共查询到20条相似文献,搜索用时 31 毫秒
1.
D. B. Shakhmatov 《Journal of Mathematical Sciences》1995,75(3):1754-1769
Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.
- Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
- Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G* and continuous homomorphisms g: G→G*, h: G*→H so that π=h o g, G*=g(G), w(G*)≤τ andind G*≤ind G. If nw(G)≤τ, then g is one-to-one.
- For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind 0 Z≤ind 0 X, whereind 0 is the dimension function defined by V.V.Filippov with the help of Gδ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
- For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G* and continuous homomorphisms g: G→G*, h:G*→H so that π=h o g, G*=g(G), w(G*)≤w(H),dimG*≤dimG, andind G*≤ind G. Bibliography: 25 titles.
2.
We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F *, thenT is an additive injective operator preserving rank-additivity onS n(F) if and only if there exists an invertible matrixU∈M n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ?UT, ?X=(xij)∈Sn(F) wherec∈F *,X ?=(?(x ij)). As applications, we determine the additive operators preserving minus-order onS n(F) over the fieldF. 相似文献
3.
The augmented penalty function is used to solve optimization problems with constraints and for faster convergence while adopting gradient techniques. In this note, an attempt is made to show that, ifx* ∈S maximizes the function $$W(x,\lambda ,{\rm K}) = f(x) - \sum\limits_{j = 1}^n {\lambda _j C_j (x)} - K\sum\limits_{j = 1}^n {C_j ^2 (x)} ,$$ thenx* maximizesf(x) over all thosex ∈S such that $$C_j (x) \leqslant C_j ,j = 1,2, \ldots ,n,$$ under the assumptions that the λ j 's andk are nonnegative, real numbers. Here,W(x, λ,K),f(x), andC j (x),j=1, 2,...,n, are real-valued functions andC j (x) ≥ 0 forj=1, 2,...,n and for allx. The above result is generalized considering a more general form of the augmented penalty function. 相似文献
4.
С. А. ТЕЛЯКОВСКИЙ 《Analysis Mathematica》1982,8(4):305-319
Let σ n 2 (f, x) be the Cesàro means of second order of the Fourier expansion of the function f. Upper bounds of the deviationf(x)-σ n 2 (f, x) are studied in the metricC, while f runs over the class \(\bar W^1 C\) , i. e., of the deviation $$F_n^2 (\bar W^1 ,C) = \mathop {\sup }\limits_{f \in \bar W^1 C} \left\| {f(x) - \sigma _n^2 (f,x)} \right\|_c$$ . It is proved that the function $$g^* (x) = \frac{4}{\pi }\mathop \sum \limits_{v = 0}^\infty ( - 1)^v \frac{{\cos (2v + 1)x}}{{(2v + 1)^2 }}$$ , for whichg *′(x)=sign cosx, satisfies the following asymptotic relation: $$F_n^2 (\bar W^1 ,C) = g^* (0) - \sigma _n^2 (g^* ,0) + O\left( {\frac{1}{{n^4 }}} \right)$$ , i.e.g * is close to the extremal function. This makes it possible to find some of the first terms in the asymptotic formula for \(F_n^2 (\bar W^1 ,C)\) asn → ∞. The corresponding problem for approximation in the metricL is also considered. 相似文献
5.
пУсть {f k; f k * ?X×X* — пОлНАь БИОРтОгОНАльНАь сИс тЕМА В БАНАхОВОМ пРОстРАН стВЕ X (X* — сОпРьжЕННОЕ пРОст РАНстВО). пУсть (?→+0) $$\begin{gathered} S_n f = \sum\limits_{k = 0}^n {f_k^* (f)f_k ,} K(f,t) = \mathop {\inf }\limits_{g \in Z} (\left\| {f - g} \right\|_x + t\left| g \right|_z ), \hfill \\ X_0 = \{ f \in X:\mathop {\lim }\limits_{n \to \infty } \left\| {S_n f - f} \right\|_x = 0\} ,X_\omega = \{ f \in X:K(f,t) = 0(\omega (t))\} , \hfill \\ \end{gathered} $$ гДЕZ?X — НЕкОтОРОЕ пОД пРОстРАНстВО с пОлУН ОРМОИ ¦·¦ И Ω — МОДУль НЕпРЕРыВНО стИ УДОВлЕтВОРьУЩИИ Усл ОВИУ sup Ω(t)/t=∞. пОслЕДОВАтЕ льНОстьΤ={Τ k} кОМплЕксНых ЧИ сЕл НАжыВАЕтсь МНОжИтЕл ЕМ сИльНОИ схОДИМОст И ДльX Τ, жАпИсьΤ?