the 38 assemblies of the general generator sets for 1421 magnetic double space groups thirty eight assemblies of general expressions of the generator sets for 1421 magnetic double space groups in terms of the 32 minimal general generator sets of the unitary space groups are provided

monotone methods for a finite difference system of reaction diffusion equation with time delay abstract the aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction diffusion equations with time delays the iterative schemes lead to computational algorithms as well as existence uniqueness and upper and lower bounds of the solution an application to a diffusive logistic equation with time delay is given

a new method of constructing symmetry adapted linear combinations for a finite group ii correspondence theorem and basis operators abstract a correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry adapted linear combination belonging to the same irreducible representation of a point group by means of basis operators the general expressions are given for the symmetry adapted linear combinations the coupling coefficients the lattice harmonics and the hybrid atomic orbitals illustrative examples are discussed to demonstrate the effectiveness of the formalism

initial value methods for discrete boundary value problems motivated by the work of angel and kalaba s and golberg 8 on twopoint boundary value problems for difference equations in this paper we shall study several initial value methods for multipoint boundary value problems for linear and nonlinear difference systems we generalize method of adjoints goodman and lance 9 method of complementary functions tifford 17 d an method of particular solutions miele 111 proposed for two point continuous problems to linear multipoint problems for difference equations next we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems to justify the results obtained here several examples are also given the methods discussed here are similar to our earlier work l 3 on continuous problems we will take up the initial adjusting method of ojika kasue and welsh 12 15 18 for the discrete case later throughout the paper we shall assume the existence uniqueness and stability properties of solutions of the problems discussed and refer to 4 for some of these questions

monotone enclosure for a class of discrete boundary value problems without monotone nonlinearities abstract a new concept of a pair of upper and lower solutions is introduced for a class of discrete boundary value problems without monotone nonlinearities some comparison results are established an existence and enclosing theorem for the solutions is given in terms of upper and lower solutions a new monotone iterative scheme is proposed the numerical example is given

models of the water retention curve for soils with a fractal pore size distribution the relationship between water content and water potential for a soil is termed its water retention curve this basic hydraulic property is closely related to the soil pore size distribution for which it serves as a conventional method of measurement in this paper a general model of the water retention curve is derived for soils whose pore size distribution is fractal in the sense of the mandelbrot number size distribution this model which contains two adjustable parameters the fractal dimension and the upper limiting value of the fractal porosity is shown to include other fractal approaches to the water retention curve as special cases application of the general model to a number of published data sets covering a broad range of soil texture indicated that unique independent values of the two adjustable parameters may be difficult to obtain by statistical analysis of water retention data for a given soil discrimination among different fractal approaches thus will require water retention data of high density and precision

boundary value problems for general discrete systems on infinite intervals this paper continues the study of boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type while the finite discrete interval case regular as well as at resonance has been discussed in 1 2 in the present paper the infinite interval case will be addressed

numerical infinitesimals in a variable metric method for convex nonsmooth optimization abstract the objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems where the objective function is assumed to be convex and not necessarily differentiable for such family of problems the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems we focus on a family of nonsmooth optimization methods based on a variable metric approach and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large as a consequence of nonsmoothness in particular we consider the case treated in the literature where the metric is defined via a diagonal matrix with positive entries we provide the computational results of our implementation on a set of benchmark test problems from scientific literature

some recent developments of numerov s method abstract this paper is a survey of some recent developments of numerov s method for solving nonlinear two point boundary value problems the survey consists of three different parts the existence uniqueness of a solution computational algorithm for computing a solution and some extensions of numerov s method the sufficient conditions for the existence and uniqueness of a solution are presented some of them are best possible various iterative methods are reviewed including picard s iterative method modified newton s iterative method monotone iterative method and accelerated monotone iterative method in particular two more direct monotone iterative methods are presented to save computational work each of these iterative methods not only gives a computational algorithm for computing a solution but also leads to an existence and uniqueness theorem the estimate on the rate of convergence of the iterative sequence is given the extensions of numerov s method to a coupled problem and a general problem are addressed the numerical results are presented to validate the theoretical analysis

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

asymptotic behavior of solutions for a lotka volterra mutualism reaction diffusion system with time delays this paper is to investigate the asymptotic behavior of solutions for a time delayed lotka volterra n species mutualism reaction diffusion system with homogeneous neumann boundary condition it is shown under a simple condition on the reaction rates that the system has a unique bounded time dependent solution and a unique constant positive steady state solution and for any nontrivial nonnegative initial function the corresponding time dependent solution converges to the constant positive steady state solution as time tends to infinity this convergence result implies that the trivial steady state solution and all forms of semitrivial steady state solutions are unstable and moreover the system has no nonconstant positive steady state solution a condition ensuring the convergence of the time dependent solution to one of nonnegative semitrivial steady state solutions is also given

accelerated monotone iterative methods for a boundary value problem of second order discrete equations abstract an accelerated monotone iterative method for a boundary value problem of second order discrete equations is presented this method leads to an existence comparison theorem as well as a computational algorithm for the solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations

numerical point of view on calculus for functions assuming finite infinite and infinitesimal values over finite infinite and infinitesimal domains abstract the goal of this paper consists of developing a new more physical and numerical in comparison with standard and non standard analysis approaches point of view on calculus with functions assuming infinite and infinitesimal values it uses recently introduced infinite and infinitesimal numbers being in accordance with the principle the part is less than the whole observed in the physical world around us these numbers have a strong practical advantage with respect to traditional approaches they are representable at a new kind of a computer the infinity computer able to work numerically with all of them an introduction to the theory of physical and mathematical continuity and differentiation including subdifferentials for functions assuming finite infinite and infinitesimal values over finite infinite and infinitesimal domains is developed in the paper this theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values as well it is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one in dependence on the wish of the researcher i e as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text numerous examples are given

a unified theory of point groups vi the projective corepresentations of the magnetic point groups of infinite order this paper provides all the p inequivalent projective irreducible unitary corepresentations of all the magnetic point groups of infinite order with full use of their isomorphisms

a new method of constructing the symmetry coordinates of molecular vibrations based on the correspondence theorem a general method of constructing the symmetry coordinates of molecular vibrations is presented via the correspondence theorem which describes the parallelism between a salc symmetry adapted linear combination and an elementary basis function belonging to the same irreducible representation of the symmetry group of the molecule

lexicographic multi objective linear programming using grossone methodology theory and algorithm abstract numerous problems arising in engineering applications can have several objectives to be satisfied an important class of problems of this kind is lexicographic multi objective problems where the first objective is incomparably more important than the second one which in its turn is incomparably more important than the third one etc in this paper lexicographic multi objective linear programming lmolp problems are considered to tackle them traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single objective function the latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single objective one and the search of correct weights can be very time consuming in this work a new approach for solving lmolp problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed it is shown that a smart application of infinitesimal weights allows one to construct a single objective problem avoiding the necessity to determine finite weights the equivalence between the original multi objective problem and the new single objective one is proved a simplex based algorithm working with finite and infinitesimal numbers is proposed implemented and discussed results of some numerical experiments are provided

numerosities of labelled sets a new way of counting abstract the notions of labelled set and numerosity are introduced to generalize the counting process of finite sets the resulting numbers called numerosities are then used to develop nonstandard analysis the existence of a numerosity function is equivalent to the existence of a selective ultrafilter hence it is independent of the axioms of zfc

the 32 minimal general generator sets of 230 double space groups the 32 minimal general generator sets mggs for the three dimensional double space groups are provided each mggs has a very limited number of parameters that describe the translational parts of the space groups belonging to a crystal class the method of construction is based on the algebraic defining relations of the abstract generators of the point groups the equivalence or inequivalence criteria for the space groups with respect to lattice transformations are established by introducing the definite set u of the unimodular matrices that leaves a lattice type l of each crystal class g based on the set u it will be shown that mere shifts of the lattice origin are necessary and sufficient to determine the inequivalence of almost all the space groups belonging to a crystal class of high symmetry it is simpler to construct mggs of higher symmetry than those of lower symmetry this contrasts with the existing methods which are based on the solvability of the space groups

on discrete boundary value problems arising in transport phenomena for the discrete boundary value problems arising in transport processes we provide comparison results these results are used to develop monotone iterative methods for the construction of the maximal and minimal solutions in a sector the advantage of this technique is that the successive approximations are the solutions of the initial and terminal value problems numerical illustration showing the sharpness as well the importance of the obtained results is also included

