A new approach to solve a set of nonlinear split boundary value problems

In this paper, we propose a new approach based on conjunction of the orthogonal collocation on finite elements method with decoupling and quasi-linearization technique to approximate solutions of a set of nonlinear split boundary value problems. The numerical stability, the convergence and the accuracy of the results are checked by this algorithm. The approach developed in this study is illustrated by some numerical examples. These examples are solved using a special software package which implements the proposed algorithms.     

A numerical method to solve a class of linear integro‐differential equations with weakly singular kernel

In this paper, a collocation method based on the Bessel polynomials is introduced for the approximate solution of a class of linear integro‐differential equations with weakly singular kernel under the mixed conditions. The exact solution can be obtained if the exact solution is polynomial. In other cases, increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Discrete and continuous time representations and mathematical models for large production scheduling problems: A case study from the pharmaceutical industry

The underlying time framework used is one of the major differences in the basic structure of mathematical programming formulations used for production scheduling problems. The models are either based on continuous or discrete time representations. In the literature there is no general agreement on which is better or more suitable for different types of production or business environments. In this paper we study a large real-world scheduling problem from a pharmaceutical company. The problem is at least NP-hard and cannot be solved with standard solution methods. We therefore decompose the problem into two parts and compare discrete and continuous time representations for solving the individual parts. Our results show pros and cons of each model. The continuous formulation can be used to solve larger test cases and it is also more accurate for the problem under consideration.     

A homological approach to representations of algebras I: The wild case

We prove that the pure global dimension of a polynomial ring over an integral domain k in a finite or countable number n?2 of commuting (non-commuting, resp.) variables is t + 1, provided $\text{|k| = ?}{}_{\mathrm{t}}$. As an application, we determine the pure global dimension of wild algebras of quiver type, also (in case k is an algebraically closed field) of the wild local and wild commutative algebras of finite k-dimension.     

Mathematics and science teacher educators' use of representations of practice: A mixed methods study

This study sought to explore math and science teacher educators' use of various media to represent practice within methods courses. There is little understanding of why certain media is used over other representations and the rationale for these choices. Specifically, the study focused on the prevalence and familiarity of teacher educators with comics and animations, standard videos, and 360 videos. This mixed methods study utilized a survey and interviews to ascertain math and science teacher educators' level of familiarity and perceived usefulness of representations of practice. Results indicate that standard video is by far the most used representation of practice in methods classes with three themes explaining this finding: access to representations of practice, dimensions of representation, and pertinacity of using representations. Familiarity with representations of practice relates to teacher educators' perceptions of access thereby indicating a need for teacher educators to have better access to representations. Implications of this study include supporting current literature about the relationship between the level of familiarity and perceived usefulness in media along with the potential need for a central platform that houses these representations of practice resources for teacher educators.     

Classroom notes: A geometric method for the derivation of a special case of an F-distribution

The concept of stereographic projection from a circle to one of its tangent lines is applied in obtaining an F-distribution with degrees of freedom 1 and 3. The derivation brings out clearly the relationship between circular and non-circular distributions.     

A numerical study of electromagnetic waves in periodic waveguides

Here are considered time‐harmonic electromagnetic waves in a quadratic waveguide consisting of a periodic dielectric core enclosed by conducting walls. The permittivity function may be smooth or have jumps. The electromagnetic field is given by a magnetic vector potential in Lorenz gauge, and defined on a Floquet cell. The Helmholtz operator is approximated by a Chebyshev collocation, Fourier–Galerkin method. Laurent's rule and the inverse rule are employed for the representation of Fourier coefficients of products of functions. The computations yield, for known wavenumbers, values of the first few eigenfrequencies of the field. In general, the dispersion curves exhibit band gaps. Field patterns are identified as transverse electric, TE, transverse magnetic, TM, or hybrid modes. Maxwell's equations are fulfilled. A few trivial solutions appear when the permittivity varies in the guiding direction and across it. The results of the present method are consistent with exact results and with those obtained by a low‐order finite element software. The present method is more efficient than the low‐order finite element method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 490–513, 2014     

Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory

Wavelets on closed surfaces in Euclidean space ℝ³ are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of function values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

How algebra spoiled recreational problems: A case study in the cross-cultural dissemination of mathematics

This paper deals with a sub-class of recreational problems which are solved by a simple memorized rule resulting from an elementary arithmetical or algebraic solution, called proto-algebraic rules. Their recreational aspect is derived from a surprise or trick solution which is not immediately obvious to the subjects involved. Around 1560 many such problems wane from arithmetic and algebra textbooks to reappear in the eighteenth century. Several hypotheses are investigated why popular Renaissance recreational problems lost their appeal. We arrive at the conclusion that the emergence of algebra as a general problem solving method changed the scope of what is considered recreational in mathematics.     

Speeding up the FMMR perfect sampling algorithm: A case study revisited

In a previous paper by the second author, two Markov chain Monte Carlo perfect sampling algorithms—one called coupling from the past (CFTP) and the other (FMMR) based on rejection sampling—are compared using as a case study the move‐to‐front (MTF) self‐organizing list chain. Here we revisit that case study and, in particular, exploit the dependence of FMMR on the user‐chosen initial state. We give a stochastic monotonicity result for the running time of FMMR applied to MTF and thus identify the initial state that gives the stochastically smallest running time; by contrast, the initial state used in the previous study gives the stochastically largest running time. By changing from worst choice to best choice of initial state we achieve remarkable speedup of FMMR for MTF; for example, we reduce the running time (as measured in Markov chain steps) from exponential in the length n of the list nearly down to n when the items in the list are requested according to a geometric distribution. For this same example, the running time for CFTP grows exponentially in n. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

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 probabilistic model to describe the spread of the Human Immunodeficiency Virus by intravenous drug use

In contrast to the majority of mathematical investigations into the dynamics of the spread of the human immunodeficiency virus (HIV), which have been concerned with sexual transmission, this investigation deals with its spread by intravenus drug use (IVDU). An ‘addict based’ approach is presented in which the proportion of the susceptible population to be infected is determined by Monte Carlo means. This is in contrast to a 1989 ‘shooting gallery’ approach by Kaplan.     

Multifunctional land‐use value mapping and space type classification: A case study of Puge County,China

Land exhibits diverse functions under the combined influence of natural and human forces. A production–living–ecology functional classification system was constructed by integrating land, ecosystem, and landscape functions. The land functional value was calculated by systematically integrating ecosystem service value assessments. The primary and secondary functions, as well as combinations of different land‐use types, were determined using vertical and horizontal comparison methods. The production–living–ecology ranges were then delineated in Puge County, which is a typical mountain county in China. The production–living–ecology functions identified were well connected with the current land‐use types. The “production–living–ecology” space in Puge County showed obvious multifunctionality and agglomeration. The function identification system proposed in this paper integrated multiple methods, overcame the difficulty of direct quantitative identification of land functions. The methods used to map and quantify land function will enhance our ability to understand and model land system changes and adequately inform policies and planning. Summary for Managers

• The function valuation method constructed in this paper could be used to reflect the multifunctionality and importance of land use and provide guidance and a quantitative basis for regional development planning.
• The spatial classification results provided in this paper could offer a valuable reference for the land management department to scientifically formulate land use planning.
• Under the goal of creating a group‐type urban development pattern and constructing an ecological protection pattern in Puge County, the hotspot analysis results of this paper can provide decision‐making tools and alternative spatial plans.

     

Unobserved Component models applied to the assessment of wear in railway points: A case study

Railways are experiencing a fundamental transformation. The introduction of high speed networks and the increased traffic levels on suburban routes and freight lines require new technologies for both railway infrastructure and trains, all of which must be subjected to rigorous quality control before and during operation and must be supported with effective maintenance processes during their operating lives. Safety in railway infrastructure provision must be ensured by all the main components operating reliably all the time. From an economic, quality and safety point of view, points are probably one of the most critical infrastructure elements in railway transportation.     

On the use of the geometric approach to global stability for three dimensional ODE systems: A bilinear case

In this paper, we consider a general bilinear three dimensional ODE system, whose structure generalizes many mathematical models of biological interest, including many from epidemics. Our main goal is to find sufficient conditions, expressed in terms of the parameters of the system, ensuring that the geometric approach to global stability analysis, due to [M.Y. Li, J.S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal. 27 (4) (1996) 1070-1083], may be successfully applied. We completely determine the dynamics of the general system, including thresholds and global stability of the nontrivial equilibrium. The obtained result is applied to several epidemic models. We further show how the role of new parameters on stability of well-established models may be emphasized.     

Addendum to “An extension of an inequality of Hoeffding to unbounded random variables”: The non-i.i.d. case

Let S _n = X ₁ + ⋯ + X _n be a sum of independent random variables such that 0 ⩽ X _k ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where X _k are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d. case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related Hoeffding inequalities if 0 ⩽ X _k ⩽ 1. Our conditions are X _k ⩾ 0 and {ie237-05}. In particular, X _k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of X _k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities in Hoeffding [6] and Bentkus The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08.     

Tiny families of functions with random properties: A quality-size trade-off for hashing

We present three explicit constructions of hash functions, which exhibit a trade-off between the size of the family (and hence the number of random bits needed to generate a member of the family), and the quality (or error parameter) of the pseudorandom property it achieves. Unlike previous constructions, most notably universal hashing, the size of our families is essentially independent of the size of the domain on which the functions operate. The first construction is for the mixing property—mapping a proportional part of any subset of the domain to any other subset. The other two are for the extraction property—mapping any subset of the domain almost uniformly into a range smaller than it. The second and third constructions handle, respectively, the extreme situations when the range is very large or very small. We provide lower bounds showing that our constructions are nearly optimal, and mention some applications of the new constructions. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 315–343 (1997)     

A Lagrangian relaxation approach to the mixed-product assembly line sequencing problem: A case study of a door-lock company in Taiwan

In mixed-product assembly line sequencing, the production resources required for the assembly lines should be scheduled to minimize the overall cost and meet customer demand. In this paper, we study an assembly line sequencing problem for the door-lock industry in Taiwan and develop an integer programming formulation with realistic constraints. The complex solution space makes the resulting program difficult to solve using commercial optimization packages. Therefore, a heuristic based on the Lagrangian relaxation principle is developed to solve this problem efficiently. We evaluate the efficiency of the developed Lagrangian relaxation heuristic by comparing its solutions with those obtained using a commercial optimization package: the computational results show that the developed heuristic solves the real-world problem faster than the optimization package by almost 15 times in CPU time at a comparable solution quality.     

Application of fuzzy optimization to a supply chain network design: A case study of an edible vegetable oils manufacturer

This study applies fuzzy sets to integrate the supply chain network of an edible vegetable oils manufacturer. The proposed fuzzy multi-objective linear programming model attempts to simultaneously minimize the total transportation costs. The first part of the total transportation costs is between suppliers and silos; and rest one is between manufacturer and warehouses. The approach incorporates all operating realities and actual flow patterns at production/distribution network with reference to demands of warehouses, capacities of tin and pet packaging lines. The model has been formulated as a multi objective linear programming model where data are modeled by triangular fuzzy numbers. Finally, the developed fuzzy model is applied for the case study, compiled the results and discussed.