The representation obtained as a tensor product of a rational elliptic curve with a weight 1 modular form is in general an irreducible four-dimensional representation. However, there are some instances where such representation is reducible. In this short article we give an exotic way to obtain such a reducible representation.

We are interested in the concept of quasi-compact operators allowing us to provide some advances on the theory of operators acting in Banach spaces. More precisely, our main objective is to exhibit the importance of the use of this notion to outline a new approach in the analysis of the stability problems of upper and lower semi-Fredholm, upper and lower semi-Weyl, and upper and lower semi-Browder operators, and to provide a fine description and characterization of some Browder's essential spectra involving this kind of operators.

We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of $S\mathcal{O}(4)$, and of $U(n)$, endowed with suitable bi-invariant Riemannian metrics.

Given a conjugation $C$ on a complex separable Hilbert space $H$, a bounded linear operator $A$ acting on $H$ is said to be $C$-symmetric if $A=CA^*C$. In this paper, we provide a complete description to all those maps on the algebra of linear operators acting on a finite dimensional Hilbert space that preserve the Jordan product of $C$-symmetric operators, in both directions, for every conjugation $C$ on $H$.

The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. Let $A$ be a right linear operator on a two-sided quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$, where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.

The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by Berra, Carena and Pradolini [J. Math. Anal. Appl. 479 (2019)]. We prove that, under certain conditions on the weights, there exists a constant $C$ such that \[ \Bigg\| \frac{\mathcal{G}_{\alpha}(\vec{f})}{v}\Bigg\|_{L^{q, \infty}(\nu v^q)} \leq C \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}}, \] where $\mathcal{G}_{\alpha}(\vec{f})$ is the multilinear maximal function $\mathcal{M}_{\alpha}(\vec{f})$ introduced by Moen [Collect. Math. 60 (2009)] or the multilineal fractional integral $\mathcal{I}_{\alpha}(\vec{f})$. As an application, a vector-valued weighted mixed inequality for $\mathcal{I}_{\alpha}(\vec{f})$ is provided.

We give a complete classification, up to isometric isomorphism and scaling, of 4-dimensional metric Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we distinguish those metric Lie algebras that admit such an endomorphism which is not a multiple of a complex structure, and for each of them we obtain the de Rham decomposition of the associated simply connected Lie group with the corresponding left invariant metric. On the other hand, we find that the associated simply connected Lie group is irreducible as a Riemannian manifold for those metric Lie algebras where each parallel skew-symmetric endomorphism is a multiple of a complex structure.

In the present paper, we are concerned with the determination of the cortex of semidirect product exponential Lie groups. More precisely, we consider a finite dimensional real vector space $V$ and some abelian matrix group $H=\exp\big(\sum_{i=1}^{n}\mathbb{R} A_i\bigr)$, where ${A_1,\dots, A_n}$ is a set of pairwise commuting non-singular matrices acting on $V$. We first investigate the cortex of the action of the group $H$ on $V$. As an application, we investigate the cortex of the group semidirect product $G:=V\rtimes\mathbb{R}^n$.

As a natural generalization of the classical coloring problem in graph theory, the dynamic coloring problem deals with the existence of a proper coloring $c$ of a graph so that $|c(N(v))| \geq \min{r, d(v)}$ for every vertex $v$. In this paper, we obtain the $r$-dynamic edge chromatic number of any given closed helm graph for any positive integer $r$. This coincides with the $r$-dynamic chromatic number of the line graph of a closed helm graph.

The John–Nirenberg inequality is widely studied in the field of mathematical analysis and probability theory. In this paper we study a new type of the John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting. To be precise, let $0 < q \leq \infty$ and $\Phi$ be an $N$-function with some proper restrictions. We prove that if the stochastic basis ${\mathcal{F}_n}_{n \geq 0}$ is regular, then $BMO_{\Phi,q}=BMO$, with equivalent (quasi)-norms. The result is new, which improves previous work on martingale Hardy theory.

The existence of at least two homogeneous geodesics in any homogeneous Finsler manifold was proved in a previous paper by the author. The examples of solvable Lie groups with invariant Finsler metric which admit just two homogeneous geodesics were presented in another paper. In the present work, it is shown that a homogeneous Finsler manifold with indefinite Killing form admits at least four homogeneous geodesics. Examples of invariant Randers metrics on Lie groups with definite Killing form admitting just two homogeneous geodesics and examples with indefinite Killing form admitting just four homogeneous geodesics are presented.

