Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.     

On the continuity of \text{ SLE }_{\kappa } in \kappa

We prove that for almost every Brownian motion sample, the corresponding SLE \(_\kappa \) curves parameterized by capacity exist and change continuously in the supremum norm when \(\kappa \) varies in the interval \([0,\kappa _0)\) , where \(\kappa _0=8(2-\sqrt{3})\approx 2.1\) . We estimate the \(\kappa \) -dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with \(\kappa \) .     

Alternating groups as a quotient of $${{\varvec{PSL}}}\left( \mathbf{2}, \pmb {\mathbb {Z}} \left[ {{\varvec{i}}}\right] \right) $$

In this study, we developed an algorithm to find the homomorphisms of the Picard group \(\textit{PSL}(2,Z[i])\) into a finite group G. This algorithm is helpful to find a homomorphism (if it is possible) of the Picard group to any finite group of order less than 15! because of the limitations of the GAP and computer memory. Therefore, we obtain only five alternating groups \( A_{n}\), where \(n=5,6,9,13\) and 14 are quotients of the Picard group. In order to extend the degree of the alternating groups, we use coset diagrams as a tool. In the end, we prove our main result with the help of three diagrams which are used as building blocks and prove that, for \(n\equiv 1,5,6(\mathrm { mod}\, 8)\), all but finitely many alternating groups \(A_{n}\) can be obtained as quotients of the Picard group \(\textit{PSL}(2,Z[i])\). A code in Groups Algorithms Programming (GAP) is developed to perform the calculation.     

Amenability,tubularity, and embeddings into $$\mathcal{R}^{\omega}$$

Suppose M is a tracial von Neumann algebra embeddable into (the ultraproduct of the hyperfinite II₁-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any ε >  0 there exist and γ > 0 such that if then there exists a k × k unitary u satisfying for each 1 ≤  i ≤  n. We show that the following conditions are equivalent:

On the diophantine equation F $$\left( {\left( {_n^x } \right)} \right) = b\left( {_m^y } \right)$$

As a generalization of the results of [3] and [22], we characterize those pairs (m,n) and those polynomials b{Z}[x] of prime degree for which equation (1) has only finitely many integer solutions.     

On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$\mathbb {F}_{\varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).     

$$\varvec{}$$-Classes in finite groups of conjugate type $$\varvec{(n,1)}$$(n,1)

Two elements in a group G are said to be z-equivalent or to be in the same z-class if their centralizers are conjugate in G. In a recent work, Kulkarni et al. (J. Algebra Appl., 15 (2016) 1650131) proved that a non-abelian p-group G can have at most \(\frac{p^k-1}{p-1} +1\) number of z-classes, where \(|G/Z(G)|=p^k\). Here, we characterize the p-groups of conjugate type (n, 1) attaining this maximal number. As a corollary, we characterize p-groups having prime order commutator subgroup and maximal number of z-classes.     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

On the Normality of $$\mathfrak F^{\omega}$$ -Abnormal Maximal Subgroups of Finite Groups

Mathematical Notes - In the paper, the notion of an $$\mathfrak F^{\omega}$$ -abnormal (and $$\mathfrak F^{\omega}$$ -normal) maximal subgroup of a finite group is introduced, where $$\mathfrak F$$...     

Disintegration of positive isometric group representations on $$\varvec{\mathrm {L}^{p}}$$-spaces

Let G be a Polish locally compact group acting on a Polish space \({{X}}\) with a G-invariant probability measure \(\mu \). We factorize the integral with respect to \(\mu \) in terms of the integrals with respect to the ergodic measures on X, and show that \(\mathrm {L}^{p}({{X}},\mu )\) (\(1\le p<\infty \)) is G-equivariantly isometrically lattice isomorphic to an \({\mathrm {L}^p}\)-direct integral of the spaces \(\mathrm {L}^{p}({{X}},\lambda )\), where \(\lambda \) ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of \(\mathrm {L}^{p}({{X}},\mu )\) as an \({\mathrm {L}^p}\)-direct integral of order indecomposable representations. If \(({{X}}^\prime ,\mu ^\prime )\) is a probability space, and, for some \(1\le q<\infty \), G acts in a strongly continuous manner on \(\mathrm {L}^{q}({{X}}^\prime ,\mu ^\prime )\) as isometric lattice automorphisms that leave the constants fixed, then G acts on \(\mathrm {L}^{p}({{X}}^{\prime },\mu ^{\prime })\) in a similar fashion for all \(1\le p<\infty \). Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If \(({{X}}^\prime ,\mu ^\prime )\) is separable, the representation of G on \(\mathrm {L}^p(X^\prime ,\mu ^\prime )\) can then be disintegrated into order indecomposable representations. The notions of \({\mathrm {L}^p}\)-direct integrals of Banach spaces and representations that are developed extend those in the literature.     

