nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

d split sequences and derived equivalences abstract in this paper we introduce the so called d split sequences and show that each d split sequence gives rise to a derived equivalence via a tilting module in particular we obtain derived equivalences from auslander reiten sequences via bb tilting modules or from n almost split sequences via n bb tilting modules and auslander reiten triangles further we recover n almost split sequences from n bb tilting modules over n auslander algebras

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

l 2 serre duality on domains in complex manifolds and applications an l 2 version of the serre duality on domains in complex manifolds involving duality of hilbert space realizations of the operator is established this duality is used to study the solution of the equation with prescribed support applications are given to closed extension of forms as well to bochner hartogs type extension of cr functions comp e with the quotient topology where we endow spaces of compactly supported forms with the natural inductive limit topology in fact condition that the two maps in 1 have closed range is also necessary for the duality theorem to hold see 9 also see 26 27 28 for further results of this type serre s original proof 33 is based on sheaf theory and the theory of topological vector spaces a different approach to this result in the case when is a compact complex manifold was given by kodaira using hodge theory see 23 or 7 in this note we extend kodaira s method to non compact hermitian manifolds to obtain an l 2 analog of the serre duality special cases of serre duality using l 2 methods have appeared before in many contexts see 25 or 11 theorem 5 1 7 and 19 20 for example our treatment aims to streamline and systematize these results with emphasis on non compact manifolds and point out its close relation with the choice of l 2 realizations of the cauchy riemann operator or alternatively choice of boundary conditions for the l 2 realizations of the formal complex laplacian e e e e the l 2 duality can be interpreted in many ways at one level it is a duality between the standard laplacian with neumann boundary conditions and thec laplacian with dual dirichlet boundary conditions using another approach results regarding solution of the equation in l 2 can be converted to statements regarding the solution of the c equation this leads to a solution of the cauchy problem i e solution of the equation with prescribed support at the heart of the matter lies the existence of

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

a nonequilibrium statistical model of spectrally truncated burgers hopf dynamics exact spectral truncations of the unforced inviscid burgers hopf equation are hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior for this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies in the present paper the burgers hopf dynamics truncated to n fourier modes is treated by a new statistical model reduction technique and a closed system of evolution equations for the mean values of the m lowest modes is derived for m n in the reduced model the m mode macrostates are associated with trial probability densities on the phase space of the n mode microstates and a cost functional is introduced to quantify the lack of fit of paths of these densities to the liouville equation the best fit macrodynamics is obtained by minimizing the cost functional over paths and the equations governing the closure are then derived from hamilton jacobi theory the resulting reduced equations have a fractional diffusion and modified nonlinear interactions and the explicit form of both are determined up to a single closure parameter the accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles and the predicted behavior is found to be well represented by the reduced equations 2014 wiley periodicals inc

optimal thermalization in a shell model of homogeneous turbulence we investigate the turbulence induced dissipation of the large scales in a statistically homogeneous flow using an optimal closure which one of us bt has recently exposed in the context of hamiltonian dynamics this statistical closure employs a gaussian model for the turbulent scales with corresponding vanishing third cumulant and yet it captures an intrinsic damping the key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies most easily grasped when one works directly with the liouville equation rather than the cumulant hierarchy we focus on a simple problem for which the optimal closure can be fully and exactly worked out the relaxation arbitrarily far from equilibrium of a single energy shell towards gibbs equilibrium in an inviscid shell model of 3d turbulence the predictions of the optimal closure are validated against dns and contrasted with those derived from edqnm closure

a boussinesq system for two way propagation of interfacial waves the theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion a boussinesq system for two way propagation of interfacial waves in a rigid lid configuration is derived in most cases the nonlinearity is quadratic however when the square of the depth ratio is close to the density ratio the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered the propagation as well as the collision of solitary waves and or fronts is studied numerically

