Computer Mathematics

Identification of co-expressed genes sharing similar biological behaviors is an essential step in functional genomics. Traditional clustering techniques are generally based on overall similarity of expression levels and often generate clusters with mixed profile patterns. This paper proposes a novel pattern recognition method for selecting co-expressed genes based on rate of change and modulation status of gene expression at each time interval. The proposed method is capable of identifying gene clusters consisting of highly similar shapes of expression profiles and modulation patterns. Furthermore, we develop a quality index based on the semantic similarity in gene annotations to assess the likelihood of a cluster being a co-regulated group. The effectiveness of the proposed methodology is demonstrated through its application to the well-known yeast sporulation dataset and an in-house cancer genomics dataset.

In this paper, we consider a rational algorithm for modification of a positive measure by quadratic factor, d σ (t) = (tz) 2 dσ (t), where it is allowed z to be in supp(dσ ). Also, we present an application of modified algorithm to the measures d σ (t) = T 2 2 (t) dσ (t) and dσ (t) = t 2 T 2 2 (t) dσ (t), where T 2 (t) = t 2 -1 2 is the second degree monic Chebyshev polynomial of the first kind and dσ is the Chebyshev measure of the second kind. Also, we present an application to the constrained L 2 -polynomial approximation.

In this paper, we consider a rational algorithm for modification of a positive measure by quadratic factor, d σ (t) = (tz) 2 dσ (t), where it is allowed z to be in supp(dσ ). Also, we present an application of modified algorithm to the measures d σ (t) = T 2 2 (t) dσ (t) and dσ (t) = t 2 T 2 2 (t) dσ (t), where T 2 (t) = t 2 -1 2 is the second degree monic Chebyshev polynomial of the first kind and dσ is the Chebyshev measure of the second kind. Also, we present an application to the constrained L 2 -polynomial approximation.

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Assuming data domains are partially ordered, we apply Paredaens' and Bancilhon's Theorem to examine the expressiveness of the extended relational algebra (the PORA), which allows the ordering predicate to be used in formulae of the selection operator (σ). The PORA expresses exactly the set of all possible relations which are invariant under order-preserving automorphism of databases. Our main result shows that there is a one-toone correspondence between three hierarchies of: (1) computable queries, (2) query languages and (3) partially ordered domains.

In the article we show that combining random generators by group operation improves the statistical properties of the composite. It gives an effective way of finding random generators more and more close to the uniform. Moreover we obtain an effective estimation of the speed of convergence to the uniform generator.

We present a family of multipoint methods without memory with sixth order convergence for solving systems of nonlinear equations. The methods use first order divided difference operators and are derivative free. Extending the work further, we explore a family of methods with memory with increasing order of convergence from the basic family without memory. The increase in convergence order is attained by varying a free parameter step to step using information available from the previous step. It is proved that the convergence order of the family with memory is increased from 6 to 3 + √ 10 ≈ 6.162, and in some special cases to 3 + 2 √ 3 ≈ 6.464 and 3 + √ 13 ≈ 6.606. Computational efficiency of the methods is discussed and compared with existing methods. Numerical examples, including those arise from integral equations and boundary value problems, are considered to verify the theoretical results. A comparison with the existing methods shows that the new methods are more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.

The spline collocation method is used to approximate solutions of boundary value problems. The convergence analysis is given and the method is shown to have second-order convergence. A numerical illustration is given to show the pertinent features of the technique.

String matching algorithms are used to find a particular pattern in a string. The aim of this research is to explain the basic ideas and complexity of current string matching algorithms and to perform applied comparison. There are many algorithms used for string matching. In this study, Knuth-Morris-Pratt, Rabin Karp and Boyer Moore Horspool algorithms were compared. It is aimed to increase the accuracy of the study by choosing different algorithms. The basic ideas, potential difficulties and complexities of the algorithms are explained and how these problems can be solved is emphasized. In order to better evaluate the algorithms, three different algorithms were compared in twenty-five different pattern types. As a result of the studies, it has been seen that the Knuth-Morris-Pratt algorithm outperforms other algorithms in most cases. The second best performing algorithm was Boyer Moore Horspool algorithm, and the worst performing algorithm was Rabin Karp algorithm.

In this paper, polynomial-based superconvergent degenerate kernel and Nyström methods for solving Fredholm integral equations of the second kind with the smooth kernel are studied. By using an interpolatory projection based on Legendre polynomials of degree ≤ n, we analyze the convergence of these methods and we establish superconvergence results for their iterated versions. Two numerical examples are given to illustrate the theoretical estimates.

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An efficient implicit finite-difference algorithm for the gasdynamic equations utilizing matrix reduction techniques is presented. A significant reduction in arithmetic operations is achieved while maintaining the same favorable stability characteristics and generality found in the Beam and "'farming approximate factorization algorithm. Steady-state solutions to the conservative Euler equations in generalized coordinates are obtained for transonic flows about a NACA0012 airfoil. The theoretical extension of the matrix reduction technique to the full Navier-Stokes equations in Cartesian coordinates is presented in detail. Linear stability, using a Fourier stability analysis, is demonstrated and discussed for the one-dimensional Euler equations. It is shown that the method offers advantages over the conventional Beam and Warming scheme and can retrofit existing Beam and Warming codes with minimal effort.

We study some procedures for the approximation of three-dimensional data on a grid with a hypothesis of periodicity. The first part proposes a generalization of a discrete periodic approximation defined by Dunham Jackson. The functions used have the advantage of owning an analytical explicit expression in terms of the samples (specific values) of the original function or data. In the second part we describe a continuous approximation function for the same problem, defined through an integral. Some results of the rate of convergence and bounds of the approximation error are presented, with the single hypothesis of Hölder continuity or continuity of the original function.

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators...

We present a new theorem in Threshold Logic which is a generalization of Mezei's theorem. We demonstrate one of its applications, i.e., the optimization of the size of depth-2 Multioperand Binary Adders. By applying this theorem, the weight complexity of the gates in the second layer of a previous construction is reduced from where m is the number of operands and n is the number of bits.

This paper describes the Adaptive Steepest Descent (ASD) and Optimal Fletcher-Reeves (OFR) algorithms for linear neural network training. The algorithms are applied to well-known pattern classification and function approximation problems, belonging to benchmark collection Proben1. The paper discusses the convergence behavior and performance of the ASD and OFR training algorithms by computer simulations and compares the results with those produced by linear-RPROP method.

Student academic underachievement is a concern of paramount importance in Europe, where around 15% of the students in the last courses in high school do not achieve the minimum knowledge academic requirement. In this paper, we propose a model based on a system of differential equations to study the dynamics of the students academic performance in the German region of North Rhine-Westphalia. This approach is supported by the idea that both, good and bad study habits, are a mixture of personal decisions and influence of classmates. This model allows us to forecast the student academic performance by means of confidence intervals over the next few years.

Boussinesq-type nonlinear wave equations with dispersive terms are solved via split-step Fourier methods. We decompose the equations into linear and nonlinear parts, then solve them orderly. The linear part can be projected into phase space by a Fourier transformation, and resulting in a variable separable ordinary differential system, which can be integrated exactly. Next, by an invert Fourier transformation, the classical explicit fourth-order Runge-Kutta method is adopted to solve the nonlinear subproblem. To examine the numerical accuracy and efficiency of the method, we compare the numerical solutions with exact solitary wave solutions. Additionally, various initial-value problems for all the listed Boussinesq-type system are studied numerically. In the study, we can observe that sech 2 -type waves for KdV-BBM system will split into several solitons, which is a very interesting physical phenomenon. The interaction between solitons, including overtaking and head-on collisions, is also simulated.

Boussinesq-type nonlinear wave equations with dispersive terms are solved via split-step Fourier methods. We decompose the equations into linear and nonlinear parts, then solve them orderly. The linear part can be projected into phase space by a Fourier transformation, and resulting in a variable separable ordinary differential system, which can be integrated exactly. Next, by an invert Fourier transformation, the classical explicit fourth-order Runge-Kutta method is adopted to solve the nonlinear subproblem. To examine the numerical accuracy and efficiency of the method, we compare the numerical solutions with exact solitary wave solutions. Additionally, various initial-value problems for all the listed Boussinesq-type system are studied numerically. In the study, we can observe that sech 2 -type waves for KdV-BBM system will split into several solitons, which is a very interesting physical phenomenon. The interaction between solitons, including overtaking and head-on collisions, is also simulated.

This paper considers a neighborhood broadcasting protocol in undirected de Bruijn and Kautz networks. The neighborhood broadcasting problem(NBP) is the problem of disseminating a message from an originator vertex to only its neighbors. Our protocol works under the single-port and half-duplex model and solves NBP in 5 log 2 (n + 1) + O(1) time units on the undirected de Bruijn graph UB (n, d) with n d vertices and the undirected Kautz graph UK (n, d) with n d +n d-1 vertices, where 2n is the maximum degree of these graphs. This completion time is asymptotically optimal in this model.

We develop an algebraic theory of synchronous dataflow networks. First, a basic algebraic theory of networks, called BNA (Basic Network Algebra), is introduced. This theory captures the basic algebraic properties of networks. For synchronous dataflow networks, it is subsequently extended with additional constants for the branching connections that occur between the cells of synchronous dataflow networks and axioms for these additional constants. We also give two models of the resulting theory, the one based on stream transformers and the other based on processes as considered in process algebra.

A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The Diameter-Constrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NP-complete, for all values of k; 4 k n − 2). In this paper, we investigate the behavior of the diameter of MST in randomly-weighted complete graphs (in Erdös-Rényi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is small-independent of n, we present a one-time-tree-construction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of these heuristics on sequential and parallel machines. Our extensive empirical study shows that these two heuristics produce good solutions for a wide variety of inputs.

In the recent project BENCHOP-the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.

The aim of the BENCHOP project is to provide the finance community with a common suite of benchmark problems for option pricing. We provide a detailed description of the six benchmark problems together with methods to compute reference solutions. We have implemented fifteen different numerical methods for these problems, and compare their relative performance. All implementations are available on line and can be used for future development and comparisons.

In order to solve the large sparse systems of linear equations arising from numerical solutions of twodimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050-3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.

Identification of co-expressed genes sharing similar biological behaviors is an essential step in functional genomics. Traditional clustering techniques are generally based on overall similarity of expression levels and often generate clusters with mixed profile patterns. This paper proposes a novel pattern recognition method for selecting co-expressed genes based on rate of change and modulation status of gene expression at each time interval. The proposed method is capable of identifying gene clusters consisting of highly similar shapes of expression profiles and modulation patterns. Furthermore, we develop a quality index based on the semantic similarity in gene annotations to assess the likelihood of a cluster being a co-regulated group. The effectiveness of the proposed methodology is demonstrated through its application to the well-known yeast sporulation dataset and an in-house cancer genomics dataset.

In this paper, fourth order, five-stage embedded in fifth order six-stage Singly Diagonally Implicit Runge-Kutta (SDIRK) methods are derived, and their regions of stability are determined. Numerical results of these methods are presented and compared with the results obtained when the same systems are solved using Rabeh's third order three-stage embedded in fourth order four-stage SDIRK method. Also presented are the stability regions of the methods.

In P systems working in the set derivation mode, even in the maximally parallel derivation mode, rules are only applied in at most one copy in each derivation step. We also consider the set mode in the cases of taking those sets of rules with the maximal number of applicable rules or with affecting the maximal number of objects. For many variants of P systems, the computational completeness proofs even literally still hold true for these new set derivation modes. On the other hand, we obtain new results for P systems using target selection for the rules to be chosen together with these set derivation modes.

Matrix inversion is an important requirement for many scientific and practical applications. Parallel architectures like Graphics Processing Units (GPUs) can have noticeable impacts on accelerating such computations, however, at the expense of more power consumption. Both computation speed and power become more challenging when the size of matrices gets larger. This paper proposes an accelerated version of the Gauss-Jordan matrix inversion method using GPUs. The experimental results show that the proposed method is faster than a recent baseline method. As an additional property, the configurations have been made in a manner to consume less power. Keywords–Matrix Inversion; GPU; Power Consumption.

With the most recent advances in bioinformatics, the amount of information available for analysing certain diseases has increased considerably. Specifically, the use of microarrays makes it possible to obtain information on genetic patterns. The analysis of this information requires the use of new computational models and the modification of existing models so that it becomes possible to work with such an elevated amount of data. This study will demonstrate the integration of an expression analysis in a case-based reasoning system that can apply data mining techniques to classify and obtain patterns that have been stored in a case database for leukaemia patients.

parallel algorithm for solving the multinephron model of the renal inner medulla is developed. The intrinsic nature of this problem supplies sufficient symmetry for a high-level parallelism on distributed-memory parallel machines such as the iPSC/SSO, Paragon, and CM-5. Parallelization makes it feasible to study interesting models such as the rat kidney with 30,000 nephrons. On a high-end work station, one can study systems with 100 nephrons, while on a 32-node iPSC/SSO or Paragon we can handle more than 1000 nephrons. A nearly perfect speedup is achieved by even distribution of load and minimizing the cost of communication.

In present world of electronic digital media the demand of high speed communication devices is increasing day-by-day. The security and authenticity of the information on internet is one of the most important issues of research. This paper presents a short review of Digital Watermarking and a Discrete Wavelet Transformation (DWT) based watermarking algorithm implementation using MATLAB programming. The effects of watermarking on digital image data are simulated and analysed in the present paper.

The main aim of this paper is to propose a robust and efficient numerical scheme for solving time dependent singularly perturbed Burgers-Fisher problem based on mesh adaptive strategy in the finite element framework. The Burgers-Fisher equation in one space dimension is a partial differential equation which exhibits travelling wave phenomenon. Since Burgers-Fisher equation is a non-linear problem, quasilinearization process has been used to deal with nonlinearity occurring in the problem. Time discretization has been performed using implicit Euler method. Then spatial discretization has been carried out using finite element technique based on exponentially fitted splines on piecewise uniform Shishkin mesh. The stability of the proposed numerical scheme has been discussed. At the end, it has been shown numerically that the proposed method is very much effective for capturing sharp boundary layers arising in the solution as singular perturbation parameter → 0.

The main aim of this paper is to propose a robust and efficient numerical scheme for solving time dependent singularly perturbed Burgers-Fisher problem based on mesh adaptive strategy in the finite element framework. The Burgers-Fisher equation in one space dimension is a partial differential equation which exhibits travelling wave phenomenon. Since Burgers-Fisher equation is a non-linear problem, quasilinearization process has been used to deal with nonlinearity occurring in the problem. Time discretization has been performed using implicit Euler method. Then spatial discretization has been carried out using finite element technique based on exponentially fitted splines on piecewise uniform Shishkin mesh. The stability of the proposed numerical scheme has been discussed. At the end, it has been shown numerically that the proposed method is very much effective for capturing sharp boundary layers arising in the solution as singular perturbation parameter → 0.

The model describes an EOQ model with time-varying deterioration, partial backlogging which depends on the length of the waiting time for the next replenishment, linearly time-varying demand function over a finite time horizon and variable replenishment cycle. The model is solved analytically to obtain the optimal solution of the problem. It is illustrated with the help of a numerical example.

The r-th order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed-Muller code R(r, n). In this paper we deduce the lower bounds of the second order nonlinearity of the following two types of Boolean functions: 1. f λ (x) = T r n 1 (λx d) with d = 2 2r + 2 r + 1 and λ ∈ F 2 n where n = 6r. 2. f (x, y) = T r t 1 (xy 2 i +1) where x, y ∈ F 2 t , n = 2t, n ≥ 6 and i is an integer such that 1 ≤ i < t, gcd(2 t − 1, 2 i + 1) = 1. For some λ, the functions of the first type are bent functions whereas Boolean functions of the second type are all bent functions, i.e., they possess maximum first order nonlinearity. It is demonstrated that in some cases our bounds are better than the previously obtained bounds.

Let COL k be the set of all graphs that are k-colorable. It is well known that COL k is NP-complete. It is also well known, and easy, to show that if a ≤ b then COL a ≤ COL b. If 3 ≤ a ≤ b then we also have COL b ≤ SAT ≤ COL a which is an insane reduction from COL b to COL a. In this paper we give a sane reduction from COL b to COL a .

In this paper, we characterize digraphs of 3 vertices associated with Lie algebras according to isomorphism classes of these associated Lie algebras. At this respect, we introduce and implement two algorithmic methods: the first is devoted to draw the digraph associated with a given Lie algebra and the second allows us to determine if a given digraph is associated or not with a Lie algebra.

This paper presents an innovative collocation algorithm designed to effectively handle a specific class of boundary value problems with high-order characteristics. The approach involves utilizing a novel variant of exponentialtype Chebyshev polynomials that meet all the necessary equation conditions. A key aspect of the algorithm is the transformation of both linear and nonlinear forms of the equations, along with their respective boundary conditions, into systems of algebraic equations. By solving these systems, a unique iterative technique is employed that significantly reduces the computational time required for solving these types of equations. To validate the effectiveness of the algorithm, numerous experiments using various examples with differing orders and types are conducted. The proposed technique is compared against other similar methods. The results obtained demonstrate the exceptional accuracy of the proposed approach and its potential for extension to other models in the future. Additionally, a comprehensive and detailed error analysis of the proposed method is developed, further confirming its robustness and precision in practical applications.

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We consider a monic polynomial of even degree with symbolic coefficients. We give a method for obtaining an expression in the coefficients (regarded as parameters) that is a lower bound on the value of the polynomial, or in other words a lower bound on the minimum of the polynomial. The main advantage of accepting a bound on the minimum, in contrast to an expression for the exact minimum, is that the algebraic form of the result can be kept relatively simple. Any exact result for a minimum will necessarily require parametric representations of algebraic numbers, whereas the bounds given here are much simpler. In principle, the method given here could be used to find the exact minimum, but only for low degree polynomials is this feasible; we illustrate this for a quartic polynomial. As an application, we compute rectifying transformations for integrals of trigonometric functions. The transformations require the construction of polynomials that are positive definite.

The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden.

In this manuscript, we construct a general family of optimal derivative free iterative methods by using rational interpolation. This family is further extended to a family of with-memory methods with increased order of convergence by employing two free parameters. At each iterative step, we use a suitable variation of the free parameters. These parameters are computed by using the information from current and previous iterations so that the convergence order of the existing family is increased from 2 to 2 + 2n−1 + 2n−2 without using any additional function evaluations. To check the performance of newly developed iterative schemes with and without memory, an extensive comparison with the existing withand without memory methods is done by taking some real world problems and standard nonlinear functions. Numerical experiments illustrate that the proposed family of methods with-memory retain better computational efficiency and fast convergence speed as compared to existing withand witho...

In this article, the Exp -function method is applied to construct traveling wave solutions of the fifth-order Sawada-Kotera equation. This method is one of the powerful methods that appear in recent time for establishing exact traveling wave solutions of nonlinear partial differential equations. The solution procedure of this method is implemented by symbolic software, such as, Maple. We obtain some new exact solutions including solitary and periodic wave solutions. It is shown that the Exp -function method is straightforward and effective mathematical tool for solving nonlinear evolution equations in mathematical physics and engineering sciences. In addition, some of solutions are described in the figures with the aid of commercial software Maple.

In this paper we construct new modifications of Jarratt's method for solving nonlinear equations based on the circle of curvature. A precise analysis of convergence is given to show that the new methods are of twelfth-order. Numerical examples are provided to illustrate that the presented methods achieve faster convergence in comparison with other known methods and can compete with the existing methods. MSC: 41A25 • 65D99