exact relations in the theory of developed hydrodynamic turbulence exact relations of two types in the statistical theory of fully developed homogeneous isotropic turbulence in an incompressible fluid were found the relations of the first type connect two point and three point objects of the theory which are correlation functions and susceptibilities the second type of relations are the frequency sum rules which express some frequency integrals from fully dressed many point objects like vertices via corresponding bare values our approach is based on the navier stokes equation in quasi lagrangian variables and on the generating functional technique for correlation functions and susceptibilities the derivation of these relations uses no perturbation expansions and no additional assumptions this means that the relations are exact in the framework of the statistical theory of turbulence we showed that a many point scaling gives birth to the global scaling here many point scaling is the assumption that two point three point etc objects of the theory of turbulence are uniform functions in the inertial interval and may be characterized by some scaling exponents under this assumption the only global scale invariant model of fully developed turbulence suggested by kolmogorov dokl akad nauk ssr 32 19 1941 is consistent with the exact relations deduced

temporal multiscaling in hydrodynamic turbulence on the basis of the navier stokes equations we develop the high reynolds number statistical theory of different time many point spatial correlation functions of velocity differences we find that their time dependence is not scale invariant n order correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents we derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents mathrm ensuremath zeta mathrm q of the standard structure functions we propose a representation of the time dependence using the legendre transform formalism of multifractals that automatically reproduces all the newly found bridge relationships

a new penalization tool in scalar and vector optimizations abstract in this work we employ a new method to penalize scalar and vector optimization problems by means of a functional known in the literature as the directional minimal time function we show that this new approach is useful also for the strong pareto optimality and then we prove for this kind of solutions a result concerning necessary optimality conditions expressed in terms of mordukhovich generalized differentiation

the lagrange multiplier rule in set valued optimization the known lagrange multiplier rule is extended to set valued constrained optimization problems using the contingent epiderivative as differentiability notion a necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions

variational characterization of the contingent epiderivative abstract in this paper the existence of the contingent epiderivative of a set valued map is studied from a variational perspective we give a variational characterization of the ideal minimal of a weakly compact set as a consequence we characterize the existence of the contingent epiderivative in terms of an associated family of variational systems when a set valued map takes values in r n we show that these systems can be formulated in terms of the contingent epiderivatives of scalar set valued maps by applying these results we extend some existing theorems

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

locality and stability of the cascades of two dimensional turbulence we investigate and clarify the notion of locality as it pertains to the cascades of two dimensional turbulence the mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions first introduced by l vov and procaccia as a point of departure we use a revised version of the system of hypotheses that was proposed by frisch for three dimensional turbulence we show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non perturbative statistical locality we also investigate the stability conditions for both cascades we have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade for the enstrophy cascade statistical stability requires large scale dissipation and a vanishing downscale energy dissipation a careful discussion of the subtle notion of locality is given at the end of the paper

anomalous scaling in homogeneous isotropic turbulence the anomalous scaling exponents n of the longitudinal structure functions sn for homogeneous isotropic turbulence are derived from the navier stokes equations by using field theoretic methods to develop a low energy approximation in which the kolmogorov theory is shown to act effectively as a mean field theory the corrections to the kolmogorov exponents are expressed in terms of the anomalous dimensions of the composite operators which occur in the definition of sn these are calculated from the anomalous scaling of the appropriate class of nonlinear green function using an uv fixed point of the renormalization group which thereby establishes the connection with the dynamics of the turbulence the main result is an algebraic expression for n which contains no adjustable constants it is valid at orders n below g 1 where g is the fixed point coupling constant this expression is used to calculate n for orders in the range n 2 10 and the results are shown to be in good agreement with experimental data key examples being 2 0 7 3 1 and 6 1 8

exact resummations in the theory of hydrodynamic turbulence iii scenarios for anomalous scaling and intermittency elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described we focus here on the structure functions of velocity differences that satisfy inertial range scaling laws s n r sim r zeta n and the correlation of energy dissipation k epsilon epsilon r sim r mu the goal is to understand the exponents zeta n and mu from first principles in paper ii of this series it was shown that the existence of an ultraviolet scale the dissipation scale eta is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields the leading scaling exponent in this family was denoted delta the exact resummation of ladder diagrams resulted in the calculation of delta which satisfies the scaling relation delta 2 zeta 2 in this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling i e zeta n not being linear in n with the renormalization scale being the infrared outer scale of turbulence l it is shown that deviations from k41 scaling of s n r zeta n neq n 3 must appear if the correlation of dissipation is mixing i e mu 0 we derive an exact scaling relation mu 2 zeta 2 zeta 4 we present analytic expressions for zeta n for all n and discuss their relation to experimental data one surprising prediction is that the time decay constant tau n r propto r z n of s n r scales independently of n the dynamic scaling exponent z n is the same for all n order quantities z n zeta 2

dubovitskii milyutin approach in set valued optimization by exploring the ideas around the dubovitskii milyutin approach necessary optimality conditions are given for various optimality notions in set valued optimization these optimality conditions are given by using the contingent derivative and the generalized contingent epiderivative of the objective set valued map and the set valued maps defining the constraints the notions of subgradients and scalarized subgradients for set valued maps are proposed and used to state some regularity conditions

application of generalized thermostatistics to fully developed turbulence abstract we apply the formalism of non extensive statistical mechanics to fully developed turbulent flows we obtain analytical formulas for probability density functions of velocity differences depending on distance r and reynolds number r which are in very good agreement with turbulence experiments the skewness is taken into account by analysing universal contributions due to higher order correlations of the chaotic dynamics we also calculate the moment scaling exponents m which also agree well with experimental data

computing the scaling exponents in fluid turbulence from first principles the formal setup abstract we propose a scheme for the calculation from the navier stokes equations of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence the scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the euler equation it constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n as a consequence the scaling exponents are determined by solvability conditions and not from power counting it is argued that in order to achieve such a formulation one must recognize that the many point space time correlation functions are not scale invariant in their time arguments the assumption of full scale invariance leads unavoidably to kolmogorov exponents it is argued that the determination of all the scaling exponents n requires equations for infinitely many renormalized objects one can however proceed in controlled successive approximations by successive truncations of the infinite hierarchy of equations clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory to this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the nth order correlation functions and green s function in light of this important result we can safely use finite resummations to construct successive closures of the infinite hierarchy of equations this paper presents the conceptual and technical details of the scheme the analysis of the high order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with v belinicher

λ c contingent derivatives of set valued maps abstract in this paper a new notion of differentiability of set valued maps is introduced based on ordered spaces techniques c lower and c upper contingent derivatives are defined conditions for c differentiability are given furthermore we show that these concepts are suitable for the formulation of optimality conditions for set valued optimization problems with set optimization criteria

the role of directional compactness in the existence and computation of contingent epiderivatives abstract this paper underlines the role of directional compactness in the scalarization of graphical derivatives of set valued maps taking values in infinite dimensional spaces two main theorems are given the first one states the equivalence of contingent epiderivatives and w contingent epiderivatives for directionally compact maps the second main result proves a variational characterization for the contingent epiderivative of stable and directionally compact maps taking values in general image spaces extending known results in finite dimensional and reflexive banach spaces the hypotheses given are minimal as is shown by means of several examples connections of these theorems with other results of the literature are also provided

fully developed isotropic turbulence nonperturbative renormalization group formalism and fixed point solution we investigate the regime of fully developed homogeneous and isotropic turbulence of the navier stokes ns equation in the presence of a stochastic forcing using the nonperturbative functional renormalization group nprg within a simple approximation based on symmetries we obtain the fixed point solution of the nprg flow equations that corresponds to fully developed turbulence both in d 2 and 3 dimensions deviations to the dimensional scalings kolmogorov in d 3 or kraichnan batchelor in d 2 are found for the two point functions to further analyze these deviations we derive exact flow equations in the large wave number limit and show that the fixed point does not entail the usual scale invariance thereby identifying the mechanism for the emergence of intermittency within the nprg framework the purpose of this work is to provide a detailed basis for nprg studies of ns turbulence the determination of the ensuing intermittency exponents is left for future work

on the elimination of the sweeping interactions from theories of hydrodynamic turbulence in this paper we revisit the claim that the eulerian and quasi lagrangian same time correlation tensors are equal this statement allows us to transform the results of an msr quasi lagrangian statistical theory of hydrodynamic turbulence back to the eulerian representation we define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity it is shown that both the elimination of the sweeping interactions and the derivation of the 4 5 law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity the quasi lagrangian transformation on the other hand requires an even stronger homogeneity assumption which is many time rather than one time but still weaker than many time global homogeneity we argue that it is possible to relax this stronger assumption and still preserve the conclusions derived from theoretical work based on the quasi lagrangian transformation

multilocality and fusion rules on the generalized structure functions in two dimensional and three dimensional navier stokes turbulence using the fusion rules hypothesis for three dimensional and two dimensional navier stokes turbulence we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences we call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator multilocality the resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry our main result is that the fusion rules hypothesis implies both locality and multilocality in both the ir and uv limits for the downscale energy cascade of three dimensional navier stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two dimensional navier stokes turbulence we stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two point correlation and response functions

about contingent epiderivatives abstract in this paper we analyze some questions concerning the contingent epiderivative when the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives we also investigate the relationship between the contingent derivative and the generalized epiderivative furthermore we give existence theorems with respect to daniell cones as a particular case we study set valued maps that take values in r n

optimality conditions for weak and firm efficiency in set valued optimization this paper concerns the study of weak and firm local efficiency in constrained mathematical problems governed by set valued mappings we derive optimality conditions by means of the bouligand derivative and by means of the mordukhovich coderivative as well

