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A left canonical factorization theorem for rational matrix functions relative to the unit circle is presented. The result is a time invariant version of a recent strict LU factorization theorem for certain semi-separable operators, due to Dewilde (2012) [6]. Explicit formulas for the factors are also given. The theorem is proved, first by using the state space method of Dewilde (2012) [6], and next by using an operator theory approach. In both cases the main part of the proof concerns rational matrix functions that are unitary on the unit circle. © 2012 Royal Dutch Mathematical Society (KWG).

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.

In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments cn(?) ? cn = ?10tnd?(t), n = 0, 1, 2, ..., of some singular distributions ? (with support ? 0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).

The date of publication of Helmholtz's first paper on the foundations of geometry is discussed.

As early as the 16th century, Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the Euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence.

The unpublished scientific papers of Joseph Liouville located in the Institut de France in Paris have been investigated by several scholars in recent years, and it has been pointed out on a number of occasions that the Paris collection is incomplete. The present article analyzes the history of Liouville's unpublished papers and gives an initial survey of additional papers discovered in Bordeaux in 1983–1984. Besides the material originating from Liouville, the Bordeaux collection also contains many manuscripts of Charles-Francois Sturm which had long been searched for and had therefore been assumed by most to have been lost. Furthermore, there are documents connected with the transmission and editing of the unpublished papers of Evariste Galois, which were also once in Liouville's possession.

That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.

Brief mention is made of early work in the design of experiments by Galen, Avicenna, and Laplace, after which a seemingly unknown 1815 paper of Gergonne's on polynomial regression is introduced and discussed, and a translation of the paper presented.

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

The present article discusses the life and work of Erich Bessel-Hagen (1898-1946), a mathematician and historian of mathematics at Bonn. In addition to an overview of his life and mathematical career, it presents an appreciation of his scientific merits and a list of his publications. Furthermore, his extensive unpublished scientific papers are presented in detail for the first time, as is the material from the estates of F. Hausdorff, F. London, O. Toeplitz, F. Klein, etc., which Bessel-Hagen collected and preserved from destruction. Together, these collections include more than 100 mathematical lecture notes and more than 1000 letters, and therefore deserve finally to be made known to a broader public.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincar\'e metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power., Comment: 42 pages, 2 figures; v.2 is a final update to agree with the published paper

We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1 , under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ) . This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of [28] , [29] . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I ∈ I ⊂ R d . We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [28] , [29] . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31] —notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28] , [29] .

In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λ φ initiated in the papers of Foralewski et al. (2008) [16] , [17] (cf. also Foralewski (2011) [11] , [12] ). As we will show in Example 1.1 , Example 1.2 , Example 1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ 2 λ , then λ φ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λ φ , a strong condition of δ 2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λ φ , of their subspaces of order continuous elements ( λ φ ) a as well as of finite dimensional subspaces λ φ n of λ φ are presented.

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Buckner in two- and three-dimensional Euclidean spaces. We prove that this generalized Buckner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces. The paper also addresses the study of an elegant completion procedure due to Maehara in Euclidean spaces, the natural setting of which are the spaces with a generating unit ball. We prove that, in these spaces, the Maehara completion is also locally Lipschitz continuous, besides establishing other geometric properties of this completion. The paper contains also new estimates of the (local) Lipschitz constants for the wide spherical hull.

In our paper [10] we discussed an error in the recent literature on abstract fractional differential equations and we proposed a different approach to treat these type of problems. By noting that the results by Hernandez et al. (2010) [10] are not applicable for problems with nonlocal conditions, in this paper we study the existence of mild solutions for a class of abstract fractional differential equations with nonlocal conditions. An application involving a partial differential equation with a fractional temporal derivative and nonlocal conditions is considered.

This paper deals with the semi-implicit scheme with respect to the t -variable combined with a finite element spatial approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms. We establish a convergence and a quasi-optimal L ∞ -asymptotic behavior, involving a weakly coupled system of discrete parabolic quasi-variational inequalities (PQVIs), for the solution of which an iterative discrete scheme of monotone kind is introduced and analyzed. Furthermore, the simple numerical example shows that the estimates introduced in this paper are efficient.

Invertibility in Banach algebras generated by two idempotents can be checked with the help of a theorem by Roch, Silbermann, Gohberg, and Krupnik. This theorem cannot be used to study generalized invertibility. The present paper is devoted to group invertibility in two types of finite-dimensional algebras which are generated by two idempotents, algebras generated by two tightly coupled idempotents on the one hand and algebras of dimension at most four on the other. As a side product, the paper gives the classification of all at most four-dimensional algebras which are generated by two idempotents.

In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

This paper is concerned with a class of reflected backward stochastic differential equations (RBSDEs in short) with two barriers. The first purpose of the paper is to establish existence and uniqueness results of adapted solutions for such RBSDEs. Most of existing results on adapted solutions for RBSDEs with two barriers are heavily based on either the Mokobodski condition or other restrictive regularity conditions. In this paper, the two barriers are modeled by stochastic differential equations with coefficients satisfying the local Lipschitz condition and the linear growth condition, which enables us to weaken the regularity conditions on the boundary processes. Existence is proved by a penalization scheme together with a comparison theorem under the Lipschitz condition on the coefficients of RBSDEs. As an application, it is proved that the initial value of an RBSDE with two barriers coincides with the value function of a certain Dynkin game under Knightian uncertainty.

