edge decompositions of graphs with high minimum degree abstract a fundamental theorem of wilson states that for every graph f every sufficiently large f divisible clique has an f decomposition here a graph g is f divisible if e f divides e g and the greatest common divisor of the degrees of f divides the greatest common divisor of the degrees of g and g has an f decomposition if the edges of g can be covered by edge disjoint copies of f we extend this result to graphs g which are allowed to be far from complete in particular together with a result of dross our results imply that every sufficiently large k 3 divisible graph of minimum degree at least 9 n 10 o n has a k 3 decomposition this significantly improves previous results towards the long standing conjecture of nash williams that every sufficiently large k 3 divisible graph with minimum degree at least 3 n 4 has a k 3 decomposition we also obtain the asymptotically correct minimum degree thresholds of 2 n 3 o n for the existence of a c 4 decomposition and of n 2 o n for the existence of a c 2 l decomposition where l 3 our main contribution is a general iterative absorption method which turns an approximate or fractional decomposition into an exact one in particular our results imply that in order to prove an asymptotic version of nash williams conjecture it suffices to show that every k 3 divisible graph with minimum degree at least 3 n 4 o n has an approximate k 3 decomposition

the thick thin decomposition and the bilipschitz classification of normal surface singularities we describe a natural decomposition of a normal complex surface singularity x 0 into its thick and thin parts the former is essentially metrically conical while the latter shrinks rapidly in thickness as it approaches the origin the thin part is empty if and only if the singularity is metrically conical the link of the singularity is then seifert fibered in general the thin part will not be empty in which case it always carries essential topology our decomposition has some analogy with the margulis thick thin decomposition for a negatively curved manifold however the geometric behavior is very different for example often most of the topology of a normal surface singularity is concentrated in the thin parts

variational bounds on energy dissipation in incompressible flows ii channel flow a variational principle for lower bounds on the time averaged mass flux for newtonian fluids driven by a pressure gradient in a channel is derived from the incompressible navier stokes equations when supplied with appropriate test background flow fields the variational formulation produces explicit estimates for the friction coefficient these rigorous bounds are compared with the predictions of conventional turbulence theory

the canonical sheaf of du bois singularities we prove that a cohen macaulay normal variety x has du bois singularities if and only if x g x for a log resolution x x where g is the reduced exceptional divisor of many basic theorems about du bois singularities become transparent using this characterization including the fact that cohen macaulay log canonical singularities are du bois we also give a straightforward and self contained proof that generalizations of semi log canonical singularities are du bois in the cohen macaulay case it also follows that the kodaira vanishing theorem holds for semi log canonical varieties and that cohen macaulay semi log canonical singularities are cohomologically insignificant in the sense of dolgachev

a sharp lower bound for the geometric genus and zariski multiplicity question it is well known that the geometric genus and multiplicity are two important invariants for isolated singularities in this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities in 1971 zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type this problem remains unsettled in this paper we answer positively zariski s multiplicity question for isolated hypersurface singularity if milnor number or geometric genus is small

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

the geometric genus of splice quotient singularities we prove a formula for the geometric genus of splice quotient singularities in the sense of neumann and wahl this formula enables us to compute the invariant from the resolution graph in fact it reduces the computation to that for splice quotient singularities with smaller resolution graphs we also discuss the dimension of the first cohomology groups of certain invertible sheaves on a resolution of a splice quotient singularity

fibred multilinks and singularities f overline g in this article we extend milnor s fibration theorem to the case of functions of the form f bar g with f g holomorphic defined on a complex analytic possibly singular germ x 0 we further refine this fibration theorem by looking not only at the link of f bar g 0 but also at its multi link structure which is more subtle we mostly focus on the case when x has complex dimension two our main result theorem 4 4 gives in this case the equivalence of the following three statements i the real analytic germ f bar g x p rightarrow mathbb r 2 0 has 0 as an isolated critical value ii the multilink l f cup l g is fibered and iii if pi tilde x rightarrow x is a resolution of the holomorphic germ fg x p rightarrow mathbb c 0 then for each rupture vertex j of the decorated dual graph of one has that the corresponding multiplicities of f g satisfy m f j not m g j moreover one has that if these conditions hold then the milnor le fibration psi f bar g mathcal l x backslash l f cup l g rightarrow mathbb s 1 eta of f bar g is a fibration of the multilink l f cup l g we also give a combinatorial criterium to decide whether or not the multilink l f cup l g is fibered if the meromorphic germ f g is semitame then we show that the milnor le fibration given by psi f bar g is equivalent to the usual milnor fibration given by f bar g f bar g we finish this article by discussing several realization problems

flips from 4 folds with isolated complete intersection singularities we investigate a flipping contraction g x y from a 4 fold x with at worst isolated complete intersection singularities if y has an anti bicanonical divisor bi elephant with only rational singularities then g carries an inductive structure involving a chain of blow ups which we call la torre pendente and in particular the flip exists this naturally contains reid s pagoda as an anticanonical divisor elephant and its proper transforms

on the link space of a gorenstein quasihomogeneous surface singularity let x o c c 0 be a normal quasihomogeneous surface singularity and m be its link space recall that m is a compact smooth 3 manifold obtained by interesting x with a small 2n 1 sphere centered at the origin or by dividingx x 0 by the action of the subgroup ir of it is known 8 that the analytic structure on x defines a geometric structure on m i e a complete locally homogeneous riemannian metric of finite volume on m in particular the universal cover of m is a homogeneous space it was shown by milnor 4 that in the special case where of o is given by an equation z z 0 the link space m is homogeneous itself and diffeomorphic to a coset space f g here g is a simply connected 3 dimensional lie group and 1 its discrete cocompact subgroup in this paper we prove the following generalization of this result

an asymptotic existence result on compressed sensing matrices abstract for any rational number h and all sufficiently large n we give a deterministic construction for an n h n compressed sensing matrix with l 1 t recoverability where t o n our method uses pairwise balanced designs and complex hadamard matrices in the construction of equiangular frames which we introduce as a generalisation of equiangular tight frames the method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design the l 1 t recoverability performance is specified as a simple function of the parameters of the design to obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified

topological invariance of intersection lattices of arrangements in let a l 1 l 2 l n be a line arrangement in cp 2 i e a collection of distinct lines in cp 2 let l a be the set of all intersections of elements of a partially ordered by x y y x let m a be cp 2 a where a l i 1 i n the central problem of the theory of arrangement of lines in cp 2 is the relationship between m a and l a

boundary of the milnor fiber of a newton non degenerate surface singularity abstract we give an algorithm for the description as a graph manifold of the boundary of the milnor fiber of a newton non degenerate surface singularity embedded in a germ of toric three dimensional variety starting from its local newton polyhedron this algorithm is based on a previous work by the author describing a general method for the computation of the boundary of the milnor fiber of any smoothing of a reduced non isolated singularity of complex surface

links of rational singularities l spaces and lo fundamental groups we prove that the link of a complex normal surface singularity is an l space if and only if the singularity is rational this via a result of hanselman et al taut foliations on graph manifolds 2015 arxiv 1508 0591 proving the conjecture of boyer et al math ann 356 4 1213 1245 2013 shows that a singularity link is not rational if and only if its fundamental group is left orderable if and only if it admits a coorientable taut foliation

abstract framework for the theory of statistical solutions abstract an abstract framework for the theory of statistical solutions is developed for general evolution equations extending the theory initially developed for the three dimensional incompressible navier stokes equations the motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique both concepts of statistical solution in trajectory space and in phase space are given and the corresponding results of existence of statistical solution for the associated initial value problems are proved the wide applicability of the theory is illustrated with the very incompressible navier stokes equations a reaction diffusion equation and a nonlinear wave equation all displaying the property of global existence of weak solutions without a known result of global uniqueness

characteristic classes of complex hypersurfaces abstract the milnor hirzebruch class of a locally complete intersection x in an algebraic manifold m measures the difference between the poincare dual of the hirzebruch class of the virtual tangent bundle of x and respectively the brasselet schurmann yokura homology hirzebruch class of x in this note we calculate the milnor hirzebruch class of a globally defined algebraic hypersurface x in terms of the corresponding hirzebruch invariants of vanishing cycles and singular strata in a whitney stratification of x our approach is based on schurmann s specialization property for the motivic hirzebruch class transformation of brasselet schurmann yokura the present results also yield calculations of todd chern and l type characteristic classes of hypersurfaces

fundamental groups of links of isolated singularities 2014 american mathematical society all rights reserved we study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities we prove that for every finitely presented group g there is a complex projective surface s with simple normal crossing singularities only so that the fundamental group of s is isomorphic to g we use this to construct 3 dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to g lastly we prove that a finitely presented group g is q superperfect has vanishing rational homology in dimensions 1 and 2 if and only g if is isomorphic to the fundamental group of the link of a rational 6 dimensional complex singularity

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

the topology of real suspension singularities of type f bar g z n in this article we study the topology of a family of real analytic germs f mathbb c 3 0 rightarrow mathbb c 0 with isolated critical point at the origin given by f x y z f x y overline g x y z r where f and g are holomorphic r in mathbb z and r ge 2 we describe the link l f as a graph manifold using its natural open book decomposition related to the milnor fibration of the map germ f bar g and the description of its monodromy as a quasi periodic diffeomorphism through its nielsen invariants furthermore such a germ f gives rise to a milnor fibration frac f f mathbb s 5 setminus l f rightarrow mathbb s 1 we present a join theorem which allows us to describe the homotopy type of the milnor fibre of f and we show some cases where the open book decomposition of mathbb s 5 given by the milnor fibration of f cannot come from the milnor fibration of a complex singularity in mathbb c 3

spectral hirzebruch milnor classes of singular hypersurfaces we introduce spectral hirzebruch milnor classes for singular hypersurfaces these can be identified with steenbrink spectra in the isolated singularity case and may be viewed as their global analogues in general their definition uses vanishing cycles of mixed hodge modules and the todd class transformation these are compatible with the pushforward by proper morphisms and the classes can be calculated by using resolutions of singularities formulas for hirzebruch milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1 these formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases here we use the thom sebastiani theorem for the underlying filtered d modules of vanishing cycles from which we can deduce the thom sebastiani type theorem for spectral hirzebruch milnor classes for the chern classes after specializing to y 1 we can give a relatively simple formula for the localized milnor classes which implies a new formula for the euler numbers of projective hypersurfaces using iterated hyperplane sections applications to log canonical thresholds and du bois singularities are also explained for instance the latter can be detected by using hirzebruch milnor classes in the projective hypersurface case

exponential decay of the power spectrum of turbulence the analyticity on a strip of the solutions of navier stokes equations in 2d is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field some subtleties in the application of the wiener khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow

the indices of log canonical singularities let p x be a three dimensional log canonical pair such that has only standard coefficients and that p is a center of log canonical singularities for x then we get an effective bound of the indices of these pairs and actually determine all the possible indices furthermore under certain assumptions including the log minimal model program an effective bound is also obtained in dimension n 4

nash problem for surface singularities is a topological problem abstract we address nash problem for surface singularities using wedges we give a refinement of the characterisation in a reguera lopez 2006 32 of the image of the nash map in terms of wedges our improvement consists in a characterisation of the bijectivity of the nash mapping using wedges defined over the base field which are convergent if the base field is c and whose generic arc has transverse lifting to the exceptional divisor this improves the results of m lejeune jalabert and a reguera 2008 16 for the surface case in the way to do this we find a reformulation of nash problem in terms of branched covers of normal surface singularities as a corollary of this reformulation we prove that the image of the nash mapping is characterised by the combinatorics of a resolution of the singularity or what is the same by the topology of the abstract link of the singularity in the complex analytic case using these results we prove several reductions of the nash problem the most notable being that if nash problem is true for singularities having rational homology sphere links then it is true in general

automorphism groups for quasi homogeneous gorenstein surface singularities abstract let x 0 be a non log canonical quasi homogeneous surface singularity germ and letg aut x 0 be the maximal reductive subgroup in this paper we bound the order ofg c by p p a purely topological invariant hurwitz s theorem comes out as a corollary we use the standard approach of taking the quotient space of x 0 by the action of a finite group of automorphisms acting freely onx 0

kolmogorov theory via finite time averages abstract several relations from the kolmogorov theory of fully developed three dimensional turbulence are rigorously established for finite time averages over leray hopf weak solutions of the navier stokes equations the navier stokes equations are considered with periodic boundary conditions and an external forcing term the main parameter is the grashof number associated with the forcing term the relations rigorously proved in this article include estimates for the energy dissipation rate the kolmogorov wavenumber the taylor wavenumber the reynolds number and the energy cascade process for some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone while for others such as the kolmogorov energy dissipation law and the energy cascade the estimates depend also on the grashof number as compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before

