The Braided product of Ockham algebras

In this paper we introduce an algebraic concept of the product of Ockham algebras called the Braided product. We show that ifL _i MS(i=1, 2, ,n) then the Braided product ofL _i(i=1, 2, ,n) exists if and only ifL ₁, ,L _n have isomorphic skeletons.     

A distance constrainedp-facility location problem on the real line

LetV = {v ₁,, v _n } be a set ofn points on the real line (existing facilities). The problem considered is to locatep new point facilities,F ₁,, F _p , inV while satisfying distance constraints between pairs of existing and new facilities and between pairs of new facilities. Fori = 1, , p, j = 1, , n, the cost of locatingF _i at pointv _j isc _ij . The objective is to minimize the total cost of setting up the new facilities. We present anO(p ³ n ² logn) algorithm to solve the model.     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

The complete figure for second order multiple integral problems in the calculus of variations

Notation Throughout this paper Greek indices, , , and Latin indicesi, j, h, k, assume the values 1, ,m, and 1, ,n respectively. The summation convention is operative in respect of both sets of indices.This work was supported by the South African Council for Scientific and Industrial Research.At time of writing Professor Grässer was Visiting Scholar at the University of Arizona, Tucson, Arizona.     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

On the convexity of the sum of the first eigenvalues of operators depending on a real parameter

Let ₁ (k) ₂ (k) be the eigenvalues of an operator of a certain type depending on a real parameterk. The paper shows that under certain requirements on the operator and on the nature of its dependence onk, the sum ₁ (k)++ _N (k) is a concave function ofk, for any positive integerN.

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Zusammenfassung Seien ₁ (k) ₂ (k) die Eigenwerte eines von einem reellen Parameterk abhängigen Operators. Man zeigt, daß unter gewissen Voraussetzungen über den Operator und seine Abhängigkeit vonk die Summe ₁ (k)++ _N (k) für jedesN eine konkave Funktion vonk ist.

     

Monomial gorenstein curves in A4 as set-theoretic complete intersections

It is shown that if the prime ideal ,, x₄], k an arbitrary field, has generic zero x_i=tⁿ _i, n_i positive integers with g.c.d. equal l, l i 4, then P(S) is a set-theoretic complete intersection if the numerical semigroup S=1,, n₄> is symmetric (i.e. if the extension of P(S) in k[[x₁,, x₄]] is a Gorenstein ideal).     

A basis for the type 〈n〉 hyperidentities of the medial variety of semigroups

Summary The medical varietyMV of semigroups is the variety defined by the medial identityxyzw = xzyw. This variety is known to satisfy the medial hyperidentitiesF(G(x ₁₁ ,, x _1n ),, G(x _n1 ,, x _nn )) = G(F(x ₁₁ ,, x _n1 ),, F(x _1n ,, x _nn )), forn 1. Taylor has observed in [2] thatMV also satisfies some other hyperidentities, which are not consequences of the medial ones. In [4] the author introduced a countably infinite family of binary hyperidentities called transposition hyperidentities, which are natural generalizations of then = 2 medial hyperidentity. It was shown that this family is irredundant, and that no finite basis is possible for theMV hyperidentities with one binary operation symbol.In this paper, we generalize the concept of a transposition hyperidentity, and extend it to cover arbitrary arityn 2. We show that theMV hyperidentities with onen-ary operation symbol have no finite basis, but do have a countably infinite basis consisting of these transposition hyperidentities.Research supported by NSERC of Canada.     

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

On a characterization of polynomials by divided differences

Summary We consider the functional equationf[x ₁,x ₂,, x _n ] =h(x ₁ + +x _n ) (x ₁,,x _n K, x _j x _k forj k), (D) wheref[x ₁,x ₂,,x _n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants a_j K, 0 j n (a := a_n, b := a_{n – 1}) such thatf = ax ⁿ +bx ^{n – 1} + +a ₀ and h = ax + b.     