extended self similarity in turbulent flows we report on the existence of a hitherto undetected form of self similarity which we call extended self similarity ess ess holds at high as well as at low reynolds number and it is characterized by the same scaling exponents of the velocity differences of fully developed turbulence

epidifferentiability of the map of infima in hilbert spaces this article deals with derivatives for set valued maps that take values in ordered vector spaces in particular it concerns about the relationship between the epiderivatives of a set valued map and its associated map of infima when the image space is a real separable hilbert space ordered by an orthonormal basis by using a variational technique based on a decoupling of the ordering cone into half spaces we show that both epiderivatives coincide under certain hypothesis of compactness and stability furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set valued maps

a new approach to computing the scaling exponents in fluid turbulence from first principles in this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence starting from first principles the closure procedure is constructed to respect the fundamental rescaling symmetry of the euler equation the starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n this hierarchy was discussed in detail in a recent publication v s l vov and i procaccia physica a 1998 in press chao dyn 9707015 the scaling exponents in this set of equations cannot be found from power counting in this short paper we discuss in detail low order non trivial closures of this infinite set of equations and prove that these closures lead to the determination of the scaling exponents from solvability conditions the equations under consideration after this closure are nonlinear integro differential equations reflecting the nonlinearity of the original navier stokes equations nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations in which amplitudes are determined by fitting to the boundary conditions in the space of scales the renormalization scale that is necessary for any anomalous scaling appears at this point the holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l

computation formulas and multiplier rules for graphical derivatives in separable banach spaces abstract in this paper we present new computation formulas for the contingent epiderivative and hypoderivative of a set valued map taking values in a banach space with a shrinking schauder basis these formulas are established in terms of the fourier coefficients and in particular in terms of the derivatives of the component maps associated with the schauder basis as an application we obtain multiplier rules for vector optimization problems in terms of the derivatives of the component maps extending classical results from smooth multiobjective optimization problems

earthquakes on riemann surfaces and on measured geodesic laminations let s be a closed orientable surface of genus at least 2 we study properties of its techmuller space f s namely of the space of isotopy classes of conformal structures on s w p thurston introduced a certain compactification of f s by that he called the space of projective measured geodesic laminations he also introduced some transformations of teichmuller space called earthquakes which are intimately related to the geometry of f s a general problem is to understand which geometric properties of techmuller space subsist at infinity on thurston s boundarty in particular it is natural to ask whether earthquakes continuously extend at certain points of thurston s boundary and at precisely which points the do so this is the principal question addressed in this paper

continued fraction representation of temporal multiscaling in turbulence it was shown recently that the anomalous scaling of simultaneous correlation functions in turbulence is intimately related to the breaking of temporal scale invariance which is equivalent to the appearance of infinitely many times scales in the time dependence of time correlation functions in this paper we derive a continued fraction representation of turbulent time correlation functions which is exact and in which the multiplicity of time scales is explicit we demonstrate that this form yields precisely the same scaling laws for time derivatives and time integrals as the multi fractal representation that was used before truncating the continued fraction representation yields the best estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain finite order it is worth noting that the derivation of a continued fraction representation obtained here for a time evolution operator which is not hermitian or anti hermitian may be of independent interest

exponents of the structure functions in a low temperature helium experiment abstract we report new measurements of longitudinal structure function exponents in fully developed turbulence the measurements are performed in low temperature helium for a wide range of microscale reynolds numbers from 150 to 5040 using large velocity records up to 1010 points for the largest one i e 3 106 integral time scales exponents of the longitudinal structure function of the velocity denoted by p are determined in general up to tenth order and in one case up to twelfth order they are slightly smaller than previously reported values and it is safe to say that within the accuracy of the present measurements they do not seem to evolve significantly with the reynolds number high order values of p obtained by extrapolating the distributions of the velocity increments are determined for the largest record they are found to decrease with p above p 18

computing the scaling exponents in fluid turbulence from first principles demonstration of multiscaling we develop a consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydro dynamic turbulence starting from first principles the closure procedure is constructed to respect the fundamental rescaling symmetry of the euler equation the starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n this hierarchy was discussed in detail in a recent publication by v s l vov and i procaccia the scaling exponents in this set of equations cannot be found from power counting in this paper we present in detail the lowest non trivial closure of this infinite set of equations and prove that this closure leads to the determination of the scaling exponents from solvability conditions the equations under consideration after this closure are nonlinear integro differential equations reflecting the nonlinearity of the original navier stokes equations nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations in which amplitudes are determined by fitting to the boundary conditions in the space of scales the renormalization scale that is necessary for any anomalous scaling appears at this point the holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l we demonstrate that the solvability condition of our equations leads to non kolmogorov values of the scaling exponents n finally we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence