mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

brauer groups and quotient stacks a natural question is to determine which algebraic stacks are quotient stacks in this paper we give some partial answers and relate it to the question of whether for a scheme x the natural map from the brauer group equivalence classes of azumaya algebras to the cohomological brauer group the torsion subgroup of h 2 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i m is surjective

a new green s function method for solving linear pde s in two variables a new spectral method for solving linear and integrable nonlinear pde s in two variables has recently been developed this method is based on a lax pair formulation and generates an elegant integral representation of the solution to a given initial boundary value problem that is particularly useful for determining long time asymptotics in this paper it is shown that the integral representation obtained using lax pairs can also be obtained using green s function techniques this new green s function method is illustrated by solving a variety of initial boundary value problems for the diffusion equation the linearized schrodinger equation and the linearized kdv equation

some results on a starlike log harmonic mapping of order alpha let h d be the linear space of all analytic functions defined on the open unit disc email protected c z 0 g 0 1 on the other hand if f vanishes at z 0 but it is not identically zero then f admits the following representation 0 3 f z z z 2 bh z g z where email protected 12 h z and g z are analytic in the open disc d with the normalization h 0 0 g 0 1 abdulhadi and bshouty 1988 2 abdulhadi and hengartner 1996 3 in the present paper we will give the extent of the idea which was introduced by abdulhadi and bshouty 1988 2 one of the interesting applications of this extent idea is an investigation of the subclass of log harmonic mappings for starlike log harmonic mappings of order alpha

harmonic univalent mappings into a half plane with nonreal vertical slits a class of harmonic univalent mappings is constructed by applying the method of clunie and sheil small these mappings assuming their values in a half plane with a vertical boundary omit two vertical half lines symmetric w r t the real axis several basic properties are proved

a sum over paths approach to diffusion on trees we use a sum over paths method to derive an explicit expression for the green s function of the diffusion equation on an arbitrary tree making connection with the conventional laplace approach but emphasizing the real time nature of the technique this provides a general framework for studying both discrete space and continuous space diffusion processes on complex one dimensional topologies

a certain class of starlike log harmonic mappings in this paper we investigate some properties of log harmonic starlike mappings for this aim we use the subordination principle or lindelof principle lewandowski 1961 7

degeneration of l adic weight spectral sequences the degeneracy of the l adic weight spectral sequence over any trait with no assumptions on its residue field is proved by the method of log geometry in the sense of fontaine illusie

bounded log harmonic functions with positive real part abstract let h d be the linear space of all analytic functions defined on the open unit disc d z z 1 and let b be the set of all functions w z h d such that w z 1 for all z d a log harmonic mapping is a solution of the non linear elliptic partial differential equation f z w z f f f z where w z is the second dilatation of f and w z b in the present paper we investigate the set of all log harmonic mappings r defined on the unit disc d which are of the form r h z g z where h z and g z are in h d h 0 g 0 1 and r e r 0 for all z d the class of such functions is denoted by p l h

a constructive method for computing the tension parameters in convexity preserving spline in tension interpolation the idea initiated by opfer for constructing conformal mappings from the disk onto starlike domains is generalized for univalent harmonic mappings this is of some interest since such mappings are not characterized by analytic means

on meromorphically harmonic starlike functions with respect to symmetric conjugate points abstract in a previous paper m p chen z r wu and z z zou m p chen z r wu z z zou on functions starlike with respect to symmetric conjugate points j math anal appl 201 1996 25 34 developed a method using some operators to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc then the same method is employed to functions meromorphic by z z zou and z r wu zhong zhu zou zhuo ren wu on meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points j math anal appl 261 2001 17 27 now the method can be employed to functions meromorphic harmonic in the punctured disc 0 z 1 especially a sharp coefficient estimate and a structural representation of such functions are obtained

landau s theorem for functions with logharmonic laplacian abstract in this paper we show the existence of landau constant for functions with logharmonic laplacian of the form f z z 2l z k z z

on the univalence of the log biharmonic mappings in this paper we establish the univalence and starlikeness connection between log biharmonic mappings and logharmonic mappings

directed intermittent search for a hidden target on a dendritic tree motivated by experimental observations of active motor driven intracellular transport in neuronal dendrites we analyze a stochastic model of directed intermittent search on a tree network a particle injected from the cell body or soma into the primary branch of the dendritic tree randomly switches between a stationary search phase and a mobile nonsearch phase that is biased in the forward direction a synaptic target is presented somewhere within the tree which the particle can locate if it is within a certain range and in the searching phase we approximate the moment generating function using green s function methods the moment generating function is then used to compute the hitting probability and conditional mean first passage time to the target we show that in contrast to a previously explored finite interval case there is a range of parameters for which a bidirectional search strategy is more efficient than a unidirectional one in finding the target

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

ordinary k3 surfaces over a finite field we give a description of the category of ordinary k3 surfaces over a finite field in terms of linear algebra data over z this gives an analogue for k3 surfaces of deligne s description of the category of ordinary abelian varieties over a finite field and refines earlier work by n o nygaard and j d yu our main result is conditional on a conjecture on potential semi stable reduction of k3 surfaces over p adic fields we give unconditional versions for k3 surfaces of large picard rank and for k3 surfaces of small degree

on harmonic univalent mappings with nonzero pole abstract in this article we consider the class a h p of all sense preserving harmonic functions f in the open unit disc d having a simple pole at z p 0 1 with the normalizations f 0 f z 0 1 0 we first derive a sufficient condition for univalence of such functions next we consider the class s h 0 p which consists of all functions belonging to a h p and are univalent with the additional normalization f z 0 0 we study the class s h 0 p in the geometric function theoretic viewpoint as a byproduct of our investigation we see that consideration of nonzero pole yields nontrivial lower bounds for the taylor coefficients of such functions we also prove a growth result for a non vanishing subclass associated with s h 0 p

complete intersections moduli torelli and good reduction we study the arithmetic of complete intersections in projective space over number fields our main results include arithmetic torelli theorems and versions of the shafarevich conjecture as proved for curves and abelian varieties by faltings for example we prove an analogue of the shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics

on the shafarevich conjecture for enriques surfaces enriques surfaces are minimal surfaces of kodaira dimension 0 with b 2 10 if we work with a field of characteristic away from 2 enriques surfaces admit double covers which are k3 surfaces in this paper we prove the shafarevich conjecture for enriques surfaces by reducing the problem to the case of k3 surfaces in our formulation of the shafarevich conjecture we use the notion admitting a cohomological good k3 cover which includes not only good reduction but also flower pot reduction

perverse t structure on artin stacks in this paper we develop the theory of perverse sheaves on artin stacks continuing our earlier study of lisse etale sheaves on stacks in laszlo and olsson the six operations for sheaves on artin stacks i finite coefficients publ math ihes 2008 the six operations for sheaves on artin stacks ii adic coefficients publ math ihes 2008

c integrable nonlinear partial differential equations iii a technique based on a change of dependent variables used in a previous paper to generate c integrable nonlinear partial differential equations pdes i e nonlinear pdes linearizable by an appropriate change of variables in 1 1 dimensions one time and one space variables is extended to the case of more than one space dimension several examples of evolution c integrable pdes in 1 2 dimensions one time and two space variables are exhibited

support points of families of analytic functions described by subordination we determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination the families we study include the functions with a positive real part the typically real functions and the functions which are subordinate to a given majorant if the majorant f is univalent then each support point has the form f 0 where p is a finite blaschke product and o 0 o this completely characterizes the set of support points when f is convex the set of support points is found for some specific majorants including f z 1 z l z p where p 1 let k and st denote the set of normalized convex and starlike mappings respectively we find the support points of the families k and st defined by the property of being subordinate to some member of k or st respectively introduction let a z e c i z i 1 and let e denote the set of functions analytic in a then 6 is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of la by a continuous linear functional on e we mean a complex valued functional defined on 6f that is linear and continuous in other words if j is such a functional then j af bg aj f bj g whenever a and b belong to c and f and g belong to b also if fn e 6b n 1 2 and fn f then j fn j f each continuous linear funcn tional j on ef is given by a sequence bn n 0 1 which satisfies limn n i n i 1 and

solving complex approximation problems by semiinfinite finite optimization techniques a study on convergence we transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity 0 2 i we study the effect of disturbing the problem by replacingi by a compact subsetm i which includes as special case the discrete case wherem consists only of finitely many points we introduce a measure for the deviation ofm fromi and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with 0 to the minimal distance of the undisturbed problem which is a generalization of a result by streit and nuttall we also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear there are some graphical representations of best complex approximations computed with the described method

univalent functions in h _ h d etude des fonctions univalentes dans h h d solutions d une equation aux derivees partielles non lineaire elliptique

the tate conjecture for k3 surfaces in odd characteristic we show that the classical kuga satake construction gives rise away from characteristic 2 to an open immersion from the moduli of primitively polarized k3 surfaces of any fixed degree to a certain regular integral model for a shimura variety of orthogonal type this allows us to attach to every polarized k3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety in turn this reduces the tate conjecture for k3 surfaces over finitely generated fields of odd characteristic to a version of the tate conjecture for certain endomorphisms on the attached kuga satake abelian variety which we prove as a by product of our methods we also show that the moduli stack of primitively polarized k3 surfaces of degree 2d is quasi projective and when d is not divisible by p 2 is geometrically irreducible in characteristic p we indicate how the same method applies to prove the tate conjecture for co dimension 2 cycles on cubic fourfolds

the nitsche conjecture the nitsche conjecture is deeply rooted in the theory of doubly connected minimal surfaces however it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli h a a r r onto a r r a in the early 1960s while attempting to describe all doubly connected minimal graphs over a given annulus a j c c nitsche observed that their conformal modulus cannot be too large then he conjectured in terms of isothermal coordinates even more a harmonic homeomorphism h a onto a exists if an only if r r 1 2 r r r r this fascinating and engaging problem remained open for almost a half of a century in the present paper we provide among further generalizations an affirmative answer to his conjecture

one pointed univalent logharmonic mappings abstract a complete characterization of univalent logharmonic mappings from the exterior of the unit disku onto c 1 is given

c integrable nonlinear partial differentiation equations i a technique to generate c integrable nonlinear partial differentiation equations pdes i e nonlinear pdes linearizable by an appropriate change of variables is reported and several examples of such pdes are exhibited

a p adic nonabelian criterion for good reduction of curves let k be a complete discrete valuation field of characteristic 0 with valuation ring ok and perfect residue field k of positive characteristic p we prove that a proper and smooth curve over k admitting a semistable model over ok has good reduction if and only if its unipotent p adic etale fundamental group is crystalline

the burgers equation on the semiline with general boundary conditions at the origin a technique is given to solve the initial boundary value problem for the burgers equation ut x t uxx x t 2 ux x t u x t on the semiline 0 x with the general boundary condition at the origin h u 0 t ux 0 t t 0 here to solve means to reduce to an equation in one variable only this equation is generally nonlinear and integrodifferent ial it comes in several equivalent avatars which contain nontrivially a free parameter whose value can be assigned arbitrarily since the solution of the equation is independent of it in the special case when h y z t a t y b t z y2 f t which is the case relevant for most applications the equations reduce to linear integral equations of volterra type which can in fact be solved by quadratures if a t f t c1 and b t f t c2 are time independent

hypersurfaces in projective schemes and a moving lemma let x s be a quasi projective morphism over an affine base we develop in this article a technique for proving the existence of closed subschemes h s of x s with various favorable properties we offer several applications of this technique including the existence of finite quasi sections in certain projective morphisms and the existence of hypersurfaces in x s containing a given closed subscheme c and intersecting properly a closed set f assume now that the base s is the spectrum of a ring r such that for any finite morphism z s pic z is a torsion group this condition is satisfied if r is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field we prove in this context a moving lemma pertaining to horizontal 1 cycles on a regular scheme x quasi projective and flat over s we also show the existence of a finite surjective s morphism to psd for any scheme x projective over s when x s has all its fibers of a fixed dimension d

good reduction criterion for k3 surfaces we prove a neron ogg shafarevich type criterion for good reduction of k3 surfaces which states that a k3 surface over a complete discrete valuation field has potential good reduction if its l adic cohomology group is unramified we also prove a p adic version of the criterion these are analogues of the criteria for good reduction of abelian varieties the model of the surface will be in general not a scheme but an algebraic space as a corollary of the criterion we obtain the surjectivity of the period map of k3 surfaces in positive characteristic

burger s equation in the quarter plane a formula for the weak limit on etudie la limite pour e 0 des solutions du probleme aux valeurs limites et initiales u t e ue 2 2 x eu xx e 2 ue x 0 u 0 x ue 0 t u b t dans d x t x 0 t 0 on suppose que u 0 x l 1 0 et u b c 0

univalent harmonic functions several families of complex valued univalent harmonic functions are studied from the point of view of geometric function theory one class consists of mappings of a simply connected domain onto an infinite horizontal strip with a normalization at the origin extreme points and support points are determined as well as sharp estimates for fourier coefficients and distortion theorems next mappings in izi 1 are considered that leave infinity fixed some coefficient estimates distortion theorems and covering properties are obtained for such mappings with real boundary values many extremal problems are solved explicitly

hom stacks and restriction of scalars fix an algebraic space s and let x and y be separated artin stacks of finite presentation over s with finite diagonals over s we define a stack homs x y classifying morphisms between x and y assume that x is proper and flat over s and assume fppf locally on s that there exists a finite finitely presented flat cover z x with z an algebraic space then we show that homs x y is an artin stack with quasi compact and separated diagonal