Extension of quasi-Lipschitz set functions

In this article we consider so-calledN -triangular and quasi-Lipschitz set functions. In terms ofN -semimeasures, we establish necessary and sufficient conditions for extending a quasi-Lipschitz set function which is continuous from above at zero from a ring of sets to the σ-ring generated by these sets, and also conditions for the uniqueness of the extension. As simple corollaries we obtain analogous results for vector-valued measures, continuous triangular measures, and real-valued finiteN -triangular set functions which are continuous from above at zero.     

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m∈{3,5} only, and that the algebras arising from {1,2}-perfect m-cycle systems form a variety for m∈{3,5,7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety.     

The space–time variety of the hyperradiance from phase-locked soliton oscillators

The hyperradiance from phase-locked soliton oscillators is investigated by using the numerical simulation method for the perturbed sine-Gordon equation. Space–time variety for the emitted power from phase-locked soliton oscillators have been diffusely exhibited for the two magnetically coupled long Josephson junctions, operated in singlefluxon modes and involving the family of solutions. We derive some simulation results of space–time character, having the extensive physics meaning, for the theory for superradiance from phase-locked oscillators.     

The weight-decay technique in learning from data: an optimization point of view

The technique known as “weight decay” in the literature about learning from data is investigated using tools from regularization theory. Weight-decay regularization is compared with Tikhonov’s regularization of the learning problem and with a mixed regularized learning technique. The accuracies of suboptimal solutions to weight-decay learning are estimated for connectionistic models with a-priori fixed numbers of computational units. The Authors were partially supported by a PRIN grant from the Italian Ministry for University and Research, project “Models and Algorithms for Robust Network Optimization”.     

On the greedy algorithm by the Haar system

The paper investigates the uniform and almost everywhere convergence of the greedy algorithm by the Haar system. Necessary and sufficient conditions for norming the functions of Haar system are obtained, which guarantee the uniformconvergence for functions from C[0, 1] and almost everywhere convergence for functions from L ¹[0, 1].     

The interplay of various Scandinavian mathematical journals (1859–1953) and the road towards internationalization

The merger of various Nordic mathematical journals in 1953 into Mathematica Scandinavica (for research) and into Nordisk Matematisk Tidsskrift (for the more elementary topics, from 1979 NORMAT) confirmed increasing cooperation between matured Scandinavian mathematical communities. The merger originated from practical considerations including the wish to have a critical mass for economically viable publications.The present paper presents the basic steps in the development of several Scandinavian mathematical journals from 1859, the year of the foundation of the first general mathematical journal in a Scandinavian language, the Danish Mathematisk Tidsskrift, through various converging and diverging tendencies, towards the events of the year 1953.     

Applications of?the Cosecant and Related Numbers

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c _k , converge rapidly to zero as k????. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for tcsc?(at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d _k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.     

Embeddings and hyperplanes of the Lie incidence geometry of type E7,1

This paper continues a program to show that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily absolutely universal embedding, by addingE _7,1 to the list. It follows from [5, 12] that any projective embedding of this point line geometry is a homomorphic image of the one afforded by the 56-dimensional module for the groupE ₇(K).This work was supported by a grant from the National Science Foundation.     

A more precise definition of the method of determining the fatigue life of polymer materials from the reheat temperature

Conclusions 1. The rate of increase in the reheat temperature on the linear segment remains constant (for a periodic loading with higher stress levels) until a critical damage is sustained for the stress in question.2. The magnitude of T_c for a given stress is independent of the loading history, provided the critical damage has not been sustained for this stress.3. With decreasing stress T_c decreases slightly. A method for the accelerated determination of T_c from a single specimen for different stress levels has been developed.4. A method of predicting the fatigue life from the results of short-term tests, which take into account the variation of T_c due to stress level, has been developed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 906–913, September–October, 1977.     

The asymptotic behavior of solutions for a class of differential equations

A class of (n+1)th-order homogeneous linear differential equations (n > 1) is found for which fundamental solutions can be constructed from fundamental solutions to second-order differential equations. Asymptotic formulas for solutions to equations from this class are presented. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Mixtures of Global and Local Edgeworth Expansions and Their Applications

Edgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive “priors,” whenever the third cumulant of the “prior” is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution.     

On weakly harmonic maps from Finsler to Riemannian manifolds

We prove global C0,α$C^{0,\alpha }$-estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, and Hildebrandt, Jost and Widman from Riemannian to Finsler domains. As consequences we obtain a Liouville theorem for entire harmonic maps on simple Finsler manifolds, and an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian target.     

Minimax estimation of a variance

The nonparametric problem of estimating a variance based on a sample of sizen from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.     

Middle dimension homologies of a codimension 2 submanifold. II

This paper is devoted to the estimates for the middle dimension homologies of codimension 2 submanifoldA of a smooth manifold X. Proofs are based on the information about homologies of the ramified covering of X branched alongA. This information is extracted from the Atiyah-Singer signature formulas and from some corollaries of the Smith theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 182–185, 1976.The author takes this oppotunity to express his gratitude to V. A. Rokhlin for the formulation of the problem and for his interest in this paper.     

Hybrid estimators for stochastic differential equations from reduced data

We treat parametric inference for unknown parameters of stochastic differential equations from discrete observations from the viewpoint of computational cost. Following Kamatani et al. (Bull Inf Cybern 48:19–35, 2016) and Kaino and Uchida (Hybrid estimators for ergodic diffusion processes from thinned data, 2018), we present the asymptotic results of the multi-step estimators with the initial Bayes type estimators for both ergodic and non-ergodic diffusion type processes. The initial Bayes type estimators are constructed by means of both the reduced data and the thinned data obtained from the full data. Some examples and simulation results are also given.     

Crisis-induced chaos in the Rose-Hindmarsh model for neuronal activity

The bifurcation diagrams for the Rose-Hindmarsh model are obtained from the Poincaré maps which govern the dynamics of this differential system. The Lyapunov spectra for this model are estimated from time series. The transition from periodicity to crisis-induced chaos. and back to periodicity is presented for I [2.5, 2.69]. and is qualitatively different from the transitions described for different parameter regions [A. V. Holden and Yinshui Fan, Chaos, Solitons & Fractals 2, 221–236 (1992); Chaos, Solitons & Fractals 2, 349–369 (1992)]. A piecewise smooth, one-dimensional map is constructed to simulate the dynamics of the model and to reproduce the process of crisis-induced chaos.     

Feasible Method for Generalized Semi-Infinite Programming

In this paper, we analyze the outer approximation property of the algorithm for generalized semi-infinite programming from Stein and Still (SIAM J. Control Optim. 42:769–788, 2003). A simple bound on the regularization error is found and used to formulate a feasible numerical method for generalized semi-infinite programming with convex lower-level problems. That is, all iterates of the numerical method are feasible points of the original optimization problem. The new method has the same computational cost as the original algorithm from Stein and Still (SIAM J. Control Optim. 42:769–788, 2003). We also discuss the merits of this approach for the adaptive convexification algorithm, a feasible point method for standard semi-infinite programming from Floudas and Stein (SIAM J. Optim. 18:1187–1208, 2007).     

General Methods for Monitoring Convergence of Iterative Simulations

Abstract

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.     

Principles and Standards for School Mathematics and Teacher Education: Preparing and Empowering Teachers

Teachers and teacher educators play a critical role in the improvement of mathematics education. Recommendations for appropriate teacher support can be gained from examining the Teaching Principle in the document, Principles and Standards for School Mathematics ( NCTM, 2000 ), as well as earlier recommendations from the Professional Standards for Teaching Mathematics ( NCTM, 1991 ). This paper discusses implications of the recommendations for preservice teacher preparation, the continuing professional development of teachers, and mathematics teacher certification policy.     

A remark on He’s variational iteration technique for solving nonlinear equations

The variational iteration method (VIM) can be usefully applied for solving many linear and nonlinear scientific and engineering problems. In this note we show that He’s approach for solving nonlinear equations, arising from the VIM, is, actually, Schröder’s method presented in his classical work from 1870.