On the distributions of the form ∑i (δpi−δni)

We present some properties of the distributions T of the form ∑_i (δ_pi−δ_ni), with ∑_i d(p_i,n_i)<∞, which arise in the study of the 3-d Ginzburg–Landau problem; see Bourgain et al. (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). We show that there always exists an irreducible representation of T. We also extend a result of Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374) which says that T is a measure iff T can be written as a finite sum of dipoles.     

σ-core and -core of bounded sequences

In this paper we characterize matrices that map every bounded sequence into one whose σ-core is a subset of the -core of the original sequence.     

The “Error” in the Indian “Taylor Series Approximation” to the Sine

It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician M dhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by M dhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen M dhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von M dhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32.     

The SC property of an expected residual function arising from stochastic complementarity problems

The stochastic nonlinear complementarity problem has been recently reformulated as an expected residual minimization problem which minimizes an expected residual function defined by an NCP function. In this work, we show that the expected residual function defined by the Fischer–Burmeister function is an function.     

Periodic solutions of second order nonlinear difference equations with discrete -Laplacian

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second order difference equations involving discrete -Laplacian. We obtain in particular upper and lower solutions theorems, Ambrosetti–Prodi type multiplicity results, sharp existence conditions for nonlinearities which are bounded from below or from above and necessary and sufficient conditions for the existence of positive periodic solutions when the nonlinearity is singular at 0.     

Approximation theorems for localized szsz–Mirakjan operators

In the present paper, we investigate the convergence and the approximation order of the localized Szsz–Mirakjan operators, and obtain some new results to improve the results due to Omey [Note on operators of Szsz–Mirakjan type, J. Approx. Theory 47 (1986) 246–254].     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

Nonlinear structure of some classical quasi-Banach spaces and -spaces

We investigate the Lipschitz structure of ℓ_p and L_p for 0<p<1 as quasi-Banach spaces and as metric spaces (with the metric induced by the p-norm) and show that they are not Lipschitz isomorphic. We prove that the -space L₀ is not uniformly homeomorphic to any other L_p space, that the L_p spaces for 0<p<1 embed isometrically into one another, and reduce the problem of the uniform equivalence amongst L_p spaces to their Lipschitz equivalence as metric spaces.     

Projective functors for algebraic groups

We investigate the regular subquotient category introduced by Soergel in [W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (1–3) (2000) 311–335]. A detailed study of projective functors enables one to relate those categories for semi-simple algebraic group G and its subgroup schemata G_rT. As an application we derive some information about the characters of tilting modules for G.     

Representation type of the blocks of category in types and

In this paper a complete classification of the representation type of the infinitesimal blocks of the parabolic category for the complex simple Lie algebras of types F₄ and G₂ is given.     

Existence and multiplicity results for some nonlinear problems with singular -Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems

where l(u,u^′)=0 denotes the Dirichlet, periodic or Neumann boundary conditions on [0,T], is an increasing homeomorphism, (0)=0. The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.     

On nonatomic submeasures on . II

It is shown that if two submeasures on that are lim sup's of sequences of measures have the same zero sets, and one is nonatomic, so is the other, and they are (ε-δ)-equivalent. Moreover, if a submeasure η is the lim sup of a sequence of lower semi-continuous (lsc) submeasures and is 0-dominated by the so-called core γ^• of an lsc submeasure γ, then η is also (ε-δ)-dominated by γ^•. And if a submeasure η of this type is 0-dominated by a nonatomic submeasure, then it is nonatomic as well.     

A non-splitting theorem in the enumeration degrees

We complete a study of the splitting/non-splitting properties of the enumeration degrees below by proving an analog of Harrington’s non-splitting theorem for the enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.     

Finding induced trees

Let G=(V(G),E(G)) be a finite connected undirected graph and WV(G) a subset of vertices. We are searching for a subset XV(G) such that WX and the subgraph induced on X is a tree. -completeness results and polynomial time algorithms are given assuming that the cardinality of W is fixed or not. Besides we give complexity results when X has to induce a path or when G is a weighted graph. We also consider problems where the cardinality of X has to be minimized.     

Finite element analysis of an A– method for time-dependent Maxwell’s equations based on explicit-magnetic-field scheme

In this paper, error analysis of a finite element A– method for the time-dependent Maxwell’s equations is presented. An explicit-magnetic-field scheme is applied. Provided that the time-stepsize τ is sufficiently small, the proposed algorithm yields for finite time T an error of in the L²-norm for the electric field E and the magnetic field H, where h is the mesh size.     

A Constructive Proof of the Gelfand—Kolmogorov Theorem

A construction of the Stone—ech compactification of a locale L is presented in this paper as a quotient of the frame of radical ideals of the algebra C ^*(L). As a corollary, a constructive, localic version of the Gelfand—Kolmogorov theorem is obtained.