Every partition has, for some d, a Durfee square of side d. Every partition π with Durfee square of side d gives rise to a “successive rank vector” r = (r1, · · · , rd). Conversely, given a vector r = (r1, · · · , rd), there is a unique partition π0 of minimal size called the basis partition with r as its successive rank vector. We give a quick derivation of the generating function for b(n, d),...

Response of a transversely isotropic 3-D half-space subjected to a surface time-harmonic excitation is presented in analytical form. The derivation of the fundamental solutions expressed in terms of displacements is based on the prefect series of displacement potential functions that have been obtained in the companion paper by the authors. First the governing equations are uncoupled in the cyl...

There are several kinds of hyperrings, for example, Krasnerhyperrings, multiplicative hyperring, general hyperrings and$H_v$-rings. In a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf Keywords:} multiplic...

The goal of this short note is to provide a simpler derivation the effective potential surrounding Schwarzschild black hole for spherically symmetric perturbations in framework torsion bigravity than one presented [V. Nikiforova, Spherically bigravity, Phys. Rev. D 104, 024032 (2021).]. We also discuss unicity reduction process that leads potential.

We give an example of an amenable, radical Banach algebra, relying on results from non-abelian harmonic analysis due to H. Leptin, D. Poguntke and J. Boidol. Let A be a Banach algebra, and let E be a Banach A-module. A bounded linear map D : A → E is called a a derivation if D(ab) = a.Db+ (Da).b (a, b ∈ A). A derivation D : A → E is said to be inner if Da = x.a− a.x (a ∈ A) for some x ∈ E. For ...

An important theme in recent generative grammar is that linguistic operations, derivations, and representations are subject to economy conditions which guarantee that they are optimal in some sense (see Chomsky 1998b). Consider an operation OP applying in a derivation D leading to the representations (PF, LF) (phonetic form and logical form). Economy considerations suggest that OP be as small a...

In one of our recent papers, the associative and the Lie algebras of Weyl type A[D] = A⊗IF [D] were defined and studied, where A is a commutative associative algebra with an identity element over a field IF of any characteristic, and IF [D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with I...

We study p-adic Euler's series Ep(t)=∑k=0∞k!tk at a point pa, a∈Z≥1, and use Padé approximations to prove lower bound for the absolute value of expression cEp(±pa)−d, where c,d∈Z. It is interesting that same polynomials which p-adically converge Ep(t), approach Hardy integral H(t)=∫0∞e−s1−tsds on Archimedean side. This connection used with trick analytic continuation when deducing an numerator ...

In a recent paper by the authors, the associative and the Lie algebras of Weyl type A[D] = A⊗F[D] were introduced, where A is a commutative associative algebra with an identity element over a field F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with F being a ...