Gamma—rings and multiplications on abelian groups

If M and Γ are abelian groups, then M will be a Γ-ring iff there exists a group homomorphism f from Γ into the group of all multiplications of M, Mult(M), such that f(Γ) satisfies the Generalized Associativity Property on M. In this note we examine the following special cases of this result: (i) M is a Γ-ring satisfying the Nobusawa Condition, (ii) M is a cyclic group, (iii) M is a direct sum of cyclic groups and (iv) M is a Γ-ring that has unity elements.     

On ω-categorical,generically stable groups and rings

We prove that every ω-categorical, generically stable group is nilpotent-by-finite and that every ω-categorical, generically stable ring is nilpotent-by-finite.     

Algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor

Algebraic Ricci solitons on Lie groups with left-invariant (pseudo)Riemannian metric and zero Schouten–Weyl tensor are studied. The absence of nontrivial algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor and diagonalizable Ricci operator is proved.     

Isomorphisms of stable linear groups over associative rings containing ½

Stable linear groups over arbitrary associative rings with ½ are considered. The action of an isomorphism between these groups on an elementary stable subgroup is described.     

Normal zeta functions of the Heisenberg groups over number rings II — the non-split case

We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(O_K), where O_K is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local zeta functions satisfy functional equations upon inversion of the prime.     

∃-Free groups as groups with length function

It is shown that there exists a length function with values in a finitely generated group relative to which G is a -free group in any finitely generated group G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 813–822, June, 1992.     

Finite groups with given σ-embedded and σ-n-embedded subgroups

Let G be a finite group and σ = {σ _i |i∈I} be a partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ _i -subgroup of G and H contains exactly one Hall σ _i -subgroup of G for every σ _i ∈ σ(G). A subgroup H is said to be σ-permutable if G possesses a complete Hall σ-set H such that HA ^x = A ^x H for all A ∈ H and all x ∈ G. Let H be a subgroup of G. Then we say that: (1) H is σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H ^σG and H ∩ T ≤ H _σG , where H _σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G, and H ^σG is the σ-permutable closure of H, that is, the intersection of all σ-permutable subgroups of G containing H. (2) H is σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H ^G and H ∩ T ≤ H _σG . In this paper, we study the properties of the new embedding subgroups and use them to determine the structure of finite groups.     

Range projections and the Moore–Penrose inverse in rings with involution

In this article we study the existence of range projections in rings with involution, relating it to the existence of the Moore–Penrose inverse. The results are applied to the solution of the equation xbx?=?x in rings with involution, extending the results of Greville for matrices. Simpler new proofs are given of the Moore–Penrose invertibility of regular elements in rings with involution, and of the Ljance's formula.     

Infinite dimensional linear groups with many G — invariant subspaces

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.     

A linear quadratic optimal control problem with disturbances—An algebraic Riccati equation and differential games approach

A linear quadratic optimal control problem with coexisting initial and persistent disturbances is studied. Upper and lower values and relevant algebraic Riccati equations (ARE for short) are introduced. Various relations among these values are presented. The solvability of the resulting AREs is shown to be closely related to the solvability of the original optimal control problem. A formula is obtained for the solution of one of the AREs, which is of nonstandard form. Several unexpected features of the original problem are revealed from the standpoint of differential games. Some known results on the so-calledH optimal control problem are recovered.This work was partially supported by the Chinese NSF under Grant 19131050, the Chinese State Education Commission Science Foundation, the SEDC Foundation for Young Academics, and the Fok Ying Tung Education Foundation.     

Evolutionary prisoner’s dilemma games with local interaction and best-response dynamics

This paper studies the long run behavior in evolutionary prisoner’s dilemma games. All players are assumed to sit around a circle and to interact only with their neighbors. It is known that full-defection is the unique long run equilibrium as the probability of players’ experimentation (or mutation) tends to zero in the best response dynamics. Here, it is shown that full-cooperation could emerge in the long run if one also cares for his neighbors in the bestresponse dynamics.     

Forward–backward stochastic differential equations with monotone functionals and mean field games with common noise

We consider a system of forward–backward stochastic differential equations (FBSDEs) with monotone functionals. We show that such a system is well-posed by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. Lastly, we show that mean field games with common noise are uniquely solvable under a linear-convex setting and weak-monotone cost functions and prove that the optimal control is in a feedback form depending only on the current state and conditional law.