We study the three dimensional quantum many-body dynamics with a delta-type potential N^3β V(N ^β x) and a Coulomb potential. As the particle number N tends to infinity and the Planck’s constant ħ tends to zero independently, we prove the weak convergence of the quantum mass and momentum densities to the Euler–Poisson equation with the pressure before its blow-up time. The proof is based on the modulated energy method in the setting of the quantum many-body dynamics, for which the key is a quantum functional inequality according to the interaction potentials. In Golse–Paul [Comm. Pure Appl. Math., 2022, 75(6): 1332–1376], the functional inequality for the Coulomb potential is achieved based on Serfaty’s inequality [Duke Math. J., 2020, 169(15): 2887–2935]. In the mean-field regime

, we prove the quantum functional inequality for the delta-type potential under technical profile assumptions on the potential V(x).

Ahlfors’ Second Fundamental Theorem for the simply connected surfaces over the Riemann sphere S states that for any set E _q of distinct q points on S with q ≥ 3, there exists a constant h = h(E _q), such that for any covering surface

, one has where U is a Jordan domain in

\mathbb{C} , f : \overline{U} \to S

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb C}, f : {\overline U} \rightarrow S$$\end{document}

is an orientation-preserving, continuous, open and finite-to-one mapping, A(Σ) is the spherical area of Σ, L(∂Σ) is the spherical length of the boundary of Σ and

\overline{n} ( \mathrm{\Sigma } , E_q ) = \mathrm{\#} [ f^{-1} ( E_q ) \cap U ]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline n}(\Sigma, \ E_{q})=\#[f^{-1}(E_{q}) \ \cap U]$$\end{document}

means the cardinality of the set f⁻¹ (E _q) ∩ U. We denote by F the space of all simply covering surfaces over the sphere, the above pairs

( f , \overline{U} )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(f, \ {\overline U})$$\end{document}

, and write F (L) = {Σ ∈ F: L(∂Σ) ≤ L}. In a preprint by the third author Zhang of this paper, the precised bound of h is identified (see arXiv.2307.04623). The first key step of Zhang’s method is to prove the existence of extremal surfaces of the subspace

\mathcal{F} ( L , m )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal F}(L, \ m)$$\end{document}

of F (L), and Zhang asserted without proof in that paper that the extremal surface of

\mathcal{F} ( L , m )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal F}(L, \ m)$$\end{document}

can be found in the smaller subspace

{\mathcal{F}}_r ( L , m )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\cal F}_{r}}(L, \ m)$$\end{document}

such that the defining function f of each surface of the form

( f , \overline{\mathrm{\Delta }} )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(f, {\overline \Delta})$$\end{document}

in

{\mathcal{F}}_r ( L , m )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\cal F}_{r}}(L, \ m)$$\end{document}

has no branch value outside E _q. In this paper, we prove this assertion.

For any positive integer n, let σ(n) be the sum of all positive divisors of n. For any prime p and any positive integer m, let ν _p(m) be the largest integer α such that p ^α ∣ m. Recently, Amdeberhan, Moll, Sharma and Villamizar proved that for any odd prime p and any integer n ≥ 2, ν _p(σ(n) ≤ [log _p n] if n satisfies some conditions, where [x] denotes the least integer not less than x. In this paper, for any odd prime p, we prove that ν _p(σ(n) ≤ [log _p n] for all positive integers n unconditionally. Moreover, we prove that if 3 ≤ p < 10⁵ is a prime and p ≠ 31, then there are only finitely many positive integers n such that ν _p(σ(n) = [log _p n]. For p = 31 and n ≥ 2, ν _p(σ(n) = [log _p n] if and only if n = 2⁴, 5², 2⁴5² and 2⁴5²(2 · 31 ^s − 1), where s is a positive integer and 2 · 31 ^s − 1 is a prime.

Using the saddle point theorem and the mountain pass theorem with Morse index estimate, the existence of periodic solutions for second-order Hamiltonian systems with mild superquadratic growth is proved in this paper. Meanwhile the existence result of homoclinic orbits for this kind of Hamiltonian systems is also obtained by the local convergence of a sequence of subharmonic solutions.

In this paper, we compute the rational cohomology groups of the classifying space of a simply connected Kac–Moody group of infinite type. The fundamental principle is “from finite to infinite”. That is, for a Kac–Moody group G(A) of infinite type, the input data for computation are the rational cohomology of classifying spaces of parabolic subgroups of G(A)(which are of finite type), and the homomorphisms induced by inclusions of these subgroups. In some special cases, we can further determine the cohomology rings. Our method also applies to study the mod p cohomology of the classifying spaces of Kac–Moody groups.

Let

, where

P = {\mathbb{F}}_p [ {\xi }_1 , {\xi }_2 , {\xi }_3 , \dots ]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P=\mathbb{F}_{p}[\xi_{1},\xi_{2},\xi_{3},\ldots]$$\end{document}

is the polynomial part of the dual Steenrod algebra and p is an odd prime. In this paper we calculate the cohomology of P(2) in dimensions less than 4 by a May spectral sequence.

The work concerns multivalued McKean–Vlasov stochastic differential equations. First of all, we prove the existence and uniqueness of strong solutions for multivalued McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients. Then, the classical Itô’s formula is extended to that for multivalued McKean–Vlasov stochastic differential equations. Finally, the asymptotic stability of second moments and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function are shown.

A birth-death process (abbr. BDP) on trees is a time-continuous and homogeneous random walk in which the transition rate from any state to its father is called death rate and the ones to its offspring are called birth rates. In this paper, we obtain explicit uniqueness and recurrence criteria for BDPs on trees. Meanwhile, we also get an explicit and recursive representation for moments of integral-type functionals for this process. We then study the uniqueness and recurrence for some specific examples of BDPs on trees and apply integral-type functionals’ results to more general integral-type functionals. Moreover, polynomial ergodicity is derived and a sufficient condition for a central limit theorem is also obtained.