Perturbation of symmetric Markov processes

We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator $\mathcal {L}We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator of a general symmetric Markov process. An illuminating concrete example for is , where D is a bounded Euclidean domain in is the Laplace operator in D with zero Dirichlet boundary condition and is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to is a Lévy process that is the sum of Brownian motion in and an independent symmetric (2s)-stable process in killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated It? formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)]. The research of T.-S. Zhang is supported by the British EPSRC.     

On the similarity of dual operators

Let k be a field with an involution σ and a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that ΛEnd(V) and Λ^*End(W) are dual operators. We show that if Λ and Λ^* are similar, then Λ^*=Λ^-1, where :V→W is Hermitian.     

Quadratic kernel-free non-linear support vector machine

A new quadratic kernel-free non-linear support vector machine (which is called QSVM) is introduced. The SVM optimization problem can be stated as follows: Maximize the geometrical margin subject to all the training data with a functional margin greater than a constant. The functional margin is equal to W ^T X + b which is the equation of the hyper-plane used for linear separation. The geometrical margin is equal to . And the constant in this case is equal to one. To separate the data non-linearly, a dual optimization form and the Kernel trick must be used. In this paper, a quadratic decision function that is capable of separating non-linearly the data is used. The geometrical margin is proved to be equal to the inverse of the norm of the gradient of the decision function. The functional margin is the equation of the quadratic function. QSVM is proved to be put in a quadratic optimization setting. This setting does not require the use of a dual form or the use of the Kernel trick. Comparisons between the QSVM and the SVM using the Gaussian and the polynomial kernels on databases from the UCI repository are shown.     

Infinitesimal non-crossing cumulants and free probability of type B

Free probabilistic considerations of type B first appeared in the paper of Biane, Goodman and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003) 2263-2303]. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between “type B” and “infinitesimal” is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A₁,…,Ak in an infinitesimal noncommutative probability space (A,φ,φ^′) and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A,φ,φ^′), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A₁,…,Ak is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from “usual” free probability. We show that the lattices NC(B)(n) of non-crossing partitions of type B appear in the combinatorial study of (A,φ,φ^′), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A₁,…,Ak in (A,φ) can be naturally upgraded to infinitesimal freeness in (A,φ,φ^′), for a suitable choice of a “companion functional” .     

New Orlicz-Hardy spaces associated with divergence form elliptic operators

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0,∞) of strictly critical lower type pω∈(0,1] and ρ(t)=t−1/ω−1(t−1) for t∈(0,∞). In this paper, the authors study the Orlicz-Hardy space Hω,L(Rn) and its dual space BMOρ,L^*(Rn), where L^* denotes the adjoint operator of L in L²(Rn). Several characterizations of Hω,L(Rn), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(Rn) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood-Paley g-function gL map Hω,L(Rn) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L(Rn) into the classical Orlicz-Hardy space Hω(Rn) for and the corresponding fractional integral L−γ for certain γ>0 maps Hω,L(Rn) continuously into , where is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω(t)=tp for all t∈(0,∞) and p∈(0,1).     

Synthesis of ferrocenylcarbodiimide as a convenient electrochemically active labeling reagent for nucleic acids

Ferrocenylcarbodiimides carrying different redox potentials, 1 and 2, were designed and synthesized as convenient electrochemically active labeling reagents for nucleic acids, which may be used as dually labeling reagents of nucleic acids like Cy3 and Cy5 dyes. These reagents could react with the imino unit of thymine or guanine base on DNA or of uracil base on RNA under a basic buffer condition to yield a labeled product quantitatively in a short period of time. The current responses of the labeled DNAs in square wave voltammetric (SWV) measurement showed a good linear correlation with the amount of the hybridized ones. DNAs labeled with the two different reagents, 1 and 2, could be detected electrochemically at different potentials after hybridization with a DNA probe-immobilized gold electrode.     

关于人口系统妇女总和生育率的范数最优控制问题

本文讨论人口系统妇女总和生育率的范数最优控制问题。本文将妇女总和生充臃当作控制变量，在一定条件下证得最优控制的存在和唯一性，并给出其相应的优化条件。     

Bimodal fluorescence signaling based on control of the excited-state conformational twisting and the ground-state protonation processes

Amino-based fluoroionophores 1 and 2 can selectively sense alkaline earth metal ions in MeCN under both neutral and acidic conditions by different signaling mechanisms. The fluoroionophoric behavior for the neutral probes is characterized by an ‘off-on’ photoinduced electron transfer (PET)-like fluorescence intensity response due to a switching from a twisted internal charge transfer (TICT) to a planar internal charge transfer (PICT) state. For the protonated probes (i.e., 1/H⁺ and 2/H⁺), the fluorescing species is the localized stilbene fluorophores, but dual fluorescence is induced upon metal-ion recognition through a deprotonation process.     

Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval . For a given compact interval , set

     

高强度聚焦超声(HIFU)——一门多学科的研究课题

高强度聚焦超声(high intensity focused ultrasound,简称HIFU)已作为一种无创外科工具而应用于门诊治疗肿瘤。文章介绍了它的基本原理、有关的研究现状和存在的问题,以及对今后研究工作的建议。文章特别强调高强度聚焦超声是一门多学科的综合研究课题,需要各方面的科学工作者通力协作。