on second order weakly hyperbolic equations and the ultradifferentiable classes abstract we consider second order weakly hyperbolic equations with time dependent coefficients in the ultradifferentiable classes our main purpose of the present paper is an investigation the relation between the classes of the functions to be well posed and the following properties of the coefficients the order of degeneration stabilization to a monotonic function and their smoothness in the ultradifferentiable classes

oscillatory properties for semilinear degenerate hyperbolic equations of second order we consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations several sufficient conditions for the oscillation or non oscillation are presented in particular they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension

on the well posedness of weakly hyperbolic equations with time dependent coefficients in this paper we analyse the gevrey well posedness of the cauchy problem for weakly hyperbolic equations of general form with time dependent coefficients the results involve the order of lower order terms and the number of multiple roots we also derive the corresponding well posedness results in the space of gevrey beurling ultradistributions

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

on the numerical experiments of the cauchy problem for semi linear klein gordon equations in the de sitter spacetime abstract the computational analysis of the cauchy problem for semi linear klein gordon equations in the de sitter spacetime is considered several simulations are performed to show the time global behaviors of the solutions of the equations in the spacetime based on the structure preserving scheme it is remarked that the sufficiently large hubble constant yields the strong diffusion effect which gives the long and stable simulations for the defocusing semi linear terms the reliability of the simulations is confirmed by the preservation of the numerically modified hamiltonian of the equations

persistence properties of a system of coagulating and annihilating random walkers we study a d dimensional system of diffusing particles that on contact either annihilate with probability 1 q ensuremath 1 or coagulate with probability q ensuremath 2 q ensuremath 1 in one dimension the system models the zero temperature glauber dynamics of domain walls in the q state potts model we calculate overline p m t the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly m ensuremath 1 coagulations using perturbative renormalization group analysis for dl2 we show that if the number of coagulations m is much less than the typical number m t then overline p m t ensuremath sim m ensuremath zeta d t ensuremath ensuremath theta with ensuremath theta dq q q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 ensuremath zeta 2q ensuremath 1 ensuremath epsilon 2q ensuremath 1 q ensuremath 1 1 2 aq ensuremath epsilon 2 o ensuremath epsilon 3 where q q ensuremath 1 q ensuremath epsilon 2 ensuremath d and a ensuremath 0 006 dots m t is shown to scale as m t ensuremath sim t d 2 ensuremath ensuremath delta where ensuremath delta d 1 ensuremath q q ensuremath 1 q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 in two dimensions we show that overline p m t ensuremath sim mathrm ln t q 3 ensuremath 2q mathrm ln m 2q ensuremath 1 2 t ensuremath 2q for m ensuremath ll t 2q ensuremath 1 we also derive an exact nonperturbative relation between the exponents namely ensuremath delta q ensuremath theta 1 ensuremath q the one dimensional results corresponding to ensuremath epsilon 1 are compared with results from monte carlo simulations

joint probability distributions for a class of non markovian processes we consider joint probability distributions for the class of coupled langevin equations introduced by fogedby h c fogedby phys rev e 50 1657 1994 we generalize well known results for the single time probability distributions to the case of n time joint probability distributions it is shown that these probability distribution functions can be obtained by an integral transform from distributions of a markovian process the integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non markovian character of the process

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

on a regularity result for the transonic pressure gradient system in gas dynamics abstract this paper is concerned with the regularity of transonic flows originated from the two dimensional riemann problem to the pressure gradient system in gas dynamics it is verified that if the transonic solution of the pressure gradient system is c 3 in a neighborhood of a sonic point and is c in the subsonic sonic region then the solution is also c in the sonic supersonic region near the sonic point

unique existence of evolution equations of hyperbolic type with countably many singular of degenerate points soit h un espace de hilbert et soit un operateur auto adjoint a spectre borne par dessous soit t 0 un nombre reel arbitraire avec 1 t 0 1 on etudie la resolubilite unique de l equation d evolution abstraite singuliere ou degeneree de type hyperbolique u t t u t 0 sur t 0 1 u t 0 u 0 u t 0 u 1 ou est une fonction non negative sur 1 1 avec des points singuliers et degeneres denombrables s accummulant en 0

self similar solutions of semilinear wave equation with variable speed of propagation abstract we investigate the issue of existence of the self similar solutions of the generalized tricomi equation in the half space where the equation is hyperbolic we look for the self similar solutions via the cauchy problem an integral transformation suggested in k yagdjian a note on the fundamental solution for the tricomi type equation in the hyperbolic domain j differential equations 206 2004 227 252 is used to represent solutions of the cauchy problem for the linear tricomi type equation in terms of fundamental solutions of the classical wave equation this representation allows us to prove decay estimates for the linear tricomi type equation with a source term obtained in k yagdjian the self similar solutions of the tricomi type equations z angew math phys in press doi 10 1007 s00033 006 5099 2 estimates for the self similar solutions of the linear tricomi type equation are the key tools to prove existence of the self similar solutions

the coexistence problem for the discrete mathieu operator we solve the coexistence problem for the periodic discrete mathieu operator in all parametric cases the main tool in the proof will be bezout s theorem for projective plane curves as an additional result we obtain the gap opening and gap growth powers for this operator

asymptotic form of the approach to equilibrium in reversible recombination reactions the reversible reactions and are investigated from the exact langevin equations describing our model we set up a systematic approximation scheme to compute the approach of the density of c particles to its equilibrium value we show that for a sufficiently long time t this approach takes the form of a power law for any dimension d the amplitude a is also computed exactly but is expected to be model dependent for uncorrelated initial conditions the c density turns out to be a monotonic time function the cases of correlated initial conditions and unequal diffusion constants are investigated as well in the former correlations may break the monotonicity of the density or in some special cases they may change the long time behaviour for the latter the power law remains valid only the amplitude changes even in the extreme case of immobile c particles we also consider the case of segregated initial condition for which a reaction front is observed and confirm that its width is governed by mean field exponent in any dimension

universal tree structures in directed polymers and models of evolving populations by measuring or calculating coalescence times for several models of coalescence or evolution with and without selection we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes

weakly hyperbolic equations with non analytic coefficients and lower order terms in this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time dependent coefficients and lower order terms we prove the gevrey well posedness of the cauchy problem under c k regularity of coefficients of the principal part and natural levi conditions on lower order terms which may be only continuous in the case of analytic coefficients in the principal part we establish the c infty well posedness the proofs are based on using the quasi symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions the main novelty compared to the existing literature is the possibility to include lower order terms to the equation which have been untreatable until now in these problems as well as considering any space dimensions we also give results on the ultradistributional and distributional well posedness of the problem and we look at new effects for equations with discontinuous lower order terms

cluster cluster aggregation as an analogue of a turbulent cascade kolmogorov phenomenology scaling laws and the breakdown of self similarity abstract we present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers we emphasize the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean field description provided by standard smoluchowski kinetic theory the presence of a source of monomers allows the system to reach a statistically stationary state at large times this state is characterized by a constant flux of mass directed from small to large masses it therefore admits a phenomenological description based on the assumption of self similarity and constant mass flux analogous to the kolmogorov s 1941 theory of turbulence unlike turbulence the aggregation problem is analytically tractable using powerful methods of statistical field theory we explain in detail how these methods should be adapted to study the far from equilibrium flux dominated states characteristic of turbulent systems we consider multipoint correlation functions of the mass density by an exact evaluation of the scaling exponents for the one and two point correlation functions we show that the assumption of self similiarity breaks down at large masses for spatial dimensions d 2 we calculate non rigorously the exponents of the higher order correlation functions as an expansion where 2 d we show that the mass distribution exhibits non trivial multiscaling an analogy can be drawn with the case of hydrodynamic turbulence the physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events these correlations stem from the recurrence of diffusion processes in d 2 the analytic methods developed here will have more general applicability beyond the study of this specific problem

very weak solutions to hypoelliptic wave equations abstract in this paper we study the cauchy problem for the wave equations for hypoelliptic homogeneous left invariant operators on graded lie groups when the time dependent non negative propagation speed is regular holder and distributional for holder coefficients we derive the well posedness in the spaces of ultradistributions associated to rockland operators on graded groups in the case when the propagation speed is a distribution we employ the notion of very weak solutions to the cauchy problem that was already successfully used in similar contexts in 12 and 20 we show that the cauchy problem for the wave equation with the distributional coefficient has a unique very weak solution in an appropriate sense which coincides with classical or distributional solutions when the latter exist examples include the time dependent wave equation for the sub laplacian on the heisenberg group or on general stratified lie groups or p evolution equations for higher order operators on r n or on groups the results already being new in all these cases

well posedness of hyperbolic systems with multiplicities and smooth coefficients we study hyperbolic systems with multiplicities and smooth coefficients in the case of non analytic smooth coefficients we prove well posedness in any gevrey class and when the coefficients are analytic we prove c infty well posedness the proof is based on a transformation to block sylvester form introduced by d ancona and spagnolo boll umi 8 1b 169 185 1998 which increases the system size but does not change the eigenvalues this reduction introduces lower order terms for which appropriate levi type conditions are found these translate then into conditions on the original coefficient matrix this paper can be considered as a generalisation of garetto and ruzhansky math ann 357 2 401 440 2013 where weakly hyperbolic higher order equations with lower order terms were considered

a study of the ambiguity in the solutions to the diophantine equation for chern numbers the chern numbers for hofstadter models with rational flux 2 p q are partially determined by a diophantine equation a mod q ambiguity remains the resolution of this ambiguity is only known for the rectangular lattice with nearest neighbors hopping where it has the form of a window condition we study a hofstadter butterfly on the triangular lattice for which the resolution of ambiguity is open in the model many pairs p q satisfy a window condition which is shifted relative to the window of the square model however we also find pairs p q where the chern numbers do not belong to any contiguous window this shows that the rectangular model and the one we study on the triangular lattice are not adiabatically connected many gaps must close our results suggest the conjecture that the mod q ambiguity in the diophantine equation generically reduces to a sign ambiguity

zero hausdorff dimension spectrum for the almost mathieu operator we study the almost mathieu operator at critical coupling we prove that there exists a dense g delta set of frequencies for which the spectrum is of zero hausdorff dimension

spectral solutions to stochastic models of gene expression with bursts and regulation signal processing molecules inside cells are often present at low copy number which necessitates probabilistic models to account for intrinsic noise probability distributions have traditionally been found using simulation based approaches which then require estimating the distributions from many samples here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation which is linear we apply the resulting spectral method to three general models of stochastic gene expression a single gene with multiple expression states often used as a model of bursting in the limit of two states a gene regulatory cascade and a combined model of bursting and regulation in all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy in the last case we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission

non uniqueness in hyperbolic cauchy problems where t e r and x e ri we say that it possesses the uniqueness property at a point to xo if for any solution u e c v of 1 v being a neighborhood of t0o xo such that u 0 on v nf t to it results that u 0 on a neighborhood of to xo the first uniqueness results for equation 1 were supplied by the classical theorems of holmgren 12 carleman 6 and calderon 5 more refined versions of these theorems are the following ones

a renormalization group study of a class of reaction diffusion models with particles input we study a class of reaction diffusion models extrapolating continuously between the pure coagulation diffusion case and the pure annihilation diffusion one with particles input at a rate j for dimension the dynamics strongly depends on the fluctuations while for d 2 the behaviour is mean field like the models are mapped onto a field theory whose properties are studied in a renormalization group approach simple relations are found between the time dependent correlation functions of the different models of the class for the pure coagulation diffusion model the time dependent density is found to be of the form where d is the diffusion constant the critical exponent and are computed to all orders in where d is the dimension of the system while the scaling function is computed to second order in for the one dimensional case an exact analytical solution is provided whose predictions are compared with the results of the renormalization group approach for

novel decay laws for the one dimensional reaction diffusion model as consequence of initial distributions the effect of the initial particle distribution on the one dimensional coagulation and annihilation reaction diffusion models is studied analytically with this aim an exact expression for the particle number n t given explicitly in terms of the initial particle distribution is derived it is found that if the initial distribution has divergent first moments new decay laws occur for the case of fractal distributions of dimension 7 we obtain exactly the relative number of particles n t n t n 0 for long times it evolves as n t cit 2 c2r 2 w 2 1 2 with r 2dt the existence of non fractal initial distributions that lead to new decays is also discussed as examples are considered an interparticle probability distribution of the form p r r 2 which yields n t a t 1 2 ln r a2r 1 2 and a family of distributions of the form p r r 1 m 7i r0 1 o with a 0 which yields n t ln r a these results are tested against monte carlo simulations

wave equation for operators with discrete spectrum and irregular propagation speed given a hilbert space mathcal h we investigate the well posedness of the cauchy problem for the wave equation for operators with a discrete non negative spectrum acting on mathcal h we consider the cases when the time dependent propagation speed is regular holder and distributional we also consider cases when it is strictly positive strictly hyperbolic case and when it is non negative weakly hyperbolic case when the propagation speed is a distribution we introduce the notion of very weak solutions to the cauchy problem we show that the cauchy problem for the wave equation with the distributional coefficient has a unique very weak solution in an appropriate sense which coincides with classical or distributional solutions when the latter exist examples include the harmonic and anharmonic oscillators the landau hamiltonian on mathbb r n uniformly elliptic operators of different orders on domains hormander s sums of squares on compact lie groups and compact manifolds operators on manifolds with boundary and many others

wave equation for sums of squares on compact lie groups in this paper we investigate the well posedness of the cauchy problem for the wave equation for sums of squares of vector fields on compact lie groups we obtain the loss of regularity for solutions to the cauchy problem in local sobolev spaces depending on the order to which the hormander condition is satisfied but no loss in globally defined spaces we also establish gevrey well posedness for equations with irregular coefficients and or multiple characteristics as in the sobolev spaces if formulated in local coordinates we observe well posedness with the loss of local gevrey order depending on the order to which the hormander condition is satisfied

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

on the cauchy problem of degenerate hyperbolic equations in this paper we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for cauchy problems the existence result is based on a priori estimates of sobolev norms of solutions such estimates illustrate a loss of derivatives because of the degeneracy

a note on weakly hyperbolic equations with analytic principal part in this note we show how to include low order terms in the c well posedness results for weakly hyperbolic equations with analytic time dependent coefficients this is achieved by doing a different reduction to a system from the previously used one we find the levi conditions such that the c well posedness continues to hold

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

finite size scaling studies of one dimensional reaction diffusion systems part i analytical results we consider two single species reaction diffusion models on one dimensional lattices of lengthl the coagulation decoagulation model and the annihilation model for the coagulation model the system of differential equations describing the time evolution of the empty interval probabilities is derived for periodic as well as for open boundary conditions this system of differential equations grows quadratically withl in the latter case the equations are solved analytically and exact expressions for the concentration are derived we investigate the finite size behavior of the concentration and calculate the corresponding scaling functions and the leading corrections for both types of boundary conditions we show that the scaling functions are independent of the initial conditions but do depend on the boundary conditions a similarity transformation between the two models is derived and used to connect the corresponding scaling functions

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

fredholm indices and the phase diagram of quantum hall systems the quantized hall conductance in a plateau is related to the index of a fredholm operator in this paper we describe the generic phase diagram of fredholm indices associated with bounded and toeplitz operators we discuss the possible relevance of our results to the phase diagram of disordered integer quantum hall systems

on the total bandwidth for the rational harper s equation in the last years several contributions have been done around the total bandwidth of the spectrum for the harper s operator in particular an interesting conjecture has been proposed by thouless which gives also strong convincing arguments for the proof in special cases on the other hand in the study of the cantor structure of the spectrum b helffer and j sjostrand have justified an heuristic semiclassical approach proposed by m wilkinson the aim of this article is to explain how one can use the first step of this approach to give a rigorous semi classical proof of the thouless formula in some of the simplest cases we shall also indicate how one can hope with more effort to prove rigorously recent results of last and wilkinson on the same conjecture

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

doi peliti path integral methods for stochastic systems with partial exclusion abstract doi peliti methods are developed for stochastic models with finite maximum occupation numbers per site we provide a generalized framework for the different fock spaces reported in the literature paragrassmannian techniques are then utilized to construct path integral formulations of factorial moments we show that for many models of interest a magnus expansion is required to construct a suitable action meaning actions containing a finite number of terms are not always feasible however for such systems perturbative techniques are still viable and for some examples including carrying capacity population dynamics and diffusion with partial exclusion the expansions are exactly summable

scaling localization and bandwidths for equations with competing periods abstract the finite size scaling theory of the measure of the spectrum of harper s equation is reexamined for sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as log p 2 or p 2 as the denominator p is increased the question of how special are the properties of the harper equation is studied it is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from localized to extended at a value of the strength of the off diagonal term that depends on energy in contrast to the harper equation where the transition is energy independent this has a crucial effect on the measure of the spectrum

fluctuation effects in coagulation processes abstract we study diffusion controlled coagulation processes in the system with a finite density of droplets using the fock space formalism we find that the effective reaction rate has a correction to smoluchowski s classical rate constant for three dimensional systems we also predict the scaling form of the scattering function and find that our results qualitatively agree with experimental results

hyperbolic systems with non diagonalisable principal part and variable multiplicities i well posedness in this paper we analyse the well posedness of the cauchy problem for a rather general class of hyperbolic systems with space time dependent coefficients and with multiple characteristics of variable multiplicity first we establish a well posedness result in anisotropic sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal namely the terms below the diagonal at a distance k to it must be of order k this setting also allows for the jordan block structure in the system second we give conditions for the schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied we give explicit details for the appearing conditions and constructions for 2 times 2 and 3 times 3 systems complemented by several examples

scaling for the discrete mathieu equation the scaling of the total width of the band for the discrete mathieu equation is studied in the critical region near the transition between localized and extended states for the special case in which there is one period of the modulation forp lattice spacings a general expression for the bandwidthw in the critical region is found at the critical point an analytic expression forpw is found which agrees to one part in 108 with the result deduced from numerical work

path integral representation for a stochastic sandpile we introduce an operator description for a stochastic sandpile model with a conserved particle density and develop a path integral representation for its evolution the resulting exact expression for the effective action highlights certain interesting features of the model for example it is nominally massless and the dynamics is via cooperative diffusion using the path integral formalism we construct a diagrammatic perturbation theory yielding a series expansion for the activity density in powers of time

energy estimates for a class of degenerate hyperbolic equations we derive estimates in usual sobolev norms for solutions of degenerate hyperbolic equations if degenerate coefficients admit only real roots loss of derivatives occurs due to the degeneracy

spatial fluctuations in reaction diffusion systems a model for exponential growth the spatial fluctuations in an exactly soluble model for the irreversible aggregation of clusters are treated the model is characterized byrate constants k ij i j for the clustering of ani and aj mer anddiffusion constants d j d it is assumed thatd 1 reaction limited aggregation explicit expressions for the correlation functions at equal and at different times are calculated the equal time correlation functions show scaling behavior in the scaling limit the correlation length remains finite ast and the fluctuations becomelarge at large times t t d inany dimension the crossover timet d at which the mean field theory smoluchowski s equation breaks down is given byt d ind these exact results imply that the upper critical dimension of this model isd c and hence that there isno unique upper critical dimension in models for the irreversible aggregation of clusters

anomalous coalescence phenomena under stationary regime condition through thompson s method abstract in this work anomalous coalescence phenomena of type a a a 0 are considered in the case of stationary regime namely the case of an external homogeneous source h of single particles a in the limit of zero external field rate the main purpose here is to pick up new critical exponents in the stationary regime for the phenomenon of anomalous coalescence of bubbles found in one dimension by josserand and rica jr phys rev lett 78 1997 1215 the critical exponents of statical concentration relaxation time and the exponent for concentration decay are evaluated in the limit h 0 zero field rate we show that such critical exponents depend on the parameter physica a 334 2004 335 which characterizes the anomalous diffusion condition in order to explain the anomalous coalescence of jr s work as already made before by the present author physica a 334 2004 335 the parameter must display an explicit dependence on the dimensionality d 4 d for d 2 we use such a result d in order to pick up new critical exponents for the special case of anomalous coalescence found by jr we find 5 4 5 and 1 4 in 1 d we also verify that the known scaling relations among such critical indexes phys rev a 32 2 1985 1129 are obeyed so that in the special case of 2 brownian diffusion condition given to mean field regime we naturally recover those results previously obtained by racz phys rev a 32 2 1985 1129 we go further to give a better explanation for such new results taking into account another exponent physica a 334 2004 335 to explain the behavior of diffusion constant which depends on concentration d d 0 n that is to say we obtain analytically new critical indexes which controll the diffusion behavior d for long time regime in the presence of an external homogeneous source

on spectral properties of harper like models we study spectral properties of harper like models by algebraic and combinatorial methods and derive sufficient conditions for the existence of spectral gaps with qualitative estimates for this class the chambers relation holds and we obtain an analytic expression for the representation dependent part models corresponding to the rectangular and triangular lattice are studied in the second case we show that one class of spectral gaps is open for magnetic fields with rational magnetic flux per unit cell a quantitative estimate for the gap widths is given for the anisotropic case and for irrational magnetic flux fulfilling some liouville condition the spectrum is a cantor set

hyperbolic equations with non analytic coefficients we prove that the cauchy problem for a hyperbolic homogeneous equation with mathcal c infty coefficients depending on time is well posed in every gevrey class although in general it is not well posed in mathcal c infty provided the characteristic roots satisfy the condition lambda i t 2 lambda j t 2 leq m lambda i t lambda j t 2 quad i neq j

a sum rule for the dispersion relations of the rational harper equation the authors derive an exact sum rule satisfied by the dispersion relations of the commensurable case of harper s equation 1955 they use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to thouless 1990 concerning the total bandwidth when the commensurability is a high order rational

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

renormalization theoretic analysis of non equilibrium phase transitions i the becker doring equations with power law rate coefficients we study in detail the application of renormalization theory to models of cluster aggregation and fragmentation of relevance to nucleation and growth processes we investigate the becker d oring equations originally formulated to describe and analyse non equilibrium phase transitions and more recently generalized to describe a wide range of physicochemical problems in this paper we analyse how the systematic coarse graining renormalization of the becker d oring system of equations affects the aggregation and fragmentation rate coefficients we consider the case of power law size dependent cluster rate coefficients which we show lead to only three classes of system that require analysis coagulation dominated systems fragmentation dominated systems and those where coagulation and fragmentation are exactly balanced we analyse the late time asymptotics associated with each class

generalized lyapunov exponents in high dimensional chaotic dynamics and products of large random matrices we study the behavior of the generalized lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsd for products of random matrices without any particular structure the generalized lyapunov exponents become equal in this limit and the value of one of the generalized lyapunov exponents is obtained by simple arguments on the contrary for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized lyapunov exponents remains different ford indicating that high dimensionality cannot always destroy intermittency

solutions globales d un probleme de cauchy lineaire abstract on resoud le probleme de cauchy pour des operateurs aux derivees partielles a coefficients polynomiaux par rapport aux variables d espace dans des classes de gevrey projectives nos resultats sont des versions globales du theoreme de cauchy kowalewski et du theoreme de nagumo

field theoretic approach to a source enhanced aggregation process a diffusion controlled cluster cluster aggregation system with a source is analysed using the fock space formalism and diagrammatic perturbation technique the effective coagulation kernel and equation for low dimensional cases d or 2 are derived the asymptotic size distribution obeying a power law is obtained for any dimension the application to the aggregation process of particles undergoing levy flight is discussed through the analysis in this paper the author clarifies a relation between critical phenomena and a source enhanced aggregation

on hyperbolic equations and systems with non regular time dependent coefficients in this paper we study higher order weakly hyperbolic equations with time dependent non regular coefficients the non regularity here means less than holder namely bounded coefficients as for second order equations in 14 we prove that such equations admit a very weak solution adapted to the type of solutions that exist for regular coefficients the main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter classical solutions are recovered as limit of very weak solutions finally by using a reduction to block sylvester form we conclude that any first order hyperbolic system with non regular coefficients is solvable in the very weak sense

the existence and singularity structures of low regularity solutions to higher order degenerate hyperbolic equations abstract this paper is a continuation of our partial work in 21 where we established that the bounded local solution u t x exists and is piecewise smooth for the second order degenerate hyperbolic equation t 2 t m x u f t x u with the initial data of c 1 piecewise smooth u 0 x and piecewise smooth t u 0 x in the present paper we will consider the lower regularity solution of the higher order degenerate hyperbolic equation in the category of discontinuous and even unbounded functions

positive hausdorff dimensional spectrum for the critical almost mathieu operator we show that there exists a dense set of frequencies with positive hausdorff dimension for which the hausdorff dimension of the spectrum of the critical almost mathieu operator is positive

homogeneous weakly hyperbolic equations with time dependent analytic coefficients abstract this paper concerns the cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients we give a sufficient condition for the c well posedness which is also necessary if the space dimension is equal to one the main point of the paper consists in expressing our condition only in terms of the coefficients of the operator without needing to know the behavior of the characteristic roots this is made possible by using the so called standard symmetrizer of a companion hyperbolic matrix

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

continuously varying exponents in reaction diffusion systems we propose a simple reaction diffusion model describing a class of competitive multi species reactions the model is exactly solvable at its upper critical dimension du 2 the local moments of the component concentrations follow a power law decay with exponents which vary continuously with system parameters the exponents also have an underlying multifractal spectrum the main results are supported by results from a numerical integration of the model

smoluchowski and beyond a field theoretical study of coagulation fragmentation processes adopting simple field theoretical techniques the author demonstrates how the smoluchowski approach to reaction diffusion models can be systematically corrected focusing on coagulation fragmentation processes he identifies the upper critical dimension dc 2 below which finite mobility corrections are important and full universality lattice independence etc can be achieved trends seem by direct simulation of simple lattice models for d 2 are confirmed whilst problems of interpretation exist for d 2

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

scaling in reaction diffusion systems the authors study irreversible reaction diffusion models of the form px from k1 from mx to k2 to qx sx from k3 from nx m dc 2 n m 1 n 1 which they study for d dc using the renormalisation group in conjunction with then epsilon expansion and exact techniques for m 1 and any n the transition remains continuous they present epsilon expansions for the case n 3 by contrast for m 2 and any n the system shows a fluctuation induced first order transition

hyperbolic second order equations with non regular time dependent coefficients in this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients this means assuming that the coefficients are less regular than holder the characteristic roots are also allowed to have multiplicities for such equations we describe the notion of a very weak solution adapted to the type of solutions that exist for regular coefficients the construction is based on considering friedrichs type mollifiers of coefficients and corresponding classical solutions and their quantitative behaviour in the regularising parameter we show that even for distributional coefficients the cauchy problem does have a very weak solution and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist in concrete applications the dependence on the regularising parameter can be traced explicitly

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

on the loss of regularity for a class of weakly hyperbolic operators in this paper we determine bounds for the optimal loss of regularity in the sobolev scale for a class of weakly hyperbolic operators

path integral evaluation of the one loop effective potential in field theory of diffusion limited reactions abstract the well established effective action and effective potential framework from the quantum field theory domain is adapted and successfully applied to classical field theories of the doi and peliti type for diffusion controlled reactions through a number of benchmark examples we show that the direct path integral calculation of the effective potential in fixed space dimension d 2 to one loop order reduces to a small set of simple elementary functions irrespective of the microscopic details of the specific model thus the technique which allows one to obtain with little additional effort the potentials for a wide variety of different models represents an alternative to the standard model dependent diagram based calculations the renormalized effective potential effective equations of motion and the associated renormalization group equations are computed in d 2 spatial dimensions for a number of single species field theories of increasing complexity