We study the generalized translation operator associated with the deformed Hankel transform on $\mathbb{R}$. Firstly, we prove the trigonometric form of the generalized translation operator. Next, we derive the positivity of this operator on a suitable space of even functions. Making use of the positivity of the generalized translation operator we introduce and study the deformed Stockwell transform. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative uncertainty principles for this new integral transform. Firstly, we mainly establish various versions of Heisenberg's uncertainty principles. Secondly, we derive some weighted uncertainty inequalities such as Pitt's and Beckner's uncertainty inequalities for the deformed Stockwell transform. We culminate our study by formulating several concentration-based uncertainty principles, including the Amrein–Berthier–Benedicks and local inequalities for the deformed Stockwell transform.

The paper presents a detailed description of duality for braided algebras, coalgebras, bialgebras, Hopf algebras, and their modules and comodules in the infinite setting. Assuming that the dual objects exist, it is shown how a given braiding induces compatible braidings for the dual objects, and how actions (resp., coactions) can be turned into coactions (resp., actions) of the dual coalgebra (resp., algebra), with an emphasis on braided bialgebras and their braided (co)module algebras. Examples are provided by considering these structures in a graded (or filtered) setting, where each degree is finite-dimensional.

We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension 24 and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.

We study (quantum) stochastic processes with independent and stationary increments (i.e., Lévy processes), and in particular Brownian motions in braided monoidal categories. The notion of increments is based on a bialgebra or Hopf algebra structure, and positivity is taken w.r.t. an involution. We show that involutive bialgebras and Hopf algebras in the Yetter–Drinfeld categories of a quasi- or coquasi-triangular $\ast$-bialgebra admit a symmetrization (or bosonization) and that their Lévy processes are in one-to-one correspondence with a certain class of Lévy processes on their symmetrization. We classify Lévy processes with quadratic generators, i.e., Brownian motions, on several braided Hopf-$\ast$-algebras that are generated by their primitive elements (also called braided $\ast$-spaces), and on the braided $\mathrm{SU}(2)$-quantum groups.

We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models we prove that there is a minimizer of the resulting energy that is unique modulo adding a constant.

Let ${T(t)}_{t\geq0}$ be an $\alpha$-times integrated semigroup of bounded linear operators on the Banach space $\mathcal{X}$ and let $A$ be their generator. In this paper, we study the uniform convergence of the Abel averages $\mathcal{A} (\lambda) = \lambda^{\alpha +1} \int_{0}^{\infty} e^{-\lambda t} T(t)dt$ as $\lambda \to 0^+$, with $\alpha \geq 0$. More precisely, we show that the following conditions are equivalent: (i) $T(t)$ is uniformly Abel ergodic; (ii) $\mathcal{X}= \mathcal{R}(A)\oplus \mathcal{N}(A)$, with $\mathcal{R}(A)$ closed; (iii) $\|\lambda^2 R(\lambda,A)\| \longrightarrow 0$ as $\lambda\to 0^+$, and $\mathcal{R}(A^k)$ is closed for some integer $k$; (iv) $A$ is $a$-Drazin invertible and $\mathcal{R}(A^k)$ is closed for some $k\geq 1$; where $\mathcal{N}(A)$, $\mathcal{R}(A)$ and $R(\lambda,A)$ are the kernel, the range, and the resolvent function of $A$, respectively. Additionally, we show that if $T(t)$ satisfies $\lim_{t\to \infty}\|T(t)\|/ t^{\alpha+1}=0$, then $T(t)$ is uniformly Abel ergodic if and only if $\frac{1}{t^{\alpha+1}}\int_{0}^{t}T(s)ds$ converges uniformly as $t \to +\infty$. Finally, we examine simultaneously this theory with the uniform power convergence of the Abel averages $\mathcal{A} (\lambda)$ for some $\lambda>0$.

We obtain two types of inequalities for Lagrangian submanifolds in holomorphic statistical manifolds of constant holomorphic sectional curvature. One relates the Oprea invariant to the mean curvature, the other relates the Chen invariant to the mean curvature. Our results generalize the corresponding inequalities for Lagrangian submanifolds in complex space forms.