Function Space

In this paper we show that for a set X of real numbers the function space C p (X) has both a property introduced by Sakai in [Proc. Amer. Math. Soc. 104 (1988) 917-919] and a property introduced by Reznichenko (see [Topology Appl. 104 (2000) 181-190]) if and only if all finite powers of X have a property that was introduced by Gerlits and Nagy in [Topology Appl. 14 (1982) 151-161]. It follows that the minimal cardinality of a set of real numbers for which the function space does not have the properties of Sakai and Reznichenko is equal to the additivity of the ideal of first category sets of real numbers. 

An optimization-based approach to fault diagnosis for nonlinear stochastic dynamic models is developed. An optimal diagnosis problem is formulated according to a receding-horizon strategy. This approach leads to a functional optimization problem (also called 'infinite optimization problem'), whose admissible solutions belong to a function space. As in such a context, the tools from mathematical programing are either inapplicable or inefficient, a methodology of approximate solution is proposed that exploits diagnosis strategies made up of combinations of a certain number of simple basis functions, easy to implement and dependent on some parameters to be optimized. The optimization of the parameters is performed in two phases. In the first, 'a-priori' knowledge of the statistics of the stochastic variables is used to initialize (off-line) the parameter values. In the second phase, the optimization continues on-line. Both off-line and on-line phases rely upon stochastic approximation algorithms. The overall procedure turns out to be effective in high-dimensional settings such as those characterized by a large dimension of the state space and a large diagnosis window. This favorable behavior results from certain properties of the proposed methodology of approximate optimization, such as polynomial bounds on the rate of growth of the number of parameterized basis functions, which guarantees the desired accuracy of approximate optimization. The effectiveness of the approach is confirmed by simulations in the context of a complex instance of the fault-diagnosis problem. The advantages over classical approaches to fault diagnosis are discussed and pointed out by numerical results.

Cross-border regions form a specific case for transport management and policy. They have to face institutional, technical and financial obstacles caused by the frontier which can impede optimal planning of ecological and landscape-related issues. Public policy as well as managing of cross-border transport projects were affected by this. Moreover, growing mobility fosters the emergence of incongruity between spaces of living,

Quantum Topology" deals with the general quantum theory as the theory of quantum space. On the quantum level space time and energy momentum forms form a connected manifold; a functional quantum space. Many problems in quantum theory and field theory flow from not perceiving this symmetry and the functional nature of the quantum space. Both topology, groups and logic are based on the concept of sets. If properties coincide with the open sets of topology, then logic and topology will have the same structure. If transformations are continuous in topology, then we will have topological groups; we can derive fields. Therefore, quantum logic underlies the manifold and the fields and nature is based on the language of quantum logic. Quantum theory and field theory based on sets and the derived topology, group and logic structures should address the question of computation and the mind; the quantum computer and the quantum mind.

This paper is concerned with the variationally consistent incorporation of time dependent boundary conditions. The proposed methodology avoids ad hoc procedures and is applicable to both linear as well as nonlinear problems. An integral formulation of the dynamic problem serves as a basis for the imposition of the corresponding constraints, which are enforced via the consistent form of the penalty method, e.g. a form that complies with the norm and inner product of the functional space where the weak formulation is posed. Also, it is shown that well known and broadly implemented modelling techniques such as "large mass" and "large spring" methods, arise as limiting cases of the penalty formulation. Further extension of the proposed methodology is provided in the case where a generalized weak formulation of the dynamic problem with an independent velocity field is assumed. Finally, a few examples serve to illustrate the above concepts.

Assuming the compactification of 4 + K-dimensional space-time implied in Kaluza-Kleintype theories, we consider the case in which the internal manifold is a quotient space, G/H. We develop normal mode expansions on the internal manifold and show that the conventional ...

