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...

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV )), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP , APP , MA, AM ) in PV1 + dWPHP(PV ).

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 Ω ⊂ R be a bounded open set. Given 2 ≤ m ≤ n, we construct a convex function φ : Ω → R whose gradient f = ∇φ is a Hölder continuous homeomorphism, f is the identity on ∂Ω, the derivative Df has rank m − 1 a.e. in Ω and Df is in the weak L space L. The proof is based on convex integration and staircase laminates.