In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of multiple zeta functions as special cases, such as the Mordell-Tornheim type of multiple zeta values, zeta values of the root systems and so on. Moreover, we can give an explicit expression in terms of lower series by using the main theorem.

We propose a unified method for constructing higher Capelli elements for the classical Lie algebras. The higher Capelli elements are obtained as the Jacobi-Trudi determinants of central elements attached to partitions of single columns.

We study the distribution of zeros of finite linear combinations of Hurwitz zeta-functions plus an arbitrary constant, and prove a Riemann-von Mangoldt type formula for their number of non-trivial zeros. Finally, as an application, we show that two Hurwitz zeta-functions sharing a complex value are already identical, as well as a similar (more general) result about the uniqueness of linear combinations.

For the fifth Painlevé transcendents, an asymptotic representation by the Jacobi sn-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrized by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the sn-function and the ϑ-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the sn-function satisfying the Boutroux equations and to construct deductively the elliptic representation.

Let k = ℚ(³√d, ζ₃), where d > 1 is a cube-free positive integer, k₀ = ℚ(ζ₃) be the cyclotomic field containing a primitive cube root of unity ζ₃, and G ＝ Gal(k / k₀). The possible prime factorizations of d in our main result in previous work (Theorem 1.1 in Aouissi et al, Preprint, arXiv:1808.04678v2) give rise to new phenomena concerning the chain Θ = (θ_i)_i∈ℤ of lattice minima in the underlying pure cubic subfield L = ℚ(³√d) of k. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals (ν) ∈ P_k^G / P_k0 among the lattice minima Θ = (θ_i)_i∈ℤ of the underlying pure cubic field L = ℚ(³√d), and to explain the exceptional behavior of the chain Θ for certain radicands d with impact on determining the principal factorization type of L and k by means of Voronoi's algorithm.

In this paper, we study the symmetrized Talagrand inequality that was proved by Fathi and has a connection with the Blaschke-Santaló inequality in convex geometry. As corollaries of our results, we have several refined functional inequalities under some conditions. We also give an alternative proof of Fathi's symmetrized Talagrand inequality on the real line.

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We use old ideas of Bowen (Trans. Amer. Math. Soc. 164 (1972), 323-331) for clarifying the local nature of metric mean dimension. We also generalize the local formula to metric mean dimension of ℝ^D-action dynamics and apply the result to simplify a previous proof of the mean dimension estimate of the ℂ-action on the space of Brody curves.

Let a₁, a₂, . . . , a_k be positive integers with gcd(a₁, a₂, . . . , a_k) = 1. Let NR = NR(a₁, a₂, . . . , a_k) denote the set of positive integers non-representable in terms of a₁, a₂, . . . , a_k. The largest non-representable integer max NR, the number of non-representable positive integers Σ_n∈NR 1, and the sum of non-representable positive integers Σ_n∈NR n have been widely studied for a long time as related to the famous Frobenius problem. In this paper, by using Eulerian numbers, we give formulas for the weighted sum Σ_n∈NR λⁿn^μ, where μ is a non-negative integer and λ is a complex number. We also examine power sums of non-representable numbers and some formulas for three variables. Several examples illustrate and support our results.

We show a Whitney approximation theorem for a continuous map from a manifold to a smooth CW complex, which enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists a partition of unity subordinate to the open covering. In addition, we observe that there are enough many smooth functions on a smooth CW complex.

The short-time asymptotic behavior of the transition density function of the diffusion process generated by the Grushin operator with parameter γ > 0 will be investigated, by using its explicit expression in terms of expectation. Further, the dependence on γ of the asymptotics will be seen.