Cellular Neural Networks (CNNs) are nonlinear dynamic array processors with mainly local interconnections. In most of the applications, the local interconnection pattern, called cloning template, is translation invariant. In this paper, an optimal ring-coding method for rotation invariant description of given set of objects, is introduced. The design methodology of the templates based on the ri...

Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depends only polynomially on the input data.

We give a polynomial bound on the spectral density function of a matrix over the complex group ring of Zd. It yields an explicit lower bound on the Novikov-Shubin invariant associated to this matrix showing in particular that the Novikov-Shubin invariant is larger than zero.

The link between Gröbner basis and linear algebra was described by Lazard [4,5] where he realized the Gröbner basis computation could be archived by applying Gaussian elimination over Macaulay’s matrix . In this paper, we indicate how same technique may be used to SAGBIGröbner basis computations in invariant rings. . Keywords— Gröbner basis, SAGBIGröbner basis, reduction, Invariant ring, permut...

We construct an invariant of t-structures on the derived category of a Noetherian ring. This invariant is complete when restricting to the category of quasi-coherent complexes, and also gives a classification of nullity classes with the same restriction. On the full derived category of Z we show that the class of distinct t-structures do not form a set.

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundam...

For any q which is a power of 2 we describe a finite subgroup of GLqðCÞ under which the complete weight enumerators of generalized doubly-even self-dual codes over Fq are invariant. An explicit description of the invariant ring and some applications to extremality of such codes are obtained in the case q 1⁄4 4: r 2003 Elsevier Inc. All rights reserved.

We report the fabrication and characterisation of the first graphene ring micro electrodes with the addition of a miniature concentric Ag/AgCl reference electrode. The graphene ring electrode is formed by dip coating fibre optics with graphene produced by a modified Hummers method. The reference electrode is formed using an established photocatalytically initiated electroless deposition (PIED) ...

Let A be a commutative Noetherian ring, and let R = A[x1, x2, . . .] be the polynomial ring in an infinite number of variables xi, indexed by the positive integers. Let S∞ be the symmetric group on an infinite number of letters {1, 2, 3, . . .}. The group S∞ gives a natural action on R, and this in turn gives R the structure of a left module over the (left) group ring RS∞. We prove that ideals ...

We introduce the finite ring S2m = Z2m + iZ2m . We develop a theory of self-dual codes over this ring and relate self-dual codes over this ring to complex unimodular lattices. We describe a theory of shadows for these codes and lattices. We construct a gray map from this ring to the ring Z2m and relate codes over these rings, giving special attention to the case when m = 2. We construct various...