The weight of a code is the number of coordinate positions where no codeword is zero. The th minimum weight is the least weight of an -dimensional subcode. Wei and Yang gave a conjecture about the minimum weights for some product codes. In this correspondence, we will find a relation between product codes and the Segre embedding of a pair of projective systems, and we use this to prove the conj...

Given a commutative ring R with identity 1?0, let the set Z(R) denote of zero-divisors and Z*(R)=Z(R)?{0} be non-zero R. The zero-divisor graph R, denoted by ?(R), is simple whose vertex Z*(R) each pair vertices in are adjacent when their product 0. In this article, we find structure Laplacian spectrum graphs ?(Zn) for n=pN1qN2, where p<q primes N1,N2 positive integers.

We study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., ...

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on Rn as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof i...

Let G be a finite product of SL(2,Ki)′s for local fields Ki of characteristic zero. We present a discreteness criterion for non-solvable subgroups of G containing an irreducible lattice of a maximal unipotent subgroup of G. In particular such a subgroup has to be arithmetic. This extends a previous result of A. Selberg when G is a product of SL2(R)′s.

Arithmetic convolution rings provide a general and unified treatment of many rings that have been called arithmetic; the best known examples are rings of complex valued functions with domain in the set of non-negative integers and multiplication the Cauchy product or the Dirichlet product. The emphasis here is on factorization and related properties of such rings which necessitates prior result...

Single mode optical fibers having minimum non-zero dispersion over the spectral windows of 1300 and 1550 nm region are essential for the multi-channel operation of dense wavelength multiplexed systems and minimizing the nonlinear four wave mixing effects This paper demonstrates that nonzero dispersion flattening of the dispersion factor extends over a wavelength range of more than 350 nm is ach...

This paper analyses the policy implications of licensing between producers of differentiated goods. We consider and compare two-part tariff, fixed fee, royalty and collusive licensing contracts. Under the optimal licensing policy, there will be no technology transfers if the innovation size is sufficiently small and degree of product differentiation is sufficiently low. On the other hand, licen...

(a) Since A is full-rank and fat, Ax = b has at least one solution. From Problem 4 in HW#4, we know that since C is full-rank and tall, 〈x, y〉C := (Cx) TCy defines a valid inner product. So the problem reduces to one of finding the least-norm solution to Ax = b w.r.t. this inner product. Clearly, the required solution will be found by projecting the zero vector on the affine subspace defined by...