М[X Τ,X Τ], ЕслИ Д ль кАжДОгО ЁлЕМЕНтАf?X Τ сУЩЕстВ УЕт тАкОИ ЁлЕМЕНтf τ?х0, ЧтОf k * (f τ)=Τkf k * (f) Дль ВсЕхk. ДОкА жАНО сРЕДИ ДРУгИх слЕДУУЩ ЕЕ УтВЕРжДЕНИЕ. тЕОРЕМА. пУсmь {fk; f k * } —Н ЕкОтОРыИ (с, 1)-БАжИс тАк ОИ, ЧтО ВыпОлНьУтсь НЕРАВЕН стВА тИпА НЕРАВЕНстВА ДжЕ ксОНА с пОРьДкОМ O(?n) u тИ пА НЕРАВЕНстВА БЕРНшmЕИ НА с пОРьДкОМ O(1/?n). ЕслИ пОслЕДОВАтЕл ьНОсть Τ кВАжИВыпУкл А И ОгРАНИЧЕНА, тО $$\tau \in M[X_{\omega ,} X_0 ] \Leftrightarrow \omega (\varphi _n )\tau _n \left\| {S_n } \right\|_{[X,X]} = o(1).$$ ЁтОт ОБЩИИ пОДхОД НЕМ ЕДлЕННО ДАЕт клАссИЧ ЕскИЕ РЕжУльтАты, ОтНОсьЩИ Есь к ОДНОМЕРНыМ тРИгОНОМЕтРИЧЕскИМ РьДАМ. НО тЕпЕРь ВОжМО жНы ДАльНЕИшИЕ пРИлОжЕН Иь, НАпРИМЕР, к РАжлОжЕНИьМ пО пОлИ НОМАМ лЕжАНДРА, лАгЕР РА ИлИ ЁРМИтА. 相似文献
6.
Vittorino Pata 《Journal of Fixed Point Theory and Applications》2011,10(2):299-305
We discuss a fixed point theorem for a function f mapping a complete metric space X into itself. For all x ? X{x \in X} the iterates of f(x) are shown to converge to x* = f(x*){{x_{\star} = f(x_{\star})}} and an explicit estimate of the convergence rate is given. 相似文献
7.
LetP(x, A) be a transition probability on a measurable space (S, Σ) and letX n be the associated Markov chain.Theorem. Letf∈B(S, Σ). Then for anyx∈S we haveP x a.s. $$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {f(X_k ) \geqslant } \mathop {\underline {\lim } }\limits_{n \to \infty } \mathop {\inf }\limits_{x \in S} \frac{1}{n}\sum\limits_{k = 1}^n {P^k f(x)} $$ and (implied by it) a corresponding inequality for the lim. If 1/n∑ k=1 n P k f converges uniformly, then for everyx∈S, 1/n ∑ k=1 n f(X k ) convergesP x a.s. Applications are made to ergodic random walks on amenable locally compact groups. We study the asymptotic behavior of 1/n∑ k=1 n μ k *f and of 1/n∑ k=1 n f(X k ) via that ofΨ n *f(x)=m(A n )?1∫ An f(xt), where {A n } is a Følner sequence, in the following cases: (i)f is left uniformly continuous (ii) μ is spread out (iii)G is Abelian. Non-Abelian Example: Let μ be adapted and spread-out on a nilpotent σ-compact locally compact groupG, and let {A n } be a Følner sequence. If forf∈B(G, ∑) m(A n )?1∫ An f(xt)dm(t) converges uniformly, then 1/n∑ k=1 n f(X k ) converges uniformly, andP x convergesP x a.s. for everyx∈G. 相似文献
8.
周云华 《数学物理学报(B辑英文版)》2011,31(1):102-108
Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space. 相似文献
9.
Shi Yingguang 《分析论及其应用》1991,7(4):28-39
Let Sn(f,x) be the Hermite-Fejér type interpolation satisfying Sn(f,xk)=f(xk), S′n(f,xk)=0, k=1,2,…,n and Sn(f,yi)=f(yi), j=1,2,…,m. For m=0, let Hn(f,x)≔Sn(f,x). This paper investigates relationship between Sn(f,x) and Hn(f,x), as well as, the saturation of Sn(f,x). 相似文献
10.
S. E. Zhukovskii Z. T. Mingaleeva 《Moscow University Computational Mathematics and Cybernetics》2012,36(2):60-65
Solutions to the equation F(x, ??) = 0 with unknown x and the parameter ?? in the neighborhood of the solution (x *, ??*) under the additional constraint x ?? U, where U is a closed convex set, are studied. The sufficient conditions for existence of an implicit function without prior assumption of the normalcy of point x * are given. The obtained result is used to investigate the local solvability of controlled systems with mixed constraints. 相似文献
11.
