Center of the semigroup algebra of a finite inverse semigroup over the field of complex numbers

In the semigroup algebra A of a finite inverse semigroup S over the field of complex numbers to an indempotent e there is assigned the sum (e)=e+(–1)^Ke_L1e_iK, where e_i,...,e_m are maximal preidempotents of the idempotent e, and the summation goes over all nonempty subsets {i₁,...,i_k} of the set {1,...m} Then for any class K of conjugate group elements of the semigroup S the element K=a·(a^–1a) (the summation goes over all ag) is a central element of the algebra A, and the set {K} of all possible such elements is a basis for the center of the algebra A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 154–158, 1978.     

Jordan代数上的可乘Jordan导子

设A是Jordan代数,如果映射d:A→A满足任给a,b∈A,都有d(aob)=d(a)o b+aod(b),则称d为可乘Jordan导子.如果A含有一个非平凡幂等p,且A对于p的Peirce分解A=A_1⊕A_(1/2)⊕A_0满足:(1)设ai∈Ai(i=1,0),如果任给t_(1/2)∈A_(1/2),都有a_i○t_(1/2)=0,则a_i=0,则A上的可乘Jordan导子d.如果满足d(p)=0,则d是可加的.由此得到结合代数和三角代数满足一定条件时,其上的任意可乘Jordan导子是可加的.     

A boundedness theorem for a certain nth order differential equation

Summary Consider the equation(1.2) in which a₁, a₂, ..., a_n−1 are constants and the functions f_n(x) and p(t) (both continuous) together with are all bodnded. A recent investigation by Reissig[1] shows that if f_n(x)sgn x>0 (|x|≥h>0) then subject to certain conditions, which are stated explicitly in[1], the solutions of such an equation(1.2) are all ultimately bounded The object of the psesent paper is to generalize that result to the equation(1.1) in which φ_n is a bounded function depending on all the variables shown, and each coefficient ϕ_i (i=2,3,..., n−1) satisfies as |ζ|→∞ for some constant a_i. Entrata in Redazione il 20 agosto 1970.     

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

Derived varieties of semigroups and groupoids

A hypersubstitution of type is a map which assigns to every fundamental operation symbol f _i of type a term (f _i ) of the same arity as f _i . For any algebra and any hypersubstitution both of type we can form the derived algebra . In this paper we consider derived algebras and derived varieties, along with the related concepts of semisolidity and mutual solidity of varieties and derivation diagrams. In particular we present a number of examples, based on varieties of semigroups and groupoids. Received March 20, 1996; accepted in final form May 21, 1997.     

Best methods of interpolation for certain classes of differentiable functions

For the class W^(r)L_q (M;a, b), 1≤q≤∞, we construct the best method of approximation of the functionalf (x), x∈ [a, b], among all the methods using only information about the values off (^k)(x_i) (k=0, 1, ..., r?1; i=1, 2, ..., N).     

A family of maximal subalgebras of R. Robinson's algebra

All maximal sub algebras of the algebra of primitive recursive functions <〉;+,*i>, whose basic sets contain the set $$A = \{ f:f \equiv 0V(x) (x > \Rightarrow f(x) > 0)\} ,$$ are described. It is shown that they are continuum in number.     

Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius

In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system \frac?f?x₀ + ?_i=1ⁿ e_i\frac?f?x_i=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives.     

New criteria for <Emphasis Type="Italic">λ</Emphasis>-Koszul algebras

The main purpose of this paper is to provide some new criteria for a standard graded algebra A = ⊕ _i≥0 A _i to be a λ-Koszul algebra, which was first introduced in [12] and was another class of “Koszul-type” algebras including Koszul and d-Koszul algebras as special examples.     

Ockham algebras with double pseudocomplementation

The variety dpO consists of those algebras (L; ∧, ∨, f, *, ⁺, 0, 1) with ∧, ∨ binary, f, *, ⁺ unary and 0, 1 nullary, and where (L; ∧, ∨, f, 0, 1) is an Ockham algebra and the unary operations f and * commute, f and⁺ commute. We describe completely the structure of the subdirectly irreducible algebras that belong to the subclass dpK_1,1, characterised by the property f³ = f. This paper is dedicated to Walter Taylor. Received September 29, 2004; accepted in final form September 8, 2005.     

Estimate of a complete rational trigonometric sum

Supposef is a polynomial of degree n≥3 with integral coefficientsa ₀,a ₁,...,a _n; q is a natural number; (a ₁,...,a _n, q)=1,f(0) = 0. It is proved that $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ .     

