Stronger arithmetic equivalence

This paper considers two alternative strengthenings of the notion of arithmetic equivalence, which the author calls local integral equivalence and solvable equivalence. (The latter turns out to be strictly stronger than the former.) They have the advantage of being easier to check than Prasad’s notion, which the author calls integral equivalence. Furthermore, solvable equivalence, which the author shows does not imply integral equivalence, is nevertheless a sufficient condition to imply that the invariants considered by Prasad are equal. This opens the door to easier proofs of Prasad’s result, and lessens the reliance on Scott’s construction: the author finds a generalization of this construction that yields infinitely many examples of solvable equivalence. The paper also contains several examples to clarify the relationships between the various different notions of equivalence. Some of these examples (which are mainly found with the help of a computer) answer open questions from the group theory literature.

An identification system based on the explicit isomorphism problem

We 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.

Crack Growth Simulation of Functionally Graded Materials Based on Improved Bond-Based Peridynamic Model

Functionally graded materials (FGMs) are widely used in the aerospace industry, especially for the thermal protection shields of aerospace vehicles. Studies show that the initiation and expansion of micro-cracks are important factors that adversely affect the service life of these shields. Based on the peridynamic theory of bonds, an improved peridynamic model is proposed in the present study for FGMs. In the proposed model, integral equivalence is applied to calculate the required material parameters. Obtained results reveal that this method can better reflect the gradient change of material properties.

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

We 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 ) .

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

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].

Lefschetz numbers and unitary groups

We give a formula for the Euler-Poincare characteristic of the fixed point set of the Cartan involution on the set of integral equivalence classes of integral unimodular hermitian forms, in terms of a product of special values of Riemann zeta functions and Dirichlet L-functions. This is done via reduction by Galois cohomology to a volume computation using the Tamagawa measure on the unitary groups.