We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K0 be an arbitrary prime field and let r > 1 and s > 1 be integers that are coprime to one another. We show that one can find a finite extension K of K0, a degree-r extension L of K, a degr...

The factor refinement principle turns a partial factorization of integers (or polynomials) into a more complete factorization represented by basis elements and exponents, with basis elements that are pairwise coprime. There are lots of applications of this refinement technique such as simplifying systems of polynomial inequations and, more generally, speeding up certain algebraic algorithms by ...

The algebras $$Q_{n,k}(E,\tau )$$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin form a family quadratic parametrized coprime integers $$n>k\ge 1$$ , complex elliptic curve E, point $$\tau \in E$$ . main result in this paper is that has same Hilbert series polynomial ring on n variables when $$ not torsion point. We also show Koszul algebra, hence global dime...

For positive integers k; n let f(n; k) be the maximal cardinality of subsets of integers in the interval < 1; n > , which don't have k + 1 pairwise coprimes. The set E (n; k) of integers in < 1; n > , which are divisible by one of the rst k primes, certainly does not have k + 1 pairwise coprimes. Whereas we disproved in [1] an old conjecture of Erdos ([4], [5], [6], [7]) by showing that the eq...

Coprime array, which utilizes a coprime pair of uniform linear subarrays, is an attractive structure to achieve sparse array configurations. Alternatively, effective coprime array configurations can be implemented using a uniform linear array with two coprime sensing frequencies. This enables the integration of the coprime array and filter concepts to achieve high capabilities in meeting system...

We discuss the maximum number of distinct non-trivial solutions that a generalized Fermat equation Ax n + By n = Cz n might possibly have. The abc-conjecture implies that it can never have more than two solutions once n > n 0 (independent of A; B;C); and that it has no solutions for xed A; B; C once n > n A;B;C. On the other hand for any set of pairwise coprime integers p 1 ; p 2 ; :: : ; p k ,...

A bstract We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at around roots of unity $$ q={e}^{\frac{2\pi i}{K}} q = e 2 πi K <...

The article discusses the results of solving two classical problems physics and their consequences - problem radiation a hydrogen-like atom scattering by mathematical method integer theory. transition to formulation these in terms theory became possible as result hypothesis that, during interaction physical objects, exchange quantities that are conserved occurs such way have common measure befo...

This paper is concerned with the factorization approach to control systems. We consider the models in which some plants admit or do not admit coprime factorization. In the case where the plants admits only one-side coprime factorization, it is known that the plant with additional zeros admits both side coprime factorizations. However we show that in the case where the plant do not admit coprime...

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely fully coprime corings (fully coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate t...