On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30

Hardy Inequality in Variable Exponent Lebesgue Spaces

Mathematics Subject Classification: 26D10, 46E30, 47B38

Invertibility characterization of Wiener-Hopf plus Hankel operators on variable exponent Lebesgue spaces via even asymmetric factorization

We obtain invertibility and Fredholm criteria for the Wiener-Hopf plus Hankel operators acting between variable exponent Lebesgue spaces on the real line. Such characterizations are obtained via the so-called even asymmetric factorization which is applied to the Fourier symbols of the operators under study.

Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in

We study the Riemann boundary value problem , for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces with variable exponent. We consider both the case when the coefficient is piecewise continuous and it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szegö-Helson theorem to the case of variable exponents.

Existence of solutions for p(x)-Laplacian problems on a bounded domain

In this paper we study the following p(x)-Laplacian problem: -div(a(x)&VERBAR;&DEL; u&VERBAR;(p(x)-2)&DEL; u)+b(x)&VERBAR; u&VERBAR;(p(x)-2)u = f(x, u), x ε &UOmega;, u = 0, on &PARTIAL; &UOmega;, where 1< p(1) &LE; p(x) &LE; p(2) < n, &UOmega; &SUB; R-n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in W-0(1,p(x)) for the p(x)-Laplacian problems in the superlinear and sublinear cases. © 2004 Elsevier Inc. All rights reserved.

Bayesian variable selection in structured high-dimensional covariate spaces with applications in genomics

We consider the problem of variable selection in regression modeling in high-dimensional spaces where there is known structure among the covariates. This is an unconventional variable selection problem for two reasons: (1) The dimension of the covariate space is comparable, and often much larger, than the number of subjects in the study, and (2) the covariate space is highly structured, and in some cases it is desirable to incorporate this structural information in to the model building process. We approach this problem through the Bayesian variable selection framework, where we assume that the covariates lie on an undirected graph and formulate an Ising prior on the model space for incorporating structural information. Certain computational and statistical problems arise that are unique to such high-dimensional, structured settings, the most interesting being the phenomenon of phase transitions. We propose theoretical and computational schemes to mitigate these problems. We illustrate our methods on two different graph structures: the linear chain and the regular graph of degree k. Finally, we use our methods to study a specific application in genomics: the modeling of transcription factor binding sites in DNA sequences. © 2010 American Statistical Association.

Models of collective cell spreading with variable cell aspect ratio : a motivation for degenerate diffusion models

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multi-scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (pme). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models.

The geostationary orbit : a critical legal geography of space’s most valuable real estate

This chapter attends to the legal and political geographies of one of Earth's most important, valuable, and pressured spaces: the geostationary orbit. Since the first, NASA, satellite entered it in 1964, this small, defined band of Outer Space, 35,786km from the Earth's surface, and only 30km wide, has become a highly charged legal and geopolitical environment, yet it remains a space which is curiously unheard of outside of specialist circles. For the thousands of satellites which now underpin the Earth's communication, media, and data industries and flows, the geostationary orbit is the prime position in Space. The geostationary orbit only has the physical capacity to hold approximately 1500 satellites; in 1997 there were approximately 1000. It is no overstatement to assert that media, communication, and data industries would not be what they are today if it was not for the geostationary orbit. This chapter provides a critical legal geography of the geostationary orbit, charting the topography of the debates and struggles to define and manage this highly-important space. Drawing on key legal documents such as the Outer Space Treaty and the Moon Treaty, the chapter addresses fundamental questions about the legal geography of the orbit, questions which are of growing importance as the orbit’s available satellite spaces diminish and the orbit comes under increasing pressure. Who owns the geostationary orbit? Who, and whose rules, govern what may or may not (literally) take place within it? Who decides which satellites can occupy the orbit? Is the geostationary orbit the sovereign property of the equatorial states it supertends, as these states argued in the 1970s? Or is it a part of the res communis, or common property of humanity, which currently legally characterises Outer Space? As challenges to the existing legal spatiality of the orbit from launch states, companies, and potential launch states, it is particularly critical that the current spatiality of the orbit is understood and considered. One of the busiest areas of Outer Space’s spatiality is international territorial law. Mentions of Space law tend to evoke incredulity and ‘little green men’ jokes, but as Space becomes busier and busier, international Space law is growing in complexity and importance. The chapter draws on two key fields of research: cultural geography, and critical legal geography. The chapter is framed by the cultural geographical concept of ‘spatiality’, a term which signals the multiple and dynamic nature of geographical space. As spatial theorists such as Henri Lefebvre assert, a space is never simply physical; rather, any space is always a jostling composite of material, imagined, and practiced geographies (Lefebvre 1991). The ways in which a culture perceives, represents, and legislates that space are as constitutive of its identity--its spatiality--as the physical topography of the ground itself. The second field in which this chapter is situated—critical legal geography—derives from cultural geography’s focus on the cultural construction of spatiality. In his Law, Space and the Geographies of Power (1994), Nicholas Blomley asserts that analyses of territorial law largely neglect the spatial dimension of their investigations; rather than seeing the law as a force that produces specific kinds of spaces, they tend to position space as a neutral, universally-legible entity which is neatly governed by the equally neutral 'external variable' of territorial law (28). 'In the hegemonic conception of the law,' Pue similarly argues, 'the entire world is transmuted into one vast isotropic surface' (1990: 568) on which law simply acts. But as the emerging field of critical legal geography demonstrates, law is not a neutral organiser of space, but is instead a powerful cultural technology of spatial production. Or as Delaney states, legal debates are “episodes in the social production of space” (2001, p. 494). International territorial law, in other words, makes space, and does not simply govern it. Drawing on these tenets of the field of critical legal geography, as well as on Lefebvrian concept of multipartite spatiality, this chapter does two things. First, it extends the field of critical legal geography into Space, a domain with which the field has yet to substantially engage. Second, it demonstrates that the legal spatiality of the geostationary orbit is both complex and contested, and argues that it is crucial that we understand this dynamic legal space on which the Earth’s communications systems rely.

