The Generalized Multifractional Field: A Nice Tool for the Study of the Generalized Multifractional Brownian Motion

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.     

Singularity Spectrum of Generic ��-H?lder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by B₁^a{\mathcal {B}}_{1}^{\alpha} the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g ? B₁^ag\in {\mathcal {B}}_{1}^{\alpha} one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space E^a=M×B₁^a\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha} equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ ^α . In particular we determine the generic H?lder singularity spectrum of g○f.     

Partial regularity for nonlinear elliptic systems with continuous growth exponent

For weak solutions of nonlinear elliptic systems of the type ${- {\rm div}a(x, u(x), Du(x)) = 0,}$ with nonstandard p(x) growth, we show interior partial Hölder continuity for any Hölder exponent ${\alpha \in (0,1)}$ , provided that the exponent function is ‘logarithmic Hölder continuous’. The result also covers the up to now open partial regularity for systems with constant growth with exponent p less than two in the case of merely continuous dependence on the spacial variable x.     

Embedding of non-commutative <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces: <Emphasis Type="Italic">p</Emphasis> < 1

If (N,t) ({\cal N},\tau) is a finite von Neumann algebra and if (M,n) ({\cal M},\nu) is an infinite von Neumann algebra, then L_p(M,n) L_{p}({\cal M},\nu) does not Banach embed in L_p(N,t) L_{p}({\cal N},\tau) for all p ? (0,1) p\in (0,1) . We also characterize subspaces of $ L_{p}({\cal N},\tau),\ 0< p <1 $ L_{p}({\cal N},\tau),\ 0< p <1 containing a copy of l_p.     

A Spectral Gap Property for Random Walks Under Unitary Representations

Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $\pi ,\mathcal{H}Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation p,H\pi ,\mathcal{H} of G, we study spectral properties of the operator π(μ) acting on H\mathcal{H} Assume that μ is adapted and that the trivial representation 1 _G is not weakly contained in the tensor product p?[`(p)]\pi\otimes \overline\pi We show that π(μ) has a spectral gap, that is, for the spectral radius r_spec(p(m))r_{\rm spec}(\pi(\mu)) of π(μ), we have r_spec(p(m)) < 1.r_{\rm spec}(\pi(\mu))< 1. This provides a common generalization of several previously known results. Another consequence is that, if G has Kazhdan’s Property (T), then r_spec(p(m)) < 1r_{\rm spec}(\pi(\mu))< 1 for every unitary representation π of G without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.     

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of C n

Let B denote the unit ball in [(?)\tilde] \widetilde{\nabla\hskip-4pt}\hskip4pt denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces D_g{{\cal D}_{\gamma}} , $\gamma > (n-1)$\gamma > (n-1) , and weighted Bergman spaces A^p_a{A^p_{\alpha}} , 0 < p < ￥0 < p < \infty , $\alpha > n$\alpha > n , of holomorphic functions f on B for which D_g( f)D_{\gamma}(\,f) and || f||_A^pa\Vert\, f\Vert_{A^p_{\alpha}} respectively are finite, where D_g( f)=ò_B (1-|z|²)^g|[(?)\tilde]  f(z)|²dt(z),D_{\gamma}(\,f)=\int_B (1-\vert z\vert^2)^{\gamma}\vert\widetilde{\nabla\hskip-4pt}\hskip4pt f(z)\vert^2d\tau(z), and || f||^p_A^pa=ò_B(1-|z|²)^a| f(z)|^pdt(z).\Vert\, f\Vert^p_{A^p_{\alpha}}=\int_B(1-\vert z\vert^2)^{\alpha}\vert\, f(z)\vert^pd\tau(z). The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and $\alpha > n$\alpha > n .     

Precise Asymptotic Formulas for Semilinear Eigenvalue Problems

. We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | ^p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L²-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = l_n (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L² norm of the eigenfunction associated with l \lambda .     

