asymptotic completeness for n body stark hamiltonians we prove asymptotic completeness for short and long rangen body stark hamiltonians with local singularities of at most coulomb type our results include the usual models for atoms and molecules

anomalous scaling of a passive scalar advected by the synthetic compressible flow abstract the field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a gaussian nonsolenoidal velocity field with finite correlation time in the presence of large scale anisotropy the energy spectrum of the velocity in the inertial range has the form e k k1 e and the correlation time at the wave number k scales as k 2 it is shown that depending on the values of the exponents e and the model exhibits various types of inertial range scaling regimes with nontrivial anomalous exponents explicit asymptotic expressions for the structure functions and other correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal independent of the anisotropy anomalous exponents calculated to the first order in e and in any space dimension these anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators the anomalous exponents depend explicitly on the degree of compressibility

on the equilibrium state of random walkers in random environments analytical results we study equilibrium properties of random walkers in one dimensional random environments of finite length l from an exact expression for the quenched average of the free energy we derive analytical results for all cumulants and all renyi entropies of the equilibrium distribution in contrast to the finite variance of a typical non equilibrium distribution in the unbiased situation we find that in equilibrium the disorder averaged variance diverges with the size of the system as

zero beta function for a model of diffusion in potential random field a rigorous perturbative proof is given of both the renormalisability of the field theoretic model of diffusion in potential random field and the triviality of the renormalisation group beta function

number of nonzero entries of s2ns matrices and matrices with signed generalized inverses abstract a real matrix a has a signed generalized inverse or signed gi if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a the notion of matrices having signed gi s is a generalization of the well known notion of strong sns matrices or s 2 ns matrices sharp bounds and characterization of equality for the number of nonzero entries of s 2 ns matrices of order n are given then sharp bounds and characterization of equality for the number of nonzero entries of m n matrices with signed gi s are given

spectral theory for 1 body stark operators abstract we investigate spectral theory for a one body stark hamiltonian under minimum regularity and decay conditions on the potential actually allowing sub linear growth at infinity our results include rellich s theorem the limiting absorption principle radiation condition bounds and sommerfeld s uniqueness and most of these are stated and proved in sharp form employing besov type spaces for the proofs we adopt a commutator scheme by ito skibsted 13 a feature of the paper is a special choice of an escape function related to parabolic coordinates which conforms well with classical mechanics for the stark hamiltonian the whole setting of the paper such as the conjugate operator and the besov type spaces is generated by this single escape function we apply our results in the sequel paper 5

on the digraphs of sign solvable linear systems abstract a linear system of equations ax b is sign solvable if it is solvable and both its solvability and the sign pattern of its solution vector x are uniquely determined by the sign patterns of a and b it was proved in l bassett j maybee j quirk economitrica 36 1968 544 563 r a bruali b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 that the sign solvability of a linear system ax b in standard form and with a square coefficient matrix a can be described in terms of the signed digraph s a of the matrix a with a specified vertex subset depending on the sign pattern of the vector b in this paper we define the notion of w sign solvable signed digraphs which represent precisely the sign solvable linear systems in standard form we give necessary and sufficient conditions for the underlying unsigned digraphs of w sign solvable signed digraphs in terms of forbidden subdigraphs we also give several necessary and sufficient conditions for strongly connected w sign solvable signed digraphs

looking for nonnegative solutions of a leontief dynamic model let a be an n n nonnegative real matrix we know that a is monotone if it has a nonnegative inverse moreover if a has an appropriate generalized inverse then we can obtain nonnegative solutions for the linear system ax b where b 0 in this paper using properties about nonnegatives matrices we present sufficient conditions in order that the leontief price model i f p k 1 1 r k fp k gp0 k 1 a0 where f i at 1bt and g i at 1 have nonnegative solutions

on toy ageing we consider the dynamics of a simple one dimensional model and we discuss the phenomenon of ageing i e the strong dependence of the dynamical correlation functions over the waiting time our model is the so called random random walk the toy model of a directed polymer evolving in a random medium

some block matrices with signed drazin inverses abstract the sign pattern of a real matrix m is the 0 1 1 matrix obtained from m by replacing each entry by its sign let q m be the set of real matrices with the same sign pattern as m for any m q m if the drazin inverses of m and m have the same sign pattern then m is said to have signed drazin inverse in this paper we give a complete characterization for a class of anti triangular matrices with signed drazin inverse and present a complete characterization for sign symmetric bipartite matrices with signed drazin inverse