We introduce a strictly weaker version of the Daugavet property as follows: a Banach space X has this alternative Daugavet property (ADP in short) if the norm identity (aDE) max |ω|=1 Id + ωT = 1 + T holds for all rank-one operators T : X → X. In such a case, all weakly compact operators on X also satisfy (aDE). We give some geometric characterizations of the alternative Daugavet property in terms of the space and its successive duals. We prove that the ADP is stable for c 0 -, l 1 -and l∞-sums and characterize when some vector-valued function spaces have the property. Finally, we show that a C * -algebra (or the predual of a von Neumann algebra) has the ADP if and only if its atomic projection (resp. the atomic projection of the algebra) are central. We also establish some geometric properties of JB * -triples, and characterize JB * -triples possessing the ADP and the Daugavet property.

Given an operator L acting on a function space, the J-matrix method consists of finding a sequence y_n of functions such that the operator L acts tridiagonally on y_n with respect to n. Once such a tridiagonalization is obtained, a number of characteristics of such an operator L can be obtained. In particular, information on eigenvalues and eigenfunctions, bound states, spectral decompositions, etc. can be obtained in this way. We review the general set-up, and we discuss two examples in detail; the Schrodinger operator with Morse potential and the Lame equation.

We show that the inclusion of topological lagrangians in non-linear sigma models introduces certain topological non-trivial abelian background fields in the configuration space of these theories. In particular, the Hopf and the Wess-Zumino terms lead, respectively, to a vortex and a monopole in the corresponding configuration space for 2+1 and 3+1 dimensional theories. That topological solitons acquire anomalous spin and

Data analysis sometimes requires the relaxation of parametric assumptions in order to gain modeling flexibility and robustness against mis-specification of the probability model. In the Bayesian context, this is accomplished by placing a prior distribution on a function space, such as the space of all probability distributions or the space of all regression functions. Unfortunately, posterior distributions ranging over function spaces are highly complex and hence sampling methods play a key role. This paper provides an introduction to a simple, yet comprehensive, set of programs for the implementation of some Bayesian nonparametric and semiparametric models in R, DPpackage. Currently, DPpackage includes models for marginal and conditional density estimation, receiver operating characteristic curve analysis, interval-censored data, binary regression data, item response data, longitudinal and clustered data using generalized linear mixed models, and regression data using generalized additive models. The package also contains functions to compute pseudo-Bayes factors for model comparison and for eliciting the precision parameter of the Dirichlet process prior, and a general purpose Metropolis sampling algorithm. To maximize computational efficiency, the actual sampling for each model is carried out using compiled C, C++ or Fortran code.

The spaces of self-maps of spheres in both stable and unstable form are extremely important in algebraic topology and its applications to differential topology. Certain subspaces consisting of all maps which are equivariant with respect to suitable group actions have also arisen in several connections, particularly in the work of Becker [2], Rothenberg , and Browder and Petrie . For the purposes of these papers, more or less qualitative homotopy properties of the equivariant function spaces were adequate. The present paper is the first in a sequence aimed at describing the homotopy structure of these spaces more precisely and applying this structural information to the study of group actions on homotopy spheres.

The major histocompatibility complex (MHC) is highly polymorphic and more than 1500 human MHC alleles are known to date. These alleles do not bind to a given peptide with identical affinity. Although MHC alleles are functionally related, it is difficult to quantify the functional variation between them. Threedimensional structures of known MHC-peptide (MHCp) complexes suggest that specific peptide residues bind selectively to functional pockets in the binding groove. From a set of known MHCp structures we identified 21 critical polymorphic functional residue positions (CPFRP) that significantly reduced functional pocket variability to just 189 among 212 HLA-A alleles. Interestingly 101 HLA-A alleles clustered into 29 clusters such that the six functional pockets formed by the CPFRPs are identical within the cluster. Human Immunology 64, 718 -728 (2003).

In this seminar we try to explain why the Drury-Arveson space is important in operator theory, why it is interesting from the viewpoint of several complex variables, how it is related to the sub-Riemannian geometry of the Heisenberg group. We do this by ...

We consider nonparametric estimation of an object such as a probability density or a regression function. Can such an estimator achieve the minimax rate of convergence on suitable function spaces, while, at the same time, when "plugged-in", estimate efficiently (at a rate of n −1/2 with the best constant) many functionals of the object? For example, can we have a density estimator whose definite integrals are efficient estimators of the cumulative distribution function? We show that this is impossible for very large sets, e.g., expectations of all functions bounded by M < ∞.

In 1948, L. V. Kantorovich extended the Newton method for solving nonlinear equations to functional spaces. This event cannot be overestimated: the Newton-Kantorovich method became a powerful tool in numerical analysis as well as in pure mathematics. We address basic ideas of the method in historical perspective and focus on some recent applications and extensions of the method and some approaches to overcoming its local nature. Bibliography: 56 titles.

Elements based purely on completeness and continuity requirements perform erroneously in a certain class of problems. These are called the locking situations, and a variety of phenomena like shear locking, membrane locking, volumetric locking, etc., have been identified. Locking has been eliminated by many techniques, e.g. reduced integration, addition of bubble functions, use of assumed strain approaches, mixed and hybrid approaches, etc. In this paper, we review the field consistency paradigm using a function space model, and propose a method to identify field-inconsistent spaces for projections that show locking behaviour. The case of the Timoshenko beam serves as an illustrative example. Copyright © 2001 John Wiley & Sons, Ltd.

Gram Schmidt orthonormalization procedure is an important technique to get a set of orthonormal linearly independent set of vectors from a given set of linearly independent vectors, which are not orthonormal.  As an illustration, the set of Legendre polynomials is generated as the orthonormal set of functions in the function space spanned by linearly independent set of functions, {1,x,x^2,    … x^N…∞} in the interval {-1,1}.

In this contribution an empirical approach to global ocean tide and Mean Sea Level (MSL) modeling based on satellite altimetry observations is presented with all details. Considering the fact that the satellite altimetry technique can provide sea level observations in the global scale, spherical harmonics defined for the whole range of spherical coordinates (0 ≤ λ ≤ 2π, and −π/2 ≤ ϕ ≤ +π/2) could be among the possible choices for global ocean tide modeling. However, spherical harmonics when applied for modeling of global ocean tide, they lose their orthogonality due to the following reasons: (1) Observation of sea surface is made over discrete points, and not as a continuous function, which is needed for having the orthogonality property of spherical harmonics in functional space. (2) The range of application of spherical harmonics for global ocean tide modeling is limited to the sea areas covered by satellite altimetry observations and not the whole globe, which is also required for the fluffiness of the orthogonality of spherical harmonics. In this contribution we have shown how a set of orthonormal base functions at the sea areas covered by the satellite altimetry observations could be derived from spherical harmonics in order to solve the lack of the orthogonality. Using, the derived orthonormal base functions, global Mean Sea Level (MSL) model, and empirical global ocean tide models for six major semidiurnal and diurnal tidal constituents, namely, S2, M2, N2, K1, P1, and O1, as well as three long term tidal components, i.e., Mf, Mm, and Ssa, are developed based on six years of Jason-1 satellite altimetry sea level data as a numerical case study.

The functional space covered by the conjunctions and and but in English is divided between three conjunctions in Russian: i ‘and,’ a ‘and, but’ and no ‘but.’ We analyse these markers as topic management devices, i.e. they impose different kinds of constraints on the discourse topics (questions under discussion) addressed by their conjuncts. This paper gives a detailed review of the observations from descriptive literature on the distribution of these markers in light of the proposed underlying classification of questions, and shows that our theoretical approach provides a uniform explanation to a large variety of their uses, as well as to the existing equivalences and non-equivalences between the Russian and the English counterparts.

This paper describes a novel search algorithm, called dynamic hill climbing, that borrows ideas from genetic algorithms and hill climbing techniques. Unlike both genetic and hill climbing algorithms, dynamic hill climbing has the ability to dynamically change its coordinate frame during the course of an optimization. Furthermore, the algorithm moves from a coarse-grained search to a ne-grained search of the function space by changing its mutation rate and uses a diversity-based distance metric to ensure that it searches new regions of the space. Dynamic hill climbing is empirically compared to a traditional genetic algorithm using De Jong's well-known ve function test suite 4] and is shown to vastly surpass the performance of the genetic algorithm, often nding better solutions using only 1% as many function evaluations.