nonlinear programming and grossone quadratic programing and the role of constraint qualifications abstract a novel and interesting approach to infinite and infinitesimal numbers was recently proposed in a series of papers and a book by sergeyev this novel numeral system is based on the use of a new infinite unit of measure the number grossone indicated by the numeral the number of elements of the set in of natural numbers based on the use of de cosmis and de leone 2012 have then proposed a new exact differentiable penalty function for constrained optimization problems in this paper these results are specialized to the important case of quadratic problems with linear constraints moreover the crucial role of constraint qualification conditions well know in constraint minimization literature is also discussed with reference to the new proposed penalty function

a unified theory of the point groups iii classification and basis functions of improper point groups this paper introduces a new system of classification of improper point groups which is most effective for describing their general irreducible representations the complete set of the general angular momentum eigenfunctions is classified according to the general irreducible representations of the seven sets of the improper point groups corresponding to cn c and dn d for an aribtrary n

stability of a class of delay difference equations on montre que les proprietes de stabilite a la fois locales et globales de la solution d equilibre positive du systeme a retard n i 1 s i n i t 1 s i g i n t h i peuvent se deduire de celles du systeme sans retard associe n i t 1 g i n t

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

finite difference solutions of reaction diffusion equations with continuous time delays abstract this paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays the system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either dirichlet or neumann robin boundary conditions various monotone iterative schemes which depend on the property of the nonlinear reaction mechanism are developed for the computation of numerical solutions it is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below respectively to a unique solution of the finite difference system applications are given to two model problems known as the diffusive logistic equation and the fisher s diffusion equation in population genetics

a new method of constructing symmetry adapted linear combinations for a finite group i salc s of equivalent invariant functions abstract a theory of representation for a finite group g has been presented based on the basis functions defined on a set of equivalent points s n with respect to the symmetry operations of g the general expressions are given for the symmetry adapted linear combinations of equivalent invariant functions and the bonding hybrid atomic orbitals

a new method of constructing the symmetry adapted linear combinations based on the correspondence theorem and induced representations the theory of induced representations is incorporated into the method of constructing symmetry adapted linear combinations for a symmetry group based on the correspondence theorem developed previously the theory is effective for a highly symmetric molecule with a set of equivalent radicals each of which again consists of a number of equivalent atoms an illustrative example is given to construct the internal symmetry coordinates of molecular vibrations of c ch3 4 with the frozen configuration belonging to the point group td

a unified theory of the point groups iv the general corepresentations of the crystallographic and noncrystallographic shubnikov point groups a new system of classification for all the shubnikov point groups gs is presented which is best suited for describing their isomorphisms as well as their construction by the new classification and the modified form of the corepresentations there are presented the simple general expressions of the irreducible corepresentations for the 10 minimum sets of proper gs from which all coirreps of the remaining gs follow icosahedral groups are excluded

general irreducibility condition for vector and projective corepresentations of antiunitary groups a general condition for irreducibility of vector and projective corepresentations of an antiunitary group is presented it depends only on the characters of the unitary halving subgroup of the covering group it reduces to the well known type criterion of corepresentations when it is specialized to the three types of corepresentations

usage of infinitesimals in the menger s sponge model of porosity abstract the present work concerns the calculation of the infinitesimal porosity by using the menger s sponge model this computation is based on the grossone theory considering the pore volume estimation for the menger s sponge and afterwards the classical definition of the porosity given by the ratio between the volume of voids and the total volume voids plus solid phase the aim is to investigate the different solutions given by the standard characterization of the porosity and the grossone theory without the direct estimation of the fractal dimension once the utility of this procedure had been clarified the focus moves to possible practical applications in which infinitesimal parts can play a fundamental role the discussion on this matter still remains open

monotone iterative methods for a general class of discrete boundary value problems abstract in this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems the discrete system we consider includes in particular the n th order prototype systems finite as well as infinite discrete delay equations and discrete integral equations further the boundary conditions we consider include the initial terminal periodic and transport type problems numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included

