Infinitesimal index: cohomology computations

In this note we study the equivariant cohomology with compact supports of the zeroes of the moment map for the cotangent bundle of a linear representation of a torus and some of its notable subsets, using the theory of the infinitesimal index, developed in [8]. We show that, in analogy to the case of equivariant K-theory dealt with in [7] using the index of transversally elliptic operators, we obtain isomorphisms with notable spaces of splines studied in [2], [3].     

Multipoint Schur algorithm and orthogonal rational functions,I: Convergence properties

Classical Schur analysis is intimately related to the theory of orthogonal polynomials on the circle. We investigate the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szegő theory in the case that interpolation points may accumulate on the unit circle. This leads to a generalization of results in [22, 10], and new types of asymptotics.     

Entire Functions on Infinite Dimensional Spaces

In this work we study partial differential operators of infinite order on spaces of entire functions, generalizing some of the results presented in [2] and [5] to the new spaces introduced in [3].     

Measurable Categories and 2-Groups

Using the theory of measurable categories developed in [10], we provide a notion of representations of 2-groups better suited to physically and geometrically interesting examples than that using 2-VECT (cf. [8]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [6], and suggest state-integral constructions for 4-manifold invariants.     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

Gauged Floer Theory Of Toric Moment Fibers

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder [Fr1]. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya–Oh–Ohta–Ono [FuOOO3,1] and Cho–Oh [CO] on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without using virtual fundamental chains. Finally, we describe a conjectural relationship with Floer cohomology in the quotient.     

Several modifications of DPR estimator of the tail index

In this paper, we continue the investigation of an estimator proposed in [Yu. Davydov, V. Paulauskas, and A. Račkauskas, More on p-stable convex sets in Banach spaces, J. Theor. Probab., 13:39–64, 2000] and [V. Paulauskas, A new estimator for tail index, Acta Appl. Math., 79:55–67, 2003] and considered in [V. Paulauskas and M. Vaičiulis, Once more on comparison of tail index estimators, preprint, 2010]. We propose a class of modifications of the so-called DPR estimator and demonstrate that these modifications can have better asymptotic properties than the original DPR estimator.     

Phase Spaces and Deformation Theory

In the papers (Laudal in Contemporary Mathematics, vol. 391, [2005]; Geometry of time-spaces, Report No. 03, [2006/2007]), we introduced the notion of (non-commutative) phase algebras (spaces) Ph ⁿ (A), n=0,1,…,∞ associated to any associative algebra A (space), defined over a field k. The purpose of this paper is to study this construction in some more detail. This seems to give us a possible framework for the study of non-commutative partial differential equations. We refer to the paper (Laudal in Phase spaces and deformation theory, Report No. 09, [2006/2007]), for the applications to non-commutative deformation theory, Massey products and for the construction of the versal family of families of modules. See also (Laudal in Homology, Homotopy, Appl. 4:357–396, [2002]; Proceedings of NATO Advanced Research Workshop, Computational Commutative and Non-Commutative Algebraic Geometry, [2004]).      

On the Characterization of Preinvex Functions

In Refs. [J. Math. Anal. Appl. 258:287–308, [2001]; J. Math. Anal. Appl. 256:229–241, [2001]], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. This research was supported by the National Natural Science Foundation of China under Grants 70671064 and 60673177, and the Education Department Foundation of Zhejiang Province Grant 20070306. The authors thank Professor F. Giannessi for valuable comments on the original version of this paper.     

Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation

In this article, we continue our study of a system described by a class of initial boundary value problem (IBVP) of the Korteweg-de Vries (KdV) equation and the KdV Burgers (KdVB) equation posed on a finite interval with nonhomogeneous boundary conditions. While the system is known to be locally well-posed (Kramer et al. , [2010]; Rivas et al. in Math. Control Relat. Fields 1:61–81, [2011]) and its small amplitude solutions are known to exist globally, it is not clear whether its large amplitude solutions would blow up in finite time or not. This problem is addressed in this article from control theory point of view: look for some appropriate feedback control laws (with boundary value functions as control inputs) to ensure that the finite time blow-up phenomena would never occur. In this article, a simple, but nonlinear, feedback control law is proposed and the resulting closed-loop system is shown not only to be globally well-posed, but also to be locally exponentially stable for the KdV equation and globally exponentially stable for the KdVB equation.     

Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics

Before we dive in this essay into the accessibility stream of nowadays indicatory applications of octonions and quaternions to computer and other sciences and to quantum physics (see for example [50-53], [41], [33]) and to Clifford algebras (see for example [17,16], 18) let us focus for a while on the crucially relevant events for today’s revival on interest to nonassociativities while the role of associative quaternions in eight periodicity constructive classification of associative Clifford algebras is now a text-book knowledge.     

The embracing Voronoi diagram and closest embracing number

A point q is embraced by a set of points S if q is interior to the convex hull of S [8]. In some illumination applications where points of S are lights and q is a point to be illuminated, the embracing concept is related to a good illumination [4, 6], also known as the ∆-guarding [12] and the well-covering [10]. In this paper, we are not only interested in convex dependency (which is actually the embracing notion) but also in proximity. Suppose that the sites of S are lights or antennas with limited range; due to their limited power, they cover a disk of a given radius r centered at the sites of S. Only the points lying in such disks are illuminated. If we want to embrace the point q with the minimum range r, we need to know which is the closest light s _q to q such that q lies in the convex hull formed by s _q and the lights of S closer to q than s _q . This subset of S related to point q is called the closest embracing set for q in relation to S and its cardinality is the closest embracing number of q. By assigning every point q in the convex hull of S to its closest embracing site s _q , we obtain a partition of the convex hull that we call the embracing Voronoi diagram of S. This paper proves some properties of the embracing Voronoi diagrams and describes algorithms to compute such diagrams, as well as the levels in which the convex hull is decomposed regarding the closest embracing number.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

An extension theorem for slice monogenic functions and some of its consequences

Slice monogenic functions were introduced by the authors in [6]. The central result of this paper is an extension theorem, which shows that every holomorphic function defined on a suitable domain D of a complex plane can be uniquely extended to a slice monogenic function defined on a domain U _D , determined by D, in a Euclidean space of appropriate dimension. Two important consequences of the result are a structure theorem for the zero set of a slice monogenic function (with a related corollary for polynomials with coefficients in Clifford algebras), and the possibility to construct a multiplicative theory for such functions. Slice monogenic functions have a very important application in the definition of a functional calculus for n-tuples of noncommuting operators.     

An Isometric Representation of the Dual of {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)}

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.     

On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory

When constructing multivariate Padé approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Padé approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Padé approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Padé approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12, 25], or approaches that require some symbolic computations [6, 18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14, 19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.Research partly funded by FWO-Vlaanderen.     

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45–48, [2002]) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach–Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in (Bezdek, Litvak in Adv. Math. 215:626–641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59–69, [2006]) on the quantitative illumination problem.     

Extensions and renormalized traces

It has been shown by Nistor (Doc Math J DMV 2:263–295, 1997) that given any extension of associative algebras over \mathbb C{\mathbb C}, the connecting morphism in periodic cyclic homology is compatible, under the Chern–Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms Perrot (J Geom Phys 60:1441–1473, 2010), leading to “local” index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue.     

Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538–543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem. Research of Shaohua Pan was partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. Jein-Shan Chen is a member of the Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work was partially supported by National Science Council of Taiwan.