We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to di erent classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, some forms of Projection Pursuit Regression and several types of neural networks. We propose to use the term Generalized Regularization Networks for this broad class of approximation schemes that follow from an extension of regularization. In the probabilistic interpretation of regularization, the di erent classes of basis functions correspond to di erent classes of prior probabilities on the approximating function spaces, and therefore to di erent types of smoothness assumptions.

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L 2 (R d ), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.

The purpose of the theory of domains is to give models for spaces on which to define computable functions. The kinds of spaces needed for denotational sematics involve not only spaces of higher type (e.g. function spaces) but also spaces defined reeursively (e.g. reflexive domains). Also required are many special domain constructs (or functors) in order to create the desired structures. There are several choices of a suitable category of domains, but the basic one which has the simplest properties is the one sometimes called consistently complete algebraic cpo's. This category of domains is studied in this paper from a new, and it is to be hoped, simpler point of view incorporating the approaches of many authors into a unified presentation. Briefly, the domains of elements are represented set theoretically with the aid of structures called information systems. These systems are very familiar from mathematical logic, and their use seems to accord well with intuition. Many things that were done previously axiomatically can now be proved in a straightfoward way as theorems. The present paper discusses many examples in an informal way that should serve as an introduction to the subject.

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces. : 34C27, 42A75, 47H04.

We present here a method to simulate the motion of a rigid body in a fluid. The method is based on a variational formulation on the whole fluid/solid domain, with some constraints on the unknown and the test functions. These constraints are relaxed by introducing a penalty term, which leads to a minimization problem over unconstrained functional spaces. This makes the method straightforward to implement from any finite element Stokes/Navier-Stokes solver. It is shown that, as the penalty parameter goes to infinity, we recover the coupled fluid-solid equations. We apply this approach to a simplified 2D model of the aortic valve.

Let X and Y be Tychonoff spaces and C(X, Y) be the space of all continuous functions from X to Y. The coincidence of the fine topology with other function space topologies on C(X, Y) is discussed. Also cardinal invariants of the fine topology on C(X,W), where R is the space of reals, are studied. To answer some questions of Di Maio and Naimpally (1992) other function space topologies are investigated, namely, Krikorian topology, open-cover topology, graph topology, topology of uniform convergence, proximal graph topology. 0 1998 Elsevier Science B.V.

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in L p and order r in W 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

We solve the long standing problem of characterizing those Tychonov spaces X for which the function space, C κ (X), of realvalued continuous functions with the compact-open topology satisfies property (db). This property is one of a spectrum of algebraic-topological properties previously known to be distinct for locally convex spaces: In increasing strength, these properties are barrelled, Bairelike, property (db), unordered Bairelike, and Baire. Strongly Hewitt spaces, recently defined and studied by Kakol andŚliwa, are characterized here using unbounded filters on X, which shows that C κ (X) is a (db)-space. We give examples of C κ (X) distinguishing property (db) within the above spectrum and from those C κ (X) with X strongly Hewitt, answer a question of Kakol, and state some further problems.  2004 Elsevier B.V. All rights reserved. MSC: primary 54C35; secondary 54E52

Suppose that (M,d,m) is an unbounded metric measure space, which possesses two geometric properties, called &quot;isoperimetric property&quot; and &quot;approximate midpoint property&quot;, and that the measure m is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure m is not globally doubling. In this paper we define an atomic

Fusco 2]. They constructed a finite-dimensional manifold of states which have spherical interfaces within the domain and then decomposed the dynamical Cahn-Hilliard equation to a system comprised of a flow which is almost tangent to this manifold and a semiflow in the normal directions. Solutions to (1.1) are found very close to certain points on that manifold and hence have almost spherical interfaces, the location of the center being identified.

