E-optimality of some row and column designs

In this paper we consider experimental settings where treatments are being tested in b ₁ rows and b ₂ columns of sizes k _1i and k _2j , respectively, i=1,2,..., b ₁, j=1,2,..., b ₂. Some sufficient conditions for designs to be E-optimal in these classes are derived and some necessary and sufficient conditions for the E-optimality of some special classes of row and column designs are presented. Examples are also given to illustrate this theory.     

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.     

A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

Uniformly More Powerful, Two-Sided Tests for Hypotheses about Linear Inequalities

Let Xhave a multivariate, p-dimensional normal distribution (p 2) with unknown mean and known, nonsingular covariance . Consider testing H ₀ : b _i 0, for some i = 1,..., k, and b _i 0, for some i = 1,..., k, versus H ₁ : b _i < 0, for all i = 1,..., k, or b _i < 0, for all i = 1,..., k, where b ₁,..., b _k , k 2, are known vectors that define the hypotheses and suppose that for each i = 1,..., k there is an j {1,..., k} (j will depend on i) such that b _i b _j 0. For any 0 < < 1/2. We construct a test that has the same size as the likelihood ratio test (LRT) and is uniformly more powerful than the LRT. The proposed test is an intersection-union test. We apply the result to compare linear regression functions.     

Distance matrices for points on a line,on a circle,and at the vertices of ann-dimensional cube

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ ^m , the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995.     

The dependence on parameters of the modulus problem for families of several classes of curves

Let, where A={a₁,..., a_n} and B={b₁,...,b_m} are systems of distinguished points, and let H be a family of homotopic classes H_i, i=1, ..., j + m, of closed Jordan curves in C, where the classes H_j+, =1, ..., m, consist of curves that are homotopic to a point curve in b. Let ={₁,...,_j+m} be a system of positive numbers. By P=P(,A,B) we denote the extremal-metric problem for the family H and the numbers : for the modulusU=U(,A,B) of this problem we have the equality , whereD ^*={D ₁ ^* ,...,D _j+m ^* } is a system of domains realizinga maximum for the indicated sum in the family of all systemsD={D ₁,...,D _j+m } of domains, associated with the family H (byU(D _i )) we denote the modulus of the domain D_i, associated with the class H_i). In the present paper we investigate the manner in whichU=U(,A,B) and the moduliU=(D ₁ ^* ) depend on the parameters _i, a_k, b; moreover, we consider the conditions under which some of the doubly connected domains D _i ^* ,i=1,...,j, from the system D^* turn out to be degenerate (Theorems 1–3). In particular, one obtains an expression for the gradient of the function M, as function of the parameter a=a_k (Theorem 4). One gives some applications of the obtained results (Theorem 5).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 136–148, 1985.     

Rotations and Hermitian symmetric spaces

On an almost Hermitian manifold (M, g, J) one considers the naturally defined field of local diffeomorphismsj _m =exp _m J _m exp _m ^–1 ,mM, and in particular, one studies isometric, harmonic, holomorphic and symplecticj _m . This leads to some characterizations of special classes of almost Hermitian manifolds, including the class of Hermitian symmetric spaces. In addition, one treats some intrinsic and extrinsic geometrical properties of geodesic spheres relating to these local diffeomorphisms.Supported by grant 203.01.50 of the C.N.R., Italy.     

Fusion frames and distributed processing

Let {W_i}_iI be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight v_i, and let be the closed linear span of the W_is, a composite Hilbert space. {(W_i,v_i)}_iI is called a fusion frame provided it satisfies a certain property which controls the weighted overlaps of the subspaces. These systems contain conventional frames as a special case, however they reach far “beyond frame theory.” In case each subspace W_i is equipped with a spanning frame system {f_ij}_jJi, we refer to {(W_i,v_i,{f_ij}_jJi)}_iI as a fusion frame system. The focus of this article is on computational issues of fusion frame reconstructions, unique properties of fusion frames important for applications with particular focus on those superior to conventional frames, and on centralized reconstruction versus distributed reconstructions and their numerical differences. The weighted and distributed processing technique described in this article is not only a natural fit to distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. Another important component of this article is an extensive study of the robustness of fusion frame systems.     

Zeros of linear recurrence sequences

Let be an exponential polynomial over a field of zero characteristic. Assume that for each pair i,j with i≠j, α _i /α _j is not a root of unity. Define . We introduce a partition of into subsets (1≤i≤m), which induces a decomposition of f into , so that, for 1≤i≤m, , while for , the number either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most solutions x∈ℤ of f(x)= 0, we have

A generalization of the inversion formulas of systems of power series in systems of implicit functions

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx _j =_j(w)=_j(w₁,...,w_m), j=1,...,n, defined by the system of equations f_j(w, x)=F_j (w₁,...,w_m:z₁,...,x _n )=0, j=1,...,n,f _j , (0, 0)=0, F_j(0, 0)/z_k=_jk in a neighborhood of the point (0, 0)C _(w,x) ^m+n , in terms of the coefficients of the power series of the functions F_j(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978.     

Cyclic Just-In-Time Sequences Are Optimal

Consider the Product Rate Variation problem. Given n products 1,...,i,...,n, and n positive integer demands d ₁,..., d_i,...,d_n. Find a sequence =₁,...,_T, T = _i=1 ⁿ d _i, of the products, where product i occurs exactly d _i times that always keeps the actual production level, equal the number of product i occurrences in the prefix ₁,..., _t, t=1,...,T, and the desired production level, equal r _i t, where r _i=d_i/T, of each product i as close to each other as possible. The problem is one of the most fundamental problems in sequencing flexible just-in-time production systems. We show that if is an optimal sequence for d ₁,...,d_i,...,d_n, then concatenation ^m of m copies of is an optimal sequence for md ₁,..., md_i,...,md_n.     

