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1.
Summary. We investigate the bounded solutions j:[0,1]? X \varphi:[0,1]\to X of the system of functional equations¶¶j(fk(x))=Fk(j(x)),    k=0,?,n-1,x ? [0,1] \varphi(f_k(x))=F_k(\varphi(x)),\;\;k=0,\ldots,n-1,x\in[0,1] ,(*)¶where X is a complete metric space, f0,?,fn-1:[0,1]?[0,1] f_0,\ldots,f_{n-1}:[0,1]\to[0,1] and F0,...,Fn-1:X? X F_0,...,F_{n-1}:X\to X are continuous functions fulfilling the boundary conditions f0(0) = 0, fn-1(1) = 1, fk+1(0) = fk(1), F0(a) = a,Fn-1(b) = b,Fk+1(a) = Fk(b), k = 0,?,n-2 f_{0}(0) = 0, f_{n-1}(1) = 1, f_{k+1}(0) = f_{k}(1), F_{0}(a) = a,F_{n-1}(b) = b,F_{k+1}(a) = F_{k}(b),\,k = 0,\ldots,n-2 , for some a,b ? X a,b\in X . We give assumptions on the functions fk and Fk which imply the existence, uniqueness and continuity of bounded solutions of the system (*). In the case X = \Bbb C X= \Bbb C we consider some particular systems (*) of which the solutions determine some peculiar curves generating some fractals. If X is a closed interval we give a collection of conditions which imply respectively the existence of homeomorphic solutions, singular solutions and a.e. nondifferentiable solutions of (*).  相似文献   

2.
Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dimp(A) £ s-crs{{\rm {dim}_p}(A)\le s-c\varrho^s} where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dimp(A) £ dimp(X)-c(log\tfrac1r)-1rt{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .  相似文献   

3.
Given $\mathcal{X}Given X\mathcal{X}, some measurable subset of Euclidean space, one sometimes wants to construct a finite set of points, P ì X\mathcal{P}\subset\mathcal {X}, called a design, with a small energy or discrepancy. Here it is shown that these two measures of design quality are equivalent when they are defined via positive definite kernels K:X2(=X×X)?\mathbbRK:\mathcal{X}^{2}(=\mathcal{X}\times\mathcal {X})\to\mathbb{R}. The error of approximating the integral òXf(x) dm(x)\int_{\mathcal{X}}f(\boldsymbol{x})\,\mathrm{d}\mu(\boldsymbol{x}) by the sample average of f over P\mathcal{P} has a tight upper bound in terms of the energy or discrepancy of P\mathcal{P}. The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the measure μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K.  相似文献   

4.
Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G\frak XG_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak Sp\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak Sp\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak Sp\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nnn\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak Sp1?\frak Sprpi \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.  相似文献   

5.
We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we consider the universal Artin motive mapping to M and denote it w0X(M)\omega^{0}_{X}(M). We use this to define a motive \mathbbEX\mathbb{E}_{X} over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors w0X\omega^{0}_{X} and the computation of the motives \mathbbEX\mathbb{E}_{X}.  相似文献   

6.
We extend a result of ?estakov to compare the complex interpolation method [X 0, X 1]θ with Calderón-Lozanovskii’s construction ${{{{X^{1-\theta}_{0}X^{\theta}_{1}}}}}We extend a result of Šestakov to compare the complex interpolation method [X 0, X 1]θ with Calderón-Lozanovskii’s construction X1-q0Xq1{{{{X^{1-\theta}_{0}X^{\theta}_{1}}}}}, in the context of abstract Banach lattices. This allows us to prove that an operator between Banach lattices T : EF which is p-convex and q-concave, factors, for any q ? (0, 1){{{{\theta \in (0, 1)}}}}, as TT 2 T 1, where T 2 is ( (\fracpq+ (1 - q)p ){{\left({\frac{p}{{\theta + (1 - \theta)p}}} \right)}}-convex and T 1 is (\fracq1 - q ){{\left({\frac{q}{{1 - \theta }}} \right)}}-concave.  相似文献   

