Class Of Symbols Research Articles

Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields

We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a 𝔭-adic distribution interpolating the special values of the twisted Rankin–Selberg L-function attached to cuspidal automorphic representations π and σ of GLn and GLn–1 over a number field k. If π and σ are ordinary at 𝔭, our distribution is bounded and yields analyticity of the associated 𝔭-adic L-function.

Dunkl‐Sobolev spaces of exponential type and applications

We study the Sobolev spaces of exponential type associated with the Dunkl operators. Some properties including completeness and imbedding theorem are proved. Next we introduce a classes of symbols of exponential type and the associated pseudo‐differential‐difference operators, which naturally act on the Dunkl‐Sobolev spaces of exponential type. Finally using the theory of reproducing kernels some applications are given for these spaces.

On pseudodifferential operators with symbols in generalized Shubin classes and an application to Landau-Weyl operators

The relevance of modulation spaces for deformation quantization, Landau--Weyl quantization and noncommutative quantum mechanics became clear in recent work. We continue this line of research and demonstrate that $Q_s(\mathbb{R}^{2d})$ is a good class of symbols for Landau-Weyl quantization and propose that the modulation spaces $M^p_{v_s}(\mathbb{R}^{2d})$ are natural generalized Shubin classes for the Weyl calculus. This is motivated by the fact that the Shubin class $Q_s(\mathbb{R}^{2d})$ is the modulation space $M^2_{v_s}(\mathbb{R}^{2d})$. The main result gives estimates of the singular values of pseudodifferential operators with symbols in $M^p_{v_s}(\mathbb{R}^{2d})$ for the standard Weyl calculus and for the Landau--Weyl calculus.

Magnetic Pseudodifferential Operators with Coefficients in $C^*$-Algebras

In previous articles, a magnetic pseudodifferential calculus and a family of C^* -algebras associated with twisted dynamical systems were introduced and the connections between them have been established. We extend this formalism to symbol classes of Hörmander type with an x -behavior modeled by an abelian C^* -algebra. Some of these classes generate C^* -algebras associated with the twisted dynamical system. We show the relevance of these classes to the spectral analysis of pseudodifferential operators with anisotropic symbols and magnetic fields.

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

In (0,T)×Ω, Ω open subset of ℝ n , n≥2, we consider a parabolic operator P=∂ t −∇ x δ(t,x)∇ x , where the (scalar) coefficient δ(t,x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderon projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hormander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0,T)×M, M a manifold, is also derived from the local result.

Bilinear Fourier integral operators

We study the boundedness of bilinear Fourier integral operators on products of Lebesgue spaces. These operators are obtained from the class of bilinear pseudodifferential operators of Coifman and Meyer via the introduction of an oscillatory factor containing a real-valued phase of five variables $${\Phi(x,y_1,y_2,\xi_1,\xi_2)}$$ which is jointly homogeneous in the phase variables (ξ1, ξ2). For symbols of order zero supported away from the axes and the antidiagonal, we show that boundedness holds in the local-L 2 case. Stronger conclusions are obtained for more restricted classes of symbols and phases.

Commutator Criteria for Magnetic Pseudodifferential Operators

The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudodifferential operators of suitable symbol classes; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudodifferential operator.

The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C * -algebra that is associated to the algebra of symbols. There is a canonical map from this analytic K-theory group to the ordinary cohomology of the manifold, which gives a de Rham class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes' tangent groupoid proof of the index theorem.

The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part II

The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and involves analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C*-algebra of symbols. There is a canonical map from this analytic K-theory group to the deRham cohomology of the manifold, which gives a class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes' tangent groupoid proof of the index theorem.

Anisotropic classes of homogeneous pseudodifferential symbols

We define homogeneous classes of $x$-dependent anisotropic symbols $\dot{S}^{m}_{\gamma, \delta}(A)$ in the framework determined by an expansive dilation $A$, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces $${\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}$$ , where we set $${d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}$$ , with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.