holomorphic arcs on singularities the study of formal arcs was initiated by nash in a 1967 preprint arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities the nash conjecture on the irreducible components of such arc spaces was proved for surfaces by fernandez de bobadilla and pe pereira and for toric singularities by ishii and kollar but counter examples were found in higher dimensions by ishii and kollar de fernex and johnson and kollar here we start the study of holomorphic arcs these are holomorphic maps from the closed unit disk to a complex analytic space as one expects there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs however a formal deformation of an arc is a much more local object than a holomorphic deformation thus in many cases the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many for a complex analytic space we define two variants the space of arcs and the space of short arcs and we study their connected components for short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions

positive ricci curvature on highly connected manifolds for k geq 2 let m 4k 1 be a closed 2k 2 connected manifold if k equiv 1 mod 4 assume further that m is 2k 1 parallelisable then there is a homotopy sphere sigma 4k 1 such that m sharp sigma admits a ricci positive metric this follows from a new description of these manifolds as the boundaries of explicit plumbings

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

the asymptotic behavior of plurigenera for a normal isolated singularity we study here some invariants of normal isolated singularities which look like the kodaira dimensions and thereby prepare for the possible localization of the rough classification theory in algebraic geometry let x x be a normal isolated singularity on an analytic space x of dimension n 2 kn 511er and watanabe independently defined m genera y x x and 6re x x of x x by considering the space of meromorphic m pie n forms divided by those with natural growth conditions at the singularity kn w1 in this paper we introduce a third m genus d x x me 1 and investigate the basic properties of these m genera it turns out that all these three m genera ym x x 6m x x d x x have integral growth orders in m which are denoted by xr x x x6 x x and xa x x respectively the asymptotic behavior of the above three m genera are summarized as follows

a characteristic number for links of surface singularities milnor and thurston mt define a characteristic number of a closed orientable 3 manifold m to be a real valued topological invariant r m such that if p m is defined and m is a k sheeted covering of m then p m is defined and equal to k p m this multiplicative property is clear for the euler characteristic but of course this is 0 for a 3 manifold if m has an appropriate riemannian metric e g hyperbolic or with all sectional curvatures 1 then the volume is a characteristic number milnor and thurston introduce several characteristic numbers one is defined for all 3 manifolds is positive on a fixed hyperbolic manifold and has the additional property that if m n is a general map of degree k then m i kk i n another characteristic number is the simplicial volume introduced by gromov g in this paper we introduce a characteristic number for each link of a complex surface singularity it is a nonnegative rational number is 0 only in wellunderstood cases is computable from any resolution dual graph plumbing diagram and has in addition the submultiplicative property above for degree k maps arising from morphisms of the singularities themselves this is the only characteristic number for links which we know of that can be computed from the graph we call this invariant p p since it is the negative of the self intersection number of a rational cycle on a complex surface a resolution of the singularity the definition arose from our work on the generalized miyaoka inequality for normal surfaces w3 this term appeared in the inequality another ingredient is the notion of zariski decomposition of a line bundle or divisor on a surface especially on a resolution of a surface singularity as in sakai s let x o be the germ of a normal complex surface singularity necessarily isolated with x contractible and ax m the link of x m is a closed connected orientable 3 manifold let x e x o be a good resolution hence the inverse image e u ei of o is the union of nonsingular curves intersecting transversally no three through a point the resolution dual graph

on the set of l space surgeries for links abstract we study the qualitative structure of the set ls of integral l space surgery slopes for links with two components it is known that the set of l space surgery slopes for a nontrivial l space knot is always a positive half line however already for two component torus links the set ls has a very complicated structure e g in some cases it can be unbounded from below in order to probe the geometry of this set we ask if it is bounded from below for more general l space links with two components for algebraic two component links we provide three complete characterizations for the boundedness from below one in terms of the alexander polynomial one in terms of the embedded resolution graph and one in terms of the so called h function introduced by the authors in 9 it turns out that ls is bounded from below for most algebraic links if it is unbounded from below it must contain a negative half line parallel to one of the axes we also give a sufficient condition for boundedness for arbitrary l space links

carrousel in family and break non isolated hypersurface singularities in ℂ3 we study the boundary l t of the milnor fiber for the reduced holomorphic germs f bbb c 3 0 rightarrow bbb c 0 having a non isolated singularity at 0 we prove that l t is a graph manifold by using a new technique of carrousels depending on one parameter our results enable us to compare the topology of l t and of the link of the normalization of f 1 0

heegaard floer homology and concordance bounds on the thurston norm we prove that twisted correction terms in heegaard floer homology provide lower bounds on the thurston norm of certain cohomology classes determined by the strong concordance class of a 2 component link l in s 3 we then specialise this procedure to knots in s 2 times s 1 and obtain a lower bound on their geometric winding number furthermore we produce an obstruction for a knot in s 3 to have untwisting number 1 we then provide an infinite family of null homologous knots with increasing geometric winding number on which the bound is sharp

on strong vanishing property and plurigenera of isolated singularities abstract we introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities a comparison theorem about strong vanishing property is also obtained we use zariski decomposition to give a formula of plurigenera for gorenstein singularities and use it to test whether a divisor is f nef

universal abelian covers of certain surface singularities every normal complex surface singularity with open image in new window homology sphere link has a universal abelian cover it has been conjectured by neumann and wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of splice diagram equations in this paper we introduce a neumann wahl system which is an analogue of the system of splice diagram equations and prove the following

the abel map for surface singularities i generalities and examples let x o be a complex normal surface singularity we fix one of its good resolutions widetilde x rightarrow x an effective cycle z supported on the reduced exceptional curve and any possible first chern class l in h 2 widetilde x mathbb z with these data we define the variety mathrm eca l z of those effective cartier divisors d supported on z which determine a line bundles mathcal o z d with first chern class l furthermore we consider the affine space mathrm pic l z subset h 1 mathcal o z of isomorphism classes of holomorphic line bundles with chern class l and the abel map c l z mathrm eca l z rightarrow mathrm pic l z the present manuscript develops the major properties of this map and links them with the determination of the cohomology groups h 1 z mathcal l where we might vary the analytic structure x o supported on a fixed topological type resolution graph and we also vary the possible line bundles mathcal l in mathrm pic l z the case of generic line bundles of mathrm pic l z and generic line bundles of the image of the abel map will have priority roles rewriting the abel map via laufer duality based on integration of forms on divisors we can make explicit the abel map and its tangent map the case of superisolated and weighted homogeneous singularities are exemplified with several details the theory has similar goals but rather different techniques as the theory of abel map or brill noether theory of reduced smooth projective curves

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

the multiplicity of isolated two dimensional hypersurface singularities consider an isolated two dimensional complex analytic hypersurface singularity v p a relation is given between the abstract topology of v p and the multiplicity of v p yielding an upper bound for the multiplicity this relation is a necessary condition for a gorenstein singularity to be a hypersurface

universal abelian covers of quotient cusps the quotient cusp singularities are isolated complex surface singularities that are double covered by cusp singularities we show that the universal abelian cover of such a singularity branched only at the singular point is a complete intersection cusp singularity of embedding dimension 4 this supports a general conjecture that we make about the universal abelian cover of a gorenstein singularity

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

a note on 3 manifolds and complex surface singularities this article is motivated by the original casson invariant regarded as an integral lifting of the rochlin invariant we aim to defining an integral lifting of the adams e invariant of stably framed 3 manifolds perhaps endowed with some additional structure we succeed in doing so for manifolds which are links of normal complex gorenstein smoothable singularities these manifolds are naturally equipped with a canonical mathrm su 2 frame to start we notice that the set of homotopy classes of mathrm su 2 frames on the stable tangent bundle of every closed oriented 3 manifold is canonically a mathbb z torsor then we define the widehat e invariant for the manifolds in question an integer that modulo 24 is the adams e invariant the widehat e invariant for the canonical frame equals the milnor number plus 1 so this brings a new viewpoint on the milnor number of the smoothable gorenstein surface singularities

rationally convex domains and singular lagrangian surfaces in mathbb c 2 we give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in mathbb c 2 we recall some classical obstructions and prove some deeper ones related to symplectic and contact topology we explain the close connection to lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces our proof also gives a complete characterization of lagrangian surfaces with open whitney umbrellas answering a question first posed by givental in 1986

statistical estimates for channel flows driven by a pressure gradient we present rigorous estimates for some physical quantities related to turbulent and non turbulent channel flows driven by a uniform pressure gradient such results are based on the concept of stationary statistical solutions which is related to the notion of ensemble averages for flows in statistical equilibrium we provide a lower bound estimate for the mean skin friction coefficient and improve on a previous upper bound estimate for the same quantity both estimates are derived in terms of the reynolds number we also present lower and upper bound estimates for the mean rate of energy dissipation the mean longitudinal bulk velocity in the direction of the pressure gradient and the mean kinetic energy in terms of various physical parameters in particular we obtain an upper bound related to the energy dissipation law namely that the mean rate of energy dissipation is essentially bounded by a non dimensional universal constant times the cube of the mean longitudinal bulk velocity over a characteristic macro scale length finally we investigate the scale by scale energy injection due to the pressure gradient proving an upper bound estimate for the decrease of this energy injection as the scale length decreases