Regular (2, 2)-systems

Let (E,I) be an independence system over the finite setE = {e ₁, ,e _n }, whose elements are orderede ₁ e _n . (E,I) is called regular, if the independence of {e _l , ,e _{l k} },l ₁ < <l _k , implies that of {e _{m l} , ,e _{m k} }, wherem _l < ··· <m _k andl ₁ m ₁, ,l _k m _k . (E,I) is called a 2-system, if for anyI I,e E I the setI {e } contains at most 2 distinct circuitsC, C I and the number 2 is minimal with respect to this property. If, in addition, for any two independent setsI andJ the family (C J, C C (J, I)), whereC(J, I) denotes {C C:e J I C {e}}, can be partitioned into 2 subfamilies each of which possesses a transversal, then (E,I) is called a (2, 2)-system. In this paper we characterize regular 2-systems and we show that the classes of regular 2-systems resp. regular (2, 2)-systems are identical.     

On products of additive functions (a third approach)

Summary Leta ₁, , a_s : G K be additive functions from an abelian groupG into a fieldK such thata ₁(g)··a_s(g) = 0 for allg G. If char(K) =0, then it is well known that one of the functions a₁ has to vanish. We give a new proof of this result and show that, if char(K) > 0, it is only valid under additional assumptions.     

Verschärfung einer Ungleichung von Ky Fan

Summary In this paper we prove the following:IfA _n ,G _n andH _n (resp.A _n ,G _n andH _n ) denote the arithmetic, geometric and harmonic means ofa ₁,, a _n (resp. 1 –a ₁,, 1 –a _n ) and ifa _i (0, 1/2],i = 1,,n, then(G _n /G _n ) ⁿ (A _n /A _n ) ^n-1 H _n /H _n , (^*) with equality holding forn = 1,2. Forn 3 equality holds if and only ifa ₁ = =a _n . The inequality (^*) sharpens the well-known inequality of Ky Fan:G _n /G _n A _n /A _n .

     

A structure theorem for sum form functional equations

Summary It is proved that, iff _ij:]0, 1[ C (i = 1, ,k;j = 1, ,l) are measurable, satisfy the equation (1) (with some functionsg _it, h_jt:]0, 1[ C), then eachf _ij is in a linear space (called Euler space) spanned by the functionsx x ^j(logx) ^k (x ]0, 1[;j = 1, ,M;k = 0, ,m _j – 1), where ₁, , _M are distinct complex numbers andm ₁, , m_M natural numbers. The dimension of this linear space is bounded by a linear function ofN.     

Knaster's problem on continuous maps of a sphere into Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere S^n–1Rⁿ find configurations of points A₁, , A_k, such that for any continuous map fS^n–1R^m one can find a rotation a of the sphere S^n–1 such that f(a(A₁)==f(a(A_k)) and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set {a S0(n)f(a(A₁))==f(a(A_k))} of solutions of Knaster's problem for a fixed configuration of points A₁, , A_kS^n–1 and a map fS^n–1R^m in general position. Unsolved problems are posed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 169–178, 1987.     

A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints

It is shown that a Lagrange multiplier rule involving the Michel-Penot subdifferentials is valid for the problem: minimizef ₀(x) subject tof _i (x) 0,i = 1, ,m;f _i (x) = 0,i = m + 1,,n;x Q where all functionsf are Lipschitz continuous andQ is a closed convex set. The proof is based on the theory of fans.     

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

We study methods for solving the constrained and weighted least squares problem min _x by the preconditioned conjugate gradient (PCG) method. HereW = diag (₁, , _m ) with _{1 m} 0, andA ^T = [T ₁ ^T , ,T _k ^T ] with Toeplitz blocksT _l R ^{n × n} ,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A ^T = 0, whereM =W ^–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E.     

A “from scratch” proof of a theorem of Rockafellar and Fulkerson

The theorem of this paper is of the same general class as Farkas' Lemma, Stiemke's Theorem, and the Kuhn—Fourier Theorem in the theory of linear inequalities. LetV be a vector subspace ofR ⁿ , and let intervalsI ¹,, I ⁿ of real numbers be prescribed. A necessary and sufficient condition is given for existence of a vector (x ¹ ,, x ⁿ ) inV such thatx ⁱ I ⁱ (i = 1, ,n); this condition involves the elementary vectors (nonzero vectors with minimal support) ofV . The proof of the theorem uses only elementary linear algebra.The author at present holds a Senior Scientist Award of the Alexander von Humboldt Stiftung.     

An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk

If (X _n ) _n ⁼¹ is a sequence of i.i.d. random variables in the Euclidean plane such that we compute the mean of the perimeter of theconvex hull ofX ₁++X _k; 0kn}.