Logic of Determination of Objects (LDO): How to Articulate ��Extension�� with ��Intension�� and ��Objects�� with ��Concepts��

From a logical viewpoint, object is never defined, even by a negative definition. This paper is a theoretical contribution about object using a new constructivist logical approach called Logic of Determination of Objects founded on a basic operation, called determination. This new logic takes into account cognitive problems such as the inheritance of properties by non typical occurrences or by indeterminate atypical objects in opposition to prototypes that are typical completely determinate objects. We show how extensional classes, intensions, more and less determined objects, more or less typical representatives of a concept and prototypes are defined and organized, using a determination operation that constructs a class of indeterminate objects from an object representation of a concept called typical object.     

��-Lines of Algebroid Functions

In this article several theorems of the theory of Γ-lines for meromorphic functions are extended to the more general setting of algebroid functions. We recall the definition of algebroid function of order k and how it can be considered as a function defined on a Riemann surface of k sheets. In this way, we prove the so called tangent variation principle for algebroid functions, previously proved for meromorphic functions by Barsegian, and we get several consequences of this result. We also extend a proposition on proximity properties of meromorphic functions.     

��-Homogeneity of Borel sets

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set C a sum of a countable number of pairwise disjoint h-homogeneous subspaces that are closed in X? It follows that every Borel set ${X \subset {\bf R}^n}$ can be partitioned into countably many h-homogeneous subspaces that are G _?? -sets in X.     

Higher asymptotics of unitarity in ��quantization commutes with reduction��

Let M be a compact K?hler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515?C538, 1982, no. 3), showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M //G. This map, though, is not in general unitary, even to leading order in ${\hslash}$ . Hall and Kirwin (Commun Math Phys 275:401?C422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed ${\hslash}$ , becomes unitary in the semiclassical limit ${\hslash\rightarrow0}$ (cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297?C302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). The unitarity of the classical Guillemin?CSternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M //G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as ${\hslash\rightarrow0}$ .     

On Newton-like methods of ��bounded deterioration�� using recurrent functions

We provide a semilocal convergence analysis for Newton-like methods of ??bounded deterioration?? in a Banach space setting. We establish tighter error bounds on the distances involved, and a more precise information on the location of the solution, under the same or weaker hypotheses than before (Argyros, Acta Math. Sin. (Engl. Ser.), 23:2087?C2096, 2007; Deuflhard, Newton methods for nonlinear problems. Affine invariance and adaptive algorithms, Springer Series in Computational Mathematics, vol. 35. Springer, Berlin, 2004; Ezquerro and Hern??ndez, IMA J. Numer. Anal., 22:187?C205, 2002) using recurrent functions. Numerical examples are also provided involving polynomial, integral, and differential equations.     

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 ? �� and 1 ? ��

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 ? ??) and (1 ? ??) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space $L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)$ .     

Diagonalization of matrices over the domain of principal ideals with minimal polynomial m(��) = (�� ? ��)(�� ? ��), �� �� ��

The necessary and sufficient conditions of diagonalization of matrices over a domain of principal ideals with minimal polynomial m(??) = (?? ? ??)(?? ? ??), ?? ?? ?? are obtained. On the basis of the obtained results, the conditions under which the matrices have common eigenvectors are indicated.     

On the Uniqueness of Hofer��s Geometry

We study the class of pseudo-norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such pseudo-norm that is continuous with respect to the C ^∞-topology, is dominated from above by the L _∞-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that is generated by an invariant pseudonorm that satisfies the aforementioned continuity assumption, is either identically zero or equivalent to Hofer’s metric.     

On the limits of Ces��ro means of polynomial powers

It is known that Cesàro means of polynomial powers of contractive operators in Hilbert spaces converge strongly. We address the question of whether the limit is a projection. We show that the only polynomials leading to projections for any operator are of degree at most one. Moreover, we find a spectral characterisation of operators in Hilbert spaces that have a projection as the limit of their polynomial Cesàro means for every reasonable polynomial.     