Qing Lin 《数学学报(英文版)》1995,11(3):225-231
In this paper, we show that, without involving theC*-bundle theory, elementary differential topology can do the classification of homogeneousC*-crossed productC(X) x
Z, whereX is a compact differentiable manifold of low dimension and is a diffeomorphism ofX. The motivation of this work is the Shultz's theorem which states that aC*-algebra can be identified with its pure state space carryingw*-topology and certain geometric structure. 相似文献
12.
A. A. Fora 《Periodica Mathematica Hungarica》1985,16(2):97-113
In this paper, we definen-segmentwise metric spaces and then we prove the following results:
- (i)|Let (X, d) be ann-segmentwise metric space. ThenX n has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C *(X) → C*(X) and for anyn-tuple of distinct points x1, x2, ?, xn ∈X, there exists anh ∈C *(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC *(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
- LetX i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX 1 × ? × Xn has the fixed point property for alln ∈N Then the product spaceX 1 × X2 × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
- There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
- There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX + does not have the fixed point property.
13.
Let e and n be positive integers and S={x1,…,xn} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (xi,xj)e of the greatest common divisor of xi and xj is called the eth power GCD matrix on S, denoted by (Se). Similarly we can define the eth power LCM matrix [Se]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with maxx∈S{|GS(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case maxx∈S{|GS(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with maxx∈S{|GS(x)|}=2 such that (Se)|[Se]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12. 相似文献
14.
Let G be a connected graph, let ${X \subset V(G)}$ and let f be a mapping from X to {2, 3, . . .}. Kaneko and Yoshimoto (Inf Process Lett 73:163–165, 2000) conjectured that if |N G (S) ? X| ≥ f (S) ? 2|S| + ω G (S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d T (x) ≥ f (x) for all ${x \in X}$ . In this paper, we show a result with a stronger assumption than this conjecture; if |N G (S) ? X| ≥ f (S) ? 2|S| + α(S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d T (x) ≥ f (x) for all ${x \in X}$ . 相似文献
15.
A. V. Arutyunov 《Computational Mathematics and Mathematical Physics》2006,46(2):195-205
The equation F(x, σ) = 0,x ∈ K, in which σ is a parameter and x is an unknown taking values in a given convex cone in a Banach space X, is considered. This equation is examined in a neighborhood of a given solution (x *, σ*) for which the Robinson regularity condition may be violated. Under the assumption that the 2-regularity condition (defined in the paper), which is much weaker than the Robinson regularity condition, is satisfied, an implicit function theorem is obtained for this equation. This result is a generalization of the known implicit function theorems even for the case when the cone K coincides with the entire space X. 相似文献
16.
Let x ~ N(μ, Z), and let S = (Σ:μ). It is shown that if x′A1x is independent of x′Bx (x′A1x is distributed as a chi-square variable), then this property is inherited by every x′A2x for which S′A2S precedes S′A1S with respect to the range preordering (with respect to the rank subtractivity partial ordering). 相似文献
17.
M. A. Wolfe 《Journal of Optimization Theory and Applications》1980,31(1):125-129
LetM 0, characterized byx k+1=G 0(x k),k?0,x 0 prescribed, be an iterative method for the solution of the operator equationF(x)=0, whereF:X → X is a given operator andX is a Banach space. Let ω:X → X be a given operator, and let the methodM mbe characterized byx x+1,m =G m(x k,m),k?0,x 0,m prescribed, where $$G_i (x) = G_0 (x) - \sum\limits_{j = 0}^{i - 1} { F'(\omega (x))^{ - 1} F(G_j (x)), i = 1, . . . ,m,} $$ in whichG 0:X → X is a given operator andF′:X → L(X) is the Fréchet derivative ofF. Sufficient conditions for the existence of a solutionx* m ofF(x)=0 to which the sequence (x k,m) generated from methodM mconverges are given, together with a rate-of-convergence estimate. 相似文献
18.
Paul Binding 《Journal of Differential Equations》1977,24(3):349-354
Differential equations ·x(t) = f(x(t), t) are exhibited in a general infinite-dimensional Banach space, failing each of the following in turn. (i) The set St of solution values x(t) from a given point x(0) is compact. (ii) St is connected. (iii) Any point on the boundary ?St of St can be reached by a solution x with x(s) ??Ss, 0 ? s ? t. 相似文献
19.
对于集合X上的任一非平凡等价关系E,本文考察了半群TE(x)上的同余C*(E),并证明了C*(E)是TE(X)的同余格的完全子格[C(E),Ca(E)]中的唯一原子. 相似文献
20.
We study the dual Dunkl-Sonine operator tSk,? on ?d, and give expression of tSk,?, using Dunkl multiplier operators on ?d. Next, we study the extremal functions f*λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f*λ} λ >0 converges uniformly to tSk,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results. 相似文献