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Given M a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum {ō p (M)} pεℕ satisfies a Weyl law that was conjectured by Gromov.

A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.

We establish a general structure theorem for the singular part of 𝓐-free Radon measures, where 𝓐 is a linear PDE operator. By applying the theorem to suitably chosen differential operators 𝓐, we obtain a simple proof of Alberti's rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents. The latter result implies that the Rademacher theorem on the differentiability of Lipschitz functions can hold only for absolutely continuous measures and that every top-dimensional Ambrosio-Kirchheim metric current in ℝd is a Federer-Fleming flat chain.

The idea of convex sets and various related results in 2-Probabilistic normed spaces were established in [7]. In this paper, we obtain the concepts of convex series closedness, convex series compactness, boundedness and their interrelationships in Menger's 2-probabilistic normed space. Finally, the idea of 𝓓- Boundedness in Menger's 2-probabilistic normed spaces and Menger's Generalized 2-Probabilistic Normed spaces are discussed.

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V(N(x i — x j )), where x = (x₁,…,x N ) denotes the positions of the particles. Let H N denote the Hamiltonian of the system and let ψ N,t be the solution to the Schrödinger equation. Suppose that the initial data ψ N,0 satisfies the energy condition $\\langle \\psi _{N,0},H_{N}^{k}\\psi _{N,0}\\rangle \\leq C^{k}N^{k}$ for k = 1,2,…. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψ N,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψ N,0 is assumed in a stronger sense.

We strengthen Gabber's l'-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme X of finite type over a quasi-excellent threefold can be desingularized by a char(X)-alteration, i.e., an alteration whose order is only divisible by primes noninvertible on X. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of X can be split into a composition of a tame Galois alteration and a char(X)-alteration. The proof of the distillation theorem is based on the following tameness theorem that we deduce from a theorem of M. Pank: if a valued field k of residue characteristic p has no nontrivial p-extensions, then any algebraic extension l/k is tame.

The traditional view that proteins possess absolute functional specificity and a single, fixed structure conflicts with their marked ability to adapt and evolve new functions and structures. We consider an alternative, \"avant-garde view\" in which proteins are conformationally dynamic and exhibit functional promiscuity. We surmise that these properties are the foundation stones of protein evolvability; they facilitate the divergence of new functions within existing folds and the evolution of entirely new folds. Packing modes of proteins also affect their evolvability, and poorly packed, disordered, and conformationally diverse proteins may exhibit high evolvability. This dynamic view of protein structure, function, and evolvability is extrapolated to describe hypothetical scenarios for the evolution of the early proteins and future research directions in the area of protein dynamism and evolution.

We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.

We study the Navier-Stokes equations governing the motion of an isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density. We establish existence of a so-called finite energy weak martingale solution under the condition that the adiabatic exponent satisfies $\\gamma \\textgreater \\frac{3}{2}$. The proof is based on a four-layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.

Consider a system of NN bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj))N^2V(N(x_i-x_j)), where x=(x1,…,xN)\\mathbf {x}=(x_1, \\ldots , x_N) denotes the positions of the particles. Let HNH_N denote the Hamiltonian of the system and let ψN,t\\psi _{N,t} be the solution to the Schrödinger equation. Suppose that the initial data ψN,0\\psi _{N,0} satisfies the energy condition \\[ ⟨ψN,0,HNψN,0⟩≤CN\\langle \\psi _{N,0}, H_N \\psi _{N,0} \\rangle \\leq C N \\] and that the one-particle density matrix converges to a projection as N→∞N \\to \\infty. Then, we prove that the kk-particle density matrices of ψN,t\\psi _{N,t} factorize in the limit N→∞N \\to \\infty. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential VV. In a recent paper, we proved the same statement under the condition that the interaction potential VV is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.