Conditional identity calculus and conditioned rational equivalence

We construct a conditional identity calculus (similar to the Birkhoff identity calculus), which complies with the concept of truth for a conditional identity on a universal algebra. The relationship is studied between the isomorphism of embedding categories of conditional varieties and the conditioned rational equivalence of these varieties. As applications, we describe invariants for the relations ‘is conditional rational equivalent’ and ‘is similar’ on finite universal algebras. Supported by RFFR grant No. 93-01-01520. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 432–459, July–August, 1998.     

Zur Theorie der Vollständigen Teilbarkeitshalbgruppen

In this paper we consider complete d-semigroups, i. e. d-semigroups with a conditionally complete lattice satisfying the implication: ⋂a_i ⇒ xsy = ⋂xa.y. As main results we present: 1. Every complete d-semigroup is archimedean. 2. A d-semigroup is complete iff its cone is complete. 3. Every archimedean d-semigroup can be embedded into an archimedean d-semigroup with identity 1. 4. Every complete d-semigroup can be embedded into a complete d-semigroup with identity.      

A generalization of the Ricci identity for a function of a field of arbitrary type given in a space of affine connectedness

For a function of a field of arbitrary type given in a space of affine connectedness a relationship is deduced which is analogous to the well-known Ricci identity for tensor fields.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 409–411, April, 1969.     

On the Self-Similar Jordan Arcs Admitting Structure Parametrization

We study the attractors γ of a finite system of contraction similarities S _j (j = 1,..., m) in ℝ^d which are Jordan arcs. We prove that if a system possesses a structure parametrization (ℐ,ϕ) and ℱ(ℐ) is the associated family of ℐ then we have one of the following cases:1. The identity mapping Id does not belong to the closure of ℱ(ℐ). Then (if properly rearranged) is a Jordan zipper.2. The identity mapping Id is a limit point of ℱ(ℐ). Then the arc γ is a straight line segment.3. The identity mapping Id is an isolated point of .We construct an example of a self-similar Jordan curve which implements the third case.Original Russian Text Copyright © 2005 Aseev V. V. and Tetenov A. V.The authors were supported by the Program “Universities of Russia” (Grant UR.04.01.456).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 733–748, July–August, 2005.     

A new proof of the Amitsur-Levitski identity

We give a simple proof of the Amitsur-Levitzki identity by analysing the powers of matrices with “differential 1-forms” as entries. Using the fact that 2-forms are central the identity is seen to follow from the Cayley-Hamilton theorem.     

Sojourn times,manifolds with infinite cylindrical ends,and an inverse problem for planar waveguides

We prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on ℝ × (−γ, γ) determine an analytic strictly convex obstacle . With an additional symmetry assumption, one entry suffices. Part of the proof is an integral identity involving an entry in the scattering matrix and a distribution related to the fundamental solution of the wave equation. This identity holds for general manifolds with infinite cylindrical ends. A consequence of this is a relationship between the singularities of the Fourier transform of an entry in the scattering matrix and the sojourn times of certain geodesics. Partially supported by NSF grant DMS 0500267.     

Identity for a spectral function connected with the Hilbert transform

We give a new proof of the identity, known as the “sum rule” in the statistical quantum mechanics, for the integral over the real axis of a nonlinear combination of a function u(x) in a certain class and its Hilbert transform. Examples where the identity fails are given Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 23–28.     

Rational orthogonal bases satisfying the Bedrosian identity

We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct orthogonal bases of L ²(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in L ^p (ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented.     

On an identity for dual fields

A new identity for dual fields is proved. Bibliography: 4 titles. Translated from,Zapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 270–281. Translated by N. A. Slavnov.     

On the quest for positivity in operator algebras

We show that in every nonzero operator algebra with a contractive approximate identity (or c.a.i.), there is a nonzero operator T such that ‖I−T‖?1. In fact, there is a c.a.i. consisting of operators T with ‖I−2T‖?1. So, the numerical range of the elements of our contractive approximate identity is contained in the closed disk center and radius . This is the necessarily weakened form of the result for C^?-algebras, where there is always a contractive approximate identity consisting of operators with 0?T?1 - the numerical range is contained in the real interval [0,1]. So, if an operator algebra has a c.a.i., it must have operators with a “certain amount” of positivity.     

On one variety of alternative algebras

We consider the variety of alternative algebras with the identity of index 5 Lie nilpotency. We prove that the variety of alternative algebras over a field of characteristic different from 2 and 3 with the identity [[[[x ₁, x ₂], x ₃], x ₄], x ₅] = 0 has the Specht property. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 23–34, 2004.     

A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

In (Can J Math 51(2):326–346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45–52, 1997) as “codes for the m-metric”) using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.      

An identity of Andrews and the Askey-Wilson integral

Andrews (Adv. Math. 41:137–172, 1981) derived a four-variable q-series identity, which is an extension of the Ramanujan ₁ ψ ₁ summation. In this paper, we shall give a simple evaluation of the Askey-Wilson integral by using this identity. The author was supported by the National Science Foundation of China, PCSIRT and Innovation Program of Shanghai Municipal Education Commission.     

Bases of the identities of products of certain varieties of groups

The following theorem is proved: The product of any variety of two-step solvable groups and a variety having a finite basis of identity relations has a finite basis of identity relations.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 137–144, January, 1969.     

Minor summation formula and a proof of Stanley’s open problem

In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.      

On the asymptotic distribution of the likelihood ratio under the regularity conditions due to doob

Summary The concepts of the asymptotic maximum likelihood estimates—AMLEs in short—and their asymptotic identity are introduced in section 1. They seem to be more adequate than the usual one for uses in the large sample theory. The AMLE is a slightly weakened version of the usual maximum likelihood estimate and therefore it should have a bit wider applicability than the original one. The asymptotic normality of a consistent AMLE and Wilks’ theorem concerning the asymptotic distribution of the statistic —2 log λ, where λ is the likelihood ratio, can be obtained under the regularity conditions due to Doob in section 2. A set of conditions which assure the existence of a unique and consistent AMLE is presented in section 3 and in the final section 4 the proof of the existence of the unique and consistent AMLE under those conditions is given. This work has been motivated by the work of Ogawa, Moustafa and Roy [3].     

On an identity of Hilbert

A simple proof of the polynomial identity used by Hilbert in the solution of the Waring problem is given. The proof is based on the continued fraction expansion of a certain formal hypergeometric series. Translated fromMatematischeskie Zametki, Vol. 66, No. 4, pp. 527–532, October, 1999.     

Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel'nikov type

One considers V. K. Mel'nikov's new class of nonlinear dynamical systems, which is a generalization of the Korteweg-de Vries dynamical system. One investigates the differential-geometric and spectral properties of dynamical systems of Mel'nikov type, one gives their Hamiltonian form, one establishes the so-called gradient identity. The class of finite-zone potentials of a Sturm-Liouville operator, satisfying the given dynamical systems, is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 655–659, May, 1990.