This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\R[t,t^{-1}]), and for the formal loop group GL_n(\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeroes of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the ``factorization problem'' using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlev\'{e} dynamical systems and Berenstein-Kazhdan's study of geometric crystals., Comment: 49 pages, 7 figures

This article extends results contained in Buzzi et al. (2006) [4] , Llibre et al. (2007, 2008) [12] , [13] concerning the dynamics of non-smooth systems. In those papers a piecewise C k discontinuous vector field Z on R n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F : U → R a polynomial function defined on the open subset U ⊂ R n . The set F − 1 ( 0 ) divides U into subdomains U 1 , U 2 , … , U k , with border F − 1 ( 0 ) . These subdomains provide a Whitney stratification on U . We consider Z i : U i → R n smooth vector fields and we get Z = ( Z 1 , … , Z k ) a discontinuous vector field with discontinuities in F − 1 ( 0 ) . Our approach combines several techniques such as e-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an e-regularization of Z (see Sotomayor and Teixeira, 1996 [18] ; Llibre and Teixeira, 1997 [15] ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16] ), in systems with hysteresis (Seidman, 2006 [17] ) and in mechanical systems with impacts (di Bernardo et al., 2008 [5] ).

In this paper, we give a classification of (regular) automorphism groups of relatively minimal rational elliptic surfaces with section over the field C which have non-constant J-maps. The automorphism group Aut ( B ) of such a surface B is the semi-direct product of its Mordell–Weil group MW ( B ) and the subgroup Aut σ ( B ) of the automorphisms preserving the zero section σ of the rational elliptic surface B. The configuration of singular fibers on the surface determines the Mordell–Weil group as has been shown by Oguiso and Shioda (1991), and Aut σ ( B ) also depends on the singular fibers. The classification of automorphism groups in this paper gives the group Aut σ ( B ) in terms of the configuration of singular fibers on the surface. In general, Aut σ ( B ) is a finite group of order less than or equal to 24 which is a Z / 2 Z extension of either Z / n Z , Z / 2 Z × Z / 2 Z , D n (the Dihedral group of order 2n) or A 4 (the Alternating group of order 12). The configuration of singular fibers does not determine the group Aut σ ( B ) uniquely; however we list explicitly all the possible groups Aut σ ( B ) and the configurations of singular fibers for which each group occurs.

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers., Comment: 46 pages, 11 figures

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and F 1 -schemes (after Kato, Deitmar and Connes–Consani). Beside this unification, the category of blueprints contains new interesting objects as “improved” cyclotomic field extensions F 1 n of F 1 and “archimedean valuation rings”. It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Titsʼ idea of Chevalley groups over F 1 , congruence schemes, sheaf cohomology, K-theory and a unified view on analytic geometry over F 1 , adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

This paper is a posthume continuation of the papers by Professor A.H. Boers about duplication of algebras, as drafted by Artibano Micali and Moussa Ouattara in collaboration with Nakelgbamba Boukary Pilabre. The aim of the paper is essentially to study the non commutative duplication of algebras. Conditions are studied in which the duplicate algebra is flexible, Lie admissible or n -associative. Connections between the dimensions of commutative and non commutative duplication of algebras are compared. In the list of references in this paper, several items have been given, which are directly connected to the subject of study. Moreover, for the interest of the reader, a (possible) complete list of publications of A.H. Boers is added to the references, i.e. known up to and including 1995 as well as a note explicative.

We exploit the geometric approach to the virtual fundamental class, due to Fukaya–Ono and Li–Tian, to compare Gromov–Witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. Various special cases of the comparison theorem in this paper have long been used in the algebraic category; some of them have also appeared in the symplectic setting. Combined with the inherent flexibility of the symplectic category, the main theorem leads to a confirmation of Pandharipandeʼs Gopakumar–Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. The proof of the main theorem uses deformations of the Cauchy–Riemann equation that respect the submanifold and Carleman Similarity Principle for solutions of perturbed Cauchy–Riemann equations. In a forthcoming paper, we apply a similar approach to relative Gromov–Witten invariants and the absolute/relative correspondence in genus 0.

We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi ε-periodic function, ε being a small parameter, while the mean thickness of the layer is of magnitude εβ, where β∈(0,1). Using the two-scale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor as defined in Vogelius and Capdeboscq [Y. Capdeboscq, M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal. 37 (2003) 159–173]. The main result of this paper is quite unexpected, and the approximate transmission conditions are not intuitive since they mix in a complex way both conductivities of the exterior medium and of the membrane. This paper extends the previous works of Poignard [C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Meth. Appl. Sci. 32 (2009) 435–453] and of Ciuperca et al. [I. Ciuperca, M. Jai, C. Poignard, Approximate transmission conditions through a rough thin layer. The case of periodic roughness, Euro. J. Appl. Math. 21 (2010) 51–75], in which β⩾1.

In this paper, we define a notion of AS-Gorenstein algebra for N -graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S= +\lambda , $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time., Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theory

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves., Comment: This a revised version (with 59 pages now) of our paper on realizations of braided categories, where we have taken into account the referee's report. Indeed, we are much indebted to the referee, whose useful observations greatly improved our manuscript

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research., Comment: 14 pages

Let Ω be a bounded domain in R 2 , u + = u if u ⩾ 0 , u + = 0 if u 0 , u − = u + − u . In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma ( P λ ) { Δ u − λ u − = 0 , in Ω , u = c , on ∂ Ω , ∫ ∂ Ω ∂ u ∂ ν d s = I , where ν is the outward unit normal of ∂Ω at x, c is a constant which is unprescribed, and I is a given positive constant. The set Ω p = { x ∈ Ω , u ( x ) 0 } is called plasma set. Existence of solutions whose plasma set consisting of one component and asymptotic behavior of plasma set were studied by Caffarelli and Friedman (1980) [3] for large λ. Under the condition that the homology of Ω is nontrivial we obtain in this paper by a constructive way that for any given integer k ⩾ 1 , there is λ k > 0 such that for λ > λ k , ( P λ ) has a solution with plasma set consisting of k components.