Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions

We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at the moment. In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one. Roswitha Hofer, Recipient of a DOC-FFORTE-fellowship of the Austrian Academy of Sciences at the Institute of Financial Mathematics at the University of Linz (Austria). Friedrich Pillichshammer, Supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. Dedicated to Prof. Robert F. Tichy on the occasion of his 50th birthday Authors’ address: Roswitha Hofer, Gerhard Larcher and Friedrich Pillichshammer, Institut für Finanzmathematik, Universit?t Linz, Altenbergerstra?e 69, A-4040 Linz, Austria     

Error bounds of the Galerkin method for singular integral equations of the second kind

For solving singular integral equations of the first kind Erdogan proposed a method of Galerkin type, and convergence was proved by Linz. In this paper we consider equations of the second kind, and it is found that the method converges also in this case. However, stronger conditions than for the first kind equations must be imposed. The computational aspect of the convergence problem is also considered.     

Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations

The block-by-block method, proposed by Linz for a kind of Volterra integral equations with nonsingular kernels, and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations (FDEs) with Caputo derivatives, is an efficient and stable scheme. We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order index $\alpha>0$.     

A new method for solving a class of mixed boundary value problems with singular coefficient

In [1], [2], [3], [4], [5], [6] and [7], it is very difficult to deal with initial boundary value conditions. In this paper, we give a new method to deal with boundary value conditions, the main contribution of this paper is to put mixed boundary value conditions into reproducing kernel Hilbert space. The numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

RESEARCH ANNOUNCEMENTS High order Zumerical Solution of Integral Transport Equation in slab Geometry

There are some common numerical methods for solving neutron transport equation, which including the well-known discrete ordinates method, PN approximation and integral transport methods[1]. There exists certain singularities in the solution of transport equation near the boundary and interface[2]. It gives rise to the difficulty in the construction of high order accurate numerical methods. The numerical solution obtained by now can not attain the second order convergent accuracy[3,4].     

Ausdehnung von Markoff-Prozessen auf erweiterte Parametermengen

Ohne ZusammenfassungGekürzte Fassung einer der philosophischen FakultÄt der UniversitÄt Innsbruck vorgelegten Dissertation.Die vorliegende Arbeit entstand wÄhrend eines Studienaufenthaltes am Mathematischen Institut der Hochschule Linz. Mein besonderer Dank gilt Herrn Prof. P.Wei\ für die Betreuung dieser Arbeit und für seine vielfÄltigen Anregungen. Weiters danke ich Herrn Ch. Kollreider für zahlreiche kritische Bemerkungen.     

A new family of iterative methods for solving system of nonlinear algebric equations

Homotopy perturbation method (HPM) is applied to construct a new iterative method for solving system of nonlinear algebric equations. Comparison of the result obtained by the present method with that obtained by revised Adomian decomposition method [Hossein Jafari, Varsha Daftardar-Gejji, Appl. Math. Comput. 175 (2006) 1–7] reveals that the accuracy and fast convergence of the new method.     

Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface,I: One Dimensional Problems

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations (following the approach proposed in [10] in the case of smooth coefficients), leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway [12] and it will be carried out by combining the discretization of [10] with the multigrid approach of [11] for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.     

On Approximation by Neural Networks with Optimized Activation Functions and Fixed Weights

Recently, Li [16] introduced three kinds of single-hidden layer feed-forward neural networks with optimized piecewise linear activation functions and fixed weights, and obtained the upper and lower bound estimations on the approximation accuracy of the FNNs, for continuous function defined on bounded intervals. In the present paper, we point out that there are some errors both in the definitions of the FNNs and in the proof of the upper estimations in [16]. By using new methods, we also give right approximation rate estimations of the approximation by Li’s neural networks.     

On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations

In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu’s equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The “spectral accuracy” convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].     

An adaptive algorithm for the Thomas–Fermi equation

A free boundary value problem is introduced to approximate the original Thomas–Fermi equation. The unknown truncated free boundary is determined iteratively. We transform the free boundary value problem to a nonlinear boundary value problem defined on [0,1]. We present an adaptive algorithm to solve the problem by means of the moving mesh finite element method. Comparison of our numerical results with those obtained by other approaches shows high accuracy of our method.     

The construction of exact solutions in the floating-plate problem

It is shown that it is possible to construct, by an inverse method, exact solutions of the problem of the flexural-gravitational oscillations of a floating elastic plate. The results obtained are used to check the accuracy of numerical solutions of the problem. It is shown that the numerical algorithm given in Ref. [Khabakhpasheva TI. The plane problem of an elastic floating plate. In Continuum Dynamics. Inst. Gidrodinamiki SO Ross Akad Nauk 2000;16:166–9.], predicts, with high accuracy, the values of the amplitudes of the oscillations of the plate and the distributions of the bending moments and hydrodynamic pressure for a wide frequency range.     

On a noisy-silent-versus-silent duel with equal accuracy functions

This paper deals with the noisy-silent-versus-silent duel with equal accuracy functions. Player I has a gun with two bullets and player II has a gun with one bullet. The first bullet of player I is noisy, the second bullet of player I is silent, and the bullet of player II is silent. Each player can fire their bullets at any time in [0, 1] aiming at his opponent. The accuracy function ist for both players. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is –1. The optimal strategies and the value of the game are obtained. Although optimal strategies in past works concerning games of timing does not depend on the firing moments of the players, the optimal strategy obtained for player II depends explicitly on the firing moment of player I's noisy bullet.     

Zur Erfolgskontrolle psychologischer Eignungsuntersuchungen

Zusammenfassung Methoden der Eignungsauslese bei den Straßenbahnfahrern der Linzer Elektrizitäts- und Straßenbahngesellschaft. Wesentlicher Rückgang der Unfälle und des Personalwechsels.

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Summary Ability-testing methods for the selection of tram drivers by the Elektrizitäts- und Straßenbahn-Gesellschaft, Linz. Considerable reduction of accidents and personnel turnover.

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Résumé Méthodes de l'examination d'aptitude au cas des conducteurs de tramway de la Linzer Elektrizitäts- und Straßenbahn Gesellschaft, Réduction considérable des accidents et du changement de personnel.

     

Refined stability theory for sandwich shells with transversally stiff core. 3. Simplest one-dimensional problems

The applicability and accuracy of stability equations of the refined theory for sandwich shells with a transversally stiff core proposed in [1] are investigated. The model problem of calculating the critical loads and stress fields in the core at mixed forms of the loss of stability is solved for an infinitely wide sandwich plate with an orthotropic core and composite load-carrying layers subjected to in-plane edge loads. The case of pure bending of the plate is considered in detail. The results obtained by variation of the physical-mechanical parameters are compared with the solutions of the three-dimensional theory for the core [2]. It is shown that the version of the refined theory [1] is more accurate than the other two-dimensional theories.For Pt. 2 see [1].Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 57–65, January–Feburary, 1998.     

Calculation of Stresses and Consistent Tangent Moduli from Automatic Differentiation of Hyerelastic Strain Energy Functions Through the Use of Hyper Dual Numbers

Many materials as e.g. engineering rubbers, polymers and soft biological tissues are often described by hyperelastic strain energy functions. For their finite element implementation the stresses and consistent tangent moduli are required and obtained mainly in terms of the first and second derivative of the strain energy function. Depending on its mathematical complexity in particular for anisotropic media the analytic derivatives may be expensive to be calculated or implemented. Then numerical approaches may be a useful alternative reducing the development time. Often-used classical finite difference schemes are however quite sensitive with respect to perturbation values and they result in a poor accuracy. The complex-step derivative approximation does never suffer from round-off errors, cf. [1], [2], but it can only provide first derivatives. A method which also provides higher order derivatives is based on hyper dual numbers [3]. This method is independent on the choice of perturbation values and does thus neither suffer from round-off errors nor from approximation errors. Therefore, here we make use of hyper dual numbers and propose a numerical scheme for the calculation of stresses and tangent moduli which are almost identical to the analytic ones. Its uncomplicated implementation and accuracy is illustrated by some representative numerical examples. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of iterative improvement techniques for schedule optimization

Due to complexity reasons of realistic scheduling applications, often iterative improvement techniques that perform a kind of local search to improve a given schedule are proposed instead of enumeration techniques that guarantee optimal solutions. In this paper we describe an experimental comparison of four iterative improvement techniques for schedule optimization that differ in the local search methodology. These techniques are iterative deepening, random search, tabu search and genetic algorithms. To compare the performance of these techniques, we use the same evaluation function, knowledge representation and data from one application. The evaluation function is defined on the gradual satisfaction of explicitly represented domain constraints and optimization functions. The satisfactions of individual constraints are weighted and aggregated for the whole schedule. We have applied these techniques on data of a steel making plant in Linz (Austria). In contrast to other applications of iterative improvement techniques reported in the literature, our application is constrained by a greater variety of antagonistic criteria that are partly contradictory.     

变分与无限维系统的高精度辛格式

1.引 言 冯康和他的研究小组提出的生成函数法[1]系统地解决了象二体问题这样地有限维Hamil-ton系统辛算法的构造问题,该方法也可以自然地推广到无限维Hamilton系统[2].首先在空间方向进行离散,例如采用差分或谱离散,得到有限维Hamilton系统,然后再采用生成函数法离散该系统.这样得到的辛格式是整个一层的格式,对于研究格式的局部性质如多辛性质[3],局部能量守恒性质[5]就相当困难.     

HIGH-ACCURACY EXPLICIT TWO-STEP METHODS WITH MINIMAL PHASE-LAG FOR y″= f（t,y）

In this paper, two families of high accuracy explicit two-step methods with minimal phase-lag are developed for the numerical integration of special second-order periodic initial-value problems. In comparison with some methods in [1-4,6], the advantage of these methods has a higher accuracy and minimal phase-lag. The methods proposed in this paper can be considered as a generalization of some methods in [1,3,4]. Numerical examples indicate that these new methods are generally more accurate than the methods used in [3,6]. second order periodic initial-value problems, phase-lag, local truncation error