convergence of the two step combined method and uniqueness of the solution of nonlinear operator equations local convergence of the two step differential difference method for solving nonlinear operator equations for generalized lipschitz conditions for frechet derivatives of the first and second order and divided differences of the first order has been proven there have been found estimations of the convergence ball s radii of this method and the uniqueness ball of solution of nonlinear equations there has been established the superquadratical order of the convergence of the two step combined method and a comparison of the results with the known ones has been made

on the local convergence of a deformed newton s method under argyros type condition abstract for the iteration which was independently proposed by king r f king tangent method for nonlinear equations numer math 18 1972 298 304 and werner w werner uber ein verfarhren der ordnung 1 2 zur nullstellenbestimmung numer math 32 1979 333 342 for solving a nonlinear operator equation in banach space we established a local convergence theorem under the condition which was introduced recently by argyros i k argyros a unifying local semilocal convergence analysis and application for two point newton like methods in banach space j math anal appl 298 2004 374 397

how long are the arms in dla the author shows that the maximal length of the arms of dla can grow at most at a rate of n2 3 in dimension 2 and at a rate of n2 d in dimension d 2 here n denotes the volume or mass of the aggregate

microstructures with finite surface energy the two well problem we study solutions of the two well problem i e maps which satisfy ueso n a so n b a c in n where a and b are n n matrices with positive determinants this problem arises in the study of microstructure in solid solid phase transitions under the additional hypothesis that the set e where the gradient lies in so n a has finite perimeter we show that u is locally only a function of one variable and that the boundary of e consists of subsets of hyperplanes which extend to and which do not intersect in this may not be the case if the assumption on e is dropped we also discuss applications of this result to magnetostrictive materials

on a counterexample of a rank 1 convex function which is not polyconvex in the case n 2 j m ball has introduced the notion of polyconvexity to study nonlinear problems in elasticity and he has shown that polyconvexity implies rank 1 convexity in this paper we prove by a counterexample that the converse of this implication is false in two dimensions

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

on the different convex hulls of sets involving singular values keywords real matrices singular values convex hull polyconvex hull rank one convex hull reference caa article 1998 004doi 10 1017 s0308210500027311view record in web of science record created on 2008 11 25 modified on 2017 05 12

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

a uniqueness theorem in kahler geometry we consider compact kahler manifolds with their kahler ricci tensor satisfying f ric constant under the nonnegative bisectional curvature assumption and certain conditions on f we prove that such metrics are in fact kahler einstein

a new semilocal convergence theorem of muller s method abstract a semilocal convergence theorem of muller s method is established under the condition an error estimation is given which matches the convergence order of the method numerical examples are provided to show that our results apply where earlier ones fail

upper bounds for the growth rate of dla abstract we show that the maximal length of the arms in dla can grow at most at a rate n 2 d 1 where n denotes the volume or mass of the aggregate and d is the dimension a logarithmic correction factor is needed if d 3 we also give a lower bound for the number of sites in a sphere which are eventually occupied by the aggregate

convergence on a deformed newton method11jointly supported by china major key project for basic researcher and provincial natural science foundation for the iteration which was independently proposed by king numer math 18 1972 298 304 and werner numer math 32 1979 333 342 and has the convergent order 1 2 to solve the operator equation f x 0 in banach space we prove that it converges when the criterion proposed by smale in the merging of disciplines new directions in pure applied and computational mathematics springer new york 1986 pp 185 196 a b c 3 22 but in the paper we only assume that f x 0 1f x 0 b f x 0 1f x 0 2 c and f x 0 1f x 6 c 2 1 c x x 0 3 where c x x 0 1 1 2

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

an example of a quasiconvex function that is not polyconvex in two dimensions we study the different notions of convexity for the function f 2 2 2 det where e 2 2 introduced by dacorogna marcellini we show that f is convex polyconvex quasiconvex rank one convex if and only if 2 3 2 1 1 for some 0 2 3 respectively

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

stationary harmonic measure and dla in the upper half plane in this paper we introduce the stationary harmonic measure in the upper half plane by bounding this measure we are able to define both the discrete and continuous time diffusion limited aggregation dla in the upper half plane with absorbing boundary conditions we prove that for the continuous model the growth rate is bounded from above by o t 2 epsilon moreover we prove that all the moments are finite for the size of the aggregation when time is discrete we also prove a better upper bound of o n 2 3 epsilon on the maximum height of the aggregate at time n an important tool developed in this paper is an interface growth process bounding any process growing according to the stationary harmonic measure together with 12 one obtains non zero growth rate for any such process

on the step by step construction of quasi monte carlo integration rules that achieve strong tractability error bounds in weighted sobolev spaces we develop and justify an algorithm for the construction of quasi monte carlo qmc rules for integration in weighted sobolev spaces the rules so constructed are shifted rank 1 lattice rules the parameters characterising the shifted lattice rule are found component by component the d 1 th component of the generator vector and the shift are obtained by successive 1 dimensional searches with the previous d components kept unchanged the rules constructed in this way are shown to achieve a strong tractability error bound in weighted sobolev spaces a search for n point rules with n prime and all dimensions 1 to d requires a total cost of o n3d2 operations this may be reduced to o n3d operations at the expense of o n2 storage numerical values of parameters and worst ease errors are given for dimensions up to 40 and n up to a few thousand the worst case errors for these rules are found to be much smaller than the theoretical bounds