Local probabilities and total variation distance for random permutations

We are concerned with permutations taken uniformly at random from the symmetric group. Firstly, we study the probability of a permutation missing short cycles. Secondly, the result is employed to establish a formula for total variation distance between the process of multiplicities of cycle lengths in a random permutation and a process of independent Poisson random variables. We apply an analytic approach originated in number theory (K. Gyory et al. (Eds.) in Number Theory in Progress, 1999).     

Dedekind Complete Commutative BCK-algebras

We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubík known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic to direct products of basic BCK-chains, generalizing a result of Cignoli for MV-algebras.     

An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions

The evolution Boussinesq equations describe the evolution of the temperature and velocity fields of viscous incompressible Newtonian fluids. Very often, they are a reasonable model to render relevant phenomena of flows in which the thermal effects play an essential role. In the paper we prescribe non-Dirichlet boundary conditions on a part of the boundary and prove the existence and uniqueness of solutions to the Boussinesq equations on a (short) time interval. The length of the time interval depends only on certain norms of the given data. In the proof we use a fixed point theorem method in Sobolev spaces with non-integer order derivatives. The proof is performed for Lipschitz domains and a wide class of data.     

Thermodynamics of shape-memory alloys under electric current

Phase transformation in shape-memory alloys is known to cause electric resistivity variation that, under electric current, may conversely influence Joule heat production and thus eventually the martensitic transformation itself. A thermodynamically consistent general continuum-mechanical model at large strains is presented. In special cases, a proof of the existence of a weak solution is outlined, using a semidiscretization in time.     

The isomorphism relation between tree-automatic Structures

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ₂¹-set nor a Π₂¹-set.