Revista Matemática Complutense (ISSN 1139-1138, ISSN-e 1988-2807), currently published three times a year, it includes original or literature reviews that are carefully selected from all areas of Mathematics including the “Conferenciante Santaló” article, written by a designated prestigious mathematician. Since January 2010, Springer has been responsible for its publication
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Navier-Stokes/darcy coupling: modeling, analysis, and numerical approximation
This paper is an overview of known and new results about the coupling of Navier-Stokes and Darcy equations to model the filtration of incompressible fluids through porous media. We discuss coupling conditions and we analyze the global coupled model in order to prove its well-posedness and to characterize effective algorithms to compute the solution of its numerical approximation.
Coincidence of some classes of universal functions
Let Ω a domain in the complex plane such that Ω satisfies appropriate geometrical and topological properties. We prove that if f is a holomorphic function in Ω, then its Taylor series, with center at any ξ Є Ω , is universal with respect to overconvergence if and only if its Cesáro (C, k)-means are universal for any real k > −1. This is an extension of the same result, proved recently by F. Bayart, for any integer k ≥ 0. As a consequence, several classes of universal functions introduced in the related literature are shown to coincide.
Monodromy zeta-functions of deformations and Newton diagrams
For a one-parameter deformation of an analytic complex function germ of several variables, there is defined its monodromy zeta-function. We give a Varchenko type formula for this zeta-function if the deformation is non-degenerate with respect to its Newton diagram.
Asymptotic uniform moduli and Kottman constant of Orlicz sequence spaces
We give lower and upper bounds, involving moduli of asymptotic uniform convexity and smoothness, for the Kottman separation constant of Orlicz sequence spaces equipped with the Luxemburg norm.
Weighted composition operators between spaces of Dirichlet type
In this work we characterize boundedness and compactness of weighted composition operators acting between Dirichlet type spaces by using Carleson measures. We also find essential norm estimates for these operators.
The SL(2,C)-character varieties of torus knots
Let G be the fundamental group of the complement of the torus knot of type (m, n). This has a presentation G = ‹x, y | xm= yⁿ›. We find the geometric description of the character variety X(G) of characters of representations of G into SL(2,C).
Symmetrization and sharp Sobolev inequalities in metric spaces
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for H¨ormander vector fields.
Erratum to “Fundamental groups of some special quadric arrangements”
This erratum relates to our work “Fundamental groups of some special quadric arrangements”. The original Theorems 2.2, 2.5, 2.8 and Propositions 2.3(ii)(iii), 2.6(ii)(iii), 2.9(ii)(iii) have wrong results. They need to be rephrased. Corollaries 2.4 and 2.7 are incomplete, and they are extended. We add a new Corollary 2.10, which does not appear in the original paper. Proposition 3.1 has a wrong result and it is rephrased and reproved. In Proposition 4.1 and its Corollary 4.2 a slight error has occurred: as the correct proofs in the paper show, the monodromy is a quadruple fulltwist.
Uniqueness of Entropy Solutions of Nonlinear Elliptic-Parabolic-Hyperbolic Problems in One Dimension Space
We consider a class of elliptic-parabolic-hyperbolic degenerate equations of the form b(u)t — a(u, φ(ux)x= f with homogeneous Dirichlet conditions and initial conditions. In this paper we prove an L1-contraction principle and the uniqueness of entropy solutions under rather general assumptions on the data.
Renormalized Solutions for Nonlinear Degenerate Elliptic Problems with L1 Data
We are interested in a class of nonlinear degenerate diffusion problems with a diffusion function a(x, u, Vu) which is not controlled with respect to u and which is not uniformly coercive on the weighted Sobolev spaces W1,p0 (Ω, w). Existence of a renormalized solution is proved in the L1-setting.
Semistability of Certain Bundles on a Quintic Calabi-Yau Threefold
In a recent paper Douglas and Zhou aim for explicit examples of string theory compactifications that have a different number of generations and can be connected. For this purpose, they provide a list of bundles on a quintic Calabi-Yau threefold. They need to show that (at least some of) these bundles are semistable and leave this as an open question. In this paper we prove the semistability of most of the bundles in the list, thus completing the result of Douglas and Zhou.
Solving Variational Inclusions by a Method Obtained Using a Multipoint Iteration Formula
This paper deals with variational inclusions of the form: 0 ε f(x)+F(x) where f is a single function admitting a second order Fréchet derivative and F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (xk) satisfying 0 ε f(xk)+∑M i=1 aiΛf(xk+βi(xk+1-xk))(xk+1-xk)+F(xk+1) where the single-valued function involved in this relation is an approximation of the function f based on a multipoint iteration formula and we show that this method is locally cubically convergent.
A Discrete Hardy-Laptev-Weidl-Type Inequality and Associated Schrödinger-Type Operators
Although the classical Hardy inequality is valid only in the three- and higher dimensional case, Laptev and Weidl established a two-dimensional Hardy-type inequality for the magnetic gradient with an Aharonov-Bohm magnetic potential. Here we consider a discrete analogue, replacing the punctured plane with a radially exponential lattice. In addition to discrete Hardy and Sobolev inequalities, we study the spectral properties of two associated self-adjoint operators. In particular, it is shown that, for suitable potentials, the discrete Schrödingertype operator in the Aharonov-Bohm field has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satisfies a CLR-type inequality.
