Properties of Codes from Difference Sets in 2-Groups

A ( v, k, λ)-difference set D in a group G can be used to create a symmetric 2-( v, k, λ) design, , from which arises a code C, generated by vectors corresponding to the characteristic function of blocks of . This paper examines properties of the code C, and of a subcode, C ^o=JC, where J is the radical of the group algebra of G over . When G is a 2-group, it is shown that C^o is equivalent to the first-order Reed-Muller code, , precisely when the 2-divisor of C^o is maximal. In addition, ifD is a non-trivial difference set in an elementary abelian 2-group, and if D is generated by a quadratic bent function, then C^o is equal to a power of the radical. Finally, an example is given of a difference set whose characteristic function is not quadratic, although the 2-divisor of C^o is maximal.     

Constructions of Nested Partial Difference Sets with Galois Rings

There have been several recent constructions of partial difference sets (PDSs) using the Galois rings for p a prime and t any positive integer. This paper presents constructions of partial difference sets in where p is any prime, and r and t are any positive integers. For the case where 2$$ " align="middle" border="0"> many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring , and in particular, the ring × . The paper concludes with some open related problems.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Hermitian and Positive Integrated C-cosine Functions on Banach Spaces

Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on L ^p () (1L ¹(), C₀ , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.     

Local Asymptotic Normality for Linear Homogeneous Difference Equations with Non-Gaussian Noise

This paper considers the problem of estimation of drift parameter for linear homogeneous stochastic difference equations. The Local Asymptotic Normality (LAN) for the problem is proved. LAN implies the Hajek–Le Cam minimax lower bound. In particular, it is shown that the Fisher's information matrix for the problem can be expressed in terms of the stationary distribution of an auxiliary Markov chain on the projective space P(^d).     

The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx S such that the set of ally R ⁿ satisfyingp(y–x) r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ . In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.     

The norm-strict bidual of a Banach algebra and the dual of Cu(G)

To each Banach algebra A we associate a (generally) larger Banach algebra A⁺ which is a quotient of its bidual A. It can be constructed using the strict topology on A and the Arens product on A. A⁺ has certain more pleasant properties than A, e.g. if A has a bounded right approximate identity, then A⁺ has a two-sided unit. In the special case A=L¹(G) (G a locally compact abelian group) one gets A⁺=C_u(G), the dual of the space of bounded, uniformly continuous functions on G, and we show that the center of the convolution algebra C_u(G) is precisely the space M(G) of finite measures on G.     

Geometrical Aspects of Hilbert C*-modules

The aim of the present paper is to solve some major open problems of Hilbert C^*-module theory by applying various aspects of multiplier C^*-theory. The key result is the equivalence established between positive invertible quasi-multipliers of the C^*-algebra of compact operators on a Hilbert C^*-module {, ., } and A-valued inner products on , inducing an equivalent norm to the given one. The problem of unitary isomorphism of C^*-valued inner products on a Hilbert C^*-module is considered and new criteria are formulated. Countably generated Hilbert C^*-modules turn out to be unitarily isomorphic if they are isomorphic as Banach C^*-modules. The property of bounded module operators on Hilbert C^*-modules of being compact and/or adjointable is unambiguously connected to operators with respect to any choice of the C^*-valued inner product on a fixed Hilbert C^*-module if every bounded module operator possesses an adjoint operator on the module. Every bounded module operator on a given full Hilbert C^*-module turns out to be adjointable if the Hilbert C^*-module is orthogonally complementary. Moreover, if the unit ball of the Hilbert C^*-module is complete with respect to a certain locally convex topology, then these two properties are shown to be equivalent to self-duality.     

Looking for Difference Sets in Groups with Dihedral Images

We prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p with p an odd prime, contains a nontrivial (v, k, ) difference set D with order n = k – prime to p and self-conjugate modulo p. If G has an image of order p, then 0 2a + 2 for a unique choice of = ±1, and for a = (k – )/2p. If G has an image of order 2p, then and ( – 1)/( – 1). There are further constraints on n, a and . We give examples in which these theorems imply no difference set can exist in a group of a specified order, including filling in some entries in Smith's extension to nonabelian groups of Lander's tables. A similar theorem covers the case when p|n. Finally, we show that if G contains a nontrivial (v, k, ) difference set D and has a dihedral image D _2m with either (n, m) = 1 or m = p ^t for p an odd prime dividing n, then one of the C ₂ intersection numbers of D is divisible by m. Again, this gives some non-existence results.     

On the Relation between Stability of Continuous- and Discrete-Time Evolution Equations via the Cayley Transform

In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence where A_d = (I + A) (I − A)⁻¹. We show that if A and A⁻¹ generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A_d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A_d is equivalent to the uniform boundedness of the semigroup generated by A.     

Hadamard and Conference Matrices

We discuss new constructions of Hadamard and conference matrices using relative difference sets. We present the first example of a relative -difference set where n – 1 is not a prime power.     

On the inverse of the generator of a bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let A be the generator of a uniformly bounded C ₀-semigroup in a Banach space B, and let A have a densely defined inverse A ^-1 . We present sufficient conditions on the resolvent (A-I ^-1, Re > 0, under which A ^-1 is also the generator of a uniformly bounded C ₀-semigroup.Dedicated to V. B. Lidskii on the occasion of his eightieth birthdayTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 6–12, 2004Original Russian Text Copyright © by A. M. Gomilko     

Homomorphism-Homogeneous Partially Ordered Sets

A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we characterize homomorphism-homogeneous partially ordered sets (where a homomorphism between partially ordered sets A and B is a mapping f : A →B satisfying ). We show that there are five types of homomorphism-homogeneous partially ordered sets: partially ordered sets whose connected components are chains; trees; dual trees; partially ordered sets which split into a tree and a dual tree; and X ₅-dense locally bounded partially ordered sets. Supported by the Ministry od Science and Environmental Protection of the Republic of Serbia, Grant No. 144017.     

The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A ^d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

Antiproximinal sets in spaces of continuous functions

Closed convex bounded antiproximinal bodies are constructed in the infinite-dimensional spacesC(Q), C ₀(T), L(S, S, ), andB(S), whereQ is a topological space andT is a locally compact Hausdorff space. It is shown that there are no closed bounded antiproximinal sets in Banach spaces with the Radon-Nikodym property.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 643–657, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00196.     

Computational Results for Regular Difference Systems of Sets Attaining or Being Close to the Levenshtein Bound

Difference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of m‐subsets in a finite field of prime order to be a regular DSS for an integer m, and give a lower bound on the parameter ρ of the DSS using cyclotomic numbers. We show that when we choose ‐subsets from the multiplicative group of order e, the lower bound on ρ is independent of the choice of subsets. In addition, we present some computational results for DSSs with block sizes and , whose parameter ρ attains or comes close to the Levenshtein bound for .     

Solution of elementary equations in the Minkowski geometric algebra of complex sets

The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks and with radii a and b, a solution of the linear equation in an unknown set exists if and only if ab. When it exists, the solution is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limaçon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a<b<1, the solution of the nonlinear monomial equation is shown to be the region that is bounded by a single loop of a generalized form of the ovals of Cassini. The latter result is obtained by considering the nth Minkowski root of the region bounded by the inner loop of a Cartesian oval. Preliminary consideration is also given to the problems of solving univariate polynomial equations and multivariate linear equations with complex disk coefficients.     

Quadratic Inequalities for Hilbert Space Operators

Properties of sets of solutions to inequalities of the form

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.