on the method of tangent hyperbolas on banach spaces abstract as an application of the technique employed by the author in a series of papers 15 18 some results are established concerning convergence of the method of tangent hyperbolas for solving nonlinear equations in banach spaces as well as existence and uniqueness of solution the results are compared with those obtained by doring 5

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

classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

application of iterative processes of r order at least three to operators with unbounded second derivative abstract in this paper we apply a family of newton like methods which contains the best known iterative processes to operator equations where the usual convergence conditions are relaxed we weaken these conditions by assuming f x0 and f x f y x y with a non decreasing continuous real function our results include the ones obtained when the convergence of the family is studied under lipschitz continuous or holder continuous conditions for the second derivative of the operator involved to finish we apply the study to boundary value problems

computing with groups and their character tables in this survey an attempt is made to give some impression of the capabilities of currently available programs for computations with finitely generated groups and their representations

the neumann problem in a 2 d exterior domain with cuts and singularities at the tips abstract the neumann problem for the harmonic functions in an exterior connected plane region with cuts is studied the problem is considered with different conditions at infinity which lead to different theorems on uniqueness and solvability the existence of a classical solution is proved by potential theory the problem is reduced to a fredholm equation of the second kind which is uniquely solvable explicit formulas for singularities of a gradient of the solution at the tips of the cuts are obtained the results of the paper can be used to model the flow of an ideal fluid over several obstacles including wings

minimal normalization of wiener hopf operators in spaces of bessel potentials a class of operators is investigated which results from certain boundary and transmission problems the so called sommerfeld diffraction problems in various cases these are of normal type but not normally solvable and the problem is how to normalize the operators in a physically relevant way i e not loosing the hilbert space structure of function spaces defined by a locally finite energy norm the present approach solves this question rigorously for the case where the lifted fourier symbol matrix function is holder continuous on the real line with a jump at infinity it incorporates the intuitive concept of compatibility conditions which is known from some canonical problems further it presents explicit analytical formulas for generalized inverses of the normalized operators in terms of matrix factorization

computational group theory in crystal physics a review abstract a brief account is given of the role played by three dimensional crystallographic point and space groups and their representations in computer programs in crystal physics it covers their use in specific applications for example energy band programs and lattice dynamics programs together with their use in those programs written to calculate for example irreducible characters and matrix representatives selection rules clebsch gordan coefficients etc which have general application in crystal physics

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

an extension of gander s result for quadratic equations in the study of iterative methods with high order of convergence gander provides a general expression for iterative methods with order of convergence at least three in the scalar case taking into account an extension of this result we define a family of iterations in banach spaces with r order of convergence at least four for quadratic equations

a composite third order newton steffensen method for solving nonlinear equations in this paper we suggest a third order method formed by the composition of newton and steffensen methods for finding simple and real roots of a nonlinear equation in single variable per iteration the formula requires two evaluations of the function and single evaluation of the derivative experiments show that the method is suitable in the cases where newton and steffensen methods fail

modified chebyshev halley methods with sixth order convergence abstract in this paper we present a family of modified chebyshev halley methods the new methods add one evaluation of the function at the point iterated by chebyshev halley methods analysis of convergence shows that these methods have sixth order convergence numerical results show that the new methods can be of practical interest

a general method to calculate all irreducible representations of a finite group a so called stochastic method is introduced which can be used to decompose the reducible representations of any finite group into irreducible representations in this way the regular representation of a finite group is reduced completely to obtain all of the irreducible representations of the group it also induces a new approach to calculate the character table of a finite group based on this method these calculations or decompositions can be performed by means of a computer

the improvements of chebyshev halley methods with fifth order convergence abstract in this paper we present a family of new fifth order methods which improves the classical chebyshev halley methods with cubic convergence the new methods add one evaluation of the function at the point iterated by chebyshev halley methods analysis of efficiency shows that the new methods can be of practical interest which is also corroborated by numerical tests

acoustic diffraction from the junction of two flat plates a general solution is developed for the acoustic and structural scattered response from the junction of two flat plates under unilateral fluid loading the plates are modelled by the classical theory of flexure and the solution is found using the wiener hopf technique for the dual integral equations for the unknown pressure on the plates explicit formulae are obtained for the pressure transform when the plates are in welded or clamped contact and corresponding explicit expressions are given for the various diffraction coefficients associated with the fluid structure interaction the magnitudes of the reflection and transmission coefficients of structural waves display very simple analytic forms at low frequency