lower bounds on ricci curvature and quantitative behavior of singular sets let yn denote the gromov hausdorff limit m n i stackrel d mathrm gh longrightarrow y n of v noncollapsed riemannian manifolds with mathrm ric m n i geq n 1 the singular set mathcal s subset y has a stratification mathcal s 0 subset mathcal s 1 subset cdots subset mathcal s where y in mathcal s k if no tangent cone at y splits off a factor k 1 isometrically here we define for all 0 0 r 1 the k th effective singular stratum mathcal s k eta r satisfying bigcup eta bigcap r mathcal s k eta r mathcal s k sharpening the known hausdorff dimension bound dim mathcal s k leq k we prove that for all y the volume of the r tubular neighborhood of mathcal s k eta r satisfies mathrm vol t r mathcal s k eta r cap b frac 1 2 y leq c n mathrm v eta r n k eta the proof involves a quantitative differentiation argument this result has applications to einstein manifolds let mathcal b r denote the set of points at which the c2 harmonic radius is r if also the m n i are kahler einstein with l2 curvature bound rm l 2 leq c then mathrm vol mathcal b r cap b frac 1 2 y leq c n mathrm v c r 4 for all y in the kahler einstein case without assuming any integral curvature bound on the m n i we obtain a slightly weaker volume bound on mathcal b r which yields an a priori lp curvature bound for all p 2 the methodology developed in this paper is new and is applicable in many other contexts these include harmonic maps minimal hypersurfaces mean curvature flow and critical sets of solutions to elliptic equations

on the stability of the unsmoothed fourier method for hyperbolic equations it has been a long open question whether the pseudospectral fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs in this work we answer this question with a detailed stability analysis of prototype cases of the fourier method we show that due to weighted l 2 stability the n degree fourier solution is algebraically stable in the sense that its l 2 amplification does not exceed o n yet the fourier method is weakly l 2 unstable in the sense that it does experience such o n amplification the exact mechanism of this weak instability is due the aliasing phenomenon which is responsible for an o n amplification of the fourier modes at the boundaries of the computed spectrum two practical conclusions emerge from our discussion first the fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon otherwise the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones second independent of whether smoothing was used or not the small scale information contained in the highest modes of the fourier solution will be destroyed by their o n amplification happily with enough resolution nothing worse can happen

a hierarchy of poisson brackets in non equilibrium thermodynamics abstract reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients an energy functional and a poisson bracket the goal of this paper is to elucidate how the poisson brackets can be constructed and what additional features we also gain by the construction in particular the poisson brackets governing reversible evolution in one particle kinetic theory kinetic theory of binary mixtures binary fluid mixtures classical irreversible thermodynamics and classical hydrodynamics are derived from liouville equation although the construction is quite natural a few examples where it does not work are included e g the bbgky hierarchy finally a new infinite grand canonical hierarchy of poisson brackets is proposed which leads to poisson brackets expressing non local phenomena such as turbulent motion or evolution of polymeric fluids eventually lie poisson structures standing behind some of the brackets are identified

global solvability and regularity for on an annulus between two weakly pseudoconvex domains let m1 and m2 be two bounded pseudo convex domains in c with smooth boundaries such that m1 c m2 we consider the cauchy riemann operators 8 on the annulus m m2 m1 the main result of this paper is the following given a a closed p q form a 0 3 a has l2 coefficients and satisfies 0 2 and a is orthogonal to a finite dimensional space i e the harmonic p q forms then there exists a p q 1 form u such that 0 1 holds furthermore if a is smooth up to the boundary of m then we can find a u that is smooth up to the boundary also and u satisfies 0 1 theorems 1 2 and 3 our method is to use the a neumann problem with weights which was used by hormander 3 and kohn 4 to solve the equation 0 1 on weakly pseudo convex domains the a neumann problem was a method to solve the equation a with solutions smooth up to the boundary if one can show that the subelliptic estimate holds for the a neumann problem then one can conclude that the harmonic forms are finite dimensional and one can solve 0 1 provided a has l2 coefficients and satisfies 0 2 and a is orthogonal to the harmonic space one can find a solution u received by the editors october 26 1984 1980 mathematics subject classification primary 35b45 35b65 c1985 american mathematical society 0002 9947 85 1 00 25 per page

elementary derivation of nonlinear transport equations from statistical mechanics exact closed nonlinear transport equations for a set of macroscopic variablesa are derived from classical statistical mechanics the derivation involves only simple manipulations of the liouville equation and makes no use of projection operators or graphical expansions it is based on the chapman enskog idea of separating the distribution function into a constrained equilibrium part obtained from information theory and a small remainder the resulting exact transport equations involve time convolutions over the past history of botha t anda t however if the variablesa provide a complete macroscopic description the equations may be simplified this is accomplished by a systematic expansion procedure of chapman enskog type in which the small parameter is the natural parameter of slowness relevant to the problem when carried out to second order this expansion leads to approximate nonlinear transport equations that are local in time these equations are valid far from equilibrium they contain nonlinear i e state dependent transport coefficients given by integrals of time correlation functions in the constrained equilibrium ensemble earlier results are recovered when the equations are linearized about equilibrium as an illustrative application of the formalism an expression is derived for the nonlinear i e velocity dependent friction coefficient for a heavy particle in a bath of light particles

structure of equations of macrophysics the first and the second laws of thermodynamics form the constraints on the equations describing macrophysical phenomena it is argued in this paper that there are some additional universal constraints these constraints are caused by the hamiltonian structure of microequations previously one feature of micromotion its reversibility was used by onsager to explain the observed reciprocity relations hamiltonian structure is richer than reversibility and yields richer consequences some of these consequences are a nonlinear version of onsager s relations hamiltonian structure of reversible equations of macrophysics and quasi hamiltonian structure of irreversible equations

an optimization principle for deriving nonequilibrium statistical models of hamiltonian dynamics a general method for deriving closed reduced models of hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation given a vector of resolved variables selected to describe the macroscopic state of the system a family of quasi equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities specifically a cost function is constructed to quantify the lack of fit to the microscopic dynamics of any feasible path of densities from the statistical model it is an ensemble averaged weighted squared norm of the residual that results from submitting the path of densities to the liouville equation the path that minimizes the time integral of the cost function determines the best fit evolution of the mean resolved vector the closed reduced equations satisfied by the optimal path are derived by hamilton jacobi theory when expressed in terms of the macroscopic variables these equations have the generic structure of governing equations for nonequilibrium thermodynamics in particular the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes the adjustable closure parameters in the best fit reduced equations depend explicitly on the arbitrary weights that enter into the lack of fit cost function two particular model reductions are outlined to illustrate the general method in each example the set of weights in the optimization principle contracts into a single effective closure parameter

partial tilting modules over m replicated algebras abstract let a be a hereditary algebra over an algebraically closed field k and a m be the m replicated algebra of a given an a m module t we denote by t the number of non isomorphic indecomposable summands of t in this paper we prove that a partial tilting a m module t is a tilting a m module if and only if t a m and that every partial tilting a m module has complements as an application we deduce that the tilting quiver k a m of a m is connected moreover we investigate the number of complements to almost tilting modules over duplicated algebras

a path integral formalism for non equilibrium hamiltonian statistical systems a path integral formalism for non equilibrium systems is proposed based on a manifold of quasi equilibrium densities a generalized boltzmann principle is used to weight manifold paths with the exponential of minus the information discrepancy of a particular manifold path with respect to full liouvillean evolution the likelihood of a manifold member at a particular time is termed a consistency distribution and is analogous to a quantum wavefunction the lagrangian here is of modified generalized onsager machlup form for large times and long slow timescales the thermodynamics is of ottinger form the proposed path integral has connections with those occuring in the quantum theory of a particle in an external electromagnetic field it is however entirely of a wiener form and so practical to compute finally it is shown that providing certain reasonable conditions are met then there exists a unique steady state consistency distribution

the lp boundedness of the bergman projection for a class of bounded hartogs domains abstract we generalize the hartogs triangle to a class of bounded hartogs domains and we prove that the corresponding bergman projection is bounded on l p if and only if p is in the range 2 n n 1 2 n n 1

essential spectra of tensor product hilbert complexes and the neumann problem on product manifolds abstract we investigate tensor products of hilbert complexes in particular the essential spectrum of their laplacians it is shown that the essential spectrum of the laplacian associated to the tensor product complex is computable in terms of the spectra of the factors applications are given for the neumann problem on the product of two or more hermitian manifolds especially regarding non compactness of the associated neumann operator

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

sobolev regularity of the equation on the hartogs triangle the regularity of the overline partial problem on the domain left z 1 right left z 2 right 1 in mathbb c 2 is studied using l 2 methods estimates are obtained for the canonical solution in weighted l 2 sobolev spaces with a weight that is singular at the point 0 0 in particular the singularity of the bergman projection for the hartogs triangle is contained at the singular point and it does not propagate

hamiltonian structure and statistically relevant conserved quantities for the truncated burgers hopf equation in dynamical systems with intrinsic chaos many degrees of freedom and many conserved quantities a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes the galerkin truncation of the burgers hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments here it is established that the truncated burgers hopf model is a hamiltonian system with hamiltonian given by the integral of the third power this additional conserved quantity beyond the energy has been ignored in previous statistical mechanics studies of this equation thus the question arises of the statistical significance of the hamiltonian beyond that of the energy first an appropriate statistical theory is developed that includes both the energy and hamiltonian then a convergent monte carlo algorithm is developed for computing equilibrium statistical distributions the probability distribution of the hamiltonian on a microcanonical energy surface is studied through the monte carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the hamiltonian empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy the predictions of the theory for relevant and irrelevant values for the hamiltonian are confirmed through systematic numerical simulations for statistically relevant values of the hamiltonian these simulations show a surprising spectral tilt rather than equipartition of energy this spectral tilt is predicted and confirmed independently by monte carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt on the other hand the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the hamiltonian with excellent accuracy the results established here for the burgers hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications several interesting mathematical problems suggested by this study are mentioned in the final section 2002 wiley periodicals inc

endomorphism algebras of tilting modules over m replicated algebras abstract let a be a finite dimensional hereditary algebra over an algebraically closed field k and let a m be the m replicated algebra of a in this paper we investigate the structure properties of the endomorphism algebras of tilting modules of a m and prove that all the endomorphism algebras of tilting modules of a m can be realized as the iterated endomorphism algebras of bb tilting modules that is for each pair of basic tilting a m modules t 1 and t 2 there exists a series of finite dimensional algebras 0 1 s which are the endomorphism algebras of some basic tilting a m modules and for each i there is a bb tilting i module m i such that 0 end a m t 1 i end i 1 m i 1 for 1 i s and end a m t 2 end s m s

gorenstein projective modules and symmetric recollements we introduce compatible bimodules if m is a compatible a b bimodule then the gorenstein projective modules over algebra am0b are explicitly described and if is gorenstein then this description implies that m is compatible as an application if m is compatible then there is a left recollement of the stable category gp and if is gorenstein and ma is projective then there is a symmetric recollement of the singularity category dsgb

the transpose of modules relative to subcategories abstract let r be a left noetherian ring and s a right noetherian ring and let x be a subcategory of finitely generated left r modules and b s r a finite r s bimodule as a generalization of the auslander transpose replacing a projective presentation by an x presentation and the functor hom r r by the functor hom r b we introduce the x b transpose of an r module m admitting an x presentation for a suitable subcategory x some useful properties of x b transposes are obtained it is shown that the x b transpose shares many nice properties with the auslander transpose some known results are obtained as corollaries

the bar partial neumann problem on product domains in mathbb c n let omega omega 1 times cdots times omega n subset mathbb c n where omega j subset mathbb c is a bounded domain with smooth boundary we study the solution operator to the overline partial neumann problem for 0 1 forms on in particular we construct singular functions which describe the singular behavior of the solution as a corollary our results carry over to the overline partial neumann problem for 0 q forms despite the singularities we show that the canonical solution to the overline partial equation obtained from the neumann operator does not exhibit singularities when given smooth data

maximal rigid subcategories and cluster tilting subcategories in 2 cy categories abstract let c be a connected hom finite 2 cy triangulated category we prove that if t is a functorially finite maximal rigid subcategory of c without loops in its quiver then t is cluster tilting in particular this gives a positive answer to a conjecture proposed in 5

dispersive approximations in fluid dynamics we have compared the results of a von neumann richtmyer calculation with the weak limit of calculations performed by von neumann s original method although there are discernible differences they are surprisingly small we analyze von newmann s algorithm for solving the initial value problem for the lagrangian equations of compressible flow

wavelet methods to eliminate resonances in the galerkin truncated burgers and euler equations it is well known that solutions to the fourier galerkin truncation of the inviscid burgers equation and other hyperbolic conservation laws do not converge to the physically relevant entropy solution after the formation of the first shock this loss of convergence was recently studied in detail ray et al phys rev e 84 016301 2011 and traced back to the appearance of a spatially localized resonance phenomenon perturbing the solution in this work we propose a way to remove this resonance by filtering a wavelet representation of the galerkin truncated equations a method previously developed with a complex valued wavelet frame is applied and expanded to embrace the use of real valued orthogonal wavelet bases which we show to yield satisfactory results only under the condition of adding a safety zone in wavelet space we also apply the complex valued wavelet based method to the two dimensional euler equation problem showing that it is able to filter the resonances in this case as well

double centralizer properties dominant dimension and tilting modules double centralizer properties play a central role in many parts of algebraic lie theory soergel s double centralizer theorem relates the principal block of the bernstein gelfand gelfand category o of a semisimple complex lie algebra with the coinvariant algebra i e the cohomology algebra of the corresponding flag manifold schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic or via the quantized version in nondefining characteristic in this paper we exhibit algebraic structures behind these double centralizer properties we show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective injective or tilting modules general arguments which combine methods from ring theory qf 3 rings and dominant dimension with tools from representation theory approximations tilting modules then imply the validity of these double centralizer properties as well as new ones in contrast to the traditional proofs e g by the fundamental theorems of invariant theory no computations are necessary

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

defining an m cluster category we show that a certain orbit category considered by keller encodes the combinatorics of the m clusters of fomin and reading in a fashion similar to the way the cluster category of buan marsh reineke reiten and todorov encodes the combinatorics of the clusters of fomin and zelevinsky this allows us to give type uniform proofs of certain results of fomin and reading in the simply laced cases

weighted bergman projections on the hartogs triangle abstract we prove the l p regularity of the weighted bergman projections on the hartogs triangle where the weights are powers of the distance to the singularity at the boundary the restricted range of p is proved to be sharp by using a two weight inequality on the upper half plane with muckenhoupt weights we can consider a slightly wider class of weights

product domains multi cauchy transforms and the equation abstract solution operators for the equation u f are constructed on general product domains in c n when the factors are one dimensional the operator is a simple integral operator specific derivatives of f are integrated against iterated cauchy kernels for higher dimensional factors the solution is constructed by solving sub equations with modified data on the factors estimates in several norms are proved

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

stability and spectral convergence of fourier method for nonlinear problems on the shortcomings of the 2 3 2 3 de aliasing method the high order accuracy of fourier method makes it the method of choice in many large scale simulations we discuss here the stability of fourier method for nonlinear evolution problems focusing on the two prototypical cases of the inviscid burgers equation and the multi dimensional incompressible euler equations the fourier method for such problems with quadratic nonlinearities comes in two main flavors one is the spectral fourier method the other is the 2 3 2 3 pseudo spectral fourier method where one removes the highest 1 3 1 3 portion of the spectrum this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors two main themes are discussed in this paper first we prove that as long as the underlying exact solution has a minimal c 1 alpha c 1 spatial regularity then both the spectral and the 2 3 2 3 pseudo spectral fourier methods are stable consequently we prove their spectral convergence for smooth solutions of the inviscid burgers equation and the incompressible euler equations on the other hand we prove that after a critical time at which the underlying solution lacks sufficient smoothness then both the spectral and the 2 3 2 3 pseudo spectral fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations in particular after shock formation in inviscid burgers equation the total variation of bounded pseudo spectral fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3d incompressible euler equations the limiting fourier solution is shown to enforce l 2 l 2 energy conservation and the contrast with energy dissipating onsager solutions is reflected through spurious oscillations

the cauchy riemann equations on product domains we establish the l 2 theory for the cauchy riemann equations on product domains provided that the cauchy riemann operator has closed range on each factor we deduce regularity of the canonical solution on p 1 forms in special sobolev spaces represented as tensor products of sobolev spaces on the factors of the product this leads to regularity results for smooth data

on modules with trivial self extensions throughout this note all rings will be self basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated homomorphisms will be written on the opposite side of the scalars by mod a resp a mod we denote the category of all right resp left a modules for an artin algebra a the ordinary duality functor is denoted by d mod a 2 a mod i e d hom f with the minimal injective cogenerator i over k we are interested in the bimodules bta with the following properties

approximations and the little finitistic dimension of artinian rings let r be a ring and let simp r be a representative set of all simple right r modules denote by p the class of all modules which are finitely generated and have finite projective dimension the little finitistic dimension of r is defined by fdim r sup proj dim m m p let a b be the complete cotorsion theory cogenerated by p for each s simp r let fs xs s be a special a precover of s we prove that fdim r max proj dim xs s simp r provided that r is right artinian as a corollary we extend to right artinian rings the well known auslander reiten sufficient condition for finiteness of the little finitistic dimension

new path integral representation of the quantum mechanical propagator a new path integral representation of the quantum mechanical propagator is proposed which is even more intimately connected with classical mechanics than the feynman path integral the emergence of the semiclassical propagator is discussed in detail it is argued that those paths which actually contribute in the new path integral have a very simple relation with the classical dynamical trajectories in configuration space

