Delannoy Orthants of Legendre Polytopes

We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x−1)/2 in the nth Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open generalized orthant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago (Good in Proc. Camb. Philos. Soc. 54:39–42, 1958; Lawden in Math. Gaz. 36:193–196, 1952; Moser and Zayachkowski in Scr. Math. 26:223–229, 1963). The polytopes we construct are closely related to the root polytopes introduced by Gelfand et al. (Arnold–Gelfand mathematical seminars: geometry and singularity theory, pp. 205–221. Birkhauser, Boston, 1996).     

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ^∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).     

Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom. Différentielle Catég., 44: 281–316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81–108, 2004). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517, Springer, Berlin Heidelberg New York, 2005), corresponding to partially ordered subspaces of R ⁿ minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.      

Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work (Bryc et al., Ann. Probab. 34(1):1–38, 2006; Hammond and Miller, J. Theor. Probab. 18(3):537–566, 2005) showed that the spectral measures (the density of normalized eigenvalues) converge almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see Massey et al., J. Theor. Probab. 20(3):637–662, 2007) by making the first row palindromic. In this paper we study the case where there is more than one palindrome in the first row of real symmetric Toeplitz matrices. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the spectral measures converge almost surely to a universal distribution. Assuming a conjecture on the resulting Diophantine sums (which is supported by numerics and some theoretical arguments), we prove that the limiting distribution has a fatter tail than any previously seen limiting spectral measure.     

Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves

Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow of closed curves in \mathbbR^d, d 3 2{\mathbb{R}^d, d\geq2} . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.     

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.     

Organizational learning: effects of (network) structure and (individual) strategy

Earlier theoretical accounts of collective learning relied on rules and operating procedures as the organizational memory (March in Organ. Sci. 2(1):71–87, 1991; Rodan in Scand. J. Manag. 21:407–428, 2005). This paper builds on this tradition drawing on ideas from social network theory. Learning is modeled as a social-psychological process (Darr and Kurtzberg in Organ. Behav. Hum. Decis. Process. 82(1):28–44, 2000; Rulke et al. in Organ. Behav. Hum. Decis. Process. 82(1):134–149, 2000), in which organizations learn by exchanging information internally between their members (Argote et al. in Organ. Behav. Hum. Decis. Process. 82(1):1–8, 2000; Carley in Am. Soc. Rev. 56(3):331–354, 1991; Carley in Soc. Perspect. 48(4):547–571, 1995). Learning is also characterized as stochastic and creative (Gruenfeld et al. in Organ. Behav. Hum. Decis. Process. 82(1):45–59, 2000). This model is used to explore predictions about the effect social networks have on idea generation and learning and alternative strategies for choosing from whom to seek information.

On covers of cyclic acts over monoids

In (Bull. Lond. Math. Soc. 33:385–390, 2001) Bican, Bashir and Enochs finally solved a long standing conjecture in module theory that all modules over a unitary ring have a flat cover. The only substantial work on covers of acts over monoids seems to be that of Isbell (Semigroup Forum 2:95–118, 1971), Fountain (Proc. Edinb. Math. Soc. (2) 20:87–93, 1976) and Kilp (Semigroup Forum 53:225–229, 1996) who only consider projective covers. To our knowledge the situation for flat covers of acts has not been addressed and this paper is an attempt to initiate such a study. We consider almost exclusively covers of cyclic acts and restrict our attention to strongly flat and condition (P) covers. We give a necessary and sufficient condition for the existence of such covers and for a monoid to have the property that all its cyclic right acts have a strongly flat cover (resp. (P)-cover). We give numerous classes of monoids that satisfy these conditions and we also show that there are monoids that do not satisfy this condition in the strongly flat case. We give a new necessary and sufficient condition for a cyclic act to have a projective cover and provide a new proof of one of Isbell’s classic results concerning projective covers. We show also that condition (P) covers are not unique, unlike the situation for projective covers.     

A Note on the Subcritical Two Dimensional Keller-Segel System

The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π, is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691–721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is fully utilized by adapting Delort’s theory on 2D incompressible Euler equation (Delort in J. Am. Math. Soc. 4:553–386, 1991).     

Spiral Anchoring in Media with Multiple Inhomogeneities: A Dynamical System Approach

The spiral is one of nature’s more ubiquitous shapes: It can be seen in various media, from galactic geometry to cardiac tissue. Mathematically, spiral waves arise as solutions to reaction–diffusion partial differential equations (RDS). In the literature, various experimentally observed dynamical states and bifurcations of spiral waves have been explained using the underlying Euclidean symmetry of the RDS—see for example (Barkley in Phys. Rev. Lett. 68:2090–2093, 1992; Phys. Rev. Lett. 76:164–167, 1994; Sandstede et al. in C. R. Acad. Sci. 324:153–158, 1997; J. Differ. Equ. 141:122–149, 1997; J. Nonlinear Sci. 9:439–478, 1999), or additionally using the concept of forced Euclidean symmetry-breaking for situations where an inhomogeneity or anisotropy is present—see (LeBlanc in Nonlinearity 15:1179–1203, 2002; LeBlanc and Wulff in J. Nonlinear Sci. 10:569–601, 2000). In this paper, we further investigate the role of medium inhomogeneities on spiral wave dynamics by considering the effects of several localized sites of inhomogeneity. Using a model-independent approach based on n>1 simultaneous translational symmetry-breaking perturbations of the dynamics near rotating waves, we fully characterize the local anchoring behavior of the spiral wave in the n-dimensional parameter space of relative “amplitudes” of the individual perturbations. For the case n=2, we supplement the local anchoring results with a classification of the generic one-parameter bifurcation diagrams of anchored states which can be obtained by circling the origin of the two-dimensional amplitude parameter space. Numerical examples are given to illustrate our various results.     

Feynman’s Operational Calculi: Disentangling Away from the Origin

In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ ⁿ . The method of disentangling follows Feynman’s heuristic rules from in (Feynman in Phys. Rev. 84:18–128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ ⁿ while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.     

Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy ⁻¹) =  2f(x) and f(xy) + f(y ⁻¹ x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GL_n(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S _n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S _n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

Generalized Priestley Quasi-Orders

We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras of bounded distributive lattices by means of Priestley quasi-orders (Adams, Algebra Univers 3:216–228, 1973; Cignoli et al., Order 8(3):299–315, 1991; Schmid, Order 19(1):11–34, 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually characterized by Vietoris families. We show that this generalizes the well-known characterization (Priestley, Proc Lond Math Soc 24(3):507–530, 1972) of homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147–151, 1974).     

Time-dependent properties of symmetric queues

We settle a conjecture of Kella et al. (J. Appl. Probab. 42:223–234, 2005): the distribution of the number of jobs in the system of a symmetric M/G/1 queue at a fixed time is independent of the service discipline if the system starts empty. Our derivations are based on a time-reversal argument for regenerative processes and a connection with a clearing model.     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

RETRACTED ARTICLE: Convergence of Weighted Sums for Arrays of Negatively Dependent Random Variables and Its Applications

We discuss the complete convergence of weighted sums for arrays of rowwise negatively dependent random variables (ND r.v.’s) to linear processes. As an application, we obtain the complete convergence of linear processes based on ND r.v.’s which extends the result of Li et al. (Stat. Probab. Lett. 14:111–114, 1992), including the results of Baum and Katz (Trans. Am. Math. Soc. 120:108–123, 1965), from the i.i.d. case to a negatively dependent (ND) setting. We complement the results of Ahmed et al. (Stat. Probab. Lett. 58:185–194, 2002) and confirm their conjecture on linear processes in the ND case.     

A Numerical Investigation on Configurational Distortions in Nematic Liquid Crystals

When subjected to magnetic or electric fields, nematic liquid crystals confined between two parallel glass plates and initially uniformly oriented may undergo homogeneous one-dimensional spatial distortions (Fréedericksz and Zolina, Trans. Faraday Soc. 29:919, 1933) or periodic distortions (Lonberg and Meyer, Phys. Rev. Lett. 55(7):718–721, 1985; and Srajer et al., Phys. Rev. Lett. 67(9):1102–1105, 1991). According to the experimental observations, periodic phases are stable configurations at intermediate intensity of the acting field, while homogeneous phases are stable at higher strengths.     

Generalized Eisenstein series and several modular transformation formulae

B.C. Berndt (J. Reine Angew. Math. 272:182–193, 1975; 304:332–365, 1978) has derived a number of new transformation formulas, in particular, the transformation formulae of the logarithms of the classical theta functions, by using a transformation formula for a more general class of Eisenstein series. In this paper, we continue his study. By using a transformation formula for a class of twisted generalized Eisenstein series, we generalize a transformation formula given by J. Lehner (Duke Math. J. 8:631–655, 1941) and give a new proof for transformation formulas proved by Y. Yang (Bull. Lond. Math. Soc. 36:671–682, 2004). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00003). This work also partially supported by BK21-Postech CoDiMaRo.