solving ordinary differential equations on the infinity computer by working with infinitesimals numerically there exists a huge number of numerical methods that iteratively construct approximations to the solution y x of an ordinary differential equation ode y x f x y starting from an initial value y 0 y x 0 and using a finite approximation step h that influences the accuracy of the obtained approximation in this paper a new framework for solving odes is presented for a new kind of a computer the infinity computer it has been patented and its working prototype exists the new computer is able to work numerically with finite infinite and infinitesimal numbers giving so the possibility to use different infinitesimals numerically and in particular to take advantage of infinitesimal values of h to show the potential of the new framework a number of results is established it is proved that the infinity computer is able to calculate derivatives of the solution y x and to reconstruct its taylor expansion of a desired order numerically without finding the respective derivatives analytically or symbolically by the successive derivation of the ode as it is usually done when the taylor method is applied methods using approximations of derivatives obtained thanks to infinitesimals are discussed and a technique for an automatic control of rounding errors is introduced numerical examples are given

parallel monotone iterative relaxation methods for a class of two dimensional discrete boundary value problems abstract a parallel monotone iterative relaxation method for a class of two dimensional discrete boundary value problems is established and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem this monotone convergence results yields a parallel computational algorithm as well as an existence comparison result for the solutions to compute the sequence of iterations the thomas algorithm can be used in the same fashion as for one dimensional problem the existence and comparison results of the upper and lower solutions are given the local as well as global existence uniqueness of the solution are obtained the global convergence of the iterations is investigated and the influence of the parameters on the rate of convergence of the iterations is analyzed numerical results are given to corroborate the analytical results

a unified theory of the point groups v the general projective corepresentations of the magnetic point groups and their applications to magnetic space groups this paper presents the general expressions of the projective irreducible corepresentations of the 20 homologous sets of the magnetic double point groups from which follow all those of the remaining groups of finite order through isomorphisms the icosahedral group is excluded these are explicitly given in terms of the irreducible representations of the unitary double point groups their special cases provide all the irreducible corepresentations of all magnetic space groups of wave vector through simple gauge transformations a method of determining the gauge transformations is discussed through typical examples

stability of a general class of difference systems abstract in this paper we shall investigate several notions of stability of a general ordinary difference system using the prolific second method of lyapunov

the sierpinski curve viewed by numerical computations with infinities and infinitesimals abstract the sierpinski curve is one of the most known space filling curves and one with the highest number of applications we present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the sierpinski curve we emphasize that adopting this point of view we have infinitely many sierpinski curves depending contrarily to traditional analysis on each specific starting configuration of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves

a unified theory of the point groups ii the general projective representations and their application to space groups the general expressions of all inequivalent irreducible projective representations of crystallographic and noncrystallographic point groups except for the icosahedral group are given in terms of the vector representations of the proper double point groups its application to the representations of the space group of wave vector is discussed construction of the basis sets is discussed

an algorithm for approximate solving of differential equations with maxima an algorithm for constructing two sequences of successive approximations of the solution of the initial value problem for nonlinear differential equations with maxima is given this algorithm is based on the monotone iterative technique it is proved that both sequences are monotonically convergent each term of the constructed sequences is a solution of an initial value problem for linear differential equations with maxima and it is a lower upper solution of the given problem both the scalar case and the multidimensional case are studied an example solved by computer realization of the suggested algorithm illustrates the practical application of the method

a unified theory of the point groups and their general irreducible representations point groups and their general irreducible vector and projective representations are characterized by the subgroup conditions for su 2 in a unified scheme these conditions are given by simple polynomial equations imposed on the matrix elements of the 2 2 unitary matrices of su 2 the general irreducible representations for the point groups d dn with arbitrary integer n 2 o and t are given by four simple and effective tables

optimal estimation of direction in regression models with large number of parameters we consider the problem of estimating the optimal direction in regression by maximizing the probability that the scalar product between the vector of unknown parameters and the chosen direction is positive the estimator maximizing this probability is simple in form and is especially useful for situations where the number of parameters is much larger than the number of observations we provide examples which show that this estimator is superior to state of the art methods such as the lasso for estimating the optimal direction

the use of grossone in mathematical programming and operations research the concepts of infinity and infinitesimal in mathematics date back to anciens greek and have always attracted great attention very recently a new methodology has been proposed by sergeyev 10 for performing calculations with infinite and infinitesimal quantities by introducing an infinite unit of measure expressed by the numeral grossone an important characteristic of this novel approach is its attention to numerical aspects in this paper we will present some possible applications and use of in operations research and mathematical programming in particular we will show how the use of can be beneficial in anti cycling procedure for the well known simplex method for solving linear programming problems and in defining exact differentiable penalty functions in nonlinear programming