On the Evaluation of the Integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$

The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G.     

Markov subshifts and partial representation of $$\mathbb{F}_{\text n}$$

In this paper we fix a set * of positive elements of the free group (e. g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S : which we prove to be a partial representation of on H under certain conditions studied by Matsumoto.*Supported by Capes.     

On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$Z/nZ

A unit u in a commutative ring with unity R is called exceptional if $$u-1$$ is also a unit. We introduce the notion of a polynomial version of this (abbreviated as $$f\hbox {-exunits}$$) for any $$f(X) \in \mathbb {Z}[X]$$. We find the number of representations of a non-zero element of $$\mathbb {Z}/n\mathbb {Z}$$ as a sum of two f-exunits for an infinite family of polynomials f of each degree $$\ge 1$$. We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).     

The Slice Hyperholomorphic Bergman Space on $$\varvec{\mathbb {B}_{R}}$$: Integral Representation and Asymptotic Behavior

The aim of the present paper is threefolds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius R centred at the origin. The explicit expression of its Bergman kernel is given and can be written in terms of special hypergeometric functions of two non-commuting (quaternionic) variables. Secondly, we introduce and study some basic properties of an associated integral transform, the quaternionic analogue of the so-called second Bargmann transform for the holomorphic Bergman space. Finally, we establish the asymptotic behavior as R goes to infinity. We show in particular that the reproducing kernel of the weighted slice hyperholomorphic Bergman space gives rise to its analogue for the slice hyperholomorphic Bargamann–Fock space.     

On the diophantine equation $$\varvec{y^{2} = \prod _{i \le 8}(x + k_i)}$$

This paper improves the result of Tengely (Periodica Math. Hung., 72(1) (2016) 23–28).     

A note on the size of the set $$\varvec{A^2+A}$$

Let \(F(x,y,z)=xy+z\). We consider some properties of expansion of the polynomial F in different settings, namely in the integers and in prime fields. The main results concern the question of covering \(\{0,1,\ldots , N\}\) (resp. \(\mathbf {F}_p\)) by \(A^2+A\) with some thin sets A.     

An additive ($${\alpha, \beta}$$)-functional equation and linear mappings in Banach spaces

In this paper, we investigate the additive (\({\alpha, \beta}\))-functional equation \({f(x+y) + \bar{\alpha}f({\alpha}z) = \beta^{-1}f(\beta(x+y+z))}\) for all complex numbers \({\alpha}\) with \({|\alpha| = 1}\) and for a fixed nonzero complex number \({\beta}\). Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of this additive (\({\alpha, \beta}\))-functional equation in complex Banach spaces.     

On the functional equation $${\varvec{f}}({\varvec{x}})+{\varvec{f}}({\varvec{y}})=\mathbf{max} \{{\varvec{f}}({\varvec{xy}}),{\varvec{f}}({\varvec{xy}}^{-{\varvec{1}}})\}$$ on groups

We analyse the functional equation

$$\begin{aligned} f(x)+f(y)=\max \{f(xy),f(xy^{-1})\} \end{aligned}$$

for a function \(f:G\rightarrow \mathbb R\) where G is a group. Without further assumption it characterises the absolute value of additive functions. In addition \(\{z\in G\mid f(z)=0\}\) is a normal subgroup of G with abelian factor group.

     

On Compactness of Hankel and the $$\overline{\partial }$$-Neumann Operators on Hartogs Domains in $$\mathbb {C}^2$$C2

We prove that on smooth bounded pseudoconvex Hartogs domains in \(\mathbb {C}^2\) compactness of the \(\overline{\partial }\)-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain.