Description of the attainable sets of one-dimensional differential inclusions

The role played by the attainable set of a differential inclusion, in the study of dynamic control systems and fuzzy differential equations, is widely acknowledged. A procedure for estimating the attainable set is rather complicated compared to the numerical methods for differential equations. This article addresses an alternative approach, based on an optimal control tool, to obtain a description of the attainable sets of differential inclusions. In particular, we obtain an exact delineation of the attainable set for a large class of nonlinear differential inclusions.

Differential cross section measurements for the production of a W boson in association with jets in proton-proton collisions at √s=7 TeV

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Measurements of jet multiplicity and differential production cross sections of Z plus jets events in proton-proton collisions at root s=7 TeV

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Differential orthogonality: Laguerre and Hermite cases with applications

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A new deformation of W-infinity and applications to the two-loop WZNW and conformal affine Toda models

We construct a centerless W-infinity type of algebra in terms of a generator of a centerless Virasoro algebra and an abelian spin 1 current. This algebra conventionally emerges in the study of pseudo-differential operators on a circle or alternatively within KP hierarchy with Watanabe's bracket. Construction used here is based on a spherical deformation of the algebra W ∞ of area preserving diffeomorphisms of a 2-manifold. We show that this deformation technique applies to the two-loop WZNW and conformal affine Toda models, establishing henceforth W ∞ invariance of these models.

Dynamical algebra of quasi-exactly-solvable potentials

We show that by using second-order differential operators as a realization of the so(2,1) Lie algebra, we can extend the class of quasi-exactly-solvable potentials with dynamical symmetries. As an example, we dynamically generate a potential of tenth power, which has been treated in the literature using other approaches, and discuss its relation with other potentials of lowest orders. The question of solvability is also studied. © 1991 The American Physical Society.

Differential Expression of Immune Response Genes in Steller Sea Lions (Eumetopias jubatus): An Indicator of Ecosystem Health?

Characterization of the polygenic and polymorphic features of the Steller sea lion major histocompatibility complex (MHC) provides an ideal window for evaluating immunologic vigor of the population and identifying emergence of new genotypes that reflect ecosystem pressures. MHC genotyping can be used to measure the potential immunologic vigor of a population. However, since ecosystem-induced changes to MHC genotype can be slow to emerge, measurement of differential expression of these genes can potentially provide real-time evidence of immunologic perturbations. MHC DRB genes were cloned and sequenced using peripheral blood mononuclear leukocytes derived from 10 Steller sea lions from Southeast Alaska, Prince William Sound, and the Aleutian Islands. Nine unique DRB gene sequences were represented in each of 10 animals. MHC DRB gene expression was measured in a subset of six sea lions. Although DRB in genomic DNA was identical in all individuals, relative levels of expressed DRB mRNA was highly variable. Selective suppression of MHC DRB genes could be indicative of geographically disparate environmental pressures, thereby serving as an immediate and sensitive indicator of population and ecosystem health.

Virasoro Action on Imaginary Verma Modules and the Operator Form of the KZ-Equation

We define the Virasoro algebra action on imaginary Verma modules for affine and construct an analogue of the Knizhnik-Zamolodchikov equation in the operator form. Both these results are based on a realization of imaginary Verma modules in terms of sums of partial differential operators.

Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [I.M. Gelfand, V.A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Funct. Anal. Appl. 3 (1969) 325-326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators. (C) 2011 Elsevier Inc. All rights reserved.

Beyond the diffraction limit of optical/IR interferometers I. Angular diameter and rotation parameters of Achernar from differential phases

Context. Spectrally resolved long-baseline optical/IR interferometry of rotating stars opens perspectives to investigate their fundamental parameters and the physical mechanisms that govern their interior, photosphere, and circumstellar envelope structures. Aims. Based on the signatures of stellar rotation on observed interferometric wavelength-differential phases, we aim to measure angular diameters, rotation velocities, and orientation of stellar rotation axes. Methods. We used the AMBER focal instrument at ESO-VLTI in its high-spectral resolution mode to record interferometric data on the fast rotator Achernar. Differential phases centered on the hydrogen Br gamma line (K band) were obtained during four almost consecutive nights with a continuous Earth-rotation synthesis during similar to 5h/night, corresponding to similar to 60 degrees position angle coverage per baseline. These observations were interpreted with our numerical code dedicated to long-baseline interferometry of rotating stars. Results. By fitting our model to Achernar's differential phases from AMBER, we could measure its equatorial radius R-eq = 11.6 +/- 0.3 R-circle dot, equatorial rotation velocity V-eq = 298 +/- 9 km s(-1), rotation axis inclination angle i = 101.5 +/- 5.2 degrees, and rotation axis position angle (from North to East) PA(rot) = 34.9 +/- 1.6 degrees. From these parameters and the stellar distance, the equatorial angular diameter circle divide(eq) of Achernar is found to be 2.45 +/- 0.09 mas, which is compatible with previous values derived from the commonly used visibility amplitude. In particular, circle divide(eq) and PA(rot) measured in this work with VLTI/AMBER are compatible with the values previously obtained with VLTI/VINCI. Conclusions. The present paper, based on real data, demonstrates the super-resolution potential of differential interferometry for measuring sizes, rotation velocities, and orientation of rotating stars in cases where visibility amplitudes are unavailable and/or when the star is partially or poorly resolved. In particular, we showed that differential phases allow the measurement of sizes up to similar to 4 times smaller than the diffraction-limited angular resolution of the interferometer.

Hypoellipticity in spaces of ultradistributions-Study of a model case

In this work we study C (a)-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying Hormander's condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.