We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). Two results (Cannon & Holt, J. Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces to the following: given groups G,H with characteristic subgroups of the same type and isomorphic to Zp, and given the coset of isomorphisms Iso(G/Zp, H/Z d p), comp...

Birkhoff’s theorem [1] asserts that a class of algebras of a given type τ : T → N is a variety if and only if it is closed under operation of passing to subalgebras, homomorphic images and also arbitrary direct products. Motivated by applications of the theory of automata Eilenberg and Schützenberger considered in [2], [3] classes of finite monoids closed under subalgebras, homomorphic images a...

Grossberg and VanDieren have started a program to develop a stability theory for tame classes (see [GrVa1]). We name some variants of tameness (Definitions 1.4 and 1.7) and prove the following. Theorem 0.1. Let K be an AEC with Löwenheim-Skolem number ≤ κ. Assume that K satisfies the amalgamation property and is κ-weakly tame and Galois-stable in κ. Then, K is Galois-stable in κ for all n < ω. ...

We say a tame Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that OL is a free OK [G]-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame Galois field extensio...

Grossberg and VanDieren have started a program to develop a stability theory for tame classes (see [GrVa1]). We name some variants of tameness (Definitions 1.4 and 1.7) and prove the following. Theorem 0.1. Let K be an AEC with Löwenheim-Skolem number ≤ κ. Assume that K satisfies the amalgamation property and is κ-weakly tame and Galois-stable in κ. Then, K is Galois-stable in κ for all n < ω. ...

Superlinear convergence of the Newton method for nonsmooth equations requires a “semismoothness” assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismo...

We reuaew the field of result-checking, dascussang simple checkers and self-correctors. We argue that such checkers could profitably be ancorporated an software as an aad to eficaent debuggang and relaable functaonalaty. We consader how to modafy tradataonal checkang methodologaes to make them more appropraate for use an real-tame, real-number computer systems In partacular, we suggest that che...