We introduce non-associative skew Laurent polynomial rings and characterize when they are simple. Thereby, we generalize results by Jordan, Voskoglou, and Nystedt and Öinert.

We provide an algorithm for constructing a Kirby diagram of a 4-dimensional open book given a Heegaard diagram of the page. As an application, we show that any open book with trivial monodromy is diffeomorphic to an open book constructed with a punctured handlebody as page and a composition of torus twists and sphere twists as monodromy.

In this paper, we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu’s algebras and bimodules (and their g-twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study.

We show that the loop space of a moment-angle complex associated to a two-dimensional simplicial complex decomposes as a finite type product of spheres, loops on spheres and certain indecomposable spaces which appear in the loop space decomposition of Moore spaces. We also give conditions on certain subcomplexes under which, localised away from sufficiently many primes, the loop space of a moment-angle complex decomposes as a finite type product of spheres and loops on spheres.

Here we consider the discrete time dynamics described by a transformation $T:M \to M$, where T is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2, \ldots,d\}^{\mathbb{N}}$, or, T describes the action of a d to 1 expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (for example $x \to T(x) =\mathrm{d} x $ (mod 1)), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a certain normalized Hölder potential A denote by $\mu_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors X (functions $X: M \to \mathbb{R}$) to the manifold $\mathcal{N}$ at the point µA (a subspace of the Hilbert space $L^2(\mu_A)$) coincides with the kernel of the Ruelle operator for the normalized potential A. The Riemannian norm $|X|=|X|_A$ of the vector X, which is tangent to $\mathcal{N}$ at the point µA, is described via the asymptotic variance, that is, satisfies

$ |X|^2 = \langle X, X \rangle = \lim_{n \to \infty}\frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,\mathrm{d} \mu_A$.

Consider an orthonormal basis Xi, $i \in \mathbb{N}$, for the tangent space at µA. For any two orthonormal vectors X and Y on the basis, the curvature $K(X,Y)$ is

\begin{equation*}K(X,Y) = \frac{1}{4}[ \sum_{i=1}^\infty (\int X Y X_i \,\mathrm{d} \mu_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,\mathrm{d} \mu_A \int Y^2 X_i \,\mathrm{d} \mu_A ].\end{equation*}

When the equilibrium probabilities µA is the set of invariant Markov probabilities on $\{0,1\}^{\mathbb{N}}\subset \mathcal{N}$, introducing an orthonormal basis $\hat{a}_y$, indexed by finite words y, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$, which is a finite sum. These values can be positive or negative depending on A and the words x and z. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$. If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$.

Let S be an orientable, connected surface of finite topological type, with genus $g \leqslant 2$, empty boundary and complexity at least 2; we prove that any graph endomorphism of the curve graph of S is actually an automorphism. Also, as a complement of the author’s previous results, we prove that under mild conditions on the complexity of the underlying surfaces, any graph morphism between curve graphs is induced by a homeomorphism of the surfaces.

To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leininger’s finite rigid sets. We prove as a consequence that Aramayona and Leininger’s rigid set also exhausts $\mathcal{C}(S)$ via rigid expansions. The combinatorial rigidity results follow as an immediate consequence, based on the author’s previous results.

For $r\in(0,1)$, let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$, where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that

\begin{equation*}\mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{2n}r^{2n},\end{equation*}

with $\theta _{n}\in \left( 0,1\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.

We give a criterion for separability of subgroups of certain outer automorphism groups. This answers questions of Hagen and Sisto, by strengthening and generalizing a result of theirs on mapping class groups.

It is shown that if $\{H_n\}_{n \in \omega}$ is a sequence of groups without involutions, with $1 \lt |H_n| \leq 2^{\aleph_0}$, then the topologist’s product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in \omega}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in \omega} A_n/\bigoplus_{n \in \omega} A_n$ is dependent on the sequence.

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi–Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.

Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree $\displaystyle |Syl_r(G)|$ is said to be weak r-Steinberg if it vanishes on the r-singular elements of $G.$ In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group $GL(n,q).$ Then we characterize the quasi r-Steinberg characters of $GL(2,q)$ and $GL(3,q).$ Finally, we obtain a classification of the weak r-Steinberg characters of $GL(n,q).$

We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.

Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

Let $H\le F$ be two finitely generated free groups. Given $g\in F$, we study the ideal $\mathfrak I_g$ of equations for g with coefficients in H, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in F. The ideal $\mathfrak I_g$ is a normal subgroup of $H*\langle x\rangle$, and it’s possible to algorithmically compute a finite normal generating set for $\mathfrak I_g$; we give a description of one such algorithm, based on Stallings folding operations. We provide an algorithm to find an equation in w(x)\in$\mathfrak I_g$ with minimum degree, i.e.  such that its cyclic reduction contains the minimum possible number of occurrences of x and x−1; this answers a question of A. Rosenmann and E. Ventura. More generally, we show how to algorithmically compute the set Dg of all integers d such that $\mathfrak I_g$ contains equations of degree d; we show that Dg coincides, up to a finite set, with either $\mathbb N$ or $2\mathbb N$. Finally, we provide examples to illustrate the techniques introduced in this paper. We discuss the case where ${\text{rank}}(H)=1$. We prove that both kinds of sets Dg can actually occur. We show that the equations of minimum possible degree aren’t in general enough to generate the whole ideal $\mathfrak I_g$ as a normal subgroup.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by

\begin{equation*}L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0.\end{equation*}

$ L_k[f] $

$ f^n+g^n=1 $

Let G be the Lie group ${\mathbb{R}}^2\rtimes {\mathbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of G, and consider the Laplacian $\Delta=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_i\Delta^{-1/2}$, $i=0,1,2$. We first show that the atomic Hardy space H1 in G introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from H1 to H1.

The focus of this paper is to introduce an alternating inertial Tseng-type method for approximating singularity point of an inclusion problem which is defined by means of sum of a single-valued vector and a multi-valued vector field in the setting of a Hadamard manifold. Using our iterative method, we prove that the sequence generated by our method converges to a singularity point under some mild conditions. We also establish a linear convergence result when the operator is strongly monotone. As far as we are concerned, there are no results on alternating inertial steps for solving inclusion problems in the settings of Hadamard manifolds. Lastly, we present a numerical example to show the performance of our method. The result present in this article extends and generalizes many related results in the literature.

An element of a group is called strongly reversible or strongly real if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.

We prove the existence of a vector-valued cusp form for the full modular group for which the nth derivative of its L-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen’s plus space and prove an analogous result for Jacobi forms.

We show that dualising transfer maps in Hochschild cohomology of symmetric algebras over complete discrete valuations rings commutes with Tate duality. This is analogous to a similar result for Tate cohomology of symmetric algebras over fields. We interpret both results in the broader context of Calabi–Yau triangulated categories.