Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal κ. We show the consistency of E ++,λ++ λ-club , the relation of equivalence modulo the non-stationary ideal restricted to Sλ ++ λ in the space (λ++)λ ++ , being continuously reducible to E ++ λ+-club , the relation of equi...

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated monomials with leading coefficients 1. Then we constructed space~$X_A$ such that is isomorphic to $H^*(X_A)$ modulo torsion elements.

This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R : g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (Karush-KuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x) > 0 on S; further...

Finding the key of symmetric cipher takes computing common zero of polynomials, which de ne ideal and corresponding variety, usually considered over algebraically closed eld. The solution is the point of the variety over prime eld; it is de ned by a sum of the polynomial ideal and the eld ideal that de nes prime eld. Some authors use partitioning of this sum and reducing syzygies of polynomial ...

This paper studies the representation of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its KKT (Karush-KuhnTucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K; we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if f ≥ 0 on K.

A complete classification is obtained for all second-order linear recurring sequences uniformly distributed modulo an ideal of a Dedekind domain.

Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...