Possibility and necessity measures and integral equivalence

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

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Homotopy types of gauge groups of $$\mathrm {PU}(p)$$-bundles over spheres

Journal of Homotopy and Related Structures ◽

10.1007/s40062-020-00274-0 ◽

2021 ◽

Vol 16 (1) ◽

pp. 61-74

Author(s):

Simon Rea

Keyword(s):

Gauge Groups ◽

Homotopy Types ◽

Special Cases ◽

Integral Equivalence ◽

Local Equivalence

AbstractWe examine the relation between the gauge groups of $$\mathrm {SU}(n)$$ SU ( n ) - and $$\mathrm {PU}(n)$$ PU ( n ) -bundles over $$S^{2i}$$ S 2 i , with $$2\le i\le n$$ 2 ≤ i ≤ n , particularly when n is a prime. As special cases, for $$\mathrm {PU}(5)$$ PU ( 5 ) -bundles over $$S^4$$ S 4 , we show that there is a rational or p-local equivalence $$\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}$$ G 2 , k ≃ ( p ) G 2 , l for any prime p if, and only if, $$(120,k)=(120,l)$$ ( 120 , k ) = ( 120 , l ) , while for $$\mathrm {PU}(3)$$ PU ( 3 ) -bundles over $$S^6$$ S 6 there is an integral equivalence $$\mathcal {G}_{3,k}\simeq \mathcal {G}_{3,l}$$ G 3 , k ≃ G 3 , l if, and only if, $$(120,k)=(120,l)$$ ( 120 , k ) = ( 120 , l ) .

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An identification system based on the explicit isomorphism problem

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00529-0 ◽

2021 ◽

Author(s):

Sándor Z. Kiss ◽

Péter Kutas

Keyword(s):

Quadratic Forms ◽

Isomorphism Problem ◽

Identification System ◽

Division Algebras ◽

Central Simple Algebras ◽

Zero Knowledge ◽

Algorithmic Problems ◽

Simple Algebras ◽

Central Simple ◽

Integral Equivalence

AbstractWe propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

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