Existence and multiplicity of weak solutions for a weighted $(p,q)$-Laplacian problem consisting of a singular term are proved, where the weight function satisfies Muckenhoupt's condition. The method is based on the variational methods.

It is well-known that there exist holomorphic functions $f$ on the open unit disc $D$ for which the sequence $(S_n(f))$ of partial sums of its Taylor series with center 0 approximate all the polynomials on all compact subsets $K\subset\mathbb{C}\setminus D$ with connected complement. This result fails if we consider the sequence of the weighted partial sums $(2^nS_n(f))$ instead of $(S_n(f))$. In the present paper we show that there exist holomorphic functions $f$ on $D$ for which the sequence $(2^nS_n(f))$ approximate all the polynomials on {\em only some specific compact sets} $K$. The geometry of $K$ plays a crucial role. We also generalize these results for weights other than $2^n$ and on arbitrary simply connected domains.

Ribbon concordance, which was introduced by Gordon, has given us a new perspective in knot concordance theory. Recently, Agol proved that ribbon concordance forms a partial order. In this note, we focus on the partially ordered set given by Gordon and Agol. In particular, we prove that a two-bridge knot is minimal in the poset if and only if it is not smoothly concordant to any torus knot or it is a torus knot.

In this paper, we define two variants of the Arakawa-Kaneko zeta function called the parametric Arakawa-Kaneko zeta function and the parametric Kaneko-Tsumura $\eta$-function, which are natural parametric generalizations of the Arakawa-Kaneko zeta function and the Kaneko-Tsumura $\eta$-function. We exhibit nice properties similar to the ordinary Arakawa-Kaneko zeta values and Kaneko-Tsumura $\eta$-values such as duality relation and special values by using the method developed by Luo-Si and Xu. In particular, we establish some explicit formulas of parametric Arakawa-Kaneko zeta function and parametric Kaneko-Tsumura $\eta$-function via multiple polylogarithm functions.

The magnitude of finite categories is a generalization of the Euler characteristic. It is defined using the coarse incidence algebra of rational-valued functions on the given finite category, and a distinguished element in this algebra: the Dirichlet zeta function. The incidence algebra may be identified with the algebra of $n \times n$ matrices over the rational numbers, where $n$ is the cardinality of the underlying object set. The Moore-Penrose pseudoinverse of a matrix is a generalization of the inverse; it exists and is unique for any given matrix over the complex numbers. In this article, we derive a new method for calculating the magnitude of a finite category, using the pseudoinverse of the matrix that corresponds to the zeta function. The magnitude equals the sum of the entries of this pseudoinverse.

We examine the existence of Ricci solitons on $(\kappa, \mu)$-almost cosymplectic (acs) manifolds. Our study shows that on $(\kappa \prec 0, \mu)$-acs manifolds, Ricci solitons are actually expanding, and $(\kappa = 0, \mu)$-acs manifolds admit arbitrary Ricci solitons. Additionally, we analyze the behavior of Ricci solitons on $(\kappa, \mu)$-acs manifolds with the Jacobi-type potential vector field and with the potential vector field $V$ aligned with the structure vector field $\xi$.

We give the computable conditions for the Selmer groups of elliptic curves $E_{n}$ and $E_{n}'$ related to the isogenies of degree 2, where $n$ is a square-free integer. In many cases, we can compute the Selmer groups by linear systems of equations defined over finite field $\mathbb{F}_{2}.$ As an application, we find many elliptic curves with the Mordell-Weil rank 0. Finally, we also obtain a sharp upper bound for the Mordell-Weil rank of $E_{n}$.

The aim of this article is to obtain variations on the classical theorems of Schur and Baer on finiteness of commutator subgroups, valid in the contexts of Lie algebras and Leibniz algebras over a field. Using non-abelian tensor products and exterior products, we prove Schur's Theorem for finitely generated Leibniz algebras, both Schur's Theorem and Baer's Theorem for finitely generated Lie algebras, and a version of these theorems for finitely presented Lie algebras.

We improve a theorem of Flores de Chela and Green on the determinants of braided symmetrizers appearing with quantum symmetric algebras (Nichols algebras). As an application we characterize the freeness of Nichols algebras of diagonal type. We also determine the dimensions of the kernels of certain shuffle maps considered as an operator acting on the free algebra. Our approach is based on an inequality for the number of Lyndon words and on an identity for the shuffle map. For a particular family of examples, the freeness of the Nichols algebra is characterized in terms of solutions of a quadratic diophantine equation.