules

Most rank two Drinfeld modules are known to have infinitely many supersingular primes. But how many supersingular primes of a given degree can a fixed Drinfeld module have? In this paper, a congruence between the Hasse invariant and a certain Eisenstein series is used for obtaining a bound on the number of such supersingular primes. Certain exceptional cases correspond to zeros of certain Eisenstein series with rational j-invariants.     

The value of an integral in Gradshteyn and Ryzhik’s table

The Ramanujan Journal - In 2010, V. H. Moll observed that entry 3.248.5 in the sixth edition of Gradshteyn and Ryzhik’s table of integrals was incorrect, and he asked for...     

Abel’s integral operator: sparse representation based on multiwavelets

In this work, Abel’s integral operator is represented based on Alpert’s multiwavelets as a sparse matrix and then a non-linear Abel’s integral equation of the second kind is solved by multiwavelets Galerkin method. Nonlinearity and singularity make the numerical procedure more challenging. But the proposed scheme overcomes these problems. Convergence analysis is investigated and some numerical examples validated this analysis.

     

Local representation theory and möbius inversion

Various representation-theoretic parameters of a finite group are shown to satisfy formulas similar to formulas appearing in reformulations by Kölshammer, Robinson, and Thévenaz of Alperin's Conjecture. The conjecture itself is reformulated again, now as a statement not mentioning characters or conjugacy classes.     

Wiener’s lemma for singular integral operators of Bessel potential type

In this paper, we introduce an algebra of singular integral operators containing Bessel potentials of positive order, and show that the corresponding unital Banach algebra is an inverse-closed Banach subalgebra of ${\mathcal {B}} (L^q_w)$ , the Banach algebra of all bounded operators on the weighted space $L_w^q$ , for all $1\le q<\infty $ and Muckenhoupt $A_q$ -weights $w$ .     

On representation of Stinespring’s type for n-tuples of completely positive maps in Hilbert C ⋆-modules

Here we prove an analog of the Stinespring’s theorem for n-tuples of completely positive maps in Hilbert C ^?-modules.     

Fatou’s lemma and lower epi-limits of integral functionals

In this work, with the introduction in the $\sigma$-finite case of a modulus of equi-integrability, we prove some new extensions of Fatou’s lemma and some of its consequences in the convergence theory of integral functionals. We present the case of a sequence of integral functionals using an analog of Ioffe’s criterion for a sequence of integrands.     

Grüss type and related integral inequalities in probability spaces

Aequationes mathematicae - In this paper we study Grüss type inequalities for real and complex valued functions in probability spaces. Some earlier Grüss type inequalities are extended...     

Normal forms of functional-differential equations of Beardon type and Briot–Bouquet type equations in the complex domain

We study local analytic solutions of the functional-differential equation of the form \({h(\psi(z)) = b(z) h(z) h^\prime(z) + d(z)h(z)^{2}}\) which are called Beardon type functional-differential equations. All functions involved are supposed to be holomorphic in a neighbourhood of zero. Special cases are the equations f(kz) =  kf(z) f′(z) where k is a complex number, \({k \neq 0}\), and \({f(\varphi(z)) = a(z) f(z) f'(z)}\) with given \({\varphi}\) and a. The class of these equations is invariant under transformations \({h \to \alpha h, \alpha(z) \neq 0}\) for all z in a neighbourhood of zero, of the unknown function and \({z \to T(z)}\) of the argument z. In particular, we are interested to know under which conditions a Beardon type functional-differential equation can be transformed to the simplified (normal form) \({h(kz) = k h(z) h'(z) + c(z) h(z)^2}\) where \({k \in \mathbb {C} \backslash\left\{0\right\}}\). We solve this normal form by another transfomation to a so-called Briot–Bouquet type functional-differential equation.     

The integral (orbifold) Chow ring of toric Deligne–Mumford stacks

In this paper we study the integral Chow ring of toric Deligne–Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne–Mumford stack is isomorphic to the Stanley–Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.     

Some bounds and limits in the theory of Riemann’s zeta function

For any real $a>0$ we determine the supremum of the real $\sigma$ such that $\zeta (\sigma +it)=a$ for some real $t$. For $0<a<1$, $a=1$, and $a>1$ the results turn out to be quite different.We also determine the supremum $E$ of the real parts of the ‘turning points’, that is points $\sigma +it$ where a curve $\mathrm{Im}\zeta (\sigma +it)=0$ has a vertical tangent. This supremum $E$ (also considered by Titchmarsh) coincides with the supremum of the real $\sigma$ such that ${\zeta }^{\prime }(\sigma +it)=0$ for some real $t$.We find a surprising connection between the three indicated problems: $\zeta \left(s\right)=1$, ${\zeta }^{\prime }\left(s\right)=0$ and turning points of $\zeta \left(s\right)$. The almost extremal values for these three problems appear to be located at approximately the same height.     

Weil-Petersson geometry of Teichmüller–Coxeter complex and its finite rank property

On a Teichmüller space, the Weil-Petersson metric is known to be incomplete. Taking metric and geodesic completions result in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller space as its non-linear non-homogeneous fundamental domain. We then show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups.