the generalized roper suffridge extension operator in banach spaces ii abstract in this paper we generalize the roper suffridge extension operator from c n to banach spaces it is proved that this operator preserves the biholomorphic starlikeness on some domains in banach spaces from these we may construct a lots of concrete examples about biholomorphic starlike mappings on some domains in c n or hilbert spaces or banach spaces from univalent starlike functions on the unit disc u in c meanwhile the growth theorems of the corresponding mappings are given some results of gong and liu roper and suffridge graham et al in c n are extended to hilbert spaces or banach spaces

a fast combined field volume integral equation solution to em scattering by 3 d dielectric objects of arbitrary permittivity and permeability a fast solution to the combined field volume integral equation cfvie for electromagnetic scattering by large three dimensional dielectric bodies of arbitrary permittivity and permeability is presented the cfvie is formulated in the region of the scatterers by expressing the total fields as the sum of the incident wave and the radiated wave due to both the electric and magnetic polarization currents the resultant integral equation is solved using the method of moments mom then the precorrected fast fourier transform p fft method is applied to reduce the memory requirement and accelerate the matrix vector multiplication in the mom solution in the implementation of the p fft method two sets of projection operators are constructed respectively for the projections of the electric sources and magnetic sources in addition two sets of interpolation operators are also applied respectively for the computation of the vector scalar potentials and the curl of the vector potentials in the support of the testing functions the resultant method has a memory requirement of o n and a computational complexity of o nlogn respectively where n denotes the number of unknowns

the mixed order bcgs fft method for the scattering of three dimensional inhomogeneous anisotropic magnetodielectric objects this paper presents the first fft based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects the volume integral equations are formulated by employing mixed order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders volumetric roof top basis functions are used for flux densities whereas second order curl conforming basis functions are used for vector potentials a fast volume integral equations solver namely the bcgs fft method is then applied to accelerate the solution of this mixed order weak form formulation examples show that the mixed order bcgs fft method has high accuracy compared to both analytical and numerical solutions examples also show the mixed order bcgs fft method has high computational efficiency compared to commercial software

on functions of bounded p variation abstract we obtain estimates of the total p variation 1 p and other related functionals for a periodic function f l p 0 1 in terms of its l p modulus of continuity f p these estimates are sharp for any rate of the decay of f p moreover the constant coefficients in them depend on parameters in an optimal way

solving the volume integral equations of electromagnetic scattering time harmonic electromagnetic scattering by inhomogeneous three dimensional structures within a free space environment can be described by electric and magnetic field volume integral equations involving the free space green function a comprehensive set of galerkin projection formulations also known as moment methods for the numerical solution of these equations is presented together with comparative numerical results such formulations are widely used for particle scattering analysis optical near field calculation etc results are obtained with higher order divergence and curl conforming basis functions on iso parametric tetrahedral meshes the results demonstrate that all formulations converge with similar accuracy in the case of an analytically solvable test problem when modeling flux densities as solution variables it is argued that solenoidal function spaces should be used rather than the standard divergence conforming function spaces this assertion is supported by the results some of the formulations involve solving for curl conforming fields such fields can be discretized with fewer unknowns than divergence conforming ones implying lower computational costs additionally some formulations yield system matrices which are approximately halfway sparse meaning that computational costs will be down by a factor of 2 when iterative solvers are employed which is the case for the widely used fast methods

lowner chains and a subclass of biholomorphic mappings abstract in this paper we give characterization of almost starlike functions of order respectively almost starlike mappings of order on the unit disc in c respectively the unit ball in a finite dimensional complex banach space in terms of lowner chains furthermore using the properties of lowner chains we can easily prove that two classes of generalized roper suffridge extension operators preserve almost starlikeness of order on two important classes of reinhardt domains in c n respectively

extension of convex mappings of order α of the unit disk in c to convex mappings of the unit ball in cn abstract for a convex mapping f of order 0 1 of the unit disk in the complex plane c we consider conditions on the parameter 0 1 2 so that the mapping f z f z 1 f z 1 z z z 2 z n is a convex mapping of the euclidean unit ball in c n

modular metric spaces ii application to superposition operators abstract applying the theory of modular metric spaces developed in the first part of this paper v v chistyakov 2009 1 we define a metric semigroup and an abstract convex cone of functions of finite generalized variation in the approach of schramm m schramm functions of bounded variation and riemann stieltjes integration trans amer math soc 287 1 1985 49 63 which are significantly larger as compared to the spaces of bounded variation in the sense of jordan wiener young and waterman we present a complete description of generators of lipschitz continuous bounded and some other classes of superposition nemytskii operators mapping in these semigroups and cones which extends recent results by matkowski and mi j matkowski j mi on a characterization of lipschitzian operators of substitution in the space bv a b math nachr 117 1984 155 159 maligranda and orlicz l maligranda w orlicz on some properties of functions of generalized variation monatsh math 104 1987 53 65 zawadzka g zawadzka on lipschitzian operators of substitution in the space of set valued functions of bounded variation rad mat 6 1990 279 293 and chistyakov v v chistyakov mappings of generalized variation and composition operators j math sci new york 110 2 2002 2455 2466 v v chistyakov lipschitzian nemytskii operators in the cones of mappings of bounded wiener variation folia math 11 1 2004 15 39

loewner chains associated with the generalized roper suffridge extension operator on some domains in this paper we consider the generalized roper suffridge extension operator defined by n 2 2 n n f z f z1 f z1 z1 2 f z1 2z2 f z1 z1 n f z1 nzn for z z1 z2 zn p1 p2 pn where 0 j 1 0 j 1 j pj 1 and we choose the branch of the power functions such that f z1 z1 j z1 0 1 and f z1 j z1 0 1 j 1 2 n p1 p2 pn z1 z2 zn cn j 1n zj pj 1 we prove that the set n 2 2 n n s u can be embedded in loewner chains and give the answer to the problem of liu taishun we also obtain that the operator n 2 2 n n f preserves starlikeness or spirallikeness of type on p1 p2 pn for some suitable constants j j where s u is the class of all univalent analytic functions on the unit disc u in the complex plane c with f 0 0 and f 0 1

superposition operators and functions of bounded p variation ii we continue the study of the superposition operator tf g f g on the space bvp1 i of primitives of real valued functions of bounded p variation on an interval i we give first a characterization of the functions f such that tf takes bvp1 i to itself then we characterize the functions f for which tf is continuous uniformly continuous and differentiable as a mapping of bvp1 i to itself respectively by exploiting the peetre s imbedding theorem and the fubini property we derive partial results on continuity of tf in besov spaces bp qs rn for a smoothness parameter s satisfying 0 s 1 1 p

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

extension operators and subordination chains abstract since the work of roper and suffridge in 1995 there has been considerable interest in constructing holomorphic mappings of the unit ball in c n with various geometric properties by using lower dimensional mappings with similar properties such properties include convexity starlikeness and spirallikeness it is also of interest to extend subordination chains which may be related to geometric properties such as the above recently m elin has introduced an approach to extension operators on banach spaces based on semigroups which leads to an identification of some of the essential properties of such operators in this paper we adopt elin s point of view and prove a theorem about the extension of subordination chains which extends and unifies existing results we also consider applications in finite dimensions to the study of extreme and support points under the loewner variation and of reachable sets in control theory certain examples will also be provided

coefficient bounds for biholomorphic mappings which have a parametric representation abstract let b be the unit ball in c n with respect to an arbitrary norm and let f z t be a g loewner chain such that z 0 is a zero of order k 1 of e t f z t z for each t 0 in this paper the authors obtain coefficient bounds of mappings in s g k 1 0 b these results generalize the related works of hamada honda and kohr

extension operators and automorphisms abstract many of the commonly studied extension operators that take a normalized locally univalent function f defined in the open unit disk d c to a normalized locally biholomorphic mapping of the unit ball b c n n 2 are of the form f 0 f where 0 f z f z 1 f z 1 z z b z z 2 z n is the roper suffridge extension of f and is a first coordinate shear an automorphism of c n of the form w w 1 g w w where g c n 1 c is holomorphic and contains no terms of degree 0 or 1 in its taylor expansion at the origin other studies have considered the roper suffridge extension operator with the multiplier f z 1 replaced by a different expression recently it has been shown that the set of restrictions of automorphisms of c n to b satisfying some geometric property such as convexity or starlikeness is dense in the full respective family of geometric mappings when n 2 we study large families of biholomorphic mappings on b generated by automorphisms of c n abstract generalizations of the roper suffridge extension operator unitary rotations and families of locally univalent functions on d in many cases families arising from distinct automorphisms will be seen to be disjoint despite the fact that they contain the restrictions of the respective automorphisms to b and the previously noted density result of particular interest given our consideration of more general automorphisms related to extension operators we show that the compositions of the roper suffridge extension operator with a particular class of first coordinate overshears automorphisms of c n of the form w h w w 1 g w w for g h c n 1 c holomorphic and h never 0 extend certain bounded starlike mappings of d to starlike mappings of b we conclude with several open questions inspired by this work

loewner chains associated with the generalized roper suffridge extension operator abstract in this paper we introduce two classes of generalized roper suffridge extension operators and prove that they can be embedded in loewner chains in particular our proof shows that these two classes of operators preserve starlikeness and spirallikeness of type on two important classes of reinhardt domains in c n respectively finally some other related results are given

the biconjugate gradient method for electromagnetic scattering the biconjugate gradient bcg method for solving linear systems is shown to be more efficient than the conjugate gradient cg method for several examples from electromagnetic scattering a remedy for the occasional stagnation of the algorithm is proposed the potential flaw in the bcg algorithm may be avoided when encountered by restarting the algorithm with a perturbed estimate of the solution

roper suffridge extension operator and the lower bound for the distortion abstract liczberski starkov gave a sharp lower bound for d n f z near the origin where n is the roper suffridge extension operator and f is a normalized convex mapping on the unit disk in c they gave a conjecture that the sharp lower bound holds on the euclidean unit ball b n in c n in this paper we will give a sharp lower bound on b n for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f we will give a lower bound for mappings f in a linear invariant family we will also give a similar sharp lower bound on bounded convex complete reinhardt domains in c n

aim analysis of electromagnetic scattering by arbitrarily shaped magnetodielectric object a fast solution to the electromagnetic scattering by large scale three dimensional magnetodielectric objects with arbitrary permittivity and permeability is presented the scattering problem is characterized by using coupled field volume integral equation cf vie by considering the total electric and magnetic fields i e the sum of incident fields and the radiated fields by equivalent electric and magnetic volume currents the cf vie can be established in the volume of the scatterers the resultant cf vie is discretized and solved by using the method of moments mom for large scale scattering problems the adaptive integral method aim is then applied in the mom in order to reduce the memory requirement and accelerate the matrix vector multiplication in the iterative solver the conventional aim has been modified to cope with the two sets of equivalent volume currents

hausdorff measures and functions of bounded quadratic variation to each function f of bounded quadratic variation we associate a hausdorff measure f we show that the map f f is locally lipschitz and onto the positive cone of m 0 1 we use the measures f f v 2 to determine the structure of the subspaces of v 0 2 which either contain c 0 or the square stopping time space s 2

rcs computation of large inhomogeneous objects using a fast integral equation solver in this paper we apply a fast fourier transform fft accelerated volume integral equation solver to compute the radar cross section of large scale inhomogeneous objects this method is related to bojarski s k space method and the subsequent conjugate cg and biconjugate gradient bcg fft algorithms the method developed here combines the weak form discretization with the bcg and stabilized bcg bcgs solvers we show that this method has marked improvements over related algorithms the numerical method has been validated by mie series for multilayer spheres and applied to some practical problems with this method we are currently able to solve a three dimensional problem with a volume of size 3648 spl lambda sup 3 21 23 million unknowns on a single workstation

the concept of a minimal state in viscoelasticity new free energies and applications to pdes we show here the impact on the initial boundary value problem and on the evolution of viscoelastic systems of the use of a new definition of state based on the stress response see e g 48 16 41 comparisons are made between this new approach and the traditional one which is based on the identification of histories and states we shall refer to a stress response definition of state as the minimal state 29 materials with memory and with relaxation are discussed the energetics of linear viscoelastic materials is revisited and new free energies expressed in terms of the minimal state descriptor are derived together with the related dissipations furthermore both the minimum and the maximum free energy are recast in terms of the minimal state variable and the current strain the initial boundary value problem governing the motion of a linear viscoelastic body is re stated in terms of the minimal state and the velocity field through the principle of virtual power the advantages are i the elimination of the need to know the past strain history at each point of the body and ii the fact that initial and boundary data can now be prescribed on a broader space than resulting from the classical approach based on histories these advantages are shown to lead to natural results about well posedness and stability of the motion finally we show how the evolution of a linear viscoelastic system can be described through a strongly continuous semigroup of linear contraction operators on an appropriate hilbert space the family of all solutions of the evolutionary system obtained by varying the initial data in such a space is shown to have exponentially decaying energy

towards sharp superposition theorems in besov and lizorkin triebel spaces abstract this paper is devoted to the study of the superposition operator t f g f g in the framework of lizorkin triebel spaces f p q s r and besov spaces b p q s r for the case s 1 1 p 1 p 1 q it is natural to conjecture the following the operator t f takes f p q s r to itself if and only if f 0 0 and f belongs locally to f p q s r we establish this conjecture for the following two cases 1 s s 1 p 2 s s 1 p 3 4 for the case p 4 3 and s s 1 p the conjecture is also proved but with a restriction on s namely s s 1 2 1 p 1 p 3 4 a similar result holds for besov spaces b p q s r but with some extra restrictions involving q

on e quasi convex mappings in the unit ball of a complex banach space abstract in this paper a new class of biholomorphic mappings named e quasi convex mapping is introduced in the unit ball of a complex banach space meanwhile the definition of e starlike mapping is generalized from e 0 1 to e 1 1 it is proved that the class of e quasi convex mappings is a proper subset of the class of starlike mappings and contains the class of e starlike mappings properly for some e 1 0 0 1 we give a geometric explanation for e starlike mapping with e 1 1 and prove that the generalized roper suffridge extension operator preserves the biholomorphic e starlikeness on some domains in banach spaces for e 1 1 we also give some concrete examples of e quasi convex mappings or e starlike mappings for e 1 1 in banach spaces or c n furthermore some other properties of e quasi convex mapping or e starlike mapping are obtained these results generalize the related works of some authors

loewner chains and parametric representation in several complex variables abstract let b be the unit ball of c n with respect to an arbitrary norm we study certain properties of loewner chains and their transition mappings on the unit ball b we show that any loewner chain f z t and the transition mapping v z s t associated to f z t satisfy locally lipschitz conditions in t locally uniformly with respect to z b moreover we prove that a mapping f h b has parametric representation if and only if there exists a loewner chain f z t such that the family e tf z t t 0 is a normal family on b and f z f z 0 for z b also we show that univalent solutions f z t of the generalized loewner differential equation in higher dimensions are unique when e tf z t t 0 is a normal family on b finally we show that the set s0 b of mappings which have parametric representation on b is compact

the generalized roper suffridge extension operator abstract let f z be a normalized convex starlike function on the unit disc d let z c n z 1 2 z 2 p 2 z n p n where z z1 z2 zn z1 d z 2 z n c n 1 pi 1 i 2 n are real numbers in this note we prove that f z f z1 f z1 1 p2z2 f z1 1 pnzn is a normalized convex starlike mapping on where we choose the power function such that f z1 1 pi z1 0 1 i 2 n some other related results are proved

the generalized roper suffridge extension operator for some biholomorphic mappings abstract suppose f is a spirallike function of type or starlike function of order on the unit disk d in c let n p 1 p 2 p n z z 1 z 2 z n c n j 1 n z j p j 1 where 1 p 1 2 or 0 p 1 2 p j 1 j 2 n are real numbers in this paper we prove that n 2 2 n n f z f z 1 f z 1 z 1 2 f z 1 2 z 2 f z 1 z 1 n f z 1 n z n preserves spirallikeness of type or starlikeness of order on n p 1 p 2 p n where j 0 1 j 0 1 p j and j j 1 p j is the same as above we choose the branches such that f z 1 z 1 j z 1 0 1 f z 1 j z 1 0 1 j 2 n

loewner chains and the roper suffridge extension operator abstract let f be a locally univalent function on the unit disc and let 0 1 2 we consider the family of operators extending f to a holomorphic map from the unit ball b in cn to cn given by n f z f z1 z f z1 where z z2 zn when 1 2 we obtain the roper suffridge extension operator we show that if f s then n f can be imbedded in a loewner chain our proof shows that if f s then n f is starlike and if f with 2 then n f is a spirallike map of type in particular we obtain a new proof that the roper suffridge operator preserves starlikeness we also obtain the radius of starlikeness of n s and the radius of convexity of n 1 2 s we show that if f is a normalized univalent bloch function on u then n f is a bloch mapping on b finally we show that if f belongs to a class of univalent functions which satisfy growth and distortion results then n f satisfies related growth and covering results

functional calculus in holder zygmund spaces in this paper we characterize those functions f of the real line to itself such that the nonlinear superposition operator t f defined by t f g fog maps the holder zygmund space c s r n to itself is continuous and is r times continuously differentiable our characterizations cover all cases in which s is real and s 0 and seem to be novel when s 0 is an integer

composition operators on lizorkin triebel spaces abstract this paper is devoted to the study of the composition operator t f g f g on lizorkin triebel spaces f p q s r in case s 1 1 p 1 p and 1 q we will prove the following the operator t f takes f p q s r to itself if and only if f 0 0 and f belongs locally to f p q s r

distortion theorems for bloch functions we establish various distortion theorems for both normalized locally schlicht bloch functions and normalized bloch function with branch points these distortion theorems give lower bounds on either f z or re f z most of our distortion theorems are sharp and all extremal functions identified the main tools used in establishing these distortion theorems are the classical form of julia s lemma and a new version of julia s lemma that applies to certain multiple valued analytic functions

a class of loewner chain preserving extension operators abstract we consider operators that extend locally univalent mappings of the unit disk in c to locally biholomorphic mappings of the euclidean unit ball b of c n for such an operator we seek conditions under which e t e t f t t 0 is a loewner chain on b whenever f t t 0 is a loewner chain on we primarily study operators of the form g f z f z 1 g f z 1 z f z 1 z z z 2 z n where 0 1 2 and g c n 1 c is holomorphic finding that for g to preserve loewner chains the maximum degree of terms appearing in the expansion of g is a function of further applications involving bloch mappings and radius of starlikeness are given as are elementary results concerning extreme points and support points

subordination chains and biholomorphic mappings on bounded balanced pseudoconvex domains let w be a bounded balanced pseudoconvex domain with c 1 plurisubharmonic defining functions in c n and let f z t be a univalent subordination chain such that z 0 is a zero of order k 1 of e tf z t z for each t 0 in this paper we study several properties for f 0 concerning the growth covering theorems and coefficient bounds these results generalize the related works of hamada and kohr