A Family of Semi-Regular Divisible Difference Sets

We give a construction of semi-regular divisible difference sets with parametersm = p2a(r–1)+2b (pr – 1)/(p – 1), n = pr, k = p(2a+1)(r–1)+2b (pr – 1)/(p – 1)1 = p(2a+1)(r–1)+2b (pr–1 – 1)/(p-1), 2 = p2(a+1)(r–1)–r+2b (pr – 1)/(p – 1)where p is a prime and r a + 1.     

The effect of rate of deformation on the mechanical characteristics of plastics

An experimental investigation of the effect of the rate of deformation on the strength and modulus of elasticity of vinyl plastic and glass-reinforced laminate is described. It is established that when the rate of relative tensile deformation of vinyl plastic at 25°C is reduced from 2000×10^–6 sec^–1 to 5×10^–6 sec^–1, and that for glass-reinforced laminate from 1000×10^–6 sec^–1 to 1.3×10^–6 sec^–1, the decrease in the modulus of elasticity is about 40% and the decrease in ultimate strength 30 and 48%, as the case may be.Mekhanika polimerov, Vol. 1, No. 1, pp. 76–81, 1965     

On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

On the growth of entire solutions of algebraic differential equations

This is the opening memorial plenary lecture on memorial meeting in honor of Professor Shlomo Strelitz on Conference of Differential Equations and Complex Analysis, University of Haifa, Israel, December 2000. Shlomo Strelitz: 7.1.1923–27.9.1999, teached at Vilnius University 1946–1973, professor 1967–1973.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 57–63, January–March, 2005.     

Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

Global Convergence of Conjugate Gradient Methods without Line Search

Global convergence results are derived for well-known conjugate gradient methods in which the line search step is replaced by a step whose length is determined by a formula. The results include the following cases: (1) The Fletcher–Reeves method, the Hestenes–Stiefel method, and the Dai–Yuan method applied to a strongly convex LC ¹ objective function; (2) The Polak–Ribière method and the Conjugate Descent method applied to a general, not necessarily convex, LC ¹ objective function.     

The Existence of Partitioned Generalized Balanced Tournament Designs with Block Size 3

A generalized balanced tournament design, GBTD(n, k), defined on a kn-set V, is an arrangement of the blocks of a (kn, k, k – 1)-BIBD defined on V into an n × (kn – 1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. Suppose we can partition the columns of a GBTD(n, k) into k + 1 sets B1, B2,..., Bk + 1 where |Bi| = n for i = 1, 2,..., k – 2, |Bi| = n–1 for i = k – 1, k and |Bk+1| = 1 such that (1) every element of V occurs precisely once in each row and column of Bi for i = 1, 2,..., k – 2, and (2) every element of V occurs precisely once in each row and column of Bi Bk+1 for i = k – 1 and i = k. Then the GBTD(n, k) is called partitioned and we denote the design by PGBTD(n, k). The spectrum of GBTD(n, 3) has been completely determined. In this paper, we determine the spectrum of PGBTD(n,3) with, at present, a fairly small number of exceptions for n. This result is then used to establish the existence of a class of Kirkman squares in diagonal form.     

Links Between Linear Bilevel and Mixed 0–1 Programming Problems

We study links between the linear bilevel and linear mixed 0–1 programming problems. A new reformulation of the linear mixed 0–1 programming problem into a linear bilevel programming one, which does not require the introduction of a large finite constant, is presented. We show that solving a linear mixed 0–1 problem by a classical branch-and-bound algorithm is equivalent in a strong sense to solving its bilevel reformulation by a bilevel branch-and-bound algorithm. The mixed 0–1 algorithm is embedded in the bilevel algorithm through the aforementioned reformulation; i.e., when applied to any mixed 0–1 instance and its bilevel reformulation, they generate sequences of subproblems which are identical via the reformulation.     

