An optimal multivariate spline method of recovery of twice differentiable functions

A continuous quadratic polynomial spline of several variables is constructed. It solves the optimal recovery problem studied by V.F. Babenko, S.V. Borodachov, and D.S. Skorokhodov for the class of functions defined on a convex polytope in R ^d , whose second derivatives in any direction are uniformly bounded, and for a periodic analogue of this class. The information consists of the values and gradients of the function at some finite set of nodes in R ^d .     

Symmetric functions,codes of partitions and the KP hierarchy

We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ-function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.     

Several identities for certain products of theta functions

By means of a technique used by Carlitz and Subbarao to prove the quintuple product identity (Proc. Am. Math. Soc. 32(1):42–44, 1972), we recover a general identity (Chu and Yan, Electron. J. Comb. 14:#N7, 2007) for expanding the product of two Jacobi triple products. For applications, we briefly explore identities for certain products of theta functions φ(q), ψ(q) and modular relations for the Göllnitz-Gordon functions.     

Singular values of some modular functions and their applications to class fields

Since the modular curve has genus zero, we have a field isomorphism where X ₂(z) is a product of Klein forms. We apply it to construct explicit class fields over an imaginary quadratic field K from the modular function j _Δ,25(z):=X ₂(5z). And, for every integer N≥7 we further generate ray class fields K _(N) over K with modulus N just from the two generators X ₂(z) and X ₃(z) of the function field , which are also the product of Klein forms without using torsion points of elliptic curves. J.K. Koo was supported by Korea Research Foundation Grant (KRF-2002-070-C00003).     

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group’s relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.     

P-algorithm based on a simplicial statistical model of multimodal functions

A well-recognized one-dimensional global optimization method is generalized to the multidimensional case. The generalization is based on a multidimensional statistical model of multimodal functions constructed by generalizing computationally favorable properties of a popular one-dimensional model—the Wiener process. A simplicial partition of a feasible region is essential for the construction of the model. The basic idea of the proposed method is to search where improvements of the objective function are most probable; a probability of improvement is evaluated with respect to the statistical model. Some results of computational experiments are presented.     

Two modular equations for squares of the Rogers–Ramanujan functions with applications

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R ²(q ²). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q ⁴) and ν:=R ²(q ^1/2)R(q)/R(q ²). New proofs are offered for modular relations connecting R(q) to R(−q), R(q ²), and R(q ⁴), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.      

Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points

Pointwise estimates of the deviation T _n,A,B f(⋅)−f(⋅) in terms of moduli of continuity [`(w)]_·f\bar{w}_{\cdot}f and w _⋅ f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used.     

The generating functions of the rank and crank modulo 8

Let N(i,m;n) be the number of partitions of n with rank (Dyson) congruent to i (mod m) and let M(j,m;n) be the number of partitions of n with crank (Andrews, Garvan) congruent to j (mod m). I give here the generating functions for the numbers N(i,8;n) and M(j,8;n). I suggest forms for the one hundred power series

Lipschitz and differentiability properties of quasi-concave and singular normal distribution functions

The paper provides a condition for differentiability as well as an equivalent criterion for Lipschitz continuity of singular normal distributions. Such distributions are of interest, for instance, in stochastic optimization problems with probabilistic constraints, where a comparatively small (nondegenerate-) normally distributed random vector induces a large number of linear inequality constraints (e.g. networks with stochastic demands). The criterion for Lipschitz continuity is established for the class of quasi-concave distributions which the singular normal distribution belongs to.     

A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

In this paper, we present a detailed investigation for the properties of a one-parametric class of SOC complementarity functions, which include the globally Lipschitz continuity, strong semismoothness, and the characterization of their B-subdifferential. Moreover, for the merit functions induced by them for the second-order cone complementarity problem (SOCCP), we provide a condition for each stationary point to be a solution of the SOCCP and establish the boundedness of their level sets, by exploiting Cartesian P-properties. We also propose a semismooth Newton type method based on the reformulation of the nonsmooth system of equations involving the class of SOC complementarity functions. The global and superlinear convergence results are obtained, and among others, the superlinear convergence is established under strict complementarity. Preliminary numerical results are reported for DIMACS second-order cone programs, which confirm the favorable theoretical properties of the method.     

Symmetric and quasi-symmetric functions associated to polymatroids

To every subspace arrangement X we will associate symmetric functions ℘[X] and ℋ[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant ℋ[X] specializes to the Tutte polynomial . Billera, Jia and Reiner recently introduced a quasi-symmetric function ℱ[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function for polymatroids which has this property as well. Moreover, specializes to ℘[X], ℋ[X], and ℱ[X]. The author is partially supported by the NSF, grant DMS 0349019.     

Rademacher-type formulas for restricted partition and overpartition functions

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.     

Approximation of partially smooth functions

In this paper we discuss approximation of partially smooth functions by smooth functions. This problem arises naturally in the study of laminated currents.     

Strong Normalizability of Typed Lambda-Calculi for Substructural Logics

The strong normalization theorem is uniformly proved for typed λ-calculi for a wide range of substructural logics with or without strong negation. We would like to thank the referees for their valuable comments and suggestions. This research was supported by the Alexander von Humboldt Foundation. The second author is grateful to the Foundation for providing excellent working conditions and generous support of this research. This work was also supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B) 20700015, 2008.     

Schläfli modular equations for generalized Weber functions

Sets of appropriately normalized eta quotients, that we call level n Weber functions, are defined, and certain identities generalizing Weber function identities are proved for these functions. Schläfli type modular equations are explicitly obtained for Generalized Weber Functions associated with a Fricke group Γ⁰(n)⁺, for n=2,3,5,7,11,13 and 17.     

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.     

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.