Some Notes on a Minimization Problem

Let X[a,b] be a compact set containing at least n+1 points and Kan n-dimensional Haar subspace in c[a,b]. Let F(x,y) be a nonnegativefunction, defined on X×(-∞,∞), satisfying ‖F(·,p)‖<∞ with the L_∞norm forsome∈K, where F(x,p)≡F(x,p(x)). The minimization problem discussed in this paper is to find an elementp∈K such that ‖F(·,p)‖=inf ‖F(·,q)‖, such an element p(if any) is saidto be a minimum to F in K~(q∈K). The author in [1,2] studied this problem and has given the main theoremsin the Cbebyshev theory under the following assumptions: (A) lim F(x,y)=∞, x∈X; (B) lim F(x,u)=F(x,y), x∈X,y; (C)lim F(u,υ)=F(x,y),x∈X,y; (D) For each x∈X there existtwo real numbers f~-(x) and f~+(x),f~-(x)f~+(x). such that F(x,y) is strictlydecreasing with respect to y on (-∞,f~-(x)] and strictly increasing on [f~+(x),∞), and F(x,y)=F(x):=inf F(x,υ) on [f~-(x),f~+(x)]. Denote f_1(x)=inf{y:F(x,y)‖F~*‖},f_2(x)=sup{y:F(x,) ‖F‖},f_1(x)=lim f_1(u),f_2(x)=lim f_2(u), G=(q∈K: f_1qf_2}.For pεK set X_p={     

关于集值系统x∈F(x,y),y∈G(x,y)解的存在性

Let (E, ‖·‖) be a uniformly convex Banach space, X a nonempty compact and convex subset of E. Let F be a closed mapping of X×X into 2^X, G a mapping of X×X into C(X). It is shown that if for any f∈C(X), x∈X, F(x,f(x)) is a closed and convex subset of X, and G(x,f(x)) is a continuous function and for any x,y₁,y₂∈X,H(x,G(x,y₁),G(x,y₂))≤‖y₁-y₂‖ then there exist X₀,y₀∈X such that X₀∈F(x₀,y₀) and y₀∈G(x₀, y₀).     

On moments of the maximum of partial sums of moving average processes under dependence assumptions

Let {Y i;∞ < i < ∞} be a doubly infinite sequence of identically distributed-mixing random variables and let {a i;∞ < i < ∞} be an absolutely summable sequence of real numbers.In this paper we study the moments of sup(1 ≤ r < 2,p > 0) under the conditions of some moments.     

Littlewood—Paley算子及Marcinkiewicz积分在Campanato空间εa,p上的有界性

我们证明了下述结果:若f∈ε^a,p,则适当限制参数值时,有g(f)(x)(S(f)(x),g_λ^*(f)(x),μ(f)(x))<∞a.e.,或者g(f)(x)(S(f)(x),g_λ^*(f)(x),μ(f)(x))<∞a.e.;并且在前者成立时,有g(f)(S(f),g_λ^*(f),μ(f))∈ε^a,p,以及‖g(f)‖_a,p     

半质环交换性的一点注记

Let R be a semi-prime ring, C be the center of R. Let F_i (x, y) (i = 1, 2) be a product of the m times x's and n times y's.In this paper following theorem is proved: (I ) implies (Ⅱ), where( Ⅰ )If f₁(x,y) -f₂(x,y) ∈C for every x,y in R, then R is commutative;(Ⅱ)If f₁ (x,y) + f₂(x, y) ∈C for every x,y in R, then R is commutative.Thus very short proves of some theorems of references[5], [8], [9] are be given.     

Commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of Homogeneous type of finite measure

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

质环的求导和交换性

In this paper, we generalize some corresponding results of [1-4]. We obtain the main results as the following:Theorem 1 Let R be a prime ring of characteristic not 2 with nontrivial derivations d₁,d₂ and let U be a nonzero ideal of R . If C is the center of R, then the following conditions are equivalent : (i)d₁,d₂(x)∈C for all x∈U; (ii) [d₁(x),d₂(y)]∈C for all x,y∈U; (iii) d₁(x)d₂(y)+d₂(x)d₁(y)∈C for all x,y∈U; (iv) R is commutative.Theorem 2 Let R be a prime ring with nontrivial derivations d₁,d₂,…, d_n and U be a nonzero ideal of R. Let C be the center of R. If d₁(x₁)d₂(x₂)…d_n(x_n)∈C for all x₁, x₂…x_n)∈U, then R is commutative.     

On Weak Riemann-Stieltjes Integration of Abstract Function Valued in Locally Convex Topological Algebra

Let A be a locally convex topological algebra.We denote by F the set of allnon-zero multiplicative linear continuous functionals on A and WF the weak topologyon A determined by F. Definition 1 Let x(t),y,(t)be abstract functions mapping closed interval[a,b]to A.Given a severation λ:Let     

On Jemanowicz’ Conjecture Concerning Pythagorean Triples

Let(a, b, c) be a primitive Pythagorean triple. Je′smanowicz conjectured in 1956 that for any positive integer n, the Diophantine equation(an)x+(bn)y=(cn)z has only the positive integer solution(x, y, z) =(2, 2, 2). Let p ≡ 3(mod 4) be a prime and s be some positive integer. In the paper, we show that the conjecture is true when(a, b, c) =(4p2s-1, 4p s, 4p2s+ 1) and certain divisibility conditions are satisfied.     

关于商高数的Je\'{s}manowicz猜想

Let(a, b, c) be a primitive Pythagorean triple. Je′smanowicz conjectured in 1956 that for any positive integer n, the Diophantine equation(an)x+(bn)y=(cn)z has only the positive integer solution(x, y, z) =(2, 2, 2). Let p ≡ 3(mod 4) be a prime and s be some positive integer. In the paper, we show that the conjecture is true when(a, b, c) =(4p2s-1, 4p s, 4p2s+ 1) and certain divisibility conditions are satisfied.