Correlation of the two-prime Sidel��nikov sequence

Motivated by the concepts of Sidel??nikov sequences and two-prime generator (or Jacobi sequences) we introduce and analyze some new binary sequences called two-prime Sidel??nikov sequences. In the cases of twin primes and cousin primes equivalent 3 modulo 4 we show that these sequences are balanced. In the general case, besides balancedness we also study the autocorrelation, the correlation measure of order k and the linear complexity profile of these sequences showing that they have many nice pseudorandom features.     

Syst��mes hamiltoniens compl��tement int��grables

This article is dedicated to one of the greatest mathematicians of our time: V.I. Arnold, who died suddenly Thursday, June 3, 2010 in France. Integrable hamiltonian systems are nonlinear ordinary differential equations described by a hamiltonian function and possessing sufficiently many independent constants of motion in involution. The regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following purely differential geometric fact: a compact and connected n-dimensional manifold on which there exist n vector fields which commute and are independent at every point is diffeomorphic to an n-dimensional real torus and each vector field will define a linear flow there. We make a careful study of the connection with the concept of completely integrable systems and we apply the methods to several problems.     

A new proof of Branson��s classification of elliptic generalized gradients

We give a representation theoretical proof of Branson??s classification (J Funct Anal 151(2):334?C383, 1997), of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is a local one, based on the relationship between ellipticity and optimal Kato constants and on the representation theory of ${\mathfrak{so}(n)}$ . Optimal Kato constants for elliptic operators were computed by Calderbank et?al. (J Funct Anal 173(1):214?C255, 2000). We extend their method to all generalized gradients (not necessarily elliptic) and recover Branson??s result, up to one special case. The interest of this method is that it is better suited to be applied for classifying elliptic sums of generalized gradients of G-structures, for other subgroups G of the special orthogonal group.     

Projection pencils of quadrics and Ivory��s theorem

Using selfadjoint regular endomorphisms, the authors of Stachel and Wallner (Sib Math J 45(4):785?C794, 2004) defined, for an indefinite inner product, a variant of the notion of confocality for the Euclidean space. Our aim is to give a definition that is a common generalization of the usual confocality, and the variant in Stachel and Wallner (Sib Math J 45(4):785?C794, 2004). We use this definition to prove a more general form of Ivory??s theorem.     

Local bifurcation-branching analysis of global and ��blow-up�� patterns for a fourth-order thin film equation

Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4)

On certain q-analogue of Sz��sz Kantorovich operators

In the present paper we propose q-analogue of the well known Sz??sz-Kantorovich operators. We study local approximation as well as weighted approximation properties of these new operators.     

On exact solutions of Stokes second problem for a Burgers�� fluid, II. The cases �� =?��2/4 and �� > ��2/4

In this study, the exact solutions of the Stokes second problem for a Burgers?? fluid are presented when the relaxation time satisfies the conditions ?? =???²/4 and ?? >???²/4. The velocity field and the associated tangential stress, when only one initial condition is necessary for velocity, are determined by means of the Laplace transform. The physical interpretation for the emerging parameters is discussed with the help of graphical illustrations. The similar solutions for the Stokes?? first problem are obtained as the limiting cases of our solutions.     

Shell model of turbulence perturbed by L��vy noise

In this work we prove the existence and uniqueness of the strong solution of the shell model of turbulence perturbed by Lévy noise. The local monotonicity arguments have been exploited in the proofs.     

A simple generalization of Ger?gorin��s theorem

It is well known that the spectrum of a given matrix A belongs to the Ger?gorin set ??(A), as well as to the Ger?gorin set applied to the transpose of A, ??(A ^T ). So, the spectrum belongs to their intersection. But, if we first intersect i-th Ger?gorin disk ?? _i (A) with the corresponding disk $\Gamma_i(A^T)$ , and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set ??(A)??????(A ^T ). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.     

On the well-posedness of the vacuum Einstein��s equations

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g _αβ of a spacetime with vanishing Ricci curvature R _α,β and prescribed initial data. Under the harmonic gauge condition, the equations R _α,β  = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric h _ab and a second fundamental form K _ab . A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h _ab , K _ab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of the present article is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.