Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences

Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n｜Sk｜,n≥1.Suppose limn→∞ESn^2/n=：σ^2〉0 and ∑n^∞=1 ρ^2/d（2^n）〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E｜X｜^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2（r-p）/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1（-1）^k/（2k＋1）^2（r-p）/（2-p）E｜Z｜^2（r-p）/2-p,where Z has a normal distribution with mean 0 and variance σ^2.     

On a problem of Erdös and Rothschild on edges in triangles

Erdös and Rothschild asked to estimate the maximum number, denoted by h(n, c), such that every n-vertex graph with at least cn ² edges, each of which is contained in at least one triangle, must contain an edge that is in at least h(n, c) triangles. In particular, Erdös asked in 1987 to determine whether for every c > 0 there is ε > 0 such that h(n,c) > n ^ε for all sufficiently large n. We prove that h(n,c) = n ^O(1/loglogn) for every fixed c < 1/4. This gives a negative answer to the question of Erd?s, and is best possible in terms of the range for c, as it is known that every n-vertex graph with more than n ²/4 edges contains an edge that is in at least n/6 triangles.     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.     

On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem

Letp be any odd prime number. Letk be any positive integer such that . LetS = (a ₁,a ₂,...,a _2p−k ) be any sequence in ℤ_p such that there is no subsequence of lengthp of S whose sum is zero in ℤ_p. Then we prove that we can arrange the sequence S as follows:

On Filters of Double MS—algebras

51.IntroductionThestudyofOckhamalgebraswasinitiatedbyJ-Berman[1j.AnOckhamalgebrasisanalgebra(L;V,A,O,1)oftype(2,2,1,o,o)where(L;V,AlO,1)isadistributivelatticeandfisadualendormorphism;thatis,theequationsf(o)=lf(1)=of(xVY)=f(x)Af(y),f(xAy)=f(x)Vf(x)holdidentically.AnMS-algebraisanOckhama1gebra(LiV,A,', ,o,1)whichsatisfiesthecondition:(aeL)aSa'-AnOckhamalgebrawhichsatifiesthecondictiona2P(a)willbecalledadualMS-algebraandtheunaryoperationdenotedby .AdoubleMS-algebraisanalgebra(L;V,A,…     

On a Generalization of Some Restrictive Distribution Theorems

ＯｎａＧｅｎｅｒａｌｉｚａｔｉｏｎｏｆＳｏｍｅＲｅｓｔｒｉｃｔｉｖｅＤｉｓｔｒｉｂｕｔｉｏｎ　ＴｈｅｏｒｅｍｓＴｉａｎＺｈｅｎ（田震）；ＺｈａｎｇＳｏｎｇ（张松）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ）（Ｎａ...     

On the Density of Transmission Eigenvalue for the Schrödinger Operator with the Robin Boundary Condition北大核心CSCD

In this paper, we study the transmission eigenvalue problem with the Robin boundary condition. Applying the related properties of entire function of exponential type, we show the relationship between the density of eigenvalues and the length of the support interval of the potential function. Meanwhile, we prove that the transmission eigenvalue problem is equivalent to a kind of Sturm–Liouville problem with spectral parameter in the boundary condition. © 2022 Chinese Academy of Sciences. All rights reserved.     

On the Weak＊ Drop Property for Polar of Closed Bounded Convex Sets

We define and study the weak* drop property for the polar of a closed bounded convex set in a Banach space which is both a generalization of the weak* drop property for dual norm in a Banach space and a characterization of the sub-differentialmapping x→эp(x) from S(X) into 2^S(X*) that is norm upper semi-continuous and norm compact-valued.     

Asymptotic Results for Tail Probabilities of Sums of Dependent and Heavy-Tailed Random Variables

Abstract Let X1,X2,...be a sequence of dependent and heavy-tailed random variables with distributions F1,F2,…. on (-∞,∞),and let т be a nonnegative integer-valued random variable independent of the seq...     

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line

In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin's classical results.