positive solutions of second order three point boundary value problems with change of sign in banach spaces abstract in this paper by using the fixed points of strict set contractions we study the existence of at least one or two positive solutions to the second order three point boundary value problem y t a t f y t 0 t 1 y 0 y 1 y in banach space e where is the zero element of e 0 1 0 1 and a t is allowed to change sign on 0 1 as an application we also give one example to demonstrate our results

blow up of solutions of semilinear euler poisson darboux equations with nonlocal boundary conditions this article studies the hyperbolic initial nonlocal boundary value problem u t t k t u t u x x u 0 0 u x 0 u 0 x u t x 0 0 0 0 u a t 0 an y u y t q dy t 0 where k is a real number p and q are nonnegative constants and u 0 m and n are given functions criteria for u to blow up in finite time are given

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

comparison of bootstrap percolation models new simulations involving up to nearly 109 sites for square lattice bootstrap percolation give a percolation threshold compatible with unity in this model sites in a square lattice are initially occupied randomly and then are removed if they have less than three occupied neighbours the results are compared with those of similar models such as that studied by frobose 1989 and the earlier bootstrap and diffusion percolation models in two three and four dimensions of adler and aharony 1986 8 in the latter models the effective threshold approaches its asymptotic limit one or zero as lambda ln l d 1 for samples of size ld in agreement with theoretical predictions the coefficient lambda decreases monotonically with the parameters which facilitate the removal addition of sites

a cotangent sum related to zeros of the estermann zeta function abstract we consider a cotangent sum related to estermann s zeta function we provide an elementary and self contained improvement of the error term in an asymptotic formula