the solution of a problem on matrices having signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a in this paper we solve a problem proposed in r a brualdi b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 b l shader siam j matrix anal appl 16 1995 1056 about the characterizations of the matrices with a special lower triangular blocked form to have a signed generalized inverse using this characterization and the fact that every matrix which has a signed generalized inverse and full column term rank and no zero rows is permutation equivalent to a matrix of this form we give two algorithms for determining whether or not a matrix with full column term rank or a general matrix has a signed generalized inverse

a generalization of the moore penrose inverse related to matrix subspaces of c n m a natural generalization of the classical moore penrose inverse is presented the so called s moore penrose inverse of an mxn complex matrix a denoted by a s is defined for any linear subspace s of the matrix vector space c n x m the s moore penrose inverse a s is characterized using either the singular value decomposition or for the full rank case the orthogonal complements with respect to the frobenius inner product these results are applied to the preconditioning of linear systems based on frobenius norm minimization and to the linearly constrained linear least squares problem

passive advection of a vector field anisotropy finite correlation time exact solution and logarithmic corrections to ordinary scaling in this work we study the generalization of the problem considered in phys rev e 91 013002 2015 to the case of finite correlation time of the environment velocity field the model describes a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow inertial range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian with finite correlation time and preassigned pair correlation function due to the presence of distinguished direction n all the multiloop diagrams in this model vanish so that the results obtained are exact the inertial range behavior of the model is described by two regimes the limits of vanishing or infinite correlation time that correspond to the two nontrivial fixed points of the rg equations their stability depends on the relation between the exponents in the energy spectrum e k 1 and the dispersion law k 2 in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale l

structure of a nonnegative regular matrix and its generalized inverses abstract a nonnegative matrix is called regular if it admits a nonnegative generalized inverse the structure of such matrices has been studied by several authors if a is a nonnegative regular matrix then we obtain a complete description of all nonnegative generalized inverses of a in particular it is shown that if a is a nonnegative regular matrix with no zero row or column then the zero nonzero pattern of any nonnegative generalized inverse of a is dominated by that of a t the transpose of a we also obtain the structure of nonnegative matrices which admit nonnegative least squares and minimum norm generalized inverses

matrices with signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a in this paper we give complete characterizations of the m n matrices a which have signed generalized inverses for both cases a n and a where a is the term rank of a and without loss of generality we assume n m and thus solve a problem proposed in b l shader siam j matrix anal appl 16 1995 1056 using these characterizations we are also able to show that the property of having a signed generalized inverse for a matrix a is inherited by all the submatrices b of a with b a and is also inherited by all those matrices a 1 with a 1 a which can be obtained from a by replacing some nonzero entries of a by zero we also consider several special cases of a problem proposed in r a brualdi b l shader matrices of sign solvable linear systems cambridge university press cambridge ma 1995 b l shader siam j matrix anal appl 16 1995 1056 about the characterization of the matrices in a special triangular block form to have signed generalized inverses

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

on the nonsingularity of matrices with certain sign patterns abstract a new nonsingularity criterion for an n n real matrix a based on sign distribution and some conditions on the elements is given this work was motivated by the solution of poisson s equation for doubly connected regions we consider a matrix a arising in the above application to demonstrate the theory

anomalous scaling in the n point functions of a passive scalar a recent analysis of the four point correlation function of the passive scalar advected by a time decorrelated random flow is extended to the n point case it is shown that all stationary state inertial range correlations are dominated by homogeneous zero modes of singular operators describing their evolution we compute analytically the zero modes governing the n point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the two point function of the advecting velocity field the implications of these calculations for the dissipation correlations are discussed textcopyright 1996 the american physical society

on digraphs and forbidden configurations of strong sign nonsingular matrices a square real matrix a called a strong sign nonsingular matrix or s2ns matrix if all matrices with the same sign pattern as a nonsingular and the inverses of these matrices all have the same sign pattern a digraph which is the underlying digraph of the signed digraph of an s2ns matrix with a negative main diagonal is called an s2ns digraph in 9 thomassen gave a characterization of strongly connected s2ns digraphs in terms of the forbidden subdigraphs in 2 brualdi and shader constructed minimal forbidden configurations for s2ns digraphs for the general cases where the digraphs considered are not necessarily strongly connected they also proposed the problem about the existence of new minimal forbidden configurations other than those found in 2 9 in this paper we construct infinitely many new basic minimal forbidden configurations and thus obtain the answer this problem we also obtain several necessary conditions for minimal forbidden configurations and give a generalization of thomassen s theorem

sign determinancy in lu factorization of p matrices abstract for an n by n p matrix a a ij having lu factorization a lu with u u ij we determine combinatorial circumstances for which u ij is unambiguously signed for a given pair i j or all such pairs and give corresponding results for l a qualitative p matrix is a sign nonsingular matrix with all diagonal entries positive for such a matrix a we give additional sufficient conditions for an entry u ij or the matrix u to be unambiguous if a is a qualitative p matrix with a 1 unambiguously signed we prove that the matrices l u l 1 and u 1 are all unambiguous

minimum maximum rank of sign pattern tensors and sign nonsingular tensors abstract in this paper we define the sign pattern tenors minimum maximum rank of sign pattern tenors term rank of tensors and sign nonsingular tensors the necessity and sufficiency for the minimum rank of sign pattern tenors to be 1 is given we show that the maximum rank of a sign pattern tensor is not less than the term rank and the minimum rank of the sign pattern of a sign nonsingular tensor is not less than its dimension we get some characterizations of tensors having sign left or sign right inverses

asymptotic completeness for long range many particle systems with stark effect ii we prove the existence and the asymptotic completeness of the dollard type modified wave operators for many particle stark hamiltonians with long range potentials

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

anomalous scaling nonlocality and anisotropy in a model of the passively advected vector field a model of the passive vector quantity advected by the gaussian velocity field with the covariance ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon is studied the effects of pressure and large scale anisotropy are discussed the inertial range behavior of the pair correlation function is described by an infinite family of scaling exponents which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques the exponents are organized in a hierarchical order according to their degree of anisotropy with the spectrum unbounded from above and the leading minimal exponent coming from the isotropic sector this picture extends to higher order correlation functions like in the scalar model the second order structure function appears nonanomalous and is described by the simple dimensional exponent s 2 ensuremath propto r 2 ensuremath ensuremath varepsilon for the higher order structure functions s 2n ensuremath propto r n 2 ensuremath ensuremath varepsilon ensuremath delta n the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of dangerous composite operators whose negative critical dimensions determine the anomalous exponents ensuremath delta n l0 a close formal resemblance of the model with the stirred navier stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper using the renormalization group the anomalous exponents are calculated in the o ensuremath varepsilon approximation in large d dimensions for the even structure functions up to the twelfth order

when the sign pattern of a square matrix determines uniquely the sign pattern of its inverse abstract we give a precise characterization in terms of parity digraphs of those square matrices a such that for every matrix b with the same sign pattern as a b 1 exists and has the same sign pattern as a 1

calculation of the anomalous exponents in the rapid change model of passive scalar advection to order epsilon 3 the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the gaussian velocity field with zero mean and correlation function approximately equal to delta t t k d epsilon inertial range anomalous exponents identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients are calculated within the epsilon expansion to order epsilon 3 three loop approximation including the exponents in anisotropic sectors the main goal of the paper is to give the complete derivation of this third order result and to present and explain in detail the corresponding calculational techniques the character and convergence properties of the epsilon expansion are discussed the improved inverse epsilon expansion is proposed and the comparison with the existing nonperturbative results is given

rellich s theorem for spherically symmetric repulsive hamiltonians for spherically symmetric repulsive hamiltonians we prove rellich s theorem or identify the largest weighted space of agmon hormander type where the generalized eigenfunctions are absent the proof is intensively dependent on commutator arguments our novelty here is a use of conjugate operator associated with some radial flow not with dilations and not with translations our method is simple and elementary and does not employ any advanced tools such as the operational calculus or the fourier analysis

anomalous scaling of passive scalar fields advected by the navier stokes velocity ensemble effects of strong compressibility and large scale anisotropy the field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities the density and the tracer fields advected by a random turbulent velocity field the latter is governed by the navier stokes equation for compressible fluid subject to external random force with the covariance t t k 4 d y where d is the dimension of space and y is an arbitrary exponent the original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models the corresponding renormalization group equations possess infrared attractive fixed points it is shown that various correlation functions of the scalar field its powers and gradients demonstrate anomalous scaling behavior in the inertial convective range already for small values of y the corresponding anomalous exponents identified with scaling critical dimensions of certain composite fields operators in the quantum field terminology can be systematically calculated as series in y the practical calculation is performed in the leading one loop approximation including exponents in anisotropic contributions it should be emphasized that in contrast to gaussian ensembles with finite correlation time the model and the perturbation theory presented here are manifestly galilean covariant the validity of the one loop approximation and comparison with gaussian models are briefly discussed

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

on the interpretation of 1 f noise we propose a model of 1 f noise based on a random walk in a random potential numerical support for the model is given and physical applicability discussed

a non wiener random walk in a two dimensional bernoulli environment for a degenerate random walk in a 2d bernoulli environment without local traps computer results show a non wiener behavior for a better exploitation of the memory the analysis is based on the statistics of the first exit time from a square

strong forms of nonsingularity abstract motivated by certain views of the sign nonsingular matrices we begin study of several classes of nonsingular matrices naturally intermediate between sign nonsingular matrices and the ordinary nonsingular matrices these all involve hadamard products that allow constrained changes in the magnitudes but not the signs of entries or the selection of patterns of subblocks among a wide variety of results is the fact that the nonsingular matrices a for which a a 1 t is doubly stochastic arise as one of our strong forms of nonsingularity

inverse closed ray nonsingular cones of matrices abstract a description is given of those ray patterns which will be called inverse closed ray nonsingular of complex matrices that contain invertible matrices only and are closed under inversion here two n n matrices are said to belong to the same ray pattern if in each position either the entries of both matrices are zeros or both entries are nonzero and their quotient is positive possible jordan forms of matrices in the inverse closed ray nonsingular ray patterns are characterized

inconsistent fuzzy linear systems the mxn inconsistent fuzzy linear system is studied the fuzzy least squares solution and therefore the weak fuzzy least squares solution to the fuzzy system are expressed by using the generalized inverses of the coefficient matrix the existence of the strong fuzzy least squares solution to the fuzzy system is also discussed some examples are presented to illustrate the theory

ray pattern of drazin inverses of some matrices abstract let q a be the set of complex matrices with the same ray pattern as a for any a q a if the drazin inverses of a and a have the same ray pattern then a is said to have ray drazin inverse in this paper we give some properties on matrices whose drazin inverses or m p inverses are ray unique and use these properties to obtain necessary and sufficient conditions for some block matrices to have ray drazin inverses

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

rigorous renormalisation group and disordered systems abstract we consider random walks in a non symmetric random environment we report a recent result based on a renormalisation group approach showing that for d 62 2 these walks are diffusive when the disorder is weak

the moore penrose inverse of matrices with an acyclic bipartite graph abstract the moore penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs for such a matrix the sign of every entry of the moore penrose inverse is shown to be determined uniquely by the signs of the matrix entries i e the matrix has a signed generalized inverse necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative moore penrose inverse nearly reducible matrices are proved to contain no submatrix having two or more diagonals implying that a nearly reducible matrix has a signed generalized inverse furthermore it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix

sign properties of metzler matrices with applications abstract several results about sign properties of metzler matrices are obtained it is first established that checking the sign stability of a metzler sign matrix can be either characterized in terms of the hurwitz stability of the unit sign matrix in the corresponding qualitative class or in terms the negativity of the diagonal elements of the metzler sign matrix and the acyclicity of the associated directed graph similar results are obtained for the case of metzler block matrices and metzler mixed matrices the latter being a class of metzler matrices containing both sign and real type entries the problem of assessing the sign stability of the convex hull of a finite and summable family of metzler matrices is also solved and a necessary and sufficient condition for the existence of common lyapunov functions for all the matrices in the convex hull is obtained the concept of sign stability is then generalized to the concept of ker b sign stability a problem that arises in the analysis of certain jump markov processes a sufficient condition for the ker b sign stability of metzler sign matrices is obtained and formulated using inverses of sign matrices and the concept of l matrices several applications of the results are discussed in the last section

on local isotropy of passive scalars in turbulent shear flows an assessment of local isotropy and universality in high reynolds number turbulent flows is presented the emphasis is on the behaviour of passive scalar fields advected by turbulence but a brief review of the relevant facts is given for the turbulent motion itself experiments suggest that local isotropy is not a natural concept for scalars in shear flows except perhaps at such extreme reynolds numbers that are of no practical relevance on earth yet some type of scaling exists even at moderate reynolds numbers the relation between these two observations is a theme of this paper

conditional s matrices conditional s matrices and conditional s matrices abbreviated cs matrices and cs matrices were first introduced and studied in math comput modelling 17 1993 141 matrices of sign solvable linear systems cambridge university press 1995 they aregeneralizations of the well known s matrices and s matrices cs matrices and cs matrices play an important role in the study of conditionally sign solvable linear systems we study various properties and recognition criteria of cs matrices we also study several special classes of cs matrices such as square cs matrices barely cs matrices bcs matrices and maximal cs matrices

on maximal sign nonsingular matrices abstract we derive several new results on maximal sign nonsingular sns matrices these include two methods for constructing new sns matrices from smaller sns matrices the results are used to produce a matrix definition for the c 4 cockades and to introduce new families of sns matrices not previously defined all of our results were deduced with the help of tables of maximal irreducible sns matrices of orders three through nine which we previously computed the interesting matrices in these tables for orders three through eight are reproduced at the end of an appendix in which we describe the computations

stark effect in multielectron systems non existence of bound states we prove non existence of bound states for a class ofn body systems in homogeneous electric fields this class includes atoms and born oppenheimer molecules this result in conjunction with a stability result of hs implies existence of resonances for such systems

regular matrices and their strong preservers over semirings an m n matrix a over a semiring s is called regular if there is an n m matrix g over s such that aga a we study the problem of characterizing those linear operators t on the matrices over a semiring such that t x is regular if and only if x is complete characterizations are obtained for many semirings including the boolean algebra the nonnegative reals the nonnegative integers and the fuzzy scalars

exact calculation of the anomalous dimension of the diffusion coefficient for a model of a random walk in a random potential renormalisation of a model of classical diffusion in a random potential is analysed it is shown that at two dimensions the one loop expression for the anomalous dimension of the diffusion coefficient is perturbatively exact leading to the coupling dependent value 1 nu 2 g 4 pi of the exponent nu

anomalous scaling regimes of a passive scalar advected by the synthetic velocity field the field theoretic renormalization group rg is applied to the problem of a passive scalar advected by the gaussian self similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient the energy spectrum in the inertial range has the form e k ensuremath propto k 1 ensuremath ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta it is shown that depending on the values of the exponents ensuremath varepsilon and ensuremath eta the model in the inertial convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the rg equations diffusive type regimes for which the advection can be treated within ordinary perturbation theory and three nontrivial convection type regimes for which the correlation functions exhibit anomalous scaling behavior the explicit asymptotic expressions for the structure functions and other correlation functions are obtained the anomalous exponents determined by the scaling dimensions of the scalar gradients are calculated to the first order in ensuremath varepsilon and ensuremath eta in any space dimension for the first nontrivial regime the anomalous exponents are the same as in the rapid change version of the model for the second they are the same as in the model with time independent frozen velocity field in these regimes the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator for the last regime the exponents are nonuniversal they can depend also on the amplitudes however the nonuniversality can reveal itself only in the second order of the rg expansion a brief discussion of the passive advection in the non gaussian velocity field governed by the nonlinear stochastic navier stokes equation is also given

anisotropic nonperturbative zero modes for passively advected magnetic fields an analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field the velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent xi inertial range anisotropic contributions to the scaling exponents zeta j of second order magnetic correlations are associated with zero modes and have been calculated nonperturbatively for d 3 the limit xi 0 yields zeta j j 2 xi 2j 3 j 2 5j 4 2 4j 2 1 where j j or 2 is the order in the legendre polynomial decomposition applied to correlation functions conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made

anomalous scaling of a passive vector advected by the navier stokes velocity field using the field theoretic renormalization group and the operator product expansion the model of a passive vector field a weak magnetic field in the framework of the kinematic mhd advected by the velocity field which is governed by the stochastic navier stokes equation with the gaussian random stirring force correlated in time and with the correlator proportional to k4 d 2e is investigated to the first order in e one loop approximation it is shown that the single time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case as well as in the case with the presence of large scale anisotropy the hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness dimensionless ratios of correlation functions as functions of the reynolds number re it is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than on one hand the corresponding models of a passive vector field advected by synthetic gaussian velocity fields and on the other hand than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic navier stokes equation the final one loop approximate asymptotic scaling behavior of the single time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions up to normalization

multidimensional random walks in random environments with subclassical limiting behavior in this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them our results make rigorous a heuristic argument of marinari et al 1983 and show that in anyd we can have a xn is recurrent and b xn logn 2

turbulent compressible fluid renormalization group analysis scaling regimes and anomalous scaling of advected scalar fields we study a model of fully developed turbulence of a compressible fluid based on the stochastic navier stokes equation by means of the field theoretic renormalization group in this approach scaling properties are related to the fixed points of the renormalization group equations previous analysis of this model near the real world space dimension 3 identified a scaling regime n v antonov et al theor math phys 110 305 1997 tmphah0040 577910 1007 bf02630456 the aim of the present paper is to explore the existence of additional regimes which could not be found using the direct perturbative approach of the previous work and to analyze the crossover between different regimes it seems possible to determine them near the special value of space dimension 4 in the framework of double y and expansion where y is the exponent associated with the random force and 4 d is the deviation from the space dimension 4 our calculations show that there exists an additional fixed point that governs scaling behavior turbulent advection of a passive scalar density field by this velocity ensemble is considered as well we demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial convective range the corresponding anomalous exponents identified as scaling dimensions of certain composite fields can be systematically calculated as a series in y and all calculations are performed in the leading one loop approximation

characterizations of matrices with nonzero signed row compound abstract a real square matrix a is called a sign nonsingular sns matrix if every matrix with the same sign pattern as a is not singular an m n matrix a with term rank m is called to have a nonzero signed row compound provided that each square submatrix of order m of a is an sns matrix or has an identically zero determinant as generalizations of sns matrices s matrices and totally l matrices matrices with nonzero signed row compound have a close relationship with matrices with signed null spaces which are applied to characterize linear systems with signed solutions in this paper matrices with nonzero signed row compound are characterized in terms of their signed bipartite graphs following these results characterizations of matrices with signed null spaces and full row term ranks in terms of their signed bipartite graphs are obtained too the recursive structure of m m 2 row sign balanced rsb maximal 0 1 1 matrices with nonzero signed row compound or with signed null spaces is also characterized

three loop calculation of the anomalous dimension of diffusivity for a model of a random walk in a potential random field with long range correlations the three loop contribution to the anomalous dimension of the diffusion coefficient of the model of a random walk in a potential random field with long range correlations is calculated contrary to earlier conjectures the result is not zero for logarithmic growth of the correlations but vanishes only in one and two dimensions in which the one loop contribution yields the exact value of the anomalous dimension

on the absolutely continuous spectrum of stark hamiltonians we study the spectral properties of the schrodinger operator with a constant electric field perturbed by a bounded potential it is shown that if the derivative of the potential in the direction of the electric field is smaller at infinity than the electric field then the spectrum of the corresponding stark operator is purely absolutely continuous in one dimension the absolute continuity of the spectrum is implied by just the boundedness of the derivative of the potential the sharpness of our criterion for higher dimensions is illustrated by constructing smooth potentials with bounded partial derivatives for which the corresponding stark operators have a dense point spectrum

three loop calculation of the random walk problem an application of dimensional transformation and the uniqueness method the method and details of the calculation of the three loop contribution to the anomalous dimension of the diffusion coefficient of the model of a random walk in a potential random field with long range correlations are presented contrary to earlier conjectures this contribution does not vanish identically a new method of calculation of multi loop feynman graphs with complicated numerator structure is suggested it leads to simpler integrals in a space of higher dimensionality which are computed using the recursion relations of the uniqueness method

full sign invertibility and symplectic matrices abstract an n n sign pattern h is said to be sign invertible if there exists a sign pattern h 1 called the sign inverse of h such that for all matrices a q h a 1 exists and a 1 q h 1 if in addition h 1 is sign invertible implying h 1 1 h h is said to be fully sign invertible and h h 1 is called a sign invertible pair given an n n sign pattern h a symplectic pair in q h is a pair of matrices a d such that a q h d q h and a t d i symplectic pairs are a pattern generalization of orthogonal matrices which arise from a special symplectic matrix found in n body problems in celestial mechanics 1 we discuss the digraphical relationship between a sign invertible pattern h and its sign inverse h 1 and use this to cast a necessary condition for full sign invertibility of h we proceed to develop sufficient conditions for h s full sign invertibility in terms of allowed paths and cycles in the digraph of h and conclude with a complete characterization of those sign patterns that require symplectic pairs

generalized inverses over integral domains in an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r r minors of the matrix is one in the same paper a procedure for constructing a generalized inverse from such a linear combination was also given in the present paper we show that any reflexive generalized inverse can be obtained by that procedure we also show that if the integral domain is a principal ideal domain every generalized inverse can be obtained by that procedure it is also shown that a matrix a of rank r over an integral domain has moore penrose inverse if and only if the sum of squares of all r r minors of a is an invertible element of the integral domain

matrices with doubly signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a a is said to have doubly signed generalized inverse if both a and a have a signed generalized inverse matrices with doubly signed generalized inverses are generalizations of those matrices a such that both a and a 1 are s 2 ns matrices which are studied in matrices of sign solvable linear systems cambridge university press cambridge 1995 linear algebra appl 232 1996 97 in this paper we give a complete characterization of matrices with doubly signed generalized inverses we show that if a is a real matrix with no zero rows and no zero columns then a has doubly signed generalized inverses if and only if a is permutation equivalent to a direct sum of matrices of the following types a row or a column with no zero entries fully indecomposable s 2 ns matrices of order 1 or 2 matrices of the blocked form d 1 o x d 2 where d 1 and d 2 are both invertible diagonal matrices

current noise and long time tails in biased disordered random walks we calculate the mean velocity and the velocity correlation function for a random walk with a uniform bias on a disordered chain we find a new long time tail in the velocity correlation function due to the combined effects of the bias and of the disorder in the site variables this long time tail persists to a low frequency cutoff inversely proportional to the square of the bias by associating the velocity correlation function with the spectrum of current fluctuations we calculate the excess low frequency current noise associated with this long time tail the spectrum of current fluctuations goes as i2 n f 1 2 wherei is the dc current n is the number of charge carriers andf is the frequency the possible connection to 1 f noise is discussed the calculation is done by a perturbation expansion in the strength of the disorder but is shown to be exact to all orders for weak enough bias

criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed stark type operators abstract in this paper we consider the perturbed stark operator h u h 0 u q u u x u q u where q is the power decaying perturbation the criteria for q such that h h 0 q has at most one eigenvalue finitely many infinitely many eigenvalues are obtained all the results are quantitative and are generalized to the perturbed stark type operator

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

graph theoretical approach to qualitative solvability of linear systems the structure of sign solvable and strongly sign solvable systems is studied here by a refinement of the graph theoretic approach first suggested by maybee both sign solvability and strong sign solvability are characterized in terms of an associated digraph the problem of recognizing sign solvability is reduced to two subproblems recognizing strong sign solvability and recognizing sign nonsingularity under fairly general conditions on the sign patterns of a it is possible to determine all sign patterns for b which render the system ax b sign solvable

moore penrose inverse of the incidence matrix of a distance regular graph abstract let be a graph with n vertices where each edge is given an orientation and let q be the vertex edge incidence matrix we obtain a formula for the moore penrose inverse of q when is a distance regular graph the formula is illustrated by examples

schrodinger operators with an electric field and random or deterministic potentials we prove that the schrodinger operatorh d2 dx2 v x f x has purely absolutely continuous spectrum for arbitrary constant external fieldf for a large class of potentials this result applies to many periodic almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenf 0 the spectrum is almost surely pure point with exponentially decaying eigenfunctions

spectral analysis of n body stark hamiltonians we prove that the spectrum for a large class ofn body stark hamiltonians is purely absolutely continuous we need slow decay at infinity and local singularities of at most coulomb type in particular our results include the usual models for atoms and molecules

equilibrium distributions for random walkers in random media the authors study the renyi entropies of the probability distribution of a random walker in a random medium of finite total volume in one dimension of space it is shown that they tend for large lattices to constants independent of the lattice size thus the distribution is not multifractal in contrast to a recent claim the authors conjecture that the same is true for any dimension if the random force is derived from a potential in the more interesting general case unconstrained forces they cannot make a definite statement

sign nonsingular matrices and even cycles in directed graphs a sign nonsingular matrix or l mahix a is a real m x n matrix such that the columns of any real m x n matrix with the same sign pattern as a are linearly independent the problem of recognizing square l matrices is equivalent to that of finding an even cycle in a directed graph in this paper we use graph theoretic methods to investigate l matrices in particular we determine the maximum number of nonzero elements in square l matrices and we characterize completely the semicomplete l matrices i e the square l matrices a j such that at least one of a and ajl is nonzero for any i j and those square l matrices which are combinatorially symmetric i e the main diagonal has only nonzero entries and a j 0 iff a jr 0 we also show that for any n x n l matrix there is an i such that the total number of nonzero entries in the i th row and i th column is less than n unless the matrix has a completely specified structure finally we discuss the algorithmic aspects

algebraic constructions of the minimal forbidden digraphs of strong sign nonsingular matrices a square real matrix a is called a strong sign nonsingular matrix s2ns matrix if all the matrices with the same sign pattern as a are nonsingular and all the inverses of these matrices have the same sign pattern s2ns digraphs are digraphs associated with those s2ns matrices with negative main diagonals in this paper we define the associated linear system of equations l d over the finite field f2 for each digraph d and then define an undirected graph g l d representing certain relations between the equations of l d we obtain algebraic criteria to recognize the minimal forbidden configurations of s2ns digraphs in terms of the solvability of the linear system l d and some of its subsystems and the connectedness of the undirected graph g l d these algebraic criteria together with a conjunction operation of digraphs can be used to construct infinitely many new minimal forbidden configurations

renormalization group operator product expansion and anomalous scaling in a model of advected passive scalar field theoretical renormalization group methods are applied to the obukhov kraichnan model of a passive scalar advected by the gaussian velocity field with the covariance mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or dangerous composite operators powers of the local dissipation rate whose negative critical dimensions determine anomalous exponents the main technical result is the calculation of the anomalous exponents in the order ensuremath varepsilon 2 of the ensuremath varepsilon expansion generalization of the results obtained to the case of a slow velocity field is also presented

intermittency and anomalous scaling of passive scalars in any space dimension we establish exact inequalities for the structure function scaling exponents of a passively advected scalar in both the inertial convective and viscous convective ranges these inequalities involve the scaling exponents of the velocity structure functions and in a refined form an intermittency exponent of the convective range scalar flux they are valid for three dimensional navier stokes turbulence and satisfied within errors by present experimental data the inequalities also hold for any synthetic turbulent velocity statistics with a finite correlation in time we show that for time correlation exponents of the velocity smaller than the local turnover exponent the scalar spectral exponent is strictly less than that in kraichnan s phys rev lett 72 1016 1994 phys fluids 11 945 1968 j fluid mech 64 737 1974 62 305 1974 soluble rapid change model with velocity ensuremath delta correlated in time our results include as a special case an exponent inequality derived previously by constantin and procaccia nonlinearity 7 1045 1994 but with a more direct proof the inequalities in their simplest form follow from a kolmogorov type relation for the turbulent passive scalar valid in each space dimension d our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars these are compared with the relations implied by fusion rules hypothesized for scalar gradients textcopyright 1996 the american physical society

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field

renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the gaussian non solenoidal compressible velocity field with the covariance propto delta t t x x epsilon convective range anomalous scaling for the structure functions and various pair correlators is established and the corresponding anomalous exponents are calculated to the order epsilon 2 of the epsilon expansion these exponents are non universal as a result of the degeneracy of the rg fixed point in contrast to the case of a purely solenoidal velocity field obukhov kraichnan model the correlation functions in the case at hand exhibit nontrivial dependence on both the ir and uv characteristic scales and the anomalous scaling appears already at the level of the pair correlator the powers of the scalar field without derivatives whose critical dimensions determine the anomalous exponents exhibit multifractal behaviour the exact solution for the pair correlator is obtained it is in agreement with the result obtained within the epsilon expansion the anomalous exponents for passively advected magnetic fields are also presented in the first order of the epsilon expansion