Classical past in Baudelaire's Le cygne: a reconsideration

In the the third preface to Les Fleurs du Mal, Baudelaire curiously refers to Virgil as the only 'source' for his Le Cygne. Ithas been seen that Horace's description of the living poet's metamorphosis into a swan (Carmina 2.20) is a much more obvious classical reference as far as the title character is concerned, and the mention of »l'homme d'Ovide« seems to point the reader to Ovid's narrative of the creation of man in the Metamorphoses (1.76-86) as a model e contrario for the degradation of the divine bird in Baudelaire's poem. Baudelaire's modem version of the classical symbol of the sublime at first seems to suggest an ironic response to Horace and Ovid. On a second reading, however, the basic 'negativism' orsaudelaire's swan myth reveais a hidden thread of continuity with the classical past: it reveals Ovid's experience as an exilee as the primary parallel to the situations of Andromache and the swan. Conversely, Horace's swan-metamorphosis, though essentially Platonic, provides, through its over-literal, grotesque realism, an ante litteram alternative to the Platonising aesthetics of the earlier Romantics.

Singularities of solutions to the Schrödinger equation on scattering manifold

In this paper we study microlocal singularities of solutions to Schr\"odinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in terms of the initial condition and the classical scattering maps under the nontrapping condition. Our result is closely related to a recent work by Hassell and Wunsch, though our model is more general and the method, which relies heavily on scattering theoretical ideas, is simple and quite different. In particular, we use an Egorov-type argument in the standard pseudodifferential symbol classes, and avoid using Legendre distributions. In the proof, we employ a microlocal smoothing property in terms of the {\it radially homogenous wave front set}, which is more precise than the preceding results.

A dynamical systems perspective on the relationship between symbolic and non-symbolic computation

It has been claimed that connectionist (artificial neural network) models of language processing, which do not appear to employ "rules", are doing something different in kind from classical symbol processing models, which treat "rules" as atoms (e.g., McClelland and Patterson in Trends Cogn Sci 6(11):465-472, 2002). This claim is hard to assess in the absence of careful, formal comparisons between the two approaches. This paper formally investigates the symbol-processing properties of simple dynamical systems called affine dynamical automata, which are close relatives of several recurrent connectionist models of language processing (e.g., Elman in Cogn Sci 14:179-211, 1990). In line with related work (Moore in Theor Comput Sci 201:99-136, 1998; Siegelmann in Neural networks and analog computation: beyond the Turing limit. Birkhäuser, Boston, 1999), the analysis shows that affine dynamical automata exhibit a range of symbol processing behaviors, some of which can be mirrored by various Turing machine devices, and others of which cannot be. On the assumption that the Turing machine framework is a good way to formalize the "computation" part of our understanding of classical symbol processing, this finding supports the view that there is a fundamental "incompatibility" between connectionist and classical models (see Fodor and Pylyshyn 1988; Smolensky in Behav Brain Sci 11(1):1-74, 1988; beim Graben in Mind Matter 2(2):29--51,2004b). Given the empirical successes of connectionist models, the more general, super-Turing framework is a preferable vantage point from which to consider cognitive phenomena. This vantage may give us insight into ill-formed as well as well-formed language behavior and shed light on important structural properties of learning processes.

Compactly supported analytic indices for Lie groupoids

AbstractFor any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

Solving the hypergeometric system of Okubo type in terms of a certain generalized hypergeometric function

We introduce one scalar function f of a complex variable and finitely many parameters, which allows to represent all solutions of the so-called hypergeometric system of Okubo type under the assumption that one of the two coefficient matrices has all distinct eigenvalues. In the simplest non-trivial situation, f is equal to the hypergeometric function, while in other more complicated cases it is related, but not equal, to the generalized hypergeometric functions. In general, however, this function appears to be a new higher transcendental one. The coefficients of the power series of f about the origin can be explicitly given in terms of a generalized version of the classical Pochhammer symbol, involving two square matrices that in general do not commute. The function can also be characterized by a Volterra integral equation, whose kernel is expressed in terms of the solutions of another hypergeometric system of lower dimension.

