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 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.     

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.     

Robustness of positional scoring over subsets of alternatives

Positional score vectorsw=(w ₁,,w _m ) for anm-element setA, andv=(v ₁,,v _k ) for ak-element proper subsetB ofA, agree at a profiles of linear orders onA when the restriction toB of the ranking overA produced byw operating ons equals the ranking overB produced byv operating on the restriction ofs toB. Givenw ₁>w _mandv ₁>v _k , this paper examines the extent to which pairs of nonincreasing score vectors agree over sets of profiles. It focuses on agreement ratios as the number of terms in the profiles becomes infinite. The limiting agreement ratios that are considered for (m, k) in {(3,2),(4,2),(4,3)} are uniquely maximized by pairs of Borda (linear, equally-spaced) score vectors and are minimized when (w,v) is either ((1,0,,0),(1,,1,0)) or ((1,,,1,0),(1,0,,0)).This research was supported by the National Science Foundation, Grants SOC 75-00941 and SOC 77-22941.     

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.     

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.     

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.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

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.     

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.     

Interval-dividing processes

Summary If and then P(n ^–1·[(Y ₁)++(Y _n )] converges to cnts. law on R ¹) = P(n ^–1·[(Y ₁)++(Y _n )] converges to a cnts. law on R ¹). Thus if ,n then n ^–1[(X ₁)+...+(X _n )] converges a.s. The main result here generalizes this: Let X ₍₁₎ ⁿ , X ₍₂₎ ⁿ ,..., X _(n) ⁿ be the order statistics associated with X ₁, X ₂,,X _n. Define random variables Z ₁,Z ₂, by {Z _n =i}={X _n =X _(i) ⁿ }. Then if Z ₁,Z ₂,Z ₃, are independent and P(Z_ni)i/n, and {X _i} is bounded, n ^–1·[(X ₁)++(X _n)] converges a.s.     

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 polynomial-time algorithm for a class of linear complementarity problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x 0,y 0,x _i y _i = 0 (i = 1, 2,,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n ³ L) arithmetic operations by tracing the path of centers,{(x, y) S: x _i y _i = (i = 1, 2,,n) for some > 0} of the feasible regionS = {(x, y) 0:y = Mx + q}, whereL denotes the size of the input data of the problem.     

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.     

Some Criteria for Multivalently Starlikeness

Let S_n(p)(p, n N) be the class of functions f() = ^p + a_p+n^p+n + which are p-valently starlike in the unit disk. Some sufficient conditions for a function f() to be in the class S_n(p) are given.AMS Subject Classification (2000): primary 30C45     

Optimal accuracy approximation of functions and their derivatives

We consider the approximation of the function (x) and its derivative '(x) on [a, b] given that (x)C _2,N, i.e., belongs to the class of functions f(x) that satisfy the conditions f(x)L, f(x_i)=y_i, i=1,,N, where L and y_i are given real numbers and x_i are the nodes of an arbitrary grid, a=x₁<x₂<<X_N=b. A solution algorithm on the class of functions C_2,L,N is proposed which has optimal accuracy with a constant not exceeding 2. A bound on the approximation error of the function and its derivative is derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 57–61, 1985     

Asymptotically minimax tests of composite hypotheses

Summary X ₁,,X> _n are independent, identically distributed random variables with common density function f(x¦ ₁ ,, _k , _k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that _k+1 =b against the alternative that _k+1 =b+cn ^–1/2 . ₁ ,, _k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦ ₁ -a _i ¦n ^–1/2 M(n) for i=1,,k, where a ₁ , ,a _k are known, and M(n) approaches infinity as n increases but n ^–1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to _k+1 =b, and weight 1-A to _k+1 =b+cn ^–1/2 , in each case the weight being spread uniformly over the possible values of ₁ ,, _k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of ₁ ,..., _k are substituted for a ₁ ..., a _k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a ₁ ..., a _k are.Research supported by the U.S. Air Force under Grant AF-AFOSR-68-1472.     

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.

     

Properties of balanced and perfect matrices

In this paper we define wheel matrices and characterize some properties of matrices that are perfect but not balanced. A consequence of our results is that if a matrixA is perfect then we can construct a polynomial number of submatricesA _I,,A _n ofA such thatA is balanced if and only if all the submatricesA ₁,,A _n ofA are perfect. This implies that if the problem of testing perfection is polynomially solvable, then so is the problem of testing balancedness.Partial support under NSF Grants DMS-8606188, ECS-8800281 and DDM-8800281.     

Hypergraphs and hypergroups

We associate to every hypergraph a commutative quasi-hypergroupH _qG and find a necessary and sufficient condition on so thatH is associative. For certain, any finite included, we determine a sequence= ₀, ₁,, _n of hypergraphs such that ifH ₀ ,H ₁ ,H, H _n is the sequence of the associated quasi-hypergroups,H _n is a join space.Presented by I. Rosenberg.