Radial Index and Euler Obstruction of a 1-Form on a Singular Variety

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A connection between the radial index and the local Euler obstruction of a 1-form is described. This gives an expression for the local Euler obstruction of the differential of a function in terms of Euler characteristics of some Milnor fibres.Mathematics Subject Classifications (2000). 14B05, 32S60, 58A10*Partially supported by the DFG-programme “Global methods in complex geometry” (Eb 102/4–2), grants RFBR–04–01–00762, NSh–1972.2003.1     

Unconstrained 0–1 nonlinear programming: A nondifferentiable approach

The purpose of this paper is to give new formulations for the unconstrained 0–1 nonlinear problem. The unconstrained 0–1 nonlinear problem is reduced to nonlinear continuous problems where the objective functions are piecewise linear. In the first formulation, the objective function is a difference of two convex functions while the other formulations lead to concave problems. It is shown that the concave problems we obtain have fewer integer local minima than has the classical concave formulation of the 0–1 unconstrained 0–1 nonlinear problem.     

Regularized Resolvent of Sums of Commuting Operators

For B ₁ and B ₂ commuting linear operators on a Banach space such that B ₁ generates a bounded strongly continuous semigroup and –B ₂ generates an exponentially decaying strongly continuous holomorphic semigroup, it is shown that (B ₁ – B ₂)^–1 B ₂ ^–r and (B ₁ – B ₂)^–1(–B ₁)^–r are bounded and everywhere defined, for any r > 0. Density of domains may also be removed. The results are applied to various abstract Cauchy problems.     

Application of soliton theory to the construction of pseudospherical surfaces in R3

This paper studies the geometry of pseudospherical surfaces from the point of view of Lorentz harmonic maps from the Minkowski plane into S². After giving appropriate definitions, it is shown that such a map is the Gauss map of a pseudospherical surface. A natural subclass of harmonic maps is isolated and studied using well developed techniques of soliton theory. Then follows a numerical investigation based on these techniques. Examples that fall outside of the aforementioned subclass are also considered.Supported by DFG grants Pi 158/2 – 1 & 158/2–2.Partially supported by DFG grants Pi 158/2–1 & 158/2–2.     

Superrigidity for homomorphisms into isometry groups of CAT(−1) spaces

CAT(–1) spaces are generalizations of manifolds with negative curvature. In this paper, we prove three types of rigidity results related to CAT(–1) spaces, namely the rigidity of the isometric actions on CAT(–1) spaces under the commensurability subgroups, the higher rank lattices and certain ergodic cocycles. The main idea for our approach relies on a study of the boundary theory we established for the general CAT(–1) spaces.     

Mixed Partitions of Projective Geometries

Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qⁿ–1)/(q–1) or (qⁿ–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety of PG(8,q) into caps of size q²+q+1 which are Veronese surfaces.     

M. G. Krein's ideas in the theory of orthogonal polynomials

This is a survey of M. G. Krein's ideas in the theory of orthogonal polynomials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 76–86, January–February, 1994.     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

M. G. Krein's investigations in the theory of entire and meromorphic functions and their further development

This is a survey of M. Krein's results in the theory of entire and meromorphic functions. Their further development is also discussed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 87–99, January–February, 1994.     

On the rigidity of super-Grassmannians

The rigidity of the complex super-Grassmannian Gr _m|n,k|l with 0<k<m, 0<l<n, supposing that (k, l) (1,n–1), (m–1, 1), (1,n–2), (m–2, 1), (2,n–1), (m–1, 2), is proved.Partially supported by Erwin Schrödinger International Institute for Mathematical Physics (Vienna, Austria)     

On a Commutative Class of Search Directions for Linear Programming over Symmetric Cones

The commutative class of search directions for semidefinite programming was first proposed by Monteiro and Zhang (Ref. 1). In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including linear programming, second-order cone programming, and semidefinite programming as special cases. Complexity results are established for short-step, semilong-step, and long-step algorithms. Then, we propose a subclass of the commutative class for which we can prove polynomial complexities of the interior-point method using semilong steps and long steps. This subclass still contains the Nesterov–Todd direction and the Helmberg–Rendl–Vanderbei–Wolkowicz/Kojima–Shindoh–Hara/Monteiro direction. An explicit formula to calculate any member of the class is also given.