The complex exponential weighting of Feynman formalism is seen to happen at the classical level. (Finiteness of) Feynman path integral formula is suspected then to appear as a consistency condition for the existence of certain Dirac measures over functional spaces.

The classical theory of the Weierstrass transform is extended to a generalized function space which is the dual of a testing function space consisting of purely entire functions with certain growth conditions developed by Kenneth B. Howell. An inversion formula and characterizations for this transform are obtained. A comparative study with the existing literature is also undertaken.

We give a differential geometric framework for the description of (bi)modules, morphisms and reduction of star-products in deformation quantization in terms of multidifferential operators along maps. We show that algebra morphisms deform Poisson maps and left modules on function spaces deform coisotropic maps. Let C be a closed coisotropic submanifold of a Poisson manifold M : a star-product on M is representable (as differential operators on the space of smooth functions) on C iff it is equivalent to a star-product for which the vanishing ideal I[[ν]] of C is a left ideal in the deformed algebra ("coisotropic creed" by Lu and Weinstein). Moreover, for symplectic reduction the fact is important that deformation of the vanishing ideal I as a subalgebra automatically is either a left or right ideal in case M is symplectic. We show that a symplectic star-product is always representable on a coisotropic submanifold in case the reduced space exists, and we classify all bimodule structures on the function space on C (with respect to the deformed and the reduced algebra) in terms of its first de Rham cohomology. We show that in the symplectic case there are obstructions to the representability of a star-product which are related to a differential topological invariant of the foliation of the coisotropic submanifold, the so-called Atiyah-Molino class. If the latter vanishes, we prove that star-products with suitable Deligne classes can be represented. Finally, by a Fedosovian analysis we show that a Poisson map φ : M → M ′ between two symplectic manifolds is not always deformable into an algebra morphism : the obstructions are once again related to the Atiyah-Molino class of the symplectic orthogonal bundle of the kernel bundle of the tangent map of φ : if this class vanishes, we show that φ can be quantized subject to suitable conditions on the Deligne classes. The example of the cotangent bundle of the two-dimensional torus and complex projective space are discussed.

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents.

We study BMO spaces associated with semigroup of operators on noncommutative function spaces (i.e. von Neumann algebras) and apply the results to boundedness of Fourier multipliers on non-abelian discrete groups. We prove an interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 < p < ∞, with optimal constants in p. : 46L51 (42B25 46L10 47D06)

This paper continues a study of one and two variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satis ed by q-hypergeometric functions. Here we consider the quantum algebra Uq (su 2 ). We show that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the \group operators" on these representation spaces. This \local" approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. We show that the matrix elements themselves transform irreducibly under the action of the quantum algebra. We nd an alternate and simpler derivation of a q-analog, due to Groza, Kachurik and Klimyk, of the Burchnall-Chaundy formula for the product of two hypergeometric functions 2 F 1 . It is interpreted here as the expansion of the matrix elements of a \group operator" (via the exponential mapping) in a tensor product basis in terms of the matrix elements in a reduced basis.

We study BMO spaces associated with semigroup of operators and apply the results to boundedness of Fourier multipliers. We prove a universal interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 &lt; p &lt; \infty, with optimal constants in p.

In this paper the boundedness of Hardy-Littlewood maximal and singular operators in variable exponent Morrey spaces M p(·) q(·) (X) defined on spaces of homogeneous type is established provided that p and q satisfy Dini-Lipschitz (log-Hölder continuity) condition.

A. In this article we introduce Triebel-Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we give molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions.

We present a general mathematical framework for transforming functionally defined shapes. The proposed model of extended space mappings considers transformations of a hypersurface in coordinate-function space with its projection onto geometric space. This model covers coordinate space mappings, metamorphosis, and algebraic operations on defining functions, and introduces several new types of transformations, such as function-dependent space mappings and combined mappings. The approach is illustrated by new local deformations created by means of function mappings, feature-based space mapping, offsetting along the normal, thin shell generation, 2D shape blending, and collision-free metamorphosis.