collapsing limits of the kahler ricci flow and the continuity method we consider the kahler ricci flow on certain calabi yau fibration which is a calabi yau fibration with one dimensional base or a product of two calabi yau fibrations with one dimensional bases assume the kahler ricci flow on total space admits a uniform lower bound for ricci curvature then the flow converges in gromov hausdorff topology to the metric completion of the regular part of generalized kahler einstein current on the base which is a compact length metric space homeomorphic to the base the analogue results for the continuity method on such calabi yau fibrations are also obtained moreover we show the continuity method starting from a suitable kahler metric on the total space of a fano fibration with one dimensional base converges in gromov hausdorff topology to a compact metric on the base during the proof we show the metric completion of the regular part of a generalized kahler einstein current on a riemann surface is compact

remarks on chirka s proof of slodkowski s theorem by applying schauder s fixed point theorem to an integral operator chirka de veloped a remarkably simple proof for slodkowski s theorem on extending holomorphic motions in this paper we first present our understanding of a motive for designing the integral opera tor from which we also recapitulate the main work in chirka s proof when applying chirka s operator in different ways to obtain the extensions of holomorphic motions a question on the uniqueness of extensions arises after a brief summary on some known sufficient conditions for the uniqueness of extensions we raise two questions concerning the necessary conditions for the uniqueness of the extensions of the holomorphic motions of four points at the end we present an application when the extension is unique that is we show that there is a holomorphic map from the unit open disk d to the teichmuller space of symmetric circle homeomorphisms such that it can not be lifted to a holomorphic map from d to the complex banach space of vanishing beltrami coefficients

holomorphic motions and teichmuller spaces we prove an equivariant form of slodkowski s theorem that every holomorphic motion of a subset of the extended complex plane c extends to a holomorphic motion of c as a consequence we prove that every holomorphic map of the unit disc into teichmuller space lifts to a holomorphic map into the space of beltrami forms we use this lifting theorem to study the teichmuller metric

immersing almost geodesic surfaces in a closed hyperbolic three manifold let m 3 be a closed hyperbolic three manifold we construct closed surfaces that map by immersions into m 3 so that for each one the corresponding mapping on the universal covering spaces is an embedding or in other words the corresponding induced mapping on fundamental groups is an injection

cusp geometry of fibered 3 manifolds let f be a surface and suppose that varphi colon f to f is a pseudo anosov homeomorphism fixing a puncture p of f the mapping torus m m varphi is hyperbolic and contains a maximal cusp c about the puncture p we show that the area and height of the cusp torus partial c is equal to the stable translation distance of varphi acting on the arc complex scr a f p up to an explicitly bounded multiplicative error our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces in particular the proof of this theorem does not use any deep results from teichm uller theory kleinian group theory or the coarse geometry of scr a f p a similar result holds for quasi fuchsian manifolds n cong f times bbb r in that setting we find a combinatorial estimate for the area and height of the cusp annulus in the convex core of n up to explicitly bounded multiplicative and additive error as an application we show that covers of punctured surfaces induce quasi isometric embeddings of arc complexes

harmonic maps and the schoen conjecture we show that every quasisymmetric homeomorphism of the circle h 2 admits a harmonic quasiconformal extension to the hyperbolic plane h 2 this proves the schoen conjecture

the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

wavelets properties and approximate solution of a second kind integral equation abstract in this paper we show that under some conditions a wavelet basis of l 2 r can be used as a tool for the uniform approximation in the space c r l 2 r 0 where c r denotes the holder space of exponent as a result of this property we give a numerical application of wavelets this application is a wavelet based method for the numerical solution of a redholm equation of the second kind with solution lying in c 0 r the holder space of compactly supported functions with holder exponent 1 2

numerical computational method in solving fredholm integral equations of the second kind by using coifman wavelet in this paper we bring three theorems that enable us to approximate the solution of ferdholm integral equations of the second kind then we use coifman wavelets or coiflets as scaling functions for projection that satisfied the conditions of theorems for approximation also we use this projection to convert the integral equation to a galerkin system which is the most important of the expansion methods for solving linear integral equations finally by using numerical examples we show that our estimation have a good degree of accuracy