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

Inverse Scattering at Fixed Energy in de Sitter�CReissner�CNordstr?m Black Holes

In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter–Reissner–Nordstr?m black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ ≠ 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ ≠ 0 is the fixed energy and n ? \mathbbN^*{n \in \mathbb{N}^{*}} denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M, the square of the charge Q ² and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T(λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all n ? L{n \in {\mathcal{L}}} where L{\mathcal{L}} is a subset of \mathbbN^*{\mathbb{N}^{*}} that satisfies the Müntz condition ?_{n ? L}\frac1n = +￥{\sum_{n \in{\mathcal{L}}}\frac{1}{n} = +\infty} . Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the “unphysical” scattering matrix S(λ, z) in the complex variable z. We show, in particular, that the quantities \frac1T(l,z){\frac{1}{T(\lambda,z)}}, \fracR(l,z)T(l,z){\frac{R(\lambda,z)}{T(\lambda,z)}} and \fracL(l,z)T(l,z){\frac{L(\lambda,z)}{T(\lambda,z)}} belong to the Nevanlinna class in the region ${\{z \in \mathbb{C}, Re(z) > 0 \}}${\{z \in \mathbb{C}, Re(z) > 0 \}} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstruction formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.     

Singularly perturbed 1D Cahn–Hilliard equation revisited

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors ${\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M_e}, e 3 0{\{{\mathcal M}_\epsilon\}, \epsilon\geq 0} being the perturbation parameter, such that the map e? M_e{\epsilon \mapsto {\mathcal M}_\epsilon} is H?lder continuous. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e.{\epsilon.} Continuity properties of global attractors and inertial manifolds are also examined.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

A characterization of hyperbolic potentials of rational maps

Consider a rational map f of degree at least 2 acting on its Julia set J(f), a H?lder continuous potential φ: J(f) → ℝ and the pressure P(f,φ). In the case where

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

On conjugacy of some systems of functions

Summary. We investigate the bounded solutions j:[0,1]? X \varphi:[0,1]\to X of the system of functional equations¶¶j(f_k(x))=F_k(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,?,f_n-1:[0,1]?[0,1] f_0,\ldots,f_{n-1}:[0,1]\to[0,1] and F₀,...,F_n-1:X? X F_0,...,F_{n-1}:X\to X are continuous functions fulfilling the boundary conditions f₀(0) = 0, f_n-1(1) = 1, f_k+1(0) = f_k(1), F₀(a) = a,F_n-1(b) = b,F_k+1(a) = F_k(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 f_k and F_k 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 (*).     

Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L ₂(R) with domain ${C_c^\infty({\bf R})}Let H be the symmetric second-order differential operator on L ₂(R) with domain C_c^￥(R){C_c^\infty({\bf R})} and action Hj = -(c j^￠)^￠{H\varphi=-(c\,\varphi^{\prime})^{\prime}} where c ? W^1,2_loc(R){ c\in W^{1,2}_{\rm loc}({\bf R})} is a real function that is strictly positive on R\{0}{{\bf R}\backslash\{0\}} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if n = n₊ún_-{\nu=\nu_+\vee\nu_-} where n_±(x)=±ò^±1_±x c^-1{\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L₂(0,1){\nu\not\in L_2(0,1)} and a unique submarkovian extension if and only if n ? L_￥(0,1){\nu\not\in L_\infty(0,1)}. In both cases, the corresponding semigroup leaves L ₂(0,∞) and L ₂(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W^1,￥_loc(R){ c\in W^{1,\infty}_{\rm loc}({\bf R})} the corresponding operator H has a unique submarkovian extension.     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

Generalized time-dependent conditional linear models under left truncation and right censoring

Consider the model f(S(z|X)){\phi(S(z|X))} = \pmbb(z) [(X)\vec]{\pmb{\beta}(z) {\vec{X}}}, where f{\phi} is a known link function, S(·|X) is the survival function of a response Y given a covariate X, [(X)\vec]{\vec{X}} = (1, X, X ² , . . . , X ^p ) and \pmbb(z){\pmb{\beta}(z)} is an unknown vector of time-dependent regression coefficients. The response is subject to left truncation and right censoring. Under this model, which reduces for special choices of f{\phi} to e.g. Cox proportional hazards model or the additive hazards model with time dependent coefficients, we study the estimation of the vector \pmbb(z){\pmb{\beta}(z)} . A least squares approach is proposed and the asymptotic properties of the proposed estimator are established. The estimator is also compared with a competing maximum likelihood based estimator by means of simulations. Finally, the method is applied to a larynx cancer data set.     

Gaussian upper bounds for fundamental solutions of a family of parabolic equations

We study the family of divergence-type second-order parabolic equations w_e(x)\frac?u?t=div(a(x)w_e(x) ?u), x ? \mathbbRⁿ{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and w_e=1{\omega_\varepsilon=1} for x _n  < 0 and w_e=e{\omega_\varepsilon=\varepsilon} for x _n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .