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We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes.

In this paper, delta and extreme shock models and a mixed shock model which combines delta-shock and extreme shock models are studied. In particular, the interarrival times between successive shocks are assumed to belong to a class of matrix-exponential distributions which is larger than the class of phase-type distributions. The Laplace -Stieltjes transforms of the systems’ lifetimes are obtained in a matrix form. Survival functions of the systems are approximated based on the Laplace-Stieltjes transforms. The results are applied for the reliability evaluation of a certain repairable system consisting of two components.

The problem of determining the body attitude in matrix form using the least-squares method is considered. A loss function that allows us to theoretically analyze the influence of the measurement errors on the accuracy of the attitude for an arbitrary number of vectors without the need for the mutual perpendicularity of the vectors. The proposed method allows us to use measurable quantities with weight coefficients.

The vertical lift gates are exposed to hydrodynamic forces that arise as a result of pressurized flow established by the large-scale effect of the water levels in the dam reservoirs. The most influential forces are those that are applied on the gate vertically upward due to the intensity of the flow under the gate and downward as a result of passage flow above the gate. The difference between these two forces generates a downpull force that is the most important gate stability indicator. In this research, an arbitrary hydraulic model was constructed to investigate the effects of many gate lip shapes with and without extensions on the magnitudes and distribution of bottom pressure coefficient when the values of flow and shaft gap width ratio (b2/bl) are constant. The results indicate that the bottom pressure coefficient appears to vary uniformly with gate openings and seem to be influenced effectively by the gate lip geometry. So, for the given value of (b2/bl), the top pressure coefficient regarding the downward force will mostly be with uniform changes and hence the downpull coefficient depends on the magnitudes and distribution of bottom pressure coefficient. An attempt by using the correlation coefficient of Statistical Package of Social Sciences was made to determine whether the pressure fluctuation is significantly shown at the bottom pressure distribution curve, which in turn is considered as appropriate indicator for the vibration occurrence. Full range of maximum bottom pressure coefficients in a matrix form is made to assist in design purposes. The results are analyzed and many conclusions are obtained.

In this work, we present a solution of the Boussinesq equation, which models shallow water waves. A powerful inverse scattering approach of Deift–Tomei–Trubowitz was published in 1982 for solving this equation in the Schwartz class. We rewrite the corresponding Lax pair in a matrix form and show that a recently developed theory of evolutionary nodes is applicable. As a result, we can locally solve this equation with analytic initial conditions on the line, providing a formula for the tau function, which defines where the solutions exist. More generally, arbitrary analytic solution is shown to arise from a vessel.

In this paper, we discuss the design and analysis of polynomial time algorithms for two problems associated with a linearly infeasible system of Unit Two Variable Per Inequality (UTVPI) constraints, viz., (a) the read-once refutation (ROR) problem, and (b) the literal-once refutation (LOR) problem. Recall that a UTVPI constraint is a linear relationship of the form: $$a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{ij}$$ , where $$a_{i},a_{j} \in \{0,1,-1\}$$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: $$\mathbf{A \cdot x \le b}$$ . These constraints find applications in a host of domains including but not limited to operations research and program verification. For the linear system $$\mathbf{A\cdot x \le b}$$ , a refutation is a collection of m variables $$\mathbf{y}=[y_{1},y_{2},\ldots , y_{m}]^{T} \in \mathfrak {R}^{m}_{+}$$ , such that $$\mathbf{y\cdot A =0}$$ , $$\mathbf{y \cdot b } < 0$$ . Such a refutation is guaranteed to exist for any infeasible linear program, as per Farkas’ lemma. The refutation is said to be read-once, if each $$y_{i} \in \{0,1\}$$ . Read-once refutations are incomplete in that their existence is not guaranteed for infeasible linear programs, in general. Indeed they are not complete, even for UCSs. Hence, the question of whether an arbitrary UCS has an ROR is both interesting and non-trivial. In this paper, we reduce this problem to the problem of computing a minimum weight perfect matching (MWPM) in an undirected graph. This transformation results in an algorithm that runs in in time polynomial in the size of the input UCS. Additionally, we design a polynomial time algorithm (also via a reduction to the MWPM problem) for a variant of read-once resolution called literal-once resolution. The advantage of reducing our problems to the MWPM problem is that we can leverage recent advances in algorithm design for the MWPM problem towards solving the ROR and LOR problems in UCSs. Finally, we show that another variant of read-once refutation problem called the non-literal read-once refutation (NLROR) problem is NP-complete in UCSs.

In this article, we study linear barycentric rational method (LBM) to solve one-order Volterra integro-differential equations. With the help of linear barycentric rational interpolation, the matrix form of Volterra integro-differential equations is obtained. Then the convergence rate of LBM for solving one-order integro-differential equation is proved. Further more, Volterra integro-differential equation systems is also solved by the linear barycentric rational method. At last, several examples are presented to valid our theoretical analysis.

A spectral collocation method is developed to solve a nonlinear Caputo fractional differential system. The main idea is to solve the corresponding system of weakly singular nonlinear Volterra integral equations (VIEs). The convergence analysis in matrix form shows that the presented method has spectral convergence. Numerical experiments are carried out to confirm theoretical results.

Fast design of two-dimensional FIR filters in the least $${l}_{p}$$ -norm sense is investigated in this brief. The design problem is first formulated in a matrix form and then solved by a matrix-based iterative reweighted least squares algorithm. The proposed algorithm includes two loops: one for updating the weighting function and the other for solving the weighted least squares (WLS) subproblems. These WLS subproblems are solved using an efficient matrix-based WLS algorithm, which is an iterative procedure with its initial iterative matrix being the solution matrix in the last iteration, resulting in a considerable CPU-time saving. Through analysis, the new algorithm is shown to have a lower complexity than existing methods. Three design examples are provided to illustrate the high computational efficiency and design precision of the proposed algorithm.

Research evaluation is a multi-dimensional problem as there are multiple input dimensions, multiple output dimensions and a country space of many dimensions. The data making the connections are usually available in matrix form. In this paper, we use matrix normalization and multiplication so that totalized input and output measures can be obtained. This facilitates comparative research evaluation. The US and Japan lead as the major players when patents and papers are jointly considered. Most of the countries considered register totalized output that is commensurate with totalized input. Three countries are seen to be falling short of this ideal: China, Russia and Mexico.

Regularization is used to solve the ill-posed problem of magnetotelluric inversion usually by adding a stabilizing functional to the objective functional that allows us to obtain a stable solution. Among a number of possible stabilizing functionals, smoothing constraints are most commonly used, which produce spatially smooth inversion results. However, in some cases, the focused imaging of a sharp electrical boundary is necessary. Although past works have proposed functionals that may be suitable for the imaging of a sharp boundary, such as minimum support and minimum gradient support (MGS) functionals, they involve some difficulties and limitations in practice. In this paper, we propose a minimum support gradient (MSG) stabilizing functional as another possible choice of focusing stabilizer. In this approach, we calculate the gradient of the model stabilizing functional of the minimum support, which affects both the stability and the sharp boundary focus of the inversion. We then apply the discrete weighted matrix form of each stabilizing functional to build a unified form of the objective functional, allowing us to perform a regularized inversion with variety of stabilizing functionals in the same framework. By comparing the one-dimensional and two-dimensional synthetic inversion results obtained using the MSG stabilizing functional and those obtained using other stabilizing functionals, we demonstrate that the MSG results are not only capable of clearly imaging a sharp geoelectrical interface but also quite stable and robust. Overall good performance in terms of both data fitting and model recovery suggests that this stabilizing functional is effective and useful in practical applications.

There are several methods for constructing self-dual codes over various rings. Among them, the building-up method is a powerful method, and it can be applied to self-dual codes over finite fields and several rings. Recently, Alfaro and Dhul-Qarnayn (Des Codes Cryptogr, doi: 10.1007/s10623-013-9873-9 ) proposed a method for constructing self-dual codes over $${\mathbb F}_{q}[u]/(u^{t})$$ F q [ u ] / ( u t ) . Their approach is a building-up approach that uses the matrix form. In this paper, we use the matrix form to develop a building-up approach for constructing self-dual codes over $${\mathbb Z}_{2^m} (m \ge 1)$$ Z 2 m ( m ? 1 ) , which have not been considered thus far.

The main aim of this paper is to derive homogenized equations in explicit form of the linear piezoelectricity in two-dimensional domains separated by an interface which highly oscillates between two parallel straight lines. First, the basic equations of the linear theory of piezoelectricity are written down in matrix form. Then, following the techniques presented recently by these authors, the explicit homogenized equation and the associate continuity condition, for generally anisotropic piezoelectric materials, are derived. They are then written down in component form for some specific cases. Since the obtained equations are totally explicit, they are significant in practical applications.

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

In an addendum to the recent systematic Hermitization of certain N by N matrix Hamiltonians H (N)(λ) (Znojil in J. Math. Phys. 50:122105, 2009) we propose an amendment H (N)(λ,λ) of the model. The gain is threefold. Firstly, the updated model acquires a natural mathematical meaning of Runge-Kutta approximant to a differential $\mathcal{PT}$ -symmetric square well in which $\mathcal{P}$ is parity. Secondly, the appeal of the model in physics is enhanced since the related operator $\mathcal{C}$ of the so called “charge” (the requirement of observability of which defines the most popular Bender’s metric $\Theta=\mathcal{PC}$ ) becomes also obtainable (and is constructed here) in an elementary antidiagonal matrix form at all N. Last but not least, the original phenomenological energy spectrum is not changed so that the domain of its reality (i.e., the interval of admissible couplings λ∈(−1,1)) remains the same.

The structure of a canalizing function is discussed. Using a new matrix product, namely semitensor product, the logical function is expressed in its matrix form. From its matrix expression, a criterion is obtained to test whether a logical function is a canalizing function. Then a formula is obtained to calculate the number of canalizing functions. Moreover, an algorithm is presented to generate canalizing functions. Finally, some results obtained are extended to seminested canalizing functions.

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q=0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q=0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas., changed title, references updated, minor changes matching the version to appear in Ramanujan J.; 22 pp

This article deals with the dynamic analysis of train–track–bridge interaction system using the finite element method. In this interaction system, each four-wheelset vehicle in the train is modeled by a mass–spring–damper system with 10 degrees of freedom; the rails and the bridge decks are modeled as a number of Bernoulli–Euler beam elements, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The equation of motion for the interaction system is presented in matrix form with time-dependent coefficients. The correctness of the proposed procedure is illustrated by a comparison with the numerical result from the existing literature. Several numerical examples are chosen to investigate the effect of two types of vehicle models, two types of bridge models and three damping values of bridge on the maximum dynamic responses of train, track and bridges.

In this work the equations of thermoelasticity with two relaxation times for one-dimensional problem are cast into matrix form using the state space and the Laplace transform techniques. The resulting formulation is applied to a half-space problem with thermal shock and vibrational stress. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for temperature, displacement, and stress distributions are given and illustrated graphically.

The authors use the methods of initial parameters in the matrix form to study the stability of rods under the action of the following forces. They consider rods with various fastening of the ends. The obtained theoretical results have been verified by an experiment. It is shown that the difference between the theoretical and experimental values of the critical following forces does not exceed 5%.

The generalized inverse function-valued Pade approximant was defined to solve the integral equations. However, it is difficult to compute the approximants by some high-order determinant formulas. In this paper, to simplify computation of the function-valued Pade approximants, an efficient Pfaffian formula for the determinants was extended from the matrix form to the function-valued form. As an important application, a Pfaffian formula of [4/4] type Pade approximant was established.

The equations of a polar fluid of hydromagnetic fluctuating through a porous medium are cast into matrix form using the state space and Laplace transform techniques the resulting formlation is applied to a variety of problems. The solution to a problem of an electrically conducting polar fluid in the presence of a transverse magnetic field and to a probem for the flow between two parallel fixed plates is obtained. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the velocity, angular velocity distribution and the induced magnetic field are given and illustrated graphically for each problems.

A strict proof of the equivalence of the Duffin-Kemmer-Petiau and Klein-Gordon-Fock theories is presented for physical S-matrix elements in the case of charged scalar particles minimally interacting with an external or quantized electromagnetic field. The Hamiltonian canonical approach to the Duffin-Kemmer-Petiau theory is first developed in both the component and the matrix form. The theory is then quantized through the construction of the generating functional for the Green's functions, and the physical matrix elements of the S-matrix are proved to be relativistic invariants. The equivalence of the two theories is then proved for the matrix elements of the scattered scalar particles using the reduction formulas of Lehmann, Symanzik, and Zimmermann and for the many-photon Green's functions.

The equations of unsteady free convection flow of a micropolar fluid are cast into a matrix form using the state space and Laplace-transform techniques. The results obtained are used to generate solutions in the Laplace-transform domain to a broad class of problems in free convection flow. The technique is applied to a heated vertical plate problem and to a problem pertaining to a plate under uniform heating. The inversion of Laplace-transform is performed using a numerical approach. Numerical results concerning velocity, microrotation and temperature are given and illustrated graphically for both problems.

A formula in matrix form for restoration is given and the matrix with the sampling rate near the Nyquist rate is investigated. Elements of this matrix can be expanded in Laurent series of the sampling rate parameter, which is defined by the quotient of the Nyquist rate and the sampling rate. The Nyquist rate corresponds to a pole. First terms of these Laurent series near the Nyquist rate are given.

Two new approaches for writing kinetic equations in the matrix form or directly in the integrated form are presented here. While the first method allows to derive the kinetic rate matrix of kinetic systems of any kind in a direct and straightforward way, the second approach applies to species that are consumed solely through first order steps, regardless of the complexity of their formation pathways.

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL∞ functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.

Analyzing communication data within a social structure represented by a wholesale agricultural market demonstrates the utility of network analysis techniques for identifying systems. A representation of these data in matrix form permits exploration of the stability properties of the system. The stability analyses lend some support for theoretical arguments pertaining to the conditions leading to system equilibrium.

A waste management scheme is described, which aims to utilise the ambient pressure of a disposal environment, its radiation shielding and extended time of storage to ensure reliable immobilisation of radioactive waste in a glass composite or polycrystalline matrix form. The conditions required for natural sintering of the waste form in the repository are assessed for viscous flow and grain boundary diffusion mechanisms. In situ sintering of materials in the repository creates geochemically stable materials in equilibrium with the disposal environment ensuring a higher degree of safety compared to existing approaches.

Discontinuous precipitation (DP) is associated with grain boundary migration in the wake of which alternate plates of the precipitate and the depleted matrix form. Some copper base alloys show DP while others do not. In this paper the misfit strain parameter, η, has been calculated and predicted that if 100 η > ± 0·1, DP is observed. This criterion points to diffusional coherency strain theory to be the operative mechanism for DP.

Gene action and interaction have been defined in the literature by the use of a variety of notations (Mather 1949; Hayman 1954, 1955, 1957; Jinks 1954; Kempthorne 1954, 1955). This leads to unnecessary complications in understanding the subject. This paper provides a simple convenient way of translating one parameterization into another and illustrates the simple linear relationship between them. The various notations are written in matrix form by the use of a specification matrix. This provides a convenient compact method of presentation of the relevant Equations. The linear relationship between the genetic parameters enables these to be estimated in the most convenient way and then converted to other parameters for the purposes of comparison. The generation means Equations of Hayman (1958) are derived using the matrix formulation as an illustration of the use of the specification matrix.

In the present paper the concept of screw in classical mechanics is expressed in matrix form, in order to formulate the dynamical equations of the multibody systems. The mentioned method can retain the advantages of the screw theory and avoid the shortcomings of the dual number notation. Combining the screw-matrix method with the tool of graph theory in Roberson/Wittenberg formalism. We can expand the application of the screw theory to the general case of multibody systems. For a tree system, the dynamical equations for eachj-th subsystem, composed of all the outboard bodies connected byj-th joint can be formulated without the constraint reaction forces in the joints. For a nontree system, the dynamical equations of subsystems and the kinematical consistency conditions of the joints can be derived using the loop matrix. The whole process of calculation is unified in matrix form. A three-segment manipulator is discussed as an example.

A numerical method for solution of the coupled power equations in multimode optical fibres is presented. The method is based on the assumption that the contribution of weak mode coupling can be concentrated into discrete points along the fibre. The matrix form of the solution is derived, the accuracy of the method is briefly discussed and an example of practical results is also shown.

Starting from the Chew-Goldbsrger-Low equations we derive wave-equations describing small amplitude disturbances in a horizontally stratified, continuously varying CGL-plasma. A set of equations of first-order matrix form is treated by the method of Clemmow and Heading.

We give an approximate solution to the problem of determining the orientation of a rigid body in space with respect to the angular velocity. We obtain sufficiently simple and effective approxhnation formulas for the determination of O(t), the vector of finite rotation for given O(0). 1. Following [1, p. 20], we consider the classical equations of Rodrigues-Hamflton, wri t tenin matrix form dX = A (o)X, dt (I)

A New Digital Method for Three-Dimensional Stress Analysis in Elastic Media A new method for three-dimensional stress analysis in elastic media is outlined. According to this approach the problem is formulated purely in terms of conditions prevailing at the bounding surfaces of an elastic body. Stress and displacements on these surfaces, as well as inside the body, are expressed in terms of the superimposed effect of a set of basic solutions having suitable properties. The method is developed in matrix form. Some general properties of the governing matrices are established with a view to developing suitable numerical procedures. The convergence characteristics of two iterative schemes are considered. The paper concludes with observations concerning the generality of the approach.

This paper deals with wave equations describing small-amplitude disturbances in horizontally stratified, continuously varying, viscous fluids; gradients of the static pressure and of the coefficient of viscosity are neglected. A set of equations in first-order matrix form, which describes coupled longitudinal and transverse disturbances, is treated by the methods ofClemmow andHeading and ofHeading.

The fundamental factor theorem is developed in matrix form for the case of correlated factors. The properties of the correlated factor system are discussed, and some effects of sampling error considered. The psychological meaning of correlated factors is discussed, and several mechanisms by which general factors may operate in the factorial system are indicated.

The problem of vibrations of structures with heavy static loads is a non linear one, and, as such, of difficult solution in its general formulation.

As an adjunct to developing a theory of structural roles, a mathematical model of group or organizational struc­ ture was developed (Oeser & Harary, 1962, 1964; Oeser & O'Brien, 1967). This paper briefly describes the model, the generated quantitative structural indices, and a Pas­ cal program to compute the indices. The Mathematical Model Organizational or group structure is described by three elements: persons (h), positions (p), and tasks (t), and the five relations between these elements. The relations are the informal relations between persons (H-H), the assign­ ment relations between persons and positions (H-P), the power relations between positions (P-P) , the allocation relations between positions and task (P-T), and the prece­ dence relations between tasks (T-T). The structure may be represented schematically as in Figure 1. Each element is represented by a dot, and the relations are represented by directed arrows between the elements. Figure 1 represents a three-person (h., h2 , h.), three-position (p., P2, P3), three-task (t., t 2, t3) group. Each person is as­ signed to one position, and each position is allocated to one task element. Position P2 has legitimate power over positions PI and P3 and this is represented by directed lines from P2 to PI and P3' The arrows between the task ele­ ments indicate that the group is performing a coordina­ tive or assembly-line-type task. Person hi is assigned to position PI and completes the first part of the task, t., which is then passed to person h, at position P2, and so on. The informal relations or communication network be­ tween persons is similarly represented by directed lines between the person elements (h., h2 , h.). The set of five relations may also be represented in matrix form. The matrices for the set of relations described in Figure 1 are presented in Table 1. A matrix entry of 1 indicates the presence of a relation beween two elements, and an entry of 0 indicates the absence of a re­ lation. Matrix entries can also be a decimal value greater than 0 to indicate both the presence and strength of a re­ lation. The organizational or group structure is represented in matrix form because matrix algebra is used to gener­ ate quantitative structural indices.