For any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. The condition is that every minimal ideal of the ring consist entirely of bounded elements. It is not too stringent, and is satisfied, for instance, by rings of ...

We consider in dimension four weakly convergent sequences of approximate biharmonic maps into sphere with bi-tension fields bounded in L for some p > 1. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on R.

We combine the fundamental results of Breuillard, Green, and Tao (Publ Math Inst Hautes Études Sci 116:115–221, 2012) on structure approximate groups, together with “tame” arithmetic regularity methods based work authors Terry (J Eur Soc (JEMS) 24(2):583–621, 2022), to give a theorem for finite subsets A arbitrary groups G where has “small tripling” bounded VC-dimension: Roughly speaking, up sm...

let s be a locally compact topological foundation semigroup with identity and ma(s) be its semigroup algebra. in this paper, we give necessary and sufficient conditions to have abounded approximate identity in closed codimension one ideals of the semigroup algebra $m_a(s)$ of a locally compact topological foundationsemigroup with identity.

Let A be a Banach algebra with bounded approximate identity by 1. Two new topologies ?so and ?wo are introduced on A. We study these compare them each other the norm topology. The properties of then studied further we pay attention to group L1(G) locally compact G. Various necessary sufficient conditions found for G finite.

Two distributions over n-bit strings are (k, δ)-wise indistinguishable if no statistical test that observes k of the n bits can tell the two distributions apart with advantage better than δ. Motivated by secret sharing and cryptographic leakage resilience, we study the existence of pairs of distributions that are (k, δ)-wise indistinguishable, but can be distinguished by some function f of suit...

We consider weakly convergent sequences of approximate H-surface maps defined in the plane with their tension fields bounded in Lp for p > 4/3, and establish an energy quantization that accounts for the loss of their energies by the sum of energies over finitely many nontrivial bubbles maps on R2. As a direct consequence, we establish the energy identity at finite singular time to their H-surfa...