International Journal of Mathematics and Mathematical Sciences The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Calculation of Precise Constants in a Probability Model of Zipf’s Law Generation and Asymptotics of Sums of Multinomial Coefficients Sun, 07 May 2017 00:00:00 +0000 Let be a full set of outcomes (symbols) and let positive , , be their probabilities . Let us treat as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the nonincreasing order of their probabilities. Let be the probability of the th word in this list. We prove that if at least one of the ratios , , is irrational, then the limit exists and differs from zero; here is the root of the equation . The limit constant can be expressed (rather easily) in terms of the entropy of the distribution . Vladimir Bochkarev and Eduard Lerner Copyright © 2017 Vladimir Bochkarev and Eduard Lerner. All rights reserved. A Note on the Performance of Biased Estimators with Autocorrelated Errors Mon, 30 Jan 2017 06:34:57 +0000 It is a well-established fact in regression analysis that multicollinearity and autocorrelated errors have adverse effects on the properties of the least squares estimator. Huang and Yang (2015) and Chandra and Tyagi (2016) studied the PCTP estimator and the class estimator, respectively, to deal with both problems simultaneously and compared their performances with the estimators obtained as their special cases. However, to the best of our knowledge, the performance of both estimators has not been compared so far. Hence, this paper is intended to compare the performance of these two estimators under mean squared error (MSE) matrix criterion. Further, a simulation study is conducted to evaluate superiority of the class estimator over the PCTP estimator by means of percentage relative efficiency. Furthermore, two numerical examples have been given to illustrate the performance of the estimators. Gargi Tyagi and Shalini Chandra Copyright © 2017 Gargi Tyagi and Shalini Chandra. All rights reserved. On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras Thu, 12 Jan 2017 09:34:48 +0000 The notions of the Killing form and invariant form in Lie algebras are extended to the ones in Lie-Yamaguti superalgebras and some of their properties are investigated. These notions are also -graded generalizations of the ones in Lie-Yamaguti algebras. Patricia L. Zoungrana and A. Nourou Issa Copyright © 2017 Patricia L. Zoungrana and A. Nourou Issa. All rights reserved. Modal Logic Axioms Valid in Quotient Spaces of Finite CW-Complexes and Other Families of Topological Spaces Mon, 26 Dec 2016 06:01:11 +0000 In this paper we consider the topological interpretations of , the classical logic extended by a “box” operator interpreted as interior. We present extensions of S4 that are sound over some families of topological spaces, including particular point topological spaces, excluded point topological spaces, and quotient spaces of finite CW-complexes. Maria Nogin and Bing Xu Copyright © 2016 Maria Nogin and Bing Xu. All rights reserved. Vector Spaces of New Special Magic Squares: Reflective Magic Squares, Corner Magic Squares, and Skew-Regular Magic Squares Wed, 30 Nov 2016 11:17:13 +0000 The definition of a regular magic square motivates us to introduce the new special magic squares, which are reflective magic squares, corner magic squares, and skew-regular magic squares. Combining the concepts of magic squares and linear algebra, we consider a magic square as a matrix and find the dimensions of the vector spaces of these magic squares under the standard addition and scalar multiplication of matrices by using the rank-nullity theorem. Thitarie Rungratgasame, Pattharapham Amornpornthum, Phuwanat Boonmee, Busrun Cheko, and Nattaphon Fuangfung Copyright © 2016 Thitarie Rungratgasame et al. All rights reserved. Some Remarks on Quasi-Generalized CR-Null Geometry in Indefinite Nearly Cosymplectic Manifolds Mon, 14 Nov 2016 11:09:29 +0000 Attention is drawn to some distributions on ascreen Quasi-Generalized Cauchy-Riemannian (QGCR) null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifolds. Fortuné Massamba and Samuel Ssekajja Copyright © 2016 Fortuné Massamba and Samuel Ssekajja. All rights reserved. Symmetric Integer Matrices Having Integer Eigenvalues Mon, 07 Nov 2016 09:33:15 +0000 We provide characterization of symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also make some connection between Hanlon’s conjecture and integer eigenvalue problem. Lei Cao and Selcuk Koyuncu Copyright © 2016 Lei Cao and Selcuk Koyuncu. All rights reserved. Double Laplace Transform Method for Solving Space and Time Fractional Telegraph Equations Wed, 26 Oct 2016 14:31:29 +0000 Double Laplace transform method is applied to find exact solutions of linear/nonlinear space-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditions. Furthermore, we give illustrative examples to demonstrate the efficiency of the method. Ranjit R. Dhunde and G. L. Waghmare Copyright © 2016 Ranjit R. Dhunde and G. L. Waghmare. All rights reserved. Quasi-Positive Delta Sequences and Their Applications in Wavelet Approximation Tue, 25 Oct 2016 08:10:52 +0000 A sufficient literature is available for the wavelet error of approximation of certain functions in the -norm. There is no work in context of multiresolution approximation of a function in the sense of sup-error. In this paper, for the first time, wavelet estimator for the approximation of a function belonging to class under supremum norm has been obtained. Working in this direction, four new theorems on the wavelet approximation of a function belonging to class using the projection of its wavelet expansions have been estimated. The calculated estimator is best possible in wavelet analysis. Shyam Lal and Susheel Kumar Copyright © 2016 Shyam Lal and Susheel Kumar. All rights reserved. On Degrees of Modular Common Divisors and the Big Prime Algorithm Wed, 05 Oct 2016 13:47:51 +0000 We consider a few modifications of the Big prime modular algorithm for polynomials in . Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant, and on finding preliminary bounds on degrees of common divisors using auxiliary primes. These modifications are used to suggest improved algorithms for calculation and for coprime polynomials detection. To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular on polynomials from Knuth’s example of intermediate expression swell. Vahagn Mikaelian Copyright © 2016 Vahagn Mikaelian. All rights reserved. Modelling the Impact of Government Policies on Import on Domestic Price of Indian Gold Using ARIMA Intervention Method Mon, 19 Sep 2016 09:29:59 +0000 The study attempts to determine the impact of government policies of import of gold in India on the domestic price of gold during 2013 using Autoregressive Integrated Moving Average (ARIMA) intervention model. 2013 was an amazing year for Indian gold market where the price had reached its zenith. In April 2013, to curb a record trade deficit, India imposed an import duty of 10 percent on gold and tied imports for domestic consumption to exports, creating scarce supply of the yellow metal and boosting premiums to curtail the Current Account Deficit (CAD). The objective of the paper is to model the impact of this intervention by the government on the domestic price of Indian gold. Suitable ARIMA model is fit on the preintervention period and thereafter the effects of the interventions are analysed. The results indicate that ARIMA is the most suitable model during preintervention period. Intervention analysis reveals that there is significant decrease in domestic price of gold by 56% from 2013. The model may be used by policymakers to analyse the future of gold before framing regulations and policies. Jyothi Unnikrishnan and Kodakanallur Krishnaswamy Suresh Copyright © 2016 Jyothi Unnikrishnan and Kodakanallur Krishnaswamy Suresh. All rights reserved. Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression Wed, 07 Sep 2016 16:41:19 +0000 We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov’s theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed. Wayan Somayasa, Gusti N. Adhi Wibawa, La Hamimu, and La Ode Ngkoimani Copyright © 2016 Wayan Somayasa et al. All rights reserved. Almost and Nearly Isosceles Pythagorean Triples Mon, 05 Sep 2016 14:12:51 +0000 This work is about extended pythagorean triples, called NPT, APT, and AI-PT. We generate infinitely many NPTs and APTs and then develop algorithms for infinitely many AI-PTs. Since AI-PT is of , we ask generally for PT satisfying for any . These triples are solutions of certain diophantine equations. Eunmi Choi Copyright © 2016 Eunmi Choi. All rights reserved. Application of ADM Using Laplace Transform to Approximate Solutions of Nonlinear Deformation for Cantilever Beam Thu, 25 Aug 2016 16:58:19 +0000 We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method. Ratchata Theinchai, Siriwan Chankan, and Weera Yukunthorn Copyright © 2016 Ratchata Theinchai et al. All rights reserved. Loan Transactions with Random Dates for the First and Last Periodic Instalments Thu, 25 Aug 2016 16:02:32 +0000 Usually, loan transactions contracted in practice are nonrandom; that is to say, all amounts received (principal) and paid (period instalments) by the borrower are previously agreed with the lender, as well as their respective dates. In this paper, two new alternative loan models are introduced, depending on whether the borrower survives or not to fulfil all repayment obligations. In this way, either the initial or the final date of repayments can be subject to this contingency. Additionally, the different parameters of such random transactions are determined, as well as several measures of profitability/cost for the lender/borrower, respectively. These transactions can be attractive for both the lender and the borrower, which therefore make them worthy of consideration and subsequent implementation for the benefit of both parties. María del Carmen Valls Martínez and Salvador Cruz Rambaud Copyright © 2016 María del Carmen Valls Martínez and Salvador Cruz Rambaud. All rights reserved. Natural Partial Orders on Transformation Semigroups with Fixed Sets Sun, 21 Aug 2016 07:51:00 +0000 Let be a nonempty set. For a fixed subset of , let be the set of all self-maps on which fix all elements in . Then is a regular monoid under the composition of maps. In this paper, we characterize the natural partial order on and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements. Yanisa Chaiya, Preeyanuch Honyam, and Jintana Sanwong Copyright © 2016 Yanisa Chaiya et al. All rights reserved. The Exponentiated Gumbel Type-2 Distribution: Properties and Application Thu, 04 Aug 2016 12:20:17 +0000 We introduce a generalized version of the standard Gumble type-2 distribution. The new lifetime distribution is called the Exponentiated Gumbel (EG) type-2 distribution. The EG type-2 distribution has three nested submodels, namely, the Gumbel type-2 distribution, the Exponentiated Fréchet (EF) distribution, and the Fréchet distribution. Some statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimates was proposed for estimating the model parameters. The usefulness and flexibility of the Exponentiated Gumbel (EG) type-2 distribution were illustrated with a real lifetime data set. Results based on the log-likelihood and information statistics values showed that the EG type-2 distribution provides a better fit to the data than the other competing distributions. Also, the consistency of the parameters of the new distribution was demonstrated through a simulation study. The EG type-2 distribution is therefore recommended for effective modelling of lifetime data. I. E. Okorie, A. C. Akpanta, and J. Ohakwe Copyright © 2016 I. E. Okorie et al. All rights reserved. Bessel Equation in the Semiunbounded Interval : Solving in the Neighbourhood of an Irregular Singular Point Sun, 31 Jul 2016 13:00:34 +0000 This study expresses the solution of the Bessel equation in the neighbourhood of as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval . In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution at (not to mention a so-called formal solution), but a solution in the interval with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal. Qing-Hua Zhang, Jian Ma, and Yuanyuan Qu Copyright © 2016 Qing-Hua Zhang et al. All rights reserved. A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly Thu, 28 Jul 2016 14:57:28 +0000 An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs. Nazreen Waeleh and Zanariah Abdul Majid Copyright © 2016 Nazreen Waeleh and Zanariah Abdul Majid. All rights reserved. Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition Wed, 27 Jul 2016 08:43:15 +0000 Real hypersurfaces satisfying the condition have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective plane satisfying a generalization of under an additional restriction on a specific function. Theocharis Theofanidis Copyright © 2016 Theocharis Theofanidis. All rights reserved. Univalent and Starlike Properties for Generalized Struve Function Thu, 14 Jul 2016 12:15:11 +0000 We derive conditions on the parameters , , and so that the function where is the normalized form of generalized Struve function, belongs to the class Also, some sufficient conditions for the function to be in the class are obtained. Aaisha Farzana Habibullah, Adolf Stephen Bhaskaran, and Jeyaraman Muthusamy Palani Copyright © 2016 Aaisha Farzana Habibullah et al. All rights reserved. On the Commutative Rings with At Most Two Proper Subrings Thu, 14 Jul 2016 08:47:01 +0000 The commutative rings with exactly two proper (unital) subrings are characterized. An initial step involves the description of the commutative rings having only one proper subring. David E. Dobbs Copyright © 2016 David E. Dobbs. All rights reserved. Spatial Tail Dependence and Survival Stability in a Class of Archimedean Copulas Tue, 05 Jul 2016 06:39:02 +0000 This paper investigates properties of extensions of tail dependence of Archimax copulas to high dimensional analysis in a spatialized framework. Specifically, we propose a characterization of bivariate margins of spatial Archimax processes while spatial multivariate upper and lower tail dependence coefficients are modeled, respectively, for Archimedean copulas and Archimax ones. A property of stability is given using convex transformations of survival copulas in a spatialized Archimedean family. Diakarya Barro, Moumouni Diallo, and Remi Guillaume Bagré Copyright © 2016 Diakarya Barro et al. All rights reserved. On the Computation of Degenerate Hopf Bifurcations for -Dimensional Multiparameter Vector Fields Thu, 30 Jun 2016 11:31:33 +0000 The restriction of an -dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration. Michail P. Markakis and Panagiotis S. Douris Copyright © 2016 Michail P. Markakis and Panagiotis S. Douris. All rights reserved. On Shift-Dependent Cumulative Entropy Measures Tue, 07 Jun 2016 06:02:35 +0000 Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in the studies of reliability and life testing. In this paper, cumulative distribution and survival function are used to develop weighted forms of CRE and CE. These new measures are denominated as weighted cumulative residual entropy (WCRE) and weighted cumulative entropy (WCE) and the connections of these new measures with hazard and reversed hazard rates are assessed. These information-theoretic uncertainty measures are shift-dependent and various properties of these measures are studied, including their connections with CRE, CE, mean residual lifetime, and mean inactivity time. The notions of weighted mean residual lifetime (WMRL) and weighted mean inactivity time (WMIT) are defined. The connections of weighted cumulative uncertainties with WMRL and WMIT are used to calculate the cumulative entropies of some well-known distributions. The joint versions of WCE and WCRE are defined which have the additive properties similar to those of Shannon entropy for two independent random lifetimes. The upper boundaries of newly introduced measures and the effect of linear transformations on them are considered. Finally, empirical WCRE and WCE are proposed by virtue of sample mean, sample variance, and order statistics to estimate the new measures of uncertainty. The consistency of these estimators is studied under specific choices of distributions. Farsam Misagh Copyright © 2016 Farsam Misagh. All rights reserved. Quantum Difference Langevin System with Nonlocal -Derivative Conditions Sun, 05 Jun 2016 14:33:02 +0000 We introduce a new class of boundary value problems for Langevin quantum difference systems. Some new existence and uniqueness results for coupled systems are obtained by using fixed point theorems. The existence and uniqueness of solutions are established by Banach’s contraction mapping principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. The obtained results are well illustrated with the aid of examples. Surang Sitho, Sorasak Laoprasittichok, Sotiris K. Ntouyas, and Jessada Tariboon Copyright © 2016 Surang Sitho et al. All rights reserved. More of Dedekind: His Series Test in Normed Spaces Tue, 31 May 2016 11:36:58 +0000 Dedekind’s test for infinite series has a canonical interpretation in the context of normed spaces. It is shown that his test holds in a normed space precisely when the space is complete. Robert Kantrowitz and Michael M. Neumann Copyright © 2016 Robert Kantrowitz and Michael M. Neumann. All rights reserved. A Note on Primitivity of Ideals in Skew Polynomial Rings of Automorphism Type Mon, 30 May 2016 15:47:37 +0000 We extend results about primitive ideals in polynomial rings over nil rings originally proved by Smoktunowicz (2005) for -primitive ideals in skew polynomial rings of automorphism type. Edilson Soares Miranda Copyright © 2016 Edilson Soares Miranda. All rights reserved. General Quadratic-Additive Type Functional Equation and Its Stability Sun, 22 May 2016 06:21:26 +0000 We investigate the general functional equation of the form − − , whose solutions are quadratic-additive mappings in connection with stability problems. Yang-Hi Lee and Soon-Mo Jung Copyright © 2016 Yang-Hi Lee and Soon-Mo Jung. All rights reserved. The Agreement between the Generalized Value and Bayesian Evidence in the One-Sided Testing Problem Mon, 16 May 2016 11:54:31 +0000 In the problem of testing one-sided hypotheses, a frequentist may measure evidence against the null hypothesis by the value, while a Bayesian may measure it by the posterior probability that the null hypothesis is true. In this paper, we consider the relationship between the generalized value and the Bayesian evidence in testing one-sided hypotheses in the presence of nuisance parameters. The sufficient conditions for the agreement between these two kinds of evidence are given. Some examples are provided to show the agreement of Bayesian and frequentist evidence in many classical testing problems. This is an illustration of reconcilability of evidence in a general framework where the nuisance parameters are present. Yuliang Yin and Bingbing Wang Copyright © 2016 Yuliang Yin and Bingbing Wang. All rights reserved.