The representation of Euler sums with parameters

In this paper, by choosing different kernel functions and base functions, we obtain some Euler sums with parameters. Moreover, we also obtain the new Euler sums with parameters by differentiating, limiting and elementary arithmetic. Thus, more Euler sums with parameters can be obtained. Furthermore, some Euler sums given in this paper are closed forms.     

A single-level control in Moran's problem with feedback

Moran's well-known model for a dam is generalized to the case when both the random entering flow and outflow may depend on the current magnitude of the supply of water.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> integrability of trigonometric series with special varying coefficients

Very recently, Yu, Le and Zhou introduced the so called △B1^＊ and △B2^＊ conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.     

An application of rosenthal's moment inequality to the strong law of large numbers

We apply a moment inequality of H.P. Rosenthal to get various generalizations of a theorem of Brunk-Chung-Prohoroff. In particular, we show that the strong law hols provided $\text{∑}\text{n=1}\text{∞}\text{n}{}^{\mathrm{?2p}}\text{E(X}{}_{\mathrm{n}}\text{}{}^{\mathrm{2p}}\text{) < ∞}$ and $\text{∑}\text{n=1}\text{∞}\text{n}{}^{\mathrm{?2?p}}\text{(E(}\text{X}\text{n}\text{2}\text{))}{}^{\mathrm{p}}\text{< ∞}$ for one p ? 1.     

On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

A new representation of Student's t as a function of independent t's,with a generalization to the matrix t

A new synthetic representation of Student's t statistic is given which exhibits the random variable as a nonlinear function of two independent t variables. A generalization is given to the matricvariate t with an interpretation in terms of “natural conjugate” inference for multivariate normal sampling.     

Exact nonnull distribution of Wilks’ statistic: The ratio and product of independent components

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks’ statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.     

The effects of income taxation on investor's risk-taking: A stock company case

The effects of income taxation on the investor's willingness to take risks are considered. The investor is assumed to be a risk averting one. The income taxation is assumed proportional so that considerations are restricted to investments of stock companies only, and some aspects of the indebtedness of a firm are also considered. In the empirical part of the study, the efficiency of the parameters joining to taxation as the regulators of risk-taking and indebtedness is presented.     

The chaos game revisited: Yet another, but a trivial proof of the algorithm’s correctness

The aim of this work is to explain why the most popular algorithm for approximating IFS fractals, the chaos game, works. Although there are a few proofs of the algorithm’s correctness in the relevant literature, the majority of them utilize notions and theorems of measure and ergodic theories. As a result, paradoxically, although the rules of the chaos game are very simple, the logic underlying the algorithm seems to be hard to comprehend for non-mathematicians. In contrast, the proof presented in this work uses only fundamentals of probability and can be understood by anyone interested in fractals.     

Optimal management of the machine repair problem with working vacation: Newton’s method

This paper studies the M/M/1 machine repair problem with working vacation in which the server works with different repair rates rather than completely terminating the repair during a vacation period. We assume that the server begins the working vacation when the system is empty. The failure times, repair times, and vacation times are all assumed to be exponentially distributed. We use the MAPLE software to compute steady-state probabilities and several system performance measures. A cost model is derived to determine the optimal values of the number of operating machines and two different repair rates simultaneously, and maintain the system availability at a certain level. We use the direct search method and Newton’s method for unconstrained optimization to repeatedly find the global minimum value until the system availability constraint is satisfied. Some numerical examples are provided to illustrate Newton’s method.     

Waldhausen's theory of k-fold end structures: a survey

Let R⁺ be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x_f:X → R⁺, 1 ? j ? k, yielding a proper map x:X → (R⁺)^k. The pairs (X,x) are made into the category E^k of spaces with k-fold end structure. Attachments and expansions in E^k are defined by induction on k, where elementary attachments and expansions in E⁰ have their usual meaning. The category E^k/Z consists of objects (X, i) where i: Z → X is an inclusion in E^k with an attachment of i(Z) to X, and the category E^k6Z consists of pairs (X,i) of E^k/Z that admit retractions X → Z. An infinite complex over Z is a sequence X = {X₁ ? X₂ ? … ? X_n …} of inclusions in E^k6Z. The abelian grou p S₀(Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S₁(Z) is defined as the set of equivalence classes of X₁, where X ∈ E^k/Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S₁(Z × R)→^πS₀(Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].     

Influence of others on one's decision: A model of group choice

The paper suggests a model of decision of group members in supporting or opposing a given social issue. The central assumption states that the probability of a positive decision of nth person depends on the number of positive choices among the n-1 persons who preceded him in making the decision. Formulas are given for the expectation and variance of the total group support. It is also shown how these results may be applied to optimally allocating the resources among various propaganda tasks.     

Polynomial cases of graph decomposition: A complete solution of Holyer’s problem

Let H be a fixed graph. A graph G has an H-decomposition if the edge set of G can be partitioned into subsets inducing graphs isomorphic to H. Let P_H denote the following decision problem: “Does an instance graph G admit H-decomposition?” In this paper we prove that the problem P_H is polynomial time solvable if H is a graph whose every component has at most 2 edges. This way we complete a solution of Holyer’s problem which is the problem of classifying the problems P_H according to their computational complexities.     

The distribution of product of independent beta random variables with application to multivariate analysis

Summary Schatzoff [9] obtained the forms of the probability density function (pdf) and the cumulative distribution function (cdf) of the product of independent beta random variables when their parameters had some special values. The forms, however, did not indicate the constants explicitly. In this paper his approach is modified so as to allow presentation of explicit expressions for the pdf and cdf of the product of independent beta random variables (without restriction to the values of the parameters) in neat forms. Applications in multivariate analysis are given for the central and the non-central cases. Research supported by the National Research Council of Canada, No. A-4060.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.