Existence of Renormalized Solution of Some Elliptic Problems in Orlicz Spaces
In this paper, we study the problem:—div a(x, u, ∆u) — div Φ(u) + g(x; u) = f in the framework of Orlicz spaces. The main contribution of our work is to prove the existence of a renormalized solution without any restriction on the N-function of the Orlicz space.
We consider dilation operators Tk : f → f(2k.) in the framework of Besov spaces Bsp,q (Rn) when 0 < p≤ 1. If s > n(1/p — 1) Tk is a bounded linear operator from Bsp,q (Rn) into itself and there are optimal bounds for its norm. We study the situation on the line s = n(1/p — 1), an open problem mentioned in [5, 2.3.1, 2.3.2]. It turns out that the results shed new light upon the diversity of different approaches to Besov spaces on this line, associated to definitions by differences, Fourier-analytical methods, and subatomic decompositions.
Newton Binomial Formulas in Schubert Calculus
We prove Newton’s binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.
Extremal Vector Valued Inequalities for Hankel Transforms
Boundary Sentinels with Given Sensitivity
The notion of sentinels with given sensitivity was introduced by J.-L. Lions in [10] in order to identify parameters in a problem of pollution ruled by a semilinear parabolic equation. He proves that the existence of such sentinels is reduced to the solution of exact controllability problem with constraints on the state. Reconsidering this notion of sentinels in a more general framework, we prove the existence of the new sentinels by solving a boundary null-controllability problem with constraint on the control. Our results use a Carleman inequality which is adapted to the constraint.
A Symbolic Calculus and a Parametrix Construction for Pseudodifferential Operators with Non-Smooth Negative Definite Symbols
We consider pseudodifferential operators that have non-smooth negative definite symbols and develop a corresponding symbolic calculus. Combining this symbolic calculus with the use of non-smooth symbols that are asymptotically constant in the co-variable we succeed infunding a parametrix for a certain pseudodifferential equation. This in turn allows us to show that some pseudodifferential operators with non-smooth negative definite symbols are pregenerators of Feller semigroups.
We study the Banach envelope of the quasi-Banach space l1,∞ consisting of all sequences x=(ξ k) with sn(x)=O(1/n), where (sn (x)) denotes the non-increasing rearrangement of x=(ξ k) The situation turns out to be much more complicated than that in the well-known case of the separable subspace lo1,∞, whose members are characterized by sn(x) =o(1/n).
2-Microlocal Besov and Triebel-Lizorkin Spaces of Variable Integrability
We introduce 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability and give a characterization by local means. These spaces cover spaces of variable exponent, spaces of variable smoothness and weighted spaces that have been studied in recent years.
Besov and Triebel-Lizorkin Spaces Related to Singular Integrals with Flag Kernels
Let s1, s2 Є (—1, 1) and s = (s1, s2). In this paper, the author introduces the Besov space Bspqq(R2) with p, q Є [1, ∞] and the Triebel-Lizorkin space Fspqq(R2) with p Є (1, ∞) and q Є (1, ∞] associated to singular integrals with flag kernels. Some basic properties, including their dual spaces, some equivalent norm characterizations via Littlewood-Paley functions, lifting properties and some embedding theorems, on these spaces are given. Moreover, the au thor obtains the boundedness of flag singular integrals and fractional integrals on these spaces.
Erratum to “Feller Semigroups Obtained by Variable Order Subordination" (Rev. Mat. Complut. 20 (2007), no. 2, 293-307)
Divergent Cesàro Means of Jacobi-Sobolev Expansions
Let µ be the Jacobi measure supported on the interval [­1; 1]. Let introduce the Sobolev-type inner product …. In this paper we prove that, for certain indices δ, there are functions whose Ces_aro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product are divergent almost everywhere on [-1; 1].
Strongly Invertible Knots, Rational-Fold Branched Coverings, and Hyperbolic Spatial Graphs
A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperbolic spatial graphs via the construction.
A General Hilbert Space Approach to Framelets
In arbitrary separable Hilbert spaces it is possible to deffine multiscale methods of constructive approximation based on product kernels, restricting their choice in certain ways. These wavelet techniques have already filtering and localization properties and they are applicable in many areas due to their generalized deffinition. But they lack detailed information about their stability and redundancy, which are frame properties. So in this work frame conditions are introduced for approximation methods based on product kernels. In order to provide stability and redundancy the choice of product kernel ansatz function has to be restricted. Taking into account the kernel conditions for multiscale and for frame approximations one is able to deffine wavelet frames (= framelets), inheriting the approximation properties of both techniques and providing a more precise tool for multiscale analysis than the normal wavelets.
Weighted Composition Operators between Weighted Bergman Spaces and Weighted Banach Spaces of Holomorphic Functions
We characterize boundedness and compactness of weighted composition operators acting between weighted Bergman spaces Aw;p and weighted Banach spaces H1v of holomorphic functions.
On the Distribution of the Power Generator over a Residue Ring for Parts of the Period
This paper studies the distribution of the power generator of pseudorandom numbers over a residue ring for parts of the period. These results compliment some recently obtained distribution bounds of the power generator modulo an arbitrary number for the entire period. Also, the arbitrary modulus case may have some cryptography related applications and could be of interest in other settings which require quality pseudorandom numbers.
Note on Coupled Linear Systems Related to Two Soliton Collision for the Quartic gKdV Equation
We consider linear systems related to the description of the collision of two solitons for the gKdV equation with quartic nonlinearity. The computations presented in this note are applied in Martel and Merle [10] to prove a result concerning inelastic (but almost elastic) collision for a nonintegrable equation.
Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square
While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computerassisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczy´nski [22], and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.