renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

cluster method for the ashkin teller model a cluster monte carlo algorithm for the ashkin teller at model is constructed according to the guidelines of a general scheme for such algorithms its dynamical behaviour is tested for the square lattice at model we perform simulations on the line of critical points along which the exponents vary continuously and find that critical slowing down is significantly reduced we find continuous variation of the dynamical exponent z along the line following the variation of the ratio alpha nu in a manner which satisfies the li sokal bound z cluster geq alpha nu that was so far proved only for potts models

the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

local and global bifurcation results for a semilinear boundary value problem abstract we investigate the local and global nature of the bifurcation diagrams which can occur for a semilinear elliptic boundary value problem with neumann boundary conditions involving sign changing coefficients it is shown that closed loops of positive and negative solutions occur naturally for such problems and properties of these loops are investigated

on a class of quasilinear systems with sign changing nonlinearities abstract we prove existence and nonexistence of positive solutions for the quasilinear system p u a x f u v in q v b x g u v in u v 0 on where p u div u p 2 u q v div v q 2 v p q 1 is a bounded domain in r n and the coefficients a x and b x are allowed to change sign

counting and equidistribution for quaternion algebras we aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting we prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure it leads to answering the sato tate conjectures in this case and contains the counting law of the universal family with a power savings error term

existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function abstract we investigate the existence of nodal solutions of an indefinite weight boundary value problem u r h t f u 0 0 t 1 u 0 u 1 0 where h c 0 1 changes sign the proof of our main result is based upon bifurcation techniques

central limit theorems for elliptic curves and modular forms with smooth weight functions abstract in 11 the second and third named authors established a central limit theorem for the error term in the sato tate law for families of modular forms this method was adapted to families of elliptic curves in 3 by the first and second named authors in this context a central limit theorem was established only under a strong hypothesis going beyond the riemann hypothesis in the present paper we consider a smoothed version of the sato tate conjecture which allows us to overcome several limitations in particular for the smoothed version we are able to establish a central limit theorem for much smaller families of modular forms and we succeed in proving a theorem of this type for families of elliptic curves under the riemann hypothesis for l functions associated to hecke eigenforms for the full modular group

asymptotic trace formula for the hecke operators given integers m n and k we give an explicit formula with an optimal error term with square root cancelation for the petersson trace formula involving the mth and nth fourier coefficients of an orthonormal basis of s k left n right the weight k newforms with fixed square free level n provided that 4 pi sqrt mn k o left k frac 1 3 right moreover we establish an explicit formula with a power saving error term for the trace of the hecke operator mathcal t n on s k left n right averaged over k in a short interval by bounding the second moment of the trace of mathcal t n over a larger interval we show that the trace of mathcal t n is unusually large in the range 4 pi sqrt n k o left n frac 1 6 right as an application for any fixed prime p coprime to n we show that there exists a sequence k n of weights such that the error term of weyl s law for mathcal t p is unusually large and violates the prediction of arithmetic quantum chaos in particular this generalizes the result of gamburd et al j eur math soc 1 1 51 85 1999 theorem 1 4 with an improved exponent

functional integral construction of the massive thirring model verification of axioms and massless limit we present a complete construction of a quantum field theory for the massive thirring model by following a functional integral approach this is done by introducing an ultraviolet and an infrared cutoff and by proving that if the bare parameters are suitably chosen the schwinger functions have a well defined limit satisfying the osterwalder schrader axioms when the cutoffs are removed our results which are restricted to weak coupling are uniform in the value of the mass the control of the effective coupling which is the main ingredient of the proof is achieved by using the ward identities of the massless model in the approximated form they take in the presence of the cutoffs as a byproduct we show that when the cutoffs are removed theward identities have anomalies which are not linear in the bare coupling moreover we find for the interacting propagator of the massless theory a closed equation which is different from that usually stated in the physical literature

weighted averages of modular l values using an explicit relative trace formula on gl 2 we derive a formula for averages of modular l values in the critical strip weighting by fourier coefficients hecke eigenvalues and petersson norms as an application we show that a grh holds for these averages as the weight or the level goes to we also use the formula to give explicit zero free regions of the form im s 0 for some particular modular l functions

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

ihara s zeta function for periodic graphs and its approximation in the amenable case abstract in this paper we give a more direct proof of the results by clair and mokhtari sharghi b clair s mokhtari sharghi zeta functions of discrete groups acting on trees j algebra 237 2001 591 620 on the zeta functions of periodic graphs in particular using appropriate operator algebraic techniques we establish a determinant formula in this context and examine its consequences for the ihara zeta function moreover we answer in the affirmative one of the questions raised in r i grigorchuk a uk the ihara zeta function of infinite graphs the kns spectral measure and integrable maps in v a kaimanovich et al eds proc workshop random walks and geometry vienna 2001 de gruyter berlin 2004 pp 141 180 by grigorchuk and uk accordingly we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs

extended scaling relations for planar lattice models it is widely believed that the critical properties of several planar lattice systems like the eight vertex or the ashkin teller models are well described by an effective continuum fermionic theory obtained as a formal scaling limit on the basis of this assumption several extended scaling relations among their indices were conjectured we prove the validity of some of them among which the ones predicted by kadanoff phys rev lett 39 903 905 1977 and by luther and peschel phys rev b 12 3908 3917 1975

eigenvalues of graphs and a simple proof of a theorem of greenberg abstract in his ph d thesis greenberg proved that if x is the spectral radius of the universal cover x of a finite graph x then for each 0 a positive proportion depending only on x and of the eigenvalues of x have absolute value at least x in this paper we show that the same result holds true if we remove absolute from the previous result we also prove an analogue result for the smallest eigenvalues of x

transcendence of hecke operators in the big hecke algebra we prove a transcendence result of hecke operator t l over the ring generated by diamond operators over in a non cm component of the cyclotomic ordinary big p adic hecke algebra and we discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic invariant 2014

corps de nombres peu ramifies et formes automorphes autoduales let s be a finite set of primes p in s and q s a maximal algebraic extension of q unramified outside s and infinity assume that s 2 we show that the natural maps gal q p bar q p gal q s q are injective much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of gl 2n a q with prescribed properties at all places that we study via the twisted trace formula of j arthur the techniques we develop shed also some lights on the orthogonal symplectic alternative for selfdual representations of gl 2n

variational principles for indefinite eigenvalue problems abstract let a and b be n n complex hermitian matrices where b is nonsingular but neither a nor b need be definite let sk denote the linear subspaces of c n of codimension k and let k sup inf x ax x bx 1 x s s s k 1 assuming that the real eigenvalues of the problem a b are semisimple we show how to calculate the value of each k and the corresponding inf sups in consequence we show precisely which eigenvalues of can be obtained by variational formulae of the above types

on the asymptotic isoperimetric constants for riemann surfaces and graphs we study the behavior of the cheeger isoperimetric constant on infinite fam ilies of graphs and riemann surfaces and its relationship to the first eigen value 1 of the laplacian we adapt probabilistic arguments of bollobas to the setting of riemann surfaces and then show that cheeger constants of the modular surfaces are uniformly bounded from above away from the maximum value we extend this result to the class of ramanujan surfaces defined below

on a class of quasilinear problems with sign changing nonlinearities abstract we prove existence and nonexistence of positive solutions for the quasilinear problem p u m p u a x f u in u 0 on where p u div u p 2 u p u u p 2 u p 1 m 0 is a bounded domain in r n and the coefficient a x is allowed to change sign

critical properties of the ashkin teller model from the mean field renormalisation group approach by using small size clusters which carry all the basic symmetries of the ashkin teller model the authors obtain within the mean field renormalisation group approach the complete phase diagram of the model estimates of the thermal critical exponent are also obtained

pure and dilute z n spin and generalised gauge lattice systems duality and criticality considers the pure z n spin systems including the standard ising and potts models as well as generalised gauge systems plaquettes or more complex simplex in d dimensional hypercubic lattices these models are self dual and the authors show how this duality can be thought of as a series parallel transformation the simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems including some particular asymptotic behaviours which are exact as an illustration the d 2 diluted z 4 spin system is discussed in some detail for those regions where exact results are available the agreement is satisfactory

positive solutions of second order systems of boundary value problems abstract the existence of nonnegative solutions of boundary value problems of the form x t a t x t y t 0 y t b t g x t y t 0 x 0 x 1 y 0 y 1 0 is investigated where and g are functions from r p r q into r p and r q respectively and a b are functions defined on 0 1 with values in the spaces of diagonal matrices of order p and q respectively all functions are assumed to be continuous and nonnegativity as well as monotonicity and growth conditions are imposed on and g the existence results hold for a bounded interval of values of if and g are assumed to grow fast and have only nonzero values while if 0 0 0 g 0 0 0 it is shown that under reasonable conditions there exists a nontrivial nonnegative solution for every 0

mean field renormalization group study of a modified ashkin teller model abstract we study the phase diagram of a generalization of the ashkin teller model in two dimensions using the mean field renormalization group mfrg theory we calculate critical surfaces for this three parameter hamiltonian of the model using two different approaches and we discuss the resulting phase diagram and compare it with the results from the molecular field approximation

a finite element approach for finding positive solutions of a logistic equation with a sign changing weight function in this article we describe the mathematical modeling and use finite element approximation to study the existence and multiplicity of numerical positive solutions of well known generalized logistic equation with sign changing weight function we investigate the range of the real parameter involved in this problem to achieve the numerical solutions and discuss the behavior of the branch of the solutions using matlab 7 0

critical ising lines of the d 2 ashkin teller model the universal critical point ratio q is exploited to determine positions of the critical ising transition lines on the phase diagram of the ashkin teller at model on the square lattice a leading order expansion of the ratio q in the presence of a non vanishing thermal field is found from finite size scaling and the corresponding expression is fitted to the accurate perturbative transfer matrix data calculations for the l times l square clusters with l leq 9

anomalous universality in the anisotropic ashkin teller model the ashkin teller at model is a generalization of ising 2 d to a four states spin model it can be written in the form of two ising layers in general with different couplings interacting via a four spin interaction it was conjectured long ago by kadanoff and wegner wu and lin baxter and others that at has in general two critical points and that universality holds in the sense that the critical exponents are the same as in the ising model except when the couplings of the two ising layers are equal isotropic case we obtain an explicit expression for the specific heat from which we prove this conjecture in the weakly interacting case and we locate precisely the critical points we find the somewhat unexpected feature that despite universality holds for the specific heat nevertheless nonuniversal critical indexes appear for instance the distance between the critical points rescale with an anomalous exponent as we let the couplings of the two ising layers coincide isotropic limit and so does the constant in front of the logarithm in the specific heat our result also explains how the crossover from universal to nonuniversal behaviour is realized

on linear and nonlinear fourth order eigenvalue problems with indefinite weight abstract we determine the principal eigenvalues of the linear indefinite weight problem u x r g x u x x 0 1 u 0 u 1 u 0 u 1 0 moreover we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem where g 0 1 r is a continuous function which attains both positive and negative values f c r r and r is a parameter

an explicit trace formula of jacquet zagier type for hilbert modular forms abstract we give an exact formula of the average of adjoint l functions of holomorphic hilbert cusp forms with a fixed weight and a square free level which is a generalization of zagier s formula known for the case of elliptic cusp forms on sl 2 z as an application we prove that the satake parameters of hilbert cusp forms with a fixed weight and with growing square free levels are equidistributed in an ensemble constructed by values of the adjoint l functions

positive solutions to a class of elliptic boundary value problems abstract we prove the existence of positive solutions to the boundary value problems formula where is a bounded domain inrn a rmay change sign f 0 0 and 0 is sufficiently small our approach is based on the leray schauder fixed point theorem

phase diagram of the two dimensional ferromagnetic three color ashkin teller model in the present work we study the critical properties of the ferromagnetic three color ashkin teller model 3at by means of a migdal kadanoff renormalization group approach on a diamond like hierarchical lattice the analysis of the fixed points and flux diagram of the recursion relations is used to determine the corresponding phase diagram including its symmetry properties and critical exponents our numerical results show the presence of four universality classes three of them are associated to the potts model with q 2 4 and 6 states finally a connection between our findings and some known results from the literature is presented

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

universality of one dimensional fermi systems i response functions and critical exponents the critical behavior of one dimensional interacting fermi systems is expected to display universality features called luttinger liquid behavior critical exponents and certain thermodynamic quantities are expected to be related among each other by model independent formulas we establish such relations the proof of which has represented a challenging mathematical problem for a general model of spinning fermions on a one dimensional lattice interactions are short ranged and satisfy a positivity condition which makes the model critical at zero temperature proofs are reported in two papers in the present one we demonstrate that the zero temperature response functions in the thermodynamic limit are borel summable and have anomalous power law decay with multiplicative logarithmic corrections critical exponents are expressed in terms of convergent expansions and depend on all the model details all results are valid for the special case of the hubbard model

renormalization group study of the two dimensional ising model with crossing bonds abstract the simple quadratic ising model with nearest and next nearest neighbor interactions is investigated by means of the mean field renormalization group approach by conveniently choosing the size of the blocks one obtains the expected phase diagram exhibiting ferromagnetic antiferromagnetic paramagnetic and superantiferromagnetic phases the critical lines which separate the paramagnetic phase from the ordered ones have the expected asymptotic behavior for large values of the nearest neighbor coupling

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

elliptic curves over a finite field and the trace formula we prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k points of the curves contain a chosen subgroup these formulas express the moments in terms of traces of hecke operators for certain congruence subgroups of sl 2 z as our main technical input we prove an eichler selberg trace formula for a family of congruence subgroups of sl 2 z which include as special cases the groups 1 n and n our formulas generalize results of birch and ihara the case of the trivial subgroup and the full modular group and previous work of the authors the subgroups z 2z and z 2z 2 and congruence subgroups 0 2 0 4 we use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field generalizing results of vladu gekeler howe and others

effective action for the yukawa 2 quantum field theory using a rigorous version of the renormalization group we construct the effective action for they2 model the construction starts with integrating out the bosonic field which eliminates the large fields problem studying the soobtained purely fermionic theory proceeds by a series of convergent perturbation expansions we show that the continuum limit of the effective action exists and its perturbation expansion is borel summable

phase diagram of the n colour ashkin teller model the author shows that the two dimensional n colour ashkin teller model exhibits an ising like continuous transition line in the region where the model has two phase transitions the result is also valid for the antiferromagnetic ordering in addition they present results which support the conjecture of grest and widom 1981 about the location of the disorder point

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

effective multiplicity one on gl n and narrow zero free regions for rankin selberg l functions we establish zero free regions tapering as an inverse power of the analytic conductor for rankin selberg l functions on gln gln such zero free regions are equivalent to commensurate lower bounds on the edge of the critical strip and in the case of l s on the residue at s 1 as an application we show that a cuspidal automorphic representation on gln is determined by a finite number of its dirichlet series coefficients and that this number grows at most polynomially in the analytic conductor introduction let a be the ring of adeles over a number field f and let and be two cuspidal representations of gln a with restricted tensor product decompositions v v and v

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

phase transitions in the anisotropic ashkin teller model by using mean field approximation mfa and monte carlo mc simulations we have studied the effect of the anisotropy on the phase diagrams of the ashkin teller model a t m on a hypercubic lattice by varying the strength of the anisotropic parameter partially ordered phases connected by symmetry and denoted by and s appear in all dimensions d 1 as in the three dimensional isotropic case for antiferromagnetic interactions the mc simulations reduce the varieties of phase diagrams obtained from mfa to one variety in d 2 the phase diagrams present ising like phase transitions while in d 3 a first order phase transition and then multicritical points appear

the scaling limit of the energy correlations in non integrable ising models we obtain an explicit expression for the multipoint energy correlations of a non solvable two dimensional ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory which generalizes the one originally used by schultz mattis and lieb for the nearest neighbor ising model the interacting model is then analyzed by a multiscale method first proposed by pinson and spencer if the lattice spacing is finite then the correlations cannot be computed in closed form rather they are expressed in terms of infinite convergent power series in in the scaling limit these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable ising model up to a finite renormalization of the parameters explicit bounds on the speed of convergence to the scaling limit are derived

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

equidistribution theorems for holomorphic siegel modular forms for gsp_4 hecke fields and n level density this paper is a continuation of kim et al j inst math jussieu 2018 we supplement four results on a family of holomorphic siegel cusp forms for gsp 4 mathbb q first we improve the result on hecke fields namely we prove that the degree of hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of gsp 4 second we prove simultaneous vertical sato tate theorem namely we prove simultaneous equidistribution of hecke eigenvalues at finitely many primes third we compute the n level density of degree 4 spinor l functions and thus we can distinguish the symmetry type depending on the root numbers this is conditional on certain conjecture on root numbers fourth we consider equidistribution of paramodular forms in this case we can prove the conjecture on root numbers main tools are the equidistribution theorem in our previous work and shin templier s compos math 150 12 2003 2053 2014 work

mean field renormalisation group transformations for the triangular ising antiferromagnet the mean field renormalisation group of indekeu et al 1982 is applied to the antiferromagnetic nearest neighbour ising model on the triangular lattice the resulting phase diagram in the temperature field plane is in good agreement with other calculations while the predicted specific heat index alpha is negative and thus qualitatively wrong the study indicates that the method may be a useful approach at least for determining phase diagrams for frustrated systems and systems with competing interactions where conventional mean field theory is dubious

mollification of the fourth moment of automorphic l functions and arithmetic applications we compute the asymptotics of twisted fourth power moments of modular l functions of large prime level near the critical line this allows us to prove some new non vanishing results on the central values of automorphic l functions in particular those obtained by base change from gl 2 q to gl 2 k for k a cyclic field of low degree

a positive solution branch for nonlinear eigenvalue problems in rn abstract in this note we give sufficient conditions for the existence of a positive solution branch for the problem u a x f u in r n u x 0 as x with sign changing weight a where n 3 and f is a smooth nonlinearity with f 0 0

existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function this paper is concerned with the existence multiplicity and stability of positive solutions of an indefinite weight boundary value problemu email protected t f u 0 0

eigenvalue spacings for quantized cat maps according to one of the basic conjectures in quantum chaos the eigenvalues of a quantized chaotic hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices we study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms cat maps our numerical experiments indicate that for generic choices of cat maps the unfolded consecutive spacing distribution in the irreducible components of the nth quantization given by the n dimensional weil representation approaches the goe gse law of random matrix theory for certain special arithmetic transformations related to the ramanujan graphs of lubotzky phillips and sarnak the experiments indicate that the unfolded consecutive spacing distribution follows poisson statistics we provide a sharp estimate in that direction

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

length scaling in mfrg the mean field renormalisation group mfrg has so far given poor estimates for critical indices it is argued that this can be traced to an unsatisfactory definition of the length scaling factor a new definition that improves the estimates in most cases is proposed

on positive solutions for a second order differential system with indefinite weight in this paper we consider the existence of positive solutions for a second order differential system u a t u h t f u 0 t 1 b t u 0 t 1 u 0 u 1 0 0 1 0 where a t b t h t change sign the proof is based on the well known fixed point theorem of cone expansion and compression

when kloosterman sums meet hecke eigenvalues by elaborating a two dimensional selberg sieve with asymptotics and equidistributions of kloosterman sums from ell adic cohomology as well as a bombieri vinogradov type mean value theorem for kloosterman sums in arithmetic progressions it is proved that for any given primitive hecke maass cusp form of trivial nebentypus the eigenvalue of the n th hecke operator does not coincide with the kloosterman sum mathrm kl 1 n for infinitely many squarefree n with at most 100 prime factors this provides a partial negative answer to a problem of katz on modular structures of kloosterman sums

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

averages of central l values of hilbert modular forms with an application to subconvexity we use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change l functions averaged over hilbert modular forms of a fixed weight and level we apply these formulas to the subconvexity problem for these l functions we also establish an equidistribution result for the hecke eigenvalues weighted by these l values

non universal exponents and marginal operators via a monte carlo renormalisation group the authors study the critical exponents eigenoperators and the phase diagram of the n 2 ashkin teller at model in two dimensions using a monte carlo renormalisation group they pay special attention to the marginal operator responsible for the non universal behaviour the crossover exponent is measured and used to numerically calculate the thermal exponent of the eight vertex model and of the at model near the potts four point giving better results than a direct determination since the crossover operator is not affected by the additional at marginal operators