Euler, Pisot, Prouhet–Thue–Morse, Wallis and the duplication of sines

We all know Euler’s product ${\prod(1+X^{2^n}) = (1-X)^{-1}}$ and its companion ${\prod(1-X^{2^n}) = \sum \pm X^j}$ , where the sequence of signs is the so-called Prouhet–Thue–Morse automatic sequence. Discussing generalizations of these two formulae, we are led respectively (1) to Wallis’ famous infinite product for π, (2) to a characterization of Pisot numbers, (3) to multigrade equalities and the Prouhet–Tarry–Escott problem, (4) to the product ${\prod_{0\leq j \leq n}{\rm sin}(2^j x)}$ and its sequence of signs as x runs through the intervals ${(j \pi/2^n, (j+1) \pi/2^n)}$ , ${j \in [0, 2^n-1]}$ , (5) and finally to the Gelfond and Newman-Slater product and its generalization ${\prod \sin r^j x}$ , which plays a rôle in several papers when r = 2.     

Lower bound of measure and embeddings of Sobolev,Besov and Triebel–Lizorkin spaces

In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Hajłasz–Besov spaces or Hajłasz–Triebel–Lizorkin spaces defined on a doubling and geodesic metric measure space and lower bound for measure of balls either in the whole space or in a domain inside the space.     

Moduli of sheaves,Fourier–Mukai transform,and partial desingularization

We study birational maps among (1) the moduli space of semistable sheaves of Hilbert polynomial \(4m+2\) on a smooth quadric surface, (2) the moduli space of semistable sheaves of Hilbert polynomial \(m^{2}+3m+2\) on \(\mathbb {P}^{3}\), (3) Kontsevich’s moduli space of genus-zero stable maps of degree 2 to the Grassmannian Gr(2, 4). A regular birational morphism from (1) to (2) is described in terms of Fourier–Mukai transforms. The map from (3) to (2) is Kirwan’s partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.     

Skorohod–Loynes Characterizations of Queueing,Fluid, and Inventory Processes

We consider queueing, fluid and inventory processes whose dynamics are determined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory level) is regulated to stay in a finite or infinite interval – inputs or outputs are disregarded when they would lead to a state outside the interval. The sample paths of the process satisfy an integral equation; the paths have finite local variation and may have discontinuities. We establish the existence and uniqueness of the process based on a Skorohod equation. This leads to an explicit expression for the process on the doubly-infinite time axis. The expression is especially tractable when the process is stationary with stationary input–output measures. This representation is an extension of the classical Loynes representation of stationary waiting times in single-server queues with stationary inputs and services. We also describe several properties of stationary processes: Palm probabilities of the processes at jump times, Little laws for waiting times in the system, finiteness of moments and extensions to tandem and treelike networks.     

Existence,uniqueness and stability of minimizers of generalized Tikhonov–Phillips functionals

The Tikhonov–Phillips method is widely used for regularizing ill-posed inverse problems mainly due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a variety of other methods which can be considered as “variants” of the traditional Tikhonov–Phillips method of order zero. Such is the case for instance of the Tikhonov–Phillips method of order one, the total variation regularization method, etc. In this article we find sufficient conditions on the penalizers in generalized Tikhonov–Phillips functionals which guarantee existence, uniqueness and stability of the minimizers. The particular cases in which the penalizers are given by the bounded variation norm, by powers of seminorms and by linear combinations of powers of seminorms associated to closed operators, are studied. Several examples are presented and a few results on image restoration are shown.     

On Derivation of Equations of Electrodynamics and Gravitation from the Principle of Least Action,the Hamilton–Jacobi Method,and Cosmological Solutions

Doklady Mathematics - In classical texts, equations for fields are proposed without derivation of right-hand sides. Below, the right-hand sides of the Maxwell and Einstein equations are derived...     

Constraint Preserving,Inexact Solution of Implicit Discretizations of Landau–Lifshitz–Gilbert Equations and Consequences for Convergence

The Landau-Lifshitz-Gilbert equation describes dynamics of ferromagnetism. Nonlinearity of the equation and a non-convex side constraint make it difficult to design reliable approximation schemes. In this paper, we discuss the numerical solution of nonlinear systems of equations resulting from implicit, unconditionally convergent discretizations of the problem. Numerical experiments indicate that finite-time blow-up of weak solutions can occur and thereby underline the necessity of the design of reliable discretization schemes that approximate weak solutions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrals of Lipschitz–Hankel type,Legendre functions and table errata

The complete Lipschitz–Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which find application in communication theory.     

Monomiality principle,operational methods and family of Laguerre–Sheffer polynomials

In this paper, the Laguerre–Sheffer polynomials are introduced by using the monomiality principle formalism and operational methods. The generating function for the Laguerre–Sheffer polynomials is derived and a correspondence between these polynomials and the Sheffer polynomials is established. Further, differential equation, recurrence relations and other properties for the Laguerre–Sheffer polynomials are established. Some concluding remarks are also given.     

Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words

We construct irreducible representations of affine Khovanov–Lauda–Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the ‘cuspidal modules’ are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.     

On theorems of Delsarte–McEliece and Chevalley–Warning–Ax–Katz

We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte–McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte–McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c ⁽¹⁾, . . . ,c ^(t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which ${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$ . We also present a version of the theorem that, for any list of t symbols s ₁, . . . ,s _t , gives p-adic estimates of the number of positions i such that ${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$ as these words range over a product of abelian codes.     

Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows

We define an ending lamination for a Weil–Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil–Petersson metric [Bro2], these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex C(S){\mathcal{C}(S)}. As an application, we establish fundamentals of the topological dynamics of the Weil–Petersson geodesic flow, showing density of closed orbits and topological transitivity.     

Re-formulation of inverse Gaussian,reciprocal inverse Gaussian,and Birnbaum–Saunders kernel estimators

We reveal the boundary bias problem of Birnbaum–Saunders, inverse Gaussian, and reciprocal inverse Gaussian kernel estimators (Jin and Kawczak, 2003, Scaillet, 2004) and re-formulate these estimators to solve the problem. We investigate asymptotic properties of a new class of asymmetric kernel estimators.     

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–?ojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.     

Partition,Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over F_q having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over F_q having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).     

Multiparametric Families of Solutions of the Kadomtsev–Petviashvili-I Equation,the Structure of Their Rational Representations,and Multi-Rogue Waves

We construct solutions of the Kadomtsev–Petviashvili-I equation in terms of Fredholm determinants. We deduce solutions written as a quotient of Wronskians of order 2N. These solutions, called solutions of order N, depend on 2N?1 parameters. They can also be written as a quotient of two polynomials of degree 2N(N +1) in x, y, and t depending on 2N?2 parameters. The maximum of the modulus of these solutions at order N is equal to 2(2N + 1)². We explicitly construct the expressions up to the order six and study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters.     

Identifiability issues of age–period and age–period–cohort models of the Lee–Carter type

The predominant way of modelling mortality rates is the Lee–Carter model and its many extensions. The Lee–Carter model and its many extensions use a latent process to forecast. These models are estimated using a two-step procedure that causes an inconsistent view on the latent variable. This paper considers identifiability issues of these models from a perspective that acknowledges the latent variable as a stochastic process from the beginning. We call this perspective the plug-in age–period or plug-in age–period–cohort model. Defining a parameter vector that includes the underlying parameters of this process rather than its realizations, we investigate whether the expected values and covariances of the plug-in Lee–Carter models are identifiable. It will be seen, for example, that even if in both steps of the estimation procedure we have identifiability in a certain sense it does not necessarily carry over to the plug-in models.