in this paper, our purpose is twofold. firstly, the tensor andresiduum operations on $l-$nested systems are introduced under thecondition of complete residuated lattice. then we show that$l-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. thus the new representation theorem of$l-$subsets on complete re...

Based on Weitzenböck’s theorem and Nagata’s counterexample for Hilbert’s fourteenth problem we construct two finitely generated invariant rings R,S ⊂ K[x1, x2, . . . , xn] s.t. the intersection R ∩ S is not finitely generated as a K-algebra.

In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L-$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L-$subsets on complete re...

The Blaschke–Leichtweiss theorem (Leichtweiss in Abh Math Semin Univ Hambg 75:257–284, 2005) states that the smallest area convex domain of constant width w 2-dimensional spherical space $${\mathbb {S}}^2$$ is Reuleaux triangle for all $$0<w\le \frac{\pi }{2}$$ . In this paper we extend result to family wide r-disk domains , where $$0<r\le Here a an intersection disks radius r with centers cont...

‎In this article we introduce $mu$-filtered fuzzy module with a family of fuzzy submodules.  It shows the relation between $mu$-filtered fuzzy modules and crisp filtered modules by level sets. We investigate fuzzy topology on the $mu$-filtered fuzzy module and apply that to introduce fuzzy completion. Finally we extend Krull's intersection theorem of fuzzy ideals by using concept $mu$-adic comp...

Kirszbraun’s Theorem states that every Lipschitz map S → Rn, where S ⊆ Rm, has an extension to a Lipschitz map Rm → Rn with the same Lipschitz constant. Its proof relies on Helly’s Theorem: every family of compact subsets of Rn, having the property that each of its subfamilies consisting of at most n + 1 sets share a common point, has a non-empty intersection. We prove versions of these theorem...

The O’Nan-Scott theorem weakly classifies the maximal subgroups of the symmetric group S, providing some information about the lattice Λ of subgroups of S. We wish to obtain more information about Λ. We proceed by determining the intersection H of suitable pairs of maximal subgroups of S and the sublattice of overgroups of H in S.

The theta divisor Θ of the Jacobian variety of a complex curve X is best viewed as a divisor inside the component Pic(X) consisting of (isomorphism classes of) line bundles of degree g−1. Then a line bundle L belongs to Θ if and only if it has non-zero sections. Riemann proved that the multiplicity of Θ at a point L is equal to dimH(X;L) − 1. Kempf ([Ke]) obtained a geometric proof of Riemann’s...