One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA is characterized up to isomorphism by two invariants: its first-order theory Th(^ ) and its stan...

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see wheth...

As a consequence of an AM protocol for NISO, we get the following: Theorem 1. If Graph Isomorphism is NP-complete, then PH = Σp2. Proof. Suppose that Graph Isomorphism is NP-complete. Then the problem NISO is coNPcomplete. Hence, every coNP problem is polytime reducible to NISO, which, as we know, is in AM. We use this to get the following inclusions: Σp2 = ∃ ◦ ∀ ◦ P (definition of Σ p 2) ⊆ ∃ ◦...

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (Dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see wheth...

Different properties of rings and fields are discussed. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R −→ S we have R/ker(f) ∼= Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and h...

We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. When the quiver is of type Â0, we recover the results for gln. The proof of our Chevalley restriction theorem is similar to the proof for gln; however, the proof of the double analogue uses a theo...

The review presents an analysis of results the Abelian group theory, as well rings and modules, which concern definability algebraic structures by their endomorphism related structures. In systematization results, greatest attention is paid to torsion-free groups, are particular interest due presence non-isomorphic direct decompositions in this class. This significantly expands understanding ge...

For oriented $-1$ -shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and Joyce introduced certain perverse sheaves on them which can be regarded as sheaf-theoretic categorifications of the Donaldson–Thomas invariants. In this paper, we prove that hypercohomology above sheaf cotangent stack over a quasi-smooth is isomorphic to Borel–Moore homology base up shift degree. This global...

In general, the value groups and the residue elds do not suuce to classify the algebraic henselian extensions of a valued eld K, up to isomorphism over K. We deene a stronger, yet natural structure which carries information about additive and multiplica-tive congruences in the valued eld, extending the information carried by value groups and residue elds. We discuss the cases where these \mixed...