One of the interesting conjectures in the study of multiple zeta values is that all the relations among them can be deduced from regularized double shuffle relation. In this paper, the Ohno-Zagier type of relation for interpolated multiple zeta values with full height has been deduced from regularized double shuffle relation. A decomposition of interpolated multiple zeta values has also been obtained.

The goal of this paper is to give a new non-commutative approach to Serre's ordinary density conjecture. We prove that ordinary reduction is inherited at almost all finite primes for a derived equivalent of smooth proper varieties over a number field and using the techniques of non-commutative algebraic geometry, we prove that Serre's ordinary density conjecture is true for cubic 4-folds which contain ℙ².

This paper explores extensions to the Fibonacci and Lucas sequences, examining modifications to the recursive equation and alterations to initial values. Building on previous work detailed in [J. P. Ascaño and E. N. Gueco, The Bi-periodic Fibonacci-Horadam Matrix, Integers 21 (2021), #A29], our focus centers on generalizations of this sequence, particularly the bi-periodic Horadam sequences. These sequences are characterized by initial values corresponding to bi-periodic Fibonacci and bi-periodic Lucas sequences. The investigation extends to studying limits of term ratios, thereby generalizing the concept of limits and the golden ratio seen in classical versions. To enhance comprehension, we introduce three new matrices, with one being the bi-periodic Lucas-Horadam matrix denoted as H. Subsequently, this matrix is utilized to derive various properties of the bi-periodic Horadam sequences, including the exploration of summation identities.

The principal aim of this paper is to study some differential identities on prime rings and their significant role in characterizing central elements for this class of rings. Moreover, we also present a classification of the involved additive mappings.

We deduce several q-continued fractions of order forty-two from the general continued fraction identity of Ramanujan. We demonstrate some theta function identities and offer few colored partition identities for the same.

In this paper, we give a local uncertainty principle inequality for the second q-Bargmann transform 𝓑_α,q introduced by Essadiq et al. in [6] and we derive a q-version of Heisenberg-type uncertainty principle on the q-weighted Bergman space 𝓐_α,q. Also, using this local uncertainty principle and the partial q-Bargmann integrals, we establish an uncertainty principle of concentration type for this transform.

Operator matrix pencils are very useful in many analysis problems. In particular, they are often used in studies of control systems with linear descriptors. Generalized eigenvalue problems of matrix pencils have drown gained interest for decades from mathematicians and engineers. The novelty of our work is to develop a new sufficient criteria to compute the eigenvalues of square operator matrix pencil defined with diagonal domain. Some spectral properties as well as practical criteria for computation of the resolvent form of square operator matrix pencil with diagonal domain will be discussed by exploiting this new technique.

This paper explores the four-parameter Mittag-Leffler matrix function and discusses some of its properties, including Jordan decomposition. Some graphs are plotted for scalar functions. The new extended gamma matrix function(EGMF) and extended beta matrix function(EBMF) are proposed in conjunction with the four-parameter Mittag-Leffler matrix function. Various properties of these functions, including symmetry relationships, integral representations, generating relations, summation formulas and functional relations, have been investigated.

The paper deals with the invariant for unitary irreducible representation introduced by Benson and Ratcliff in [5]. We first focus on the study of this invariant for the class of reduced semi-direct product of nilpotent Lie algebras, which leads to a way of constructing families of counter-examples to the B-R conjecture, as well as identifying another class of r-step nilpotent Lie algebras for which the conjecture holds. Finally we extend the study of the B-R invariant of a coadjoint orbit of a nilpotent Lie group to the class of almost abelian solvable (non-nilpotent) Lie group ℝⁿ ⋊ ℝ.

We present here a proof of the classical reduction of singularities based on the idea of "idealistic flowers". We follow the general ideas of Maximal Contact Theory, presented in a recent book of Aroca, Hironaka and Vicente, that recovers three old publications of Jorge Juan Institute. The concept of idealistic flowers deals with the globalization problems arising from the local nature of the maximal contact.

We mainly prove that the equilateral dimensions of the Riemannian manifolds consisting of 2 × 2 invertible density matrices and positive definite matrices with determinant 1 are respectively 4. We also give the minimum of equilateral dimension of the Cartan-Hadamard manifold consisting of positive definite (Hermitian) matrices.

Given an oriented closed C^∞-manifold M of odd dimension and a unitary representation ρ : π₁(M) → GL_n(𝔽), we define a Reidemeister torsion, even if the cohomology associated with ρ is not acyclic. As corollaries, we introduce some topological invariants of M, which include non-acyclic extensions of abelian torsions and the Alexander polynomials of links. Furthermore, we propose a volume form of the SU(n)-character varieties of M. Moreover, we compute the Reidemeister torsions of some representations of 3-manifolds and give a comparison of the works of Farber and Turaev.

In this paper, we first prove the existence of relative free models of morphisms (resp., relative commutative models) in the category of DGA(R) (resp., CDGA(R)), where R is a principal ideal domain cantaining ${\frac{1}{2}}$. Next, we restrict to the category of (r, ρ(R))-H-mild algebras and we introduce, following Carrasquel's characterization, secat(-, R), the sectional category for surjective morphisms. We then apply this to the n-fold product of the commutative model of an (r, ρ(R))-mild CW-complex of finite type to introduce TC_n(X, R), mTC_n(X, R) and HTC_n(X, R) which extend the well known rational topological complexities. We do the same for sc(-, ℚ) to introduce analogous algebraic sc(-, R) in terms of their commutative models over R and prove that it is an upper bound for secat(-, R). This also yields, for any (r, ρ(R))-mild CW-complex, the algebraic tc_n(X, R), mtc_n(X, R) and Htc_n(X, R) whose relation to the homology nilpotency is investigated. In the last section, in the same spirit, we introduce in DGA(R), Asecat(-, R), Asc(-, R) and their topological correspondents. We then prove, in particular, that ATC_n(X, R) ≤ TC_n(X, R) and Atc_n(X, R) ≤ tc_n(X, R).