7.
For log\frac1+?52 £ l* £ l* < ¥{\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty , let E*, λ*) be the set {x ? [0,1): liminfn ? ¥\fraclogqn(x)n=l*, limsupn ? ¥\fraclogqn(x)n=l*}. \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}. It has been proved in [1] and [3] that E*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that dimE(l*, l*) 3 \fracl* -log\frac1+?522l*\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}}  相似文献   

8.
We study the empirical process ${{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}We study the empirical process supf ? F|N-1?i=1N f2(Xi)-\mathbbEf2|{{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}, where F is a class of mean-zero functions on a probability space (Ω, μ), and (Xi)i = 1N{(X_{i})_{i =1}^N} are selected independently according to μ.  相似文献   

9.
Let X \hookrightarrow[`(X)]{X \hookrightarrow \overline{X}} be an open immersion of smooth varieties over a field of characteristic p > 0 such that the complement is a simple normal crossing divisor and [`(Z)] í Z í [`(X)]{\overline{Z}\subseteq Z \subseteq \overline{X}} closed subschemes of codimension at least 2. In this paper, we prove that the canonical restriction functor between the categories of overconvergent F-isocrystals F-Isocf(X,[`(X)]) ? F-Isocf(X\Z,[`(X)]\[`(Z)]){F-{\rm Isoc}^\dagger(X,\overline{X}) \longrightarrow F-{\rm Isoc}^\dagger(X{\setminus}Z, \overline{X}{\setminus}\overline{Z})} is an equivalence of categories. We also give an application of our result to the equivalence of certain categories.  相似文献   

10.
Let T X denote the full transformation semigroup on a set X. For an equivalence E on X, let $T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.$ Then $T_{E^{*}}(X)Let T X denote the full transformation semigroup on a set X. For an equivalence E on X, let
TE*(X)={a ? TX:"x,y ? X,(x,y) ? E?(xa,ya) ? E}.T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.  相似文献   

11.
For a resistance form ${(X, \mathcal{D}(\varepsilon),\varepsilon)}For a resistance form (X, D(e),e){(X, \mathcal{D}(\varepsilon),\varepsilon)} and a point x0 ? X{x_0 \in X} as boundary, on the space X0:=X \{x0}{X_0:=X {\setminus}\{x_0\}} we consider the Dirichlet space Dx0:={f ? D(e) | f(x0)=0}{\mathcal{D}_{x_0}:=\{f\in\mathcal{D}(\varepsilon)\, |\, f(x_0)=0\}} and we develop a good potential theory. For any finely open subset D of X 0 we consider a localized resistance form (DX0 \ D,eD{\mathcal{D}_{X_0 {\setminus} D},\varepsilon_{D}}) where DX0 \ D:={f ? Dx0 | f=0{\mathcal{D}_{X_0 {\setminus} D}:=\{f\in\mathcal{D}_{x_0}\, |\, f=0} on X0 \ D}, eD(f,g):=e(f,g){X_0 {\setminus} D\},\, \varepsilon_D(f,g):=\varepsilon(f,g)} for all f,g ? DX0 \ D{f,g\in\mathcal{D}_{X_0 {\setminus} D}}. The main result is the equivalence between the local property of the resistance form and the sheaf property for the excessive elements on finely open sets.  相似文献   

12.
Let ${\Phi : \mathbb{R} \to [0, \infty)}Let F: \mathbbR ? [0, ¥){\Phi : \mathbb{R} \to [0, \infty)} be a Young function and let f = (fn)n ? \mathbbZ+{f = (f_n)_n\in\mathbb{Z}_{+}} be a martingale such that F(fn) ? L1{\Phi(f_n) \in L_1} for all n ? \mathbbZ+{n \in \mathbb{Z}_{+}} . Then the process F(f) = (F(fn))n ? \mathbbZ+{\Phi(f) = (\Phi(f_n))_n\in\mathbb{Z}_{+}} can be uniquely decomposed as F(fn)=gn+hn{\Phi(f_n)=g_n+h_n} , where g=(gn)n ? \mathbbZ+{g=(g_n)_n\in\mathbb{Z}_{+}} is a martingale and h=(hn)n ? \mathbbZ+{h=(h_n)_n\in\mathbb{Z}_{+}} is a predictable nondecreasing process such that h 0 = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality ||h||XC ||F(Mf) ||X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Mf)} \|_X} is valid, and those X such that the inequality ||h||XC ||F(Sf) ||X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Sf)} \|_X} is valid, where Mf and Sf denote the maximal function and the square function of f, respectively.  相似文献   

13.
Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k £ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k £ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.  相似文献   

14.
The bigraded Frobenius characteristic of the Garsia-Haiman module M μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?p1 [(H)\tilde]m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M μ from S n to S n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c μ v occurring in the expansion ?p1 [(H)\tilde]m[X; q, t] = ?v ? mcmv [(H)\tilde]v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M μ may be calculated from the recursion Fm (q, t) = ?v ? mcmv Fv (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c μ v have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].  相似文献   

15.
We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as
{ x  ?  X|t - q\textRea y1 ( tA )xt - q\textRea y2 ( tA )x ? L*p ( ( 0,¥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}  相似文献   

16.
The perturbation classes problem for semi-Fredholm operators asks when the equalities SS(X,Y)=PF+(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)} and SC(X,Y)=PF-(X,Y){\mathcal{SC}(X,Y)=P\Phi_-(X,Y)} are satisfied, where SS{\mathcal{SS}} and SC{\mathcal{SC}} denote the strictly singular and the strictly cosingular operators, and PΦ+ and PΦ denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, SS(Y*,X*)=PF+(Y*,X*){\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)} if and only if SC(X,Y)=PF-(X,Y),{\mathcal{SC}(X,Y)=P\Phi_-(X,Y),} and SC(Y*,X*)=PF-(Y*,X*){\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)} if and only if SS(X,Y)=PF+(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}. Moreover we give examples showing that both direct implications fail in general.  相似文献   

17.
Let T X be the full transformation semigroup on a set X,
TE*(X)={a ? TX:"x,y ? X, (x,y) ? E? (xa,ya) ? E}T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}  相似文献   

18.
The Generalized Multifractional Brownian Motion (GMBM) is a continuous Gaussian process {X(t)}t ? [0,1]\{X(t)\}_{t\in [0,1]} that extends the classical Fractional Brownian Motion (FBM) and the Multifractional Brownian Motion (MBM) [15, 4, 1, 1]. Its main interest is that, its Hölder regularity can change widely from point to point. In this article we introduce the Generalized Multifractional Field (GMF), a continuous Gaussian field {Y(x,y)}(x,y) ? [0,1] 2\{Y(x,y)\}_{(x,y)\in [0,1]^{\,2}} that satisfies for every tt, X(t)=Y(t,t)X(t)=Y(t,t). Then, we give a wavelet decomposition of YY and using this nice decomposition, we show that YY is b\beta-Hölder in yy, uniformly in xx. Generally speaking this result seems to be quite important for the study of the GMBM. In this article, it will allow us to determine, without any restriction, its pointwise, almost sure, Hölder exponent and to prove that two GMBM's with the same Hölder regularity differ by a "smoother' process.  相似文献   

19.
Let F be a finite extension of ℚ p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over  [`( \mathbbF)]p\overline{ \mathbb{F}}_{p} to be supersingular. We then give the classification of irreducible admissible smooth GL n (F)-representations over  [`( \mathbbF)]p\overline{ \mathbb{F}}_{p} in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.  相似文献   

20.
Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set XωX, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X ? *\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}} . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient Mw(X)/M0(X){\cal M}_{\omega}(X)/{\cal M}_0(X) , for certain external subspaces M0(X), Mw(X){\cal M}_0(X), {\cal M}_{\omega}(X) of the hyperfinite dimensional Banach space *\Bbb CX{}{^{\ast}{\Bbb C}}^X , with the norm ‖f‖1 = ∑xX |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and Mw(G)/M0(G){\cal M}_{\omega}(G)/{\cal M}_0(G) are isometrically isomorphic as Banach algebras.  相似文献   

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