For a numerical semigroup ring K[H] we study the trace of its canonical ideal. The colength this ideal is called residue H. This invariant measures how far H from being symmetric, i.e. Gorenstein ring. We remark that contains conductor ideal, and bounds for residue. 3-generated semigroups give explicit formulas Thus, in setting can classify those whose at most one (the nearly ones), show eventu...

Elementary information about convolutional codes over finite fields is introduced, and various motivations for extension to convolutional codes over finite rings are discussed. The recent primary motivation is found to be for use over phase modulation signals. Such ring codes enjoy the special property of phase weight equal to phase distance equal to the squared Euclidean distance between phase...

Let n > 1 be an integer and let Bn denote the hyperoctahedral group of rank n. The group Bn acts on the polynomial ring Q[x1, . . . , xn, y1, . . . , yn] by signed permutations simultaneously on both of the sets of variables x1, . . . , xn and y1, . . . , yn. The invariant ring MBn := Q[x1, . . . , xn, y1, . . . , yn]n is the ring of diagonally signedsymmetric polynomials. In this article, we p...

Let X be a nonsingular algebraic variety in characteristic zero. To an eeective divisor on X Kontsevich has associated a certain`motivic integral', living in a completion of the Grotendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi{Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igus...

Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain motivic integral, living in a completion of the Grotendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi–Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igus...

Let G be a group acting via ring automorphisms on an integral domain R. A ring-theoretic property of R is said to G-invariant, if $$R^G$$ also has the property, where $$R^G=\{r\in \ | \sigma (r)=r \text {for all} \in G\},$$ fixed action. In this paper we prove following classes rings are invariant under operation $$R\rightarrow R^G:$$ locally pqr domains, Strong G-domains, Hilbert rings, S-stro...

Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain ‘motivic integral’, living in a completion of the Grotendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi–Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Ig...

Let KV ] G be the invariant ring of a nite linear group G GL(V), and let GU be the pointwise stabilizer of a subspace U V. We prove that the following numbers associated to the invariant ring decrease if one passes from KV ] G to KV ] G U : the minimal number of homogeneous generators, the maximal degree of the generators, the number of syzygies and other Betti numbers, the complete intersectio...

Let K[V ] be the invariant ring of a finite linear group G ≤ GL(V ), and let GU be the pointwise stabilizer of a subspace U ≤ V . We prove that the following numbers associated to the invariant ring do not increase if one passes from K[V ] to K[V ]U : the minimal number of homogeneous generators, the maximal degree of the generators, the number of syzygies and other Betti numbers, the complete ...