On theA 0-acceptability of rational approximations to the exponential function with only real poles

We consider rational approximations to the exponential function with real poles, ₁ ^–1 ,..., _m ^–1 , that correspond to implicit Runge-Kutta collocation methods. We show that if _i 1/2,i=1,...,m, the rational approximation isA ₀-acceptable.     

On a Turán type problem of Erdös

Let L ^k be the graph formed by the lowest three levels of the Boolean lattice B _k , i.e.,V(L ^k )={0, 1,...,k, 12, 13,..., (k–1)k} and 0is connected toi for all 1ik, andij is connected toi andj (1i<jk).It is proved that if a graph G overn vertices has at leastk ^3/2 n ^3/2 edges, then it contains a copy of L ^k .Research supported in part by the Hungarian National Science Foundation under Grant No. 1812     

多背包问题的半定松驰算法

The multiple knapsack problem denoted by MKP (B,S,rn,n) can be defined as follows. A set B of n items and a set S of rn knapsacks are given such that each item j has a profit pi and weight wj,and each knapsack i has a capacity Ci. The goal is to find a subset of items of maximum profit such that they have a feasible packing in the knapsacks. MKP (B,S,m,n) is strongly NP-Complete and no polynomial time approximation algorithm can have an approximation ratio better than 0.5. In the last ten years,semi-definite programming has been empolyed to solve some combinatorial problems successfully. This paper firstly presents a semi-definite relaxation algorithm (MKPS) for MKP (B,S,rn,n). It is proved that MKPS have a approximation ratio better than 0. 5 for a subclass of MKP (B,S,m,n) with n≤100, m≤5 and max^nj=1{wj}/min^mi=1={Ci}≤2/3.     

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

A variable projection method for solving separable nonlinear least squares problems

Consider the separable nonlinear least squares problem of findinga εR ⁿ and α εR ^k which, for given data (y _i ,t _i ),i=1,2,...m, and functions ? _j (α,t),j=1,2,...,n(m>n), minimize the functional $$r(a,\alpha ) = \left\| {y - \Phi (\alpha )a} \right\|_2^2$$ where θ(α) _ij =? _j (α,t _i ). Golub and Pereyra have shown that this problem can be reduced to a nonlinear least squares problem involvingα only, and a linear least squares problem involvinga only. In this paper we propose a new method for determining the optimalα which computationally has proved more efficient than the Golub-Pereyra scheme.     

Topologies on products of partially ordered sets I: Interval topologies

Given a certain construction principle assigning to each partially ordered setP some topology θ(P) onP, one may ask under what circumstances the topology θ(P) of a productP = ?_j∈J P _j of partially ordered setsP _i agrees with the product topology ?_j∈Jθ(P _i) onP. We shall discuss this question for several types ofinterval topologies (Part I), forideal topologies (Part II), and fororder topologies (Part III). Some of the results contained in this first part are listed below:

1. Let θ_i(P) denote thesegment topology. For any family of posetsP _j ?_j∈Jθ_s(P_j)=θ_s(?_j∈JP_i) iff at most a finite number of theP _j has more than one element (1.1).
2. Let θ_cs(P) denote theco-segment topology (lower topology). For any family of lower directed posetsP _j ?_j∈Jθ_cs(P_i)=θ_cs(?_j∈JP_i) iff eachP _j has a least element (1.5).
3. Let θ_i(P) denote theinterval topology. For a finite family of chainsP _j,P _j ?_j∈Jθ_i(P_i)=θ_i(?_j∈JP_i) iff for allj∈k, P _j has a greatest element orP _k has a least element (2.11).
4. Let θ_ni(P) denote thenew interval topology. For any family of posetsP _j,P _j ?_j∈Jθ_ni(P_j)=θ_ni(?_j∈JP_j) whenever the product space is ab-space (i.e. a space where the closure of any subsetY is the union of all closures of bounded subsets ofY) (3.13).

In the case oflattices, some of the results presented in this paper are well-known and have been shown earlier in the literature. However, the case of arbitraryposets often proved to be more difficult.     

New pseudopolynomial complexity bounds for the bounded and other integer Knapsack related problems

We consider the bounded integer knapsack problem (BKP) , subject to: , and x_j{0,1,…,m_j},j=1,…,n. We use proximity results between the integer and the continuous versions to obtain an O(n³W²) algorithm for BKP, where W=max_j=1,…,nw_j. The respective complexity of the unbounded case with m_j=∞, for j=1,…,n, is O(n²W²). We use these results to obtain an improved strongly polynomial algorithm for the multicover problem with cyclical 1’s and uniform right-hand side.     

A note on "More Operator Versions of the Schwarz Inequality"

It is shown that for any (n + 1)-positive (possibly non-linear) map and any bounded linear operators A _i ,i = 1,¨,n we have [(A _i ^* A _j )] _{i,j = 1} ^*[(A _i )^*(A _j )] _{i,j = 1} ^*, and that the statement is false if "(n + 1)-positive" is replaced by "n-positive". This resolves an issue raised by Bhatia and Davis in relation to a Schwartz inequality which can be regarded as a non-commutative variance-covariance inequality [2]