Tendency toward crossover of the effective susceptibility exponent from its doubled Ising value to its doubled mean-field value near a double critical point

The critical behavior of osmotic susceptibility in an aqueous electrolyte mixture 1-propanol (1P)+water (W)+potassium chloride is reported. This mixture exhibits re-entrant phase transitions and has a nearly parabolic critical line with its apex representing a double critical point (DCP). The behavior of the susceptibility exponent is deduced from static light-scattering measurements, on approaching the lower critical solution temperatures (TL’s) along different experimental paths (by varying t) in the one-phase region. The light-scattering data analysis substantiates the existence of a nonmonotonic crossover behavior of the susceptibility exponent in this mixture. For the TL far away from the DCP, the effective susceptibility exponent γeff as a function of t displays a nonmonotonic crossover from its single limit three-dimensional (3D)-Ising value ( ∼ 1.24) toward its mean-field value with increase in t. While for that closest to the DCP, γeff displays a sharp, nonmonotonic crossover from its nearly doubled 3D-Ising value toward its nearly doubled mean-field value with increase in t. The renormalized Ising regime extends over a relatively larger t range for the TL closest to the DCP, and a trend toward shrinkage in the renormalized Ising regime is observed as TL shifts away from the DCP. Nevertheless, the crossover to the mean-field limit extends well beyond t>10−2 for the TL’s studied. The observed crossover behavior is attributed to the presence of strong ion-induced clustering in this mixture, as revealed by various structure probing techniques. As far as the critical behavior in complex or associating mixtures with special critical points (like the DCP) is concerned, our results indicate that the influence of the DCP on the critical behavior must be taken into account not only on the renormalization of the critical exponent but also on the range of the Ising regime, which can shrink with decrease in the influence of the DCP and with the extent of structuring in the system. The utility of the field variable tUL in analyzing re-entrant phase transitions is demonstrated. The effective susceptibility exponent as a function of tUL displays a nonmonotonic crossover from its asymptotic 3D-Ising value toward a value slightly lower than its nonasymptotic mean-field value of 1. This behavior in the nonasymptotic, high tUL region is interpreted in terms of the possibility of a nonmonotonic crossover to the mean-field value from lower values, as foreseen earlier in micellar systems.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

On a Degenerate Parabolic Problem in Holder Spaces

In this paper, we study some degenerate parabolic equation with Cauchy-Dirichlet boundary conditions. This problem is considered in little Holder spaces. The optimal regularity of the solution v is obtained and is specified in terms of those of the second member when some conditions upon the Holder exponent with respect to the degeneracy are satisfied. The proofs mainly use the sum theory of linear operators with or without density of domains and the results of smoothness obtained in the study of some abstract linear differential equations of elliptic type.

Efficient decoding with generative score-spaces using the expectation semiring

State-of-the-art speech recognisers are usually based on hidden Markov models (HMMs). They model a hidden symbol sequence with a Markov process, with the observations independent given that sequence. These assumptions yield efficient algorithms, but limit the power of the model. An alternative model that allows a wide range of features, including word- and phone-level features, is a log-linear model. To handle, for example, word-level variable-length features, the original feature vectors must be segmented into words. Thus, decoding must find the optimal combination of segmentation of the utterance into words and word sequence. Features must therefore be extracted for each possible segment of audio. For many types of features, this becomes slow. In this paper, long-span features are derived from the likelihoods of word HMMs. Derivatives of the log-likelihoods, which break the Markov assumption, are appended. Previously, decoding with this model took cubic time in the length of the sequence, and longer for higher-order derivatives. This paper shows how to decode in quadratic time. © 2013 IEEE.

Semiclassical treatment of Wannier's theory when the exponent diverges

We consider a non-standard application of the Wannier model. A physical example is the single ionization of a hydrogenic beryllium ion with a fully stripped beryllium ion, where the ratio of the charge of the third particle to the charges of the escaping particles is 1/4; we investigate the single ionization by an electron of an atom comprising an electron and a nucleus of charge 1/4. An infinite exponent is obtained suggesting that this process is not tractable within the Wannier model. A modified version of Crothers' uniform semiclassical wavefunction for the outgoing particles has been adopted, since the Wannier exponents and are infinite for an effective charge of Z = 1/4. We use Bessel functions to describe the Peterkop functions u and u and derive a new turning point ?. Since u is well behaved at infinity, there exists only the singularity in u at infinity, thus we employ a one- (rather than two-) dimensional change of dependent variable, ensuring that a uniform solution is obtained that avoids semiclassical breakdown on the Wannier ridge. The regularized final-state asymptotic wavefunction is employed, along with a continuum-distorted-wave approximation for the initial-state wavefunction to obtain total cross sections on an absolute scale. © 2006 IOP Publishing Ltd.

Accumulation of entanglement in a continuous variable memory

We study entanglement accumulation in a memory built out of two continuous variable systems interacting with a qubit that mediates their indirect coupling. We show that, in contrast with the case of bidimensional Hilbert spaces, entanglement superior to one ebit can be accumulated in the memory, even though no entangled resource is used. The protocol is immediately implementable and we assess the role of the main imperfections.