kraichnan turbulence via finite time averages relations central to kraichnan s theory of fully developed two dimensional turbulence are rigorously established for finite time averages in particular we prove that if the ratio of the averages of palinstrophy to enstrophy is large then a large inertial range displaying an enstrophy cascade exists moreover if this ratio is comparable up to a logarithm to the dissipation wave number a necessary condition for turbulence then the power law for the energy spectrum until now derived only heuristically is rigorously shown to provide up to a logarithm an upper bound for the energy spectrum finally we show that deep in the dissipation range the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number the averaging times needed for these relations are bounded in terms of the generalized grashof number independent of the solution for which the time averages are taken solutions are not assumed to be on the global attractor merely within the absorbing ball an easily verified condition

numerically gorenstein surface singularities are homeomorphic to gorenstein ones consider a normal complex analytic surface singularity it is called gorenstein if the canonical line bundle is holomorphically trivial in some punctured neighborhood of the singular point and is called numerically gorenstein if this line bundle is topologically trivial the second notion depends only on the topological type of the singularity laufer proved in 1977 that given a numerically gorenstein topological type of singularity every analytical realization of it is gorenstein if and only if one has either a kleinian or a minimally elliptic topological type the question to know if any numerically gorenstein topology was realizable by some gorenstein singularity was left open we prove that this is indeed the case our method is to plumb holomorphically meromorphic 2 forms obtained by adequate pullbacks of the natural holomorphic symplectic forms on the total spaces of the canonical line bundles of complex curves more generally we show that any normal surface singularity is homeomorphic to a q gorenstein singularity whose index is equal to the smallest common denominator of the coefficients of the canonical cycle of the starting singularity

topology of smoothings of non isolated singularities of complex surfaces this thesis is dedicated to the study of the topology of smoothings of non isolated singularities of complex surfaces the question is to describe the topology of the manifold called textbf milnor fiber which appears during this process of smoothing considering the great difficulty of a description of the whole of this topology many researches have focused on the study of the textbf boundary of the milnor fiber in the case of isolated singularities it is known since the work of mumford 1961 that this boundary is a graph manifold isomorphic to the link of the singularity different results michel pichon 2003 2014 nemethi szil ard 2012 have then proved that in the case of reduced non isolated singularities ajo of surfaces the boundary of the milnor fiber is again a graph manifold while restraining to the case of a smooth total space of smoothing fern andez de bobadilla menegon neto 2014 have widened the context considering non reduced surfaces and allowing the total space to have an isolated singularity in this work we pursue the extension of this result to a larger context allowing the total space to present non isolated singularities while restraining ourselves to the study of reduced surface singularities our proof is inspired by the one of nemethi and szilard and allows us furthermore to provide a method for the computation of hsout this manifold ajo the boundary of the milnor fiber this makes possible the actual computation of a large number of examples representing a step forward in the quest for the comprehension of the manifolds that can actually appear as boundaries of milnor fibers we apply in particular the method to newton non degenerate singularities defined on 3 dimensional toric germs this is a generalization of a theorem of oka 1986 expressing the boundary of the milnor fiber in terms of the newton polyhedron of the singularity

braid monodromy and topology of plane curves in this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve

weights on cohomology invariants of singularities and dual complexes we study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size when it is the exceptional divisor of a singularity we also give bounds for the cohomology of links the invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities

on the milnor fibres of cyclic quotient singularities the oriented link of the cyclic quotient singularity xp q is orientation preserving diffeomorphic to the lens space l p q and carries the standard contact structure st lisca classified the stein fillings of l p q st up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the milnor fibres associated with the irreducible components all of them being smoothing components of the reduced miniversal space of deformations of xp q we prove this conjecture using the smoothing equations given by christophersen and stevens moreover based on a different description of the milnor fibres given by de jong and van straten we also canonically identify these fibres with lisca s fillings using these and a newly introduced additional structure the order associated with lens spaces we prove that the above milnor fibres are pairwise non diffeomorphic by diffeomorphisms which preserve the orientation and order this also implies that de jong and van straten parametrize in the same way the components of the reduced miniversal space of deformations as christophersen and stevens