Wave distribution in a variable medium with random inhomogeneities

We consider the behavior of a solution of the wave equation u_tt (t, x) – a² (t) u_xx (t, x)=f (t, x) with initial conditions u (0, x)=u₀, /t6t u (t, x) ¦_t=0 =u₁ (x), a andf being random functions; a(t) characteristizes the variable character of the medium;f(t, x) is the inhomogeneity, having the character of random walks.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 75–78, 1988.     

The solvability of a boundary-value periodic problem

In the space of functions B _a³⁺={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T₃/2)=g(x, −t)}, we establish that if the condition aT ₃ (2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u _u −a ² u _xx =g(x, t), u(0, t)=u(π, t)=0, u(x, t+T ₃ )=u(x, t), ℝ², is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997 Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Optimum initial condition of projection pursuit density approximation

LetX be ap-dimensional random vector with densityf(x). We hope to approximate it by a sequence of density functions . In this paper we consider the optimum initial conditionf _p (x) such thatE _f log (f(X)/f _p (X))=min. We also consider the problem of projection pursuit density approximation.Institute of Systems Science, Academia Sinica     

Approximation of functions of finite variation by superpositions of a Sigmoidal function

The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝ^d ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝ^d and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L₂-norm by elements of the set G_n = {Σ_i=0^staggeredn c_ig(〈a_i, x〉 + b_i) : a_iℝ^d, b_i, c_iℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n^1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝ^d6t6e^d|u(t)| > ∞     

Singularity Spectrum of Generic ��-H?lder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by B₁^a{\mathcal {B}}_{1}^{\alpha} the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g ? B₁^ag\in {\mathcal {B}}_{1}^{\alpha} one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space E^a=M×B₁^a\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha} equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ ^α . In particular we determine the generic H?lder singularity spectrum of g○f.     

Region of Variability for Exponentially Convex Univalent Functions

For ${\alpha\in\mathbb C{\setminus}\{0\}}For a ? \mathbb C\{0}{\alpha\in\mathbb C{\setminus}\{0\}} let E(a){\mathcal{E}(\alpha)} denote the class of all univalent functions f in the unit disk \mathbbD{\mathbb{D}} and is given by f(z)=z+a₂z²+a₃z³+?{f(z)=z+a_2z^2+a_3z^3+\cdots}, satisfying

${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.      19. Gersgorin variations II: On themes of Fan and Gudkov      Alan J. Hoffman 《Advances in Computational Mathematics》2006,25(1-3):1-6 Assume F={f1,. . .,fn} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |aii|>fi (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(aij), and all i, fi≥∑j|aij|{xj}/xi. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov. Dedicated to Charles A. Micchelli, in celebration of his 60th birthday and our 30 years of friendship      20. A survey of published programs for best approximation      Charles B. Dunham 《Journal of Computational and Applied Mathematics》1979,5(2):141-143 We list and discuss published programs for best approximation of functions by linear and nonlinear families in all standard forms.In this note we list and discuss the published programs for obtaining best approximations. Let X be a set on which we wish to approximate. Most sets will be finite (an equivalent terms is discrete). Let 6 6 be a norm on the continuous functions on X. Let G be a familiy of continuous functions on X. For a given basis {φ1…,φn}, the linear family G is the set of all functions of the form $\text{L(A,x)=}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{φ}{}_{\mathrm{k}}\text{(x)}$ The problem of best approximation is given a continuous function f, to find g1 to minimize e(g) = ∥f-g∥ over g?G. Such g1 is called a best approximation to f.Discrete linear approximation problems are sometimes formulated as solution of an overdetermined system of linear equations Ax = b with respect to a norm where aij = φj(xi) and bi = f(x)i).

Gersgorin variations II: On themes of Fan and Gudkov

Assume F={f₁,. . .,f_n} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |a_ii|>f_i (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(a_ij), and all i, f_i≥∑_j|a_ij|{x_j}/x_i. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov. Dedicated to Charles A. Micchelli, in celebration of his 60th birthday and our 30 years of friendship     

A survey of published programs for best approximation

We list and discuss published programs for best approximation of functions by linear and nonlinear families in all standard forms.In this note we list and discuss the published programs for obtaining best approximations. Let X be a set on which we wish to approximate. Most sets will be finite (an equivalent terms is discrete). Let 6 6 be a norm on the continuous functions on X. Let G be a familiy of continuous functions on X. For a given basis {φ₁…,φ_n}, the linear family G is the set of all functions of the form

$\text{L(A,x)=}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{φ}{}_{\mathrm{k}}\text{(x)}$

The problem of best approximation is given a continuous function f, to find g¹ to minimize e(g) = ∥f-g∥ over g?G. Such g¹ is called a best approximation to f.Discrete linear approximation problems are sometimes formulated as solution of an overdetermined system of linear equations Ax = b with respect to a norm where a_ij = φ_j(x_i) and b_i = f(x)_i).