Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostant's partition function. For $\xi$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(\xi)=\sum c_i q^i$ where $c_i$ is the number of ways $\xi$ can be written as a sum of $i$ positive roots of $\mathfrak{g}$. In this way, the evaluation of Kostant's weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight representation of $\mathfrak{g}$. In this paper, we give closed formulas for computing weight $q$-multiplicities in a highest weight representation of the exceptional\linebreak Lie algebra $\mathfrak{g}_2$.

We study the automorphism group $\Aut{G}$ of a free product $G$ of finite cyclic groups. We investigate the question in which cases $\Aut{G}$ has Serre's property FA. In the case of two or three free factors, we prove that $\Aut{G}$ does not have property FA. However, if each free factor of $G$ occurs at least four times we show that $\Aut{G}$ does have property FA.

We study $n$-dimensional complete submanifolds immersed in a weighted warped product of the type $I\times_fM^{n+p}_{\varphi}$, whose warping function $f$ has convex logarithm and weight function $\varphi$ does not depend on the real parameter $t\in I$. Assuming the constancy of an appropriate support function involving the $\varphi$-mean curvature vector field of such a submanifold $\Sigma^n$ jointly with suitable constraints on the Bakry-Émery-Ricci tensor of $\Sigma^n$, we prove that it must be contained in a slice of the ambient space. As applications, we obtain codimension reductions and Bernstein-type results for complete $\varphi$-minimal bounded multi graphs constructed over the $n$-dimensional Gaussian space. Our approach is based on the weak Omori-Yau's generalized maximum principle and Liouville-type results for the drift Laplacian.

In this paper, we prove the Mohebi-Radjabalipour Conjecture under an additional condition, and obtain a new invariant subspace theorem for subdecomposable operators. Our main results contain known results in the literature as special cases. Moreover, we give some specific examples to illustrate that the conditions in our theorem are significantly weaker, simpler and more nature than those in the known results.

In this paper, we continue the ongoing research on lineability and algebrability within the context of $p$--adic analysis. This work serves a twofold purpose. Firstly, we improve and, at times, generalize some already known results within this fields and, secondly, we present some new results which have no counterpart in the classical Archimedean framework.

Given a non-degenerate $(0,2)$-tensor field $h$ on a smooth manifold $M$, we consider a natural generalized complex and a generalized product structure on the generalized tangent bundle $TM\oplus T^*M$ of $M$ and we show that they are $\nabla$-integrable, for $\nabla$ an affine connection on $M$, if and only if $(M,h,\nabla)$ is a quasi-statistical manifold. We introduce the notion of generalized quasi-statistical structure and we prove that any quasi-statistical structure on $M$ induces generalized quasi-statistical structures on $TM\oplus T^*M$. In this context, dual connections are considered and some of their properties are established. The results are described in terms of Patterson-Walker and Sasaki metrics on $T^*M$, horizontal lift and Sasaki metrics on $TM$ and, when the connection $\nabla$ is flat, we define the prolongations of the quasi-statistical structures on the manifolds to their cotangent and tangent bundles via Generalized Geometry. Moreover, Norden and Para-Norden structures are defined on $T^*M$ and $TM$.

In this paper, we deal with the singularly perturbed problem for a linear second order differential difference equation with delay as well as advance. In order to solve the problem numerically, we construct a new difference scheme by the method of integral identities with the use interpolating quadrature rules with remainder terms in integral form. Using an appropriately non-uniform mesh of Shishkin type, we find that the method is almost first order convergent in the discrete maximum norm with respect to the perturbation parameter. Furthermore, we present the numerical experiments that their results support of the theory.

We give new and simple proofs of some classical properties of hereditarily indecomposable Banach spaces, including the result by W. T. Gowers and B. Maurey that a hereditarily indecomposable Banach space cannot be isomorphic to a proper subspace of itself. These proofs do not make use of spectral theory and therefore, they work in real spaces as well as in complex spaces. We use our method to prove some new results. For example, we give a quantitative version of the latter result by Gowers and Maurey and deduce that Banach spaces that are isometric to all of their subspaces should have an unconditional basis with unconditional constant arbitrarily close to $1$. We also study the homotopy relation between into isomorphisms from hereditarily indecomposable spaces.

A weak stability bound for the $\varepsilon$-isometry $f$ form the positive cone of a reflexive, strictly convex and Gateaux smooth Banach lattice $X$ to a Banach space $Y$ is presented. This result is used to prove the stability theorem for the $\varepsilon$-isometry $f:(\mathbb{R}^n)^+\rightarrow Y$, where $\mathbb{R}^n$ is the $n$-dimensional space equipped with a $1$-unconditional norm and $Y$ is a n-dimensional, strictly convex and Gateaux smooth space.