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

on the global convergence of schroder s iterative formulae for real roots of algebraic equations abstract schroder s formulae of the first s1 and second s2 kind of order m of convergence are generalizations of newton s m 2 and halley s s2 m 3 iterative formulae for finding zeros of functions davies and dawson show that for entire functions with only real zeros halley s formula converges globally and monotonically to their zeros independently of the initial value on the real line we show that the s2 formulae of odd order 5 enjoy the same convergence feature for polynomials with only real zeros numerical examples illustrate this we illustrate no monotonic convergence of the s1 formulae and of the s2 formulae of even order

group theory and phase transitions abstract group theoretical implications in the landau theory of phase transitions are reviewed notwithstanding the large amount of work mainly concerned with construction of the landau thermodynamic potential this review emphasizes the group theoretical aspects of the actual minimization of the potential the progress of the last few decades is reviewed and some open questions stated

harmonic polynomials invariant under a finite subgroup of o n in this paper an algorithm is described which allows a systematic computation of harmonic polynomials of a given degree invariant under a finite subgroup of the group o n an application of the algorithm to the octahedral cubic subgroup is given

on a family of second derivative free variants of chebyshev s method in this paper we present a family of new second derivative free variants of chebyshev s method in which f x 0 in some points is permitted analysis of convergence shows that the new methods are cubically convergent analysis of efficiency in term of function evaluations shows that the new methods have the great practical utility which is demonstrated by numerical examples

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

symmetry and phase transitions abstract the problem of symmetry changes at phase transitions is discussed in the light of recent developments in the theory of phase transitions as well as in terms of classical landau theory various group theoretical criteria or selection rules are formulated and discussed from the point of view of how well they comply with the basic requirements of the theory of phase transitions i e the existence of the absolute minimum of the landau potential landau ginzburg wilson hamiltonian and the continuity of the transition the physical significance of these criteria is examined

an iterative method for solving nonlinear equations an iterative method for finding a solution of the equation f x 0 is presented the method is based on some specially derived quadrature rules it is shown that the method can give better results than the newton method

little groups of irreps of o 3 so 3 and the infinite axial subgroups a new chain criterion for the determination of little groups is presented this result distinguishes massive rotationally inequivalent irrep basis functions and allows for multiple branching paths consequently it is suitable for application to lie groups applied to the groups o 3 so 3 d h d c v c h and c the result enables the enumeration of all possible little groups together with their associated basis functions these results are relevant to the determination of the symmetry of a material from its linear and nonlinear optical properties and to the choices of order parameters for symmetry breaking in liquid crystals

propagation of cracks in elastic media throughout this paper the normal n denotes the interior normal direction i e n f x1 1 p 1 jf x1 j when x approaches from above denoted by and n f x1 1 p 1 jf x1 j from below denoted by in a recent paper 2 the present authors studied the asymptotic behavior of any solution of 1 2 1 4 in a neighborhood of the tip x t taking for simplicity t 0 and assuming that

newton like methods for the computation of fixed points abstract the celebrated banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent the convergence however may take place very slowly so that it may be desirable to use a newton like method for the computation of the fixed point if newton s method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping in the present note some variants of newton s method are discussed which make use of this contraction information it turns out that the convergence of newton s method can be accelerated without any relevant additional computational labour

structural phase transitions in crystals broken symmetry isotropy groups a systematic procedure ensuring the determination of all isotropy groups of a given representation of a space group is presented for the first time isotropy groups play an important role in various areas of theoretical solid state physics for example only isotropy groups may occur in a structural phase transition driven by an order parameter belonging to a given representation the method uses the chain criterion directly on the image of the representation employing a labeling of the matrices by space group elements a notion of a substar of a wave vector associated with the representation is central to the method finally as a detailed illustration the method is applied to a structural phase transition in a15 systems driven by an x point order parameter the result agrees with previously reported ones

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 the dynamics of the euler iterative function abstract the dynamics of euler s third order iterative method which is used to find roots of non linear equations applied to complex polynomials of degrees three and four is studied the conjugacy classes of this method are found explicitly

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

a uniparametric chebyshev type method free from second derivatives in this paper we present a family of new chebyshev type methods free from second derivatives for solving non linear equations analysis of convergence shows that these new methods are cubically convergent as particular cases we introduce two efficient methods in this family of the new methods the practical utility of which is demonstrated by numerical examples

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

mixed boundary value problems of the type of sommerfeld s half plane problem on etudie des modifications du probleme de demi plan de sommerfeld gouverne par deux equations elliptiques propres dans des demi espaces complementaires de r 3 et on pose des conditions aux limites et de transmission sur les deux demi plans qui ensemble forment la frontiere commune de

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

wave diffraction by a half plane with an obstacle perpendicular to the boundary abstract we prove the unique existence of solutions for different types of boundary value problems of wave diffraction by a half plane with a screen or a crack perpendicular to the boundary representations of the solutions are also obtained upon the consideration of some associated operators this is done in a bessel potential spaces framework and for complex non real wave numbers the investigation is mostly based on the construction of explicit operator relations the potential method and a factorization technique for a certain class of oscillating fourier symbols which naturally arise from the problems

image normalization of wiener hopf operators and boundary transmission value problems for a junction of two half planes abstract the present paper deals with an application of the image normalization technique for certain classes of wiener hopf operators whos associated to ill posed boundary transmission value problems we briefly describe the method of normalization and then apply it to boundary transmission value problems issued from diffraction problems for a junction of two half planes which are relevant in mathematical physics applications for each boundary transmission value problem we analyze the conditions under which the associated operator and the equivalent who are not normally solvable and define the corresponding image normalized operators

an efficient fifth order method for solving systems of nonlinear equations in this paper we present a three step iterative method of convergence order five for solving systems of nonlinear equations the methodology is based on the two step homeier s method with cubic convergence homeier 2004 computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made the performance is tested through numerical examples moreover theoretical results concerning order of convergence and computational efficiency are verified in the examples it is shown that the present method has an edge over existing methods particularly when applied to large systems of equations

a one parameter family of third order methods to solve nonlinear equations in this paper we present a new one parameter family of iterative methods to solve nonlinear equations which includes some well known third order methods as particular ones the convergence analysis shows that the order of convergence of each method of the family is three numerical examples are given to illustrate the efficiency and performance of the presented methods

an optimal and efficient general eighth order derivative free scheme for simple roots abstract the main motivation of this study is to present an optimal scheme in a general way that can be applied to any existing optimal multipoint fourth order iterative scheme whose first substep employs steffensen s method or steffensen like method to further produce optimal eighth order iterative schemes a rational function approximation approach is used in the construction of proposed scheme in addition we also discussed the theoretical and computational properties of our scheme each member of the presented scheme satisfies the optimality conjecture for multipoint iterative methods without memory which was given by kung and traub in 1970 finally we also concluded on the basis of obtained numerical results that our methods have faster convergence in contrast to the existing methods of same order because they have minimum residual errors minimum error difference between two consecutive iterations and minimum asymptotic error constants corresponding to the considered test function

properties of poincare generating functions for polynomial covariants tensors we show that irreducible polynomial covariants tensors based on a given irreducible representation of a finite group must have a definite p phase i e degree modulo p where p is the order of the center of the image of the group we also show that the numerator of the poincare function is often a polynomial symmetric around a given degree and we derive several interesting properties of the poincare function

the modified jump problem for the laplace equation and singularities at the tips the boundary value problem for the laplace equation outside several cuts in a plane is studied the jump of the solution of the laplace equation and the boundary condition containing the jump of its normal derivative are specified of the cuts the unique solution of this problem is obtained the problem is reduced to the uniquely solvable fredholm equation of the second kind and index zero the singularities at the ends of the cuts are investigated

space groups and their isotropy subgroups the theory of space groups and their subgroups is related to that of their irreducible representations several important theorems concerning the determination of isotropy subgroups are proved in particular we show that the translation subgroups of isotropy subgroups are in one to one correspondence with certain subsets called substars of the star characterizing the irreducible representation of the space group

modified group projector technique induced representations the group projector technique is developed for the representations of the direct product form d d x d in which one of them has been induced from a representation of a subgroup it is shown that the group projectors and symmetry adapted bases are essentially determined in terms of the subgroup representations to illustrate both the technical and conceptual advantages of the method it is shown how the calculation of the normal modes of polymers polyacetylene as an example can be obtained using the symmetry of the monomer only and several results of the induction theory are reconsidered within the new framework

the mixed harmonic problem in an exterior cracked domain with dirichlet condition on cracks the mixed dirichlet neumann problem for the laplace equation in an unboundedconnected plane domain with cuts cracks is studied the neumann condition is given on closed curves making up the boundary of the domain while the dirichlet condition is specified on the cuts the existence of a classical solution is proved by means of potential theory and the boundary integral equation method the integral representation for a solution is obtained in the form of potentials the density in potentials satisfies the uniquely solvable fredholm integral equation of the second kind and index zero singularities of the gradient of the solution at the tips of cuts are investigated

generalized solvability and optimization of a parabolic system with a discontinuous solution abstract we consider the following linear parabolic system in a domain with a thin low permeable insertion imperfect interface u t q u f t k u t q 1 q 2 r n u t 0 0 u 0 k k i j i j 1 n n r n 0 u lim 0 n r n 0 t 0 q 3 q 1 q 2 we consider a new formulation of the problem where the unknowns are u and the parabolic problem is converted to a first order system of partial differential equations with distributional coefficients we also prove inequalities for negative norms for the parabolic operator with the distributional coefficients and theorems of existence and uniqueness for optimization problems for the processes we show existence of optimal controls investigate smoothness of a performance criterion and give a simple condition for controllability of the system in addition we consider applications of the obtained results to a pulse control problem and prove convergence of a control mapping regularization procedure

on method of osculating circle for solving nonlinear equations abstract in this paper we present a new algorithm for solving nonlinear equations and this method requires the use of the first and second derivatives of the function as halley s laguerre s chebyshev s and other classical methods do it has well known geometric interpretation and admits its geometric derivation from a circle further the executed comparative numerical experiments show that this new method competes well with the known classical methods and has cubic convergence

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

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

selection rules for symmetry breaking and symmetry restoration in continuous transitions abstract using landau s phenomenological theory as a framework subduction and chain subduction criteria are discussed as methods for obtaining directly selection rules for symmetry breaking for symmetry restoration induction and chain induction criteria give selection rules minimal symmetry breaking and minimal symmetry restoration are related to satisfaction of chain subduction and induction criteria

extending the applicability of the local and semilocal convergence of newton s method we present a local as well a semilocal convergence analysis for newton s method in a banach space setting using the same lipschitz constants as in earlier studies we extend the applicability of newton s method as follows local case a larger radius is given as well as more precise error estimates on the distances involved semilocal case the convergence domain is extended the error estimates are tighter and the information on the location of the solution is at least as precise as before numerical examples further justify the theoretical results

a new modified secant like method for solving nonlinear equations in this paper we present a new secant like method for solving nonlinear equations analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency

some variants of ostrowski s method with seventh order convergence in this paper we present a class of new variants of ostrowski s method with order of convergence seven per iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore this class of methods has the efficiency index equal to 1 627 numerical tests verifying the theory are given and multistep iterations based on the present methods are developed