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

computational methods for integrals involving functions and daubechies wavelets abstract when wavelets are used as basis functions in galerkin approach to solve the integral equations integrals of the form supp j k f x j k x d x occur by a change of variable these integrals can be translated into integrals involving only in this paper we find quadrature rule on the supp for the integrals of the form supp f x x d x wavelets in this article are those discovered by daubechies i daubechies orthonormal bases of compactly supported wavelets comm pure appl math 41 1988 909 996 where is the scaling function and is the wavelet function

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

quasiconformal homeomorphisms and the convex hull boundary we investigate the relationship between an open simply connected region s 2 and the boundary y of the hyperbolic convex hull in h 3 of s 2 a counterexample is given to thurston s conjecture that these spaces are related by a 2 quasiconformal homeomorphism which extends to the identity map on their common boundary in the case when the homeomorphism is required to respect any group of mobius transformations which preserves we show that the best possible universal lipschitz constant for the nearest point retraction r y is 2 we find explicit universal constants 0 c 2 c 1 such that no pleating map which bends more than c 1 in some interval of unit length is an embedding and such that any pleating map which bends less than c 2 in each interval of unit length is embedded we show that every k quasiconformal homeomorphism d 2 d 2 is a k k quasi isometry where k is an explicitly computed function the multiplicative constant is best possible and the additive constant k is best possible for some values of k

adaptive solution of partial differential equations in multiwavelet bases we construct multiresolution representations of derivative and exponential operators with linear boundary conditions in multiwavelet bases and use them to develop a simple adaptive scheme for the solution of nonlinear time dependent partial differential equations the emphasis on hierarchical representations of functions on intervals helps to address issues of both high order approximation and efficient application of integral operators and the lack of regularity of multiwavelets does not preclude their use in representing differential operators comparisons with finite difference finite element and spectral element methods are presented as are numerical examples with the heat equation and burgers equation

numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis in this paper a system of nonlinear fractional differential equations fdes are considered they have been solved by legendre wavelets method combining with its operational matrix however there are no articles about solving this system using wavelets method the main purpose of this technique is to transform the initial equations into a nonlinear system of algebraic equations which can be solved easily the convergence and error analysis are presented to show the correctness and feasibility of method proposed for solving the above mentioned problem finally the applicability and efficiency of the mentioned approach are demonstrated by three numerical examples

schwarzian derivatives projective structures and the weil petersson gradient flow for renormalized volume to a complex projective structure on a surface thurston associates a locally convex pleated surface we derive bounds on the geometry of both in terms of the norms and 2 of the quadratic differential of given by the schwarzian derivative of the associated locally univalent map we show that these give a unifying approach that generalizes a number of important well known results for convex cocompact hyperbolic structures on 3 manifolds including bounds on the lipschitz constant for the nearest point retraction and the length of the bending lamination we then use these bounds to begin a study of the weil petersson gradient flow of renormalized volume on the space cc n of convex cocompact hyperbolic structures on a compact manifold n with incompressible boundary leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of dn the double of n

teichmuller curves triangle groups and lyapunov exponents we construct a teichmuller curve uniformized by a fuchsian triangle group commensurable to m n for every m n in most cases for example when m 6 n and m or n is odd the uniformizing group is equal to the triangle group m n our construction includes the teichmuller curves constructed by veech and ward as special cases the construction essentially relies on properties of hypergeometric differential operators for small m we find billiard tables that generate these teich muller curves we interpret some of the so called lyapunov exponents of the kontsevich zorich cocycle as normalized degrees of a natural line bundle on a teichmuller curve we determine the lyapunov exponents for the teichmuller curves we construct

numerical solution of the integral equation of the second kind by using wavelet bases of hermite cubic splines in this paper we use the wavelet bases of hermite cubic splines to solve the second kind integral equationsx t 0 1k t s x s ds y t t 0 1 a pair of wavelets are constructed on the basis of hermite cubic splines this wavelets are in c 1 and supported on 0 2 moreover one wavelet is symmetric and the other is anti symmetric this spline wavelets are then adapted to the interval 0 1 the computational results demonstrate the advantage of the wavelet basis

the conformal boundary and the boundary of the convex core in this note we investigate the relationship between the conformal boundary at infinity of a hyperbolic 3 manifold and the boundary of its convex core in particular we prove that the length of a curve in the conformal boundary gives an upper bound on the length of the corresponding curve in the boundary of the convex core a hyperbolic 3 manifold n is the quotient of hyperbolic 3 space h by a group of isometries the boundary at infinity of h may be identified with the riemann sphere and extends to act on as a group of conformal automorphisms the domain of discontinuity of is the largest invariant open subset of which acts on properly discontinuously if is not abelian then inherits a hyperbolic metric called the poincare metric which acts on as a group of isometries one may then consider cn to be the conformal boundary at infinity of the hyperbolic 3 manifold n the conformal boundary is the topological boundary of n h the first important result concerning the relationship between the geometry of the conformal boundary and the geometry of n is due to bers 4 who proved that if each component of is simply connected and is a closed curve in its conformal boundary cn then the geodesic in n in the homotopy class of has length ln at most twice the length l cn of in the conformal boundary if is homotopic to arbitrarily short closed curves in n then we say ln 0 canary 8 generalized this to the setting of arbitrary kleinian groups proving that given any 0 there exists k 0 such that if has injectivity radius bounded below by at each point then if is a closed curve in c n then ln kl cn

an improved bound for sullivan s convex hull theorem sullivan showed that there exists k0 such that if is a simply connected hyperbolic domain then there exists a conformally naturalk0 quasiconformal map from to the boundary dome of the convex hull of its complement which extends to the identity on explicit upper and lower bounds on k0 were obtained by epstein marden markovic and bishop we improve on these bounds by showing that one may choose k0 7 1695

the logarithmic spiral a counterexample to the k 2 conjecture given a nonempty compact connected subset x subset of s 2 with complement a simply connected open subset omega subset of s 2 let dome omega be the boundary of the hyperbolic convex hull in h 3 of x we show that if x is a certain logarithmic spiral then we obtain a counterexample to the conjecture of thurston and sullivan that there is a 2 quasiconformal homeomorphism omega dome omega which extends to the identity map on their common boundary in s 2 this leads to related counterexamples when the boundary is real analytic or a finite union of intervals straight intervals if we take s 2 c boolean or infinity we also show how this counterexample enables us to construct a related counterexample which is a domain of discontinuity of a torsion free quasifuchsian group with compact quotient another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately 98 pi 2 which is substantially larger than that of any previously known example

using the wpg method for solving integral equations of the second kind abstract in this paper we use the wavelet petrov galerkin wpg method based on discontinuous orthogonal multiwavelets for solving fredholm integral equations of the second kind that yields linear system that the condition number of its coefficient matrix is bounded and for showing efficiency of method we use numerical examples

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

braiding of the attractor and the failure of iterative algorithms le mouvement de l attracteur dans une famille stable d applications rationnelles determine un groupe de tresses sur la sphere de riemann on montre que le groupe correspondant des classes d application est soit reductible soit fini soit fixe l un des attracteurs ceci conduit a un critere topologique d existence des bifurcations dans les algorithmes iteratifs de recherche de racines bases sur des applications rationnels

real and complex earthquakes we consider real earthquakes and by their extensions complex earthquakes of the hyperbolic plane h 2 we show that an earthquake restricted to the boundary s 1 of h 2 is a quasisymmetric map if and only if its earthquake measure is bounded multiplying an earthquake measure by a positive parameter we obtain an earthquake path consequently an earthquake path with a bounded measure is a path in the universal teichmiiller space we extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part such obtained complex earthquake or bending is holomorphic in the parameter moreover the restrictions to s 1 of a bending with complex parameter of small imaginary part is a holomorphic motion of s 1 in the complex plane in particular a real earthquake path with bounded earthquake measure is analytic in its parameter

frontiers in complex dynamics rational maps on the riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems first rational maps are complexanalytic so a broad spectrum of techniques can contribute to their study quasiconformal mappings potential theory algebraic geometry etc the rational maps of a given degree form a finite dimensional manifold so exploration of this parameter space is especially tractable finally some of the conjectures once proposed for smooth dynamical systems and now known to be false seem to have a definite chance of holding in the arena of rational maps in this article we survey a small constellation of such conjectures centering around the density of hyperbolic rational maps those which are dynamically the best behaved we discuss some of the evidence and logic underlying these conjectures and sketch recent progress towards their resolution

average bending of convex pleated planes in hyperbolic three space if p is a pleated plane in 3 dimensional hyperbolic space h 3 and a geodesic in its intrinsic metric we define b p the average bending of p in the direction we show that if p is a convex pleated plane embedded in h 3 then b p k for some universal k furthermore if p is a boundary component of the convex hull of a quasi fuchsian group then b p b almost everywhere where b is a constant times the length of the bending lamination of the pleated surface x p we use these to prove a number of results about quasi fuchsian groups including a universal bound on the lipschitz constant for the map to infinity and a bound on the length of by a constant times the euler characteristic of x

on connectivity of julia sets of yang lee zeros we prove that the julia sets of the yang lee zeros of the potts model on the diamond hierarchical lattice are connected sets on the complex plane

a criterion for flatness of hodge bundles over curves and geometric applications for the definition of the kodaira dimension as well as for a discussion of the conjecture cn m i refer to the bourbaki talk by esnault 7 here i only want to say that c n and cn 1 are true and that for applications to albanese mappings a stronger version of cn m is needed which in 7 is called conjecture c m if m 1 it merely says that in addition to the validity of cn in the special case when y is an elliptic curve and k x f 0 there should exist a surjective morphism y y such that the fibre product x x y is birational to a product y x f for u n 2 m 1 this last requirement follows from a result of arakelov 1 theorem 1 1 which implies that deg f tox r 0 and equality holds if and only if f is isotrivial i e after a finite covering of y the fibre product is birational to a product here cox r is the relative dualizing sheaf see sect 5 1 kawamata s proof of c as presented in 13 depends on a result of fujita 11 about line bundle quotients of f t o x r d i m y l very roughly fujita s result says that arakelov s inequality generalizes to the case when dimx is arbitrary but still dim y 1 here i want to look more closely to what happens if deg f o2x r 0 the natural framework for this question is griffith s theory of variations of hodge structure our main result is the following

a multilevel method for solving operator equations abstract we develop a multilevel method suitable for solving operator equations this method combines the multiresolution structure of the spaces used to solve the operator equation with a gauss seidel strategy to solve the associated matrix equations we prove that this multilevel scheme has an optimal order of convergence and provide an application of it to the solution of second kind integral equations

convergence and condition number of multi projection operators by legendre wavelets in this paper we use multi projection operators to solve the linear fredholm integral equation of the second kind with wavelets moreover a numerical convergence and computation of a bound for the condition number of the multi projection operators method is obtained

comparing poincare densities however there is no formula for the poincare density pn of a general plane domain q and so one must resort to estimates many research articles have been devoted to finding and working with such estimates see for example 4 5 15 19 24 and 25 the first tool to establish such estimates is schwarz s lemma which says that every holomorphic map is a contraction with respect to the poincare metric that is ps f ldf i pd z idzl for any holomorphic map f from a riemann surface d to another riemann surface q applying this lemma

smooth deformations of singular contractions of class vii surfaces we consider normal compact surfaces y obtained from a minimal class vii surface x by contraction of a cycle c of r rational curves with c 2 0 our main result states that if the obtained cusp is smoothable then y is globally smoothable the proof is based on a vanishing theorem for h 2 theta y if r b 2 x any smooth small deformation of y is rational and if r b 2 x i e when x is a half inoue surface any smooth small deformation of y is an enriques surface the condition the cusp is smoothable in our main theorem can be checked in terms of the intersection numbers of the cycle using the looijenga conjecture which has recently become a theorem therefore this is a decidable condition we prove that this condition is always satisfied if r b 2 x leqslant 11 therefore the singular surface y obtained by contracting a cycle c of r rational curves in a minimal class vii surface x with r b 2 x leqslant 11 is always smoothable by rational surfaces the statement holds even for unknown class vii surfaces

continuity of capacity of a holomorphic motion abstract we study the behavior of various set functions under holomorphic motions we show that under such deformations logarithmic capacity varies continuously while analytic capacity may not

on mori s theorem for quasiconformal maps in the n space r fehlmann and m vuorinen proved in 1988 that mori s constant m n k for k quasiconformal maps of the unit ball in r n onto itself keeping the origin fixed satisfies m n k 1 when k 1 we give here an alternative proof of this fact with a quantitative upper bound for the constant in terms of elementary functions our proof is based on a refinement of a method due to g d anderson and m k vamanamurthy we also give an explicit version of the schwarz lemma for quasiconformal self maps of the unit disk some experimental results are provided to compare the various bounds for the mori constant when n 2

divergence groups have the bowen property we show that every deformation of a divergence type fuchsian group has a limit set which is either a circle or has dimension strictly larger than 1 this is known to be false for all convergence type groups and hence solves bowen s problem in the general case the proof uses a theorem of dennis sullivan s about convex hulls in hyperbolic 3 space and we give a new simpler proof of this result as well