Blurred Shape Model for binary and grey-level symbol recognition

Many symbol recognition problems require the use of robust descriptors in order to obtain rich information of the data. However, the research of a good descriptor is still an open issue due to the high variability of symbols appearance. Rotation, partial occlusions, elastic deformations, intra-class and inter-class variations, or high variability among symbols due to different writing styles, are just a few problems. In this paper, we introduce a symbol shape description to deal with the changes in appearance that these types of symbols suffer. The shape of the symbol is aligned based on principal components to make the recognition invariant to rotation and reflection. Then, we present the Blurred Shape Model descriptor (BSM), where new features encode the probability of appearance of each pixel that outlines the symbols shape. Moreover, we include the new descriptor in a system to deal with multi-class symbol categorization problems. Adaboost is used to train the binary classifiers, learning the BSM features that better split symbol classes. Then, the binary problems are embedded in an Error-Correcting Output Codes framework (ECOC) to deal with the multi-class case. The methodology is evaluated on different synthetic and real data sets. State-of-the-art descriptors and classifiers are compared, showing the robustness and better performance of the present scheme to classify symbols with high variability of appearance.

On the usefulness of modulation spaces in deformation quantization

We discuss the relevance to deformation quantization of Feichtinger's modulation spaces, especially of the weighted Sjöstrand classes . These function spaces are good classes of symbols of pseudodifferential operators (observables). They have a widespread use in time–frequency analysis and related topics, but are not very well known in physics. It turns out that they are particularly well adapted to the study of the Moyal star product and of the star exponential.

On the isometric composition operators on the Bloch space in Cn

Let $\varphi$ be a holomorphic self-map of a bounded homogeneous domain $D$ in $\mathbb{C}^n$. In this work, we show that the composition operator $C_\varphi: f\mapsto f\circ \varphi$ is bounded on the Bloch space $\mathcal{B}$ of the domain and provide estimates on its operator norm. We also give a sufficient condition for $\varphi$ to induce an isometry on $\cal{B}$. This condition allows us to construct non-trivial examples of isometric composition operators in the case when $D$ has the unit disk as a factor. We then obtain some necessary conditions for $C_\varphi$ to be an isometry on $\cal{B}$ when $D$ is a Cartan classical domain. Finally, we give the complete description of the spectrum of the isometric composition operators in the case of the unit disk and for a wide class of symbols on the polydisk.

Unified formulas for arbitrary order symbolic derivatives and anti-derivatives of the power-inverse hyperbolic class 1

We continue on tackling and giving a complete solution to the problem of finding the nth derivative and the nth anti-derivative, where n can be an integer, a fraction, a real, or a symbol, of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified to less general functions. In this work, we consider two subclasses of the power-inverse hyperbolic class. Namely, the power-inverse hyperbolic sine class { f ( x ) : f ( x ) = Σ l j =1 Pj ( x α j )arcsinh(β j x γ j ), α j ∈ C, β j ∈ C\{0},γ j ∈ R\{0}, (1) and the power-inverse hyperbolic cosine class { f ( x ) : f ( x ) = Σ l j =1 Pj ( x α j )arccosh(β j x γ j ), α j ∈ C, β j ∈ C\{0},γ j ∈ R\{0}, (2) where pj's are polynomials of certain degrees. One of the key points in this work is that the approach does not depend on integration techniques The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration. The motivation of this work comes from the area of symbolic computation. The idea is that: Given a function f in a variable x , can CAS find a formula for the n th derivative, the n th anti-derivative, or both of f ? This enhances the power of integration and differentiation of CAS. In Maple, the formulas correspond to invoking the commands diff( f ( x ) for the n th derivative and int( f ( x ), x$n ) for the n th anti-derivative. A software exhibition will be given using Maple. Example: A unified formula for arcsinh(√ x ) in terms of the Meijer G-function (arcsinh(√ x )) (n) = x (1/2-- n over2√π G 1,2 over 1,2 (1/2,1/2over0, n --1/2│ x ) , │ x │ &lt; 1. (3). The above G-function reduces to the original function if n = 0. It gives derivatives of any order if n &gt; 0 and anti-derivatives of any order if n &lt; 0.