action principle in nonequilibrium statistical dynamics we introduce a variational method for approximating distribution functions of dynamics with a liouville operator l in terms of a nonequilibriumaction functional for two independent left and right trial states the method is valid for deterministic or stochastic markov dynamics and for stationary or time dependent distributions a practical rayleigh ritz procedure is advanced whose inputs are a finitely parametrized ansatz for the trial states leading to a parametric action for their evolution the euler lagrange equations of the action principle are hamiltonian in form generally noncanonical this permits a simple identification of fixed points as critical points of the parametric hamiltonian we also establish a variational principle for low order statistics such as mean values and correlation functions by means of the least effective action the latter is a functional of the given variable which is positive and convex as a consequence of h older realizability inequalities its value measures the cost for a fluctuation from the average to occur and in a weak noise limit it reduces to the onsager machlup action in general the effective action is shown to arise from the nonequilibrium action functional by a constrained variation this result provides a rayleigh ritz scheme for calculating just the desired low order statistics with internal consistency checks less demanding than for the full distribution textcopyright 1996 the american physical society

a new framework for extracting coarse grained models from time series with multiscale structure in many applications it is desirable to infer coarse grained models from observational data the observed process often corresponds only to a few selected degrees of freedom of a high dimensional dynamical system with multiple time scales in this work we consider the inference problem of identifying an appropriate coarse grained model from a single time series of a multiscale system it is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback furthermore we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

behavior of the bergman projection on the diederich fornaess worm w 1 introduction in this paper we show that the bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by sobolev norms let q be a smooth bounded pseudoconvex domain in c and let p t denote the orthogonal projection from l2 f onto the bergman subspace b ff l2 f3 r with respect to the weighted norm i fll j olf z l 2 e t lzll2dv l 2 let wk t denote the sobolev space consisting of functions whose derivatives of order to k g2 on the other hand there is a large collection of results implying that for certain types of domains the unweighted bergman projection p pr preserves w k for all k 0 see fk for the strictly pseudoconvex case for results on weakly pseudoconvex domains the reader may consult ko2 ca si and the recent bstl ch as well as the references cited therein most of these results are focused on the a neumann operator rather than p see bst2 for the connection also except for kol positive results in this area are typically valid for any choice of smooth positive weight function on 0 the question of whether or not p is similarly well behaved for all weakly pseudoconvex domains has remained open for many years in this paper we show that this is not the case in fact when q is the so called worm domain of diederich and forna ss then p does not map w to w k when k zr total amount of winding this latter quantity is explained in section 4 below where the construction of the worm is reviewed and the main result is proved the proof depends on computations for a piecewise levi flat model domain depending in turn on certain one dimensional computations these are treated in sections 3 and 2 respectively section 5 contains additional remarks and questions

condition r and proper holomorphic maps between equidimensional product domains we consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex euclidean space where both source and target can be represented as cartesian products of smoothly bounded domains it is shown that such mappings extend smoothly up to the closures of the domains provided each factor of the source satisfies condition r it also shown that the number of smoothly bounded factors in the source and target must be the same and the proper holomorphic map splits as a product of proper mappings between the factor domains c 2013 elsevier inc all rights reserved

minimum dissipation principle in stationary non equilibrium states we generalize to non equilibrium states onsager s minimum dissipation principle we also interpret this principle and some previous results in terms of optimal control theory entropy production plays the role of the cost necessary to drive the system to a prescribed macroscopic configuration

generalization of the dynamical lack of fit reduction from generic to generic the lack of fit statistical reduction developed and formulated first by bruce turkington is a general method taking liouville equation for probability density detailed level and transforming it to reduced dynamics of projected quantities less detailed level in this paper the method is generalized the hamiltonian liouville equation is replaced by an arbitrary hamiltonian evolution combined with gradient dynamics generic the boltzmann entropy is replaced by an arbitrary entropy and the kinetic energy by an arbitrary energy the gradient part is a generalized gradient dynamics generated by a dissipation potential the reduced evolution of the projected state variables is shown to preserve the generic structure of the original detailed level evolution the dissipation potential is obtained by solving a hamilton jacobi equation in summary the lack of fit reduction can start with generic and obtain generic for the reduced state variables

interplay between the beale kato majda theorem and the analyticity strip method to investigate numerically the incompressible euler singularity problem decrease to zero in order to sustain a finite time singularity consistent with the bkm theorem a simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the bkm theorem with the analyticity strip method it is shown that a finite time blowup can exist only if t vanishes sufficiently fast at the singularity time in particular if a power law is assumed for t then its exponent must be greater than some critical value thus providing a new test that is applied to our 4096 3 taylor green numerical simulation our main conclusion is that the numerical results are not inconsistent with a singularity but that higher resolution studies are needed to extend the time interval on which a well resolved power law behavior of t takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly