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

strichartz estimates for the wave and schrodinger equations with the inverse square potential abstract we prove spacetime weighted l 2 estimates for the schrodinger and wave equation with an inverse square potential we then deduce strichartz estimates for these equations

good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

on a higher order secant method in this paper we carry out a modification of the classical secant method our goal is to accelerate the convergence until order two but without using any derivative two convergence theorems are presented finally some numerical experiments are analyzed

on the ball of convergence of secant like methods for non differentiable operators in this paper we analyze the local convergence of a uniparametric family of secant like methods for solving nonlinear operators in banach spaces this new study has an important and novel feature since it is applicable to non differential operators so far the results of local convergence usually considered can be only applied to differentiable operators

the total coordinate ring of a wonderful variety abstract we study the cone of effective divisors and the total coordinate ring of wonderful varieties with applications to their automorphism group we show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings

strichartz estimates for the schrodinger equation with time periodic ln 2 potentials abstract we prove strichartz estimates for the schrodinger operator h v t x with time periodic complex potentials v belonging to the scaling critical space l x n 2 l t in dimensions n 3 this is done directly from estimates on the resolvent rather than using dispersive bounds as the latter generally require a stronger regularity condition than what is stated above in typical fashion we project onto the continuous spectrum of the operator and must assume an absence of resonances eigenvalues are permissible at any location in the spectrum including at threshold energies provided that the associated eigenfunction decays sufficiently rapidly

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

convergence analysis of the secant type methods in this paper under the hypothesis that derivative satisfies some kinds of weak lipschitz condition the radius estimates of the convergence balls of secant type methods for operator equations are established in banach space some applications and numerical experiments are also given

a unifying local semilocal convergence analysis and applications for two point newton like methods in banach space abstract we provide a local as well as a semilocal convergence analysis for two point newton like methods in a banach space setting under very general lipschitz type conditions our equation contains a frechet differentiable operator f and another operator g whose differentiability is not assumed using more precise majorizing sequences than before we provide sufficient convergence conditions for newton like methods to a locally unique solution of equation f x g x 0 in the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results in the local case a larger radius of convergence is obtained several numerical examples are provided to show that our results compare favorably with earlier ones as a special case we show that the famous newton kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the newton kantorovich theorem

on the local convergence of a newton kurchatov type method for non differentiable operators abstract by means of a nice idea a newton kurchatov type iterative process is constructed for solving nonlinear equations in banach spaces we analyze the local convergence of this iterative process this study have an important and novel feature since it is applicable to non differentiable operators so far most of the local convergence results considered by other authors may apply only to differentiable operators due to the conditions that are required on the solution of the nonlinear equation

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

rationality of euler chow series and finite generation of cox rings abstract we study the rationality of the euler chow series e 1 x of codimension one cycles on a projective variety x and its relation with the effective cone and the cox ring of x among other things we prove that e 1 x is transcendental if the cone ne 1 x of pseudo effective divisors on x has infinitely many extremal rays generated by effective divisors on the other hand we give examples showing that the converse fails in addition we give an example where e 1 x is rational and cox x is infinitely generated finally we compute e 1 x for del pezzo surfaces x

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

improved inhomogeneous strichartz estimates for the schrodinger equation abstract we study inhomogeneous strichartz estimates for the schrodinger equation for dimension n 3 using a frequency localization we obtain some improved range of strichartz estimates for the solution of inhomogeneous schrodinger equation except dimension n 3

the total coordinate ring of a smooth projective surface in 5 cox introduced the homogeneous coordinate ring of a toric variety which is a polynomial ring that allows to show that such a variety behaves like a projective space in many ways an analogous to that ring can also be defined for a smooth projective variety x over an algebraically closed field k such that linear and numerical equivalence coincide for divisors on x condition which is assumed for all varieties considered in this paper indeed let us fix li i 1 a z basis of pic x and set n n1 n2 nr z and dn ri 1 nili by regarding the vector spaces h 0 x ox dn f k divx f dn 0 0 as k subvector spaces of the function field k of x the total coordinate ring of x or the cox ring of x is defined as the graded k subalgebra of k

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

improving the accessibility of steffensen s method by decomposition of operators abstract solving equations of the form h x 0 is usually done by applying iterative methods the main interest of this paper is to improve the domain of starting points for steffensen s method in general the accessibility of iterative methods that use divided differences in their algorithms is reduced since there are difficulties in the choice of starting points to guarantee the convergence of the methods in particular by using a decomposition of the operator h and applying a special type of iterative methods which combine two iterative schemes in the algorithms we can improve the accessibility of steffensen s method moreover we analyze the local convergence of the new iterative method proposed in two cases when h is differentiable and h is non differentiable the dynamical properties show that the method also improves the region of accessibility of steffensen s method for non differentiable operators so we present an alternative for the non applicability of newton s method to non differentiable operators that improves the accessibility of steffensen s method the theoretical results are illustrated with numerical experiments

uniform resolvent estimates for schrodinger operator with an inverse square potential abstract we study the uniform resolvent estimates for schrodinger operator with a hardy type singular potential let l v v x where is the usual laplacian on r n and v x v 0 r 2 where r x x x and v 0 c 1 s n 1 is a real function such that the operator v 0 n 2 2 4 is a strictly positive operator on l 2 s n 1 we prove some new uniform weighted resolvent estimates and also obtain some uniform sobolev estimates associated with the operator l v

on images of mori dream spaces the purpose of this paper is to study the geometry of images of morphisms from mori dream spaces first we prove that a variety which admits a surjective morphism from a mori dream space is again a mori dream space secondly we introduce a natural fan structure on the effective cone of mori dream spaces we show that it encodes the information of zariski decompositions which in turn is equivalent to the information of the variation of git quotients of their cox rings finally we show that under a surjective morphism between mori dream spaces the fan of the target space coincides with the restriction of the fan of the source

endpoint strichartz estimates for the magnetic schrodinger equation abstract we prove strichartz estimates for the schrodinger equation with an electromagnetic potential in dimension n 3 the decay and regularity assumptions on the potentials are almost critical i e close to the coulomb case in addition we require repulsivity and a nontrapping condition which are expressed as smallness of suitable components of the potentials while the potentials themselves can be large the proof is based on smoothing estimates and new sobolev embeddings for spaces associated to magnetic potentials

the cox ring of a spherical embedding abstract let g be a connected reductive group and g h a spherical homogeneous space we show that the ideal of relations between a natural set of generators of the cox ring of a g embedding of g h can be obtained by homogenizing certain equations which depend only on the homogeneous space using this result we describe some examples of spherical homogeneous spaces such that the cox ring of any of their g embeddings is defined by one equation

the convergence ball of newton s method and the uniqueness ball of equations under holder type continuous derivatives abstract under the hypotheses that nonlinear operators have k p holder type continuous derivatives exact estimates of the radius of the convergence ball of newton s method and of the uniqueness ball of solution of equations are obtained

sharp low frequency resolvent estimates on asymptotically conical manifolds on a class of asymptotically conical manifolds we prove two types of low frequency estimates for the resolvent of the laplace beltrami operator the first result is a uniform l 2 rightarrow l 2 bound for scal r 1 delta g z 1 scal r 1 when mbox re z is small with the optimal weight scal r 1 the second one is about powers of the resolvent for any integer n we prove uniform l 2 rightarrow l 2 bounds for scal epsilon r n epsilon 2 delta g z n scal epsilon r n when mbox re z belongs to a compact subset of 0 infty and 0 epsilon ll 1 these results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity

picard graded betti numbers and the defining ideals of cox rings abstract we introduce a complex of vector spaces whose homology determines the picard graded betti numbers of the cox ring of a mori dream space as an application of this construction we prove a conjecture of batyrev and popov which describes presentations for the cox rings of del pezzo surfaces of degree at least two

rapidly decaying solutions of the nonlinear schrodinger equation we consider global solutions of the nonlinear schrodinger equation iu t delta u lambda u alpha u in r n nls where r and 0 0 we prove asymptotic completeness for alpha 0 leqq alpha frac 4 n 2

dynamics and local convergence of a family of derivative free iterative processes abstract in this paper we establish the construction of a family of free derivative of point to point iterative processes with quadratic convergence from two known data in each previous iteration besides we study the accessibility of this family by means of the basins of attraction and the convergence balls we provide a local convergence analysis for the family of iterative processes free of derivatives when the operator f is not necessarily frechet differentiable the sufficient convergence conditions are weaker and more flexible than in earlier studies an application is provided involving mixed hammerstein nonlinear integral equation with application in real world problems finally we show also a dynamical study and the convergence regions of some members of the family

secant methods for semismooth equations some generalizations of the secant method to semismooth equations are presented in the one dimensional case the superlinear convergence of the classical secant method for general semismooth equations is proved moreover a new quadratically convergent method is proposed that requires two function values per iteration for the n dimensional cases we discuss secant methods for two classes of composite semismooth equations most often studied semismooth equations are of such form

strichartz estimates and wave equation in a conic singular space consider the metric cone x c y 0 infty r times y with metric g dr 2 r 2h where the cross section y is a compact n 1 dimensional riemannian manifold y h let delta g be the positive friedrichs extension laplacian on x and let delta h be the positive laplacian on y and consider the operator mathcal l v delta g v 0 r 2 where v 0 in mathcal c infty y such that delta h v 0 n 2 2 4 is a strictly positive operator on l 2 y in this paper we prove global in time strichartz estimates without loss regularity for the wave equation associated with the operator mathcal l v it verifies a conjecture in wang remark 2 4 in ann inst fourier 56 1903 1945 2006 for wave equation the range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of delta h v 0 n 2 2 4 to prove the result we show a sobolev inequality and a boundedness of a generalized riesz transform in this setting in addition as an application we study the well posed theory and scattering theory for energy critical wave equation with small data on this setting of dimension n ge 3

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

concerning the terra incognita between convergence regions of two newton methods abstract the majorizing principle is used to show local and semilocal convergence of newton methods to a locally unique solution of a nonlinear operator in a banach space when the frechet derivative of the operator involved satisfies a center holder and a holder continuity condition then we investigate an unknown area terra incognita between the convergence regions of newton s method and the corresponding modified newton s method our approach compares favorably with other corresponding ones in this direction

on embeddings of homogeneous spaces with small boundary we study equivariant embeddings with small boundary of a given homogeneous space g h where g is a connected linear algebraic group with trivial picard group and only trivial characters and h g is an extension of a connected grosshans subgroup by a torus under certain maximality conditions like completeness we obtain finiteness of the number of isomorphism classes of such embeddings and we provide a combinatorial description of the embeddings and their morphisms the latter allows a systematic treatment of examples and basic statements on the geometry of the equivariant embeddings of a given homogeneous space g h

non trivial l q solutions to the ginzburg landau equation it is shown that the contour problem for the stationary ginzburg landau equation where x r with r x is well posed in l 4 n for a class of small data f l 2 s n 1

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

the convergence ball of the secant method under holder continuous divided differences under the hypothesis that nonlinear operators have holder continuous divided differences of order one an estimate of the radius of the convergence ball of the secant method is obtained error estimate is also established

a lefschetz hyperplane theorem for mori dream spaces let x be a smooth mori dream space of dimension 4 we show that if x satisfies a suitable git condition which we call small unstable locus then every smooth ample divisor y of x is also a mori dream space moreover the restriction map identifies the neron severi spaces of x and y and under this identification every mori chamber of y is a union of some mori chambers of x and the nef cone of y is the same as the nef cone of x this lefschetz type theorem enables one to construct many examples of mori dream spaces by taking mori dream hypersurfaces of an ambient mori dream space provided that it satisfies the git condition to facilitate this we then show that the git condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles and in the case when x is toric we show that the condition is equivalent to the fan of x being 2 neighborly

cox rings and combinatorics given a variety x with a finitely generated total coordinate ring we describe basic geometric properties of x in terms of certain combinatorial structures living in the divisor class group of x for example we describe the singularities we calculate the ample cone and we give simple fano criteria as we show by means of several examples the results allow explicit computations as immediate applications we obtain an effective version of the kleiman chevalley quasiprojectivity criterion and the following observation on surfaces a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non factorial singularities admit a common affine neighbourhood

agmon kato kuroda theorems for a large class of perturbations we prove asymptotic completeness for operators of the form h l on l2 rd d 2 where l is an admissible perturbation our class of admissible perturbations contains multiplication operators defined by real valued potentials v lq rd q d 2 d 1 2 if d 2 then we require q 1 3 2 as well as real valued potentials v satisfying a global kato condition the class of admissible perturbations also contains first order differential operators of the form a a for suitable vector potentials a our main technical statement is a new limiting absorption principle which we prove using techniques from harmonic analysis related to the stein tomas restriction theorem

inhomogeneous strichartz estimates for the schrodinger equation we study strichartz estimates for the solution of the cauchy problem associated with the inhomogeneous free schrodinger equation in the case when the inital data is equal to zero proving some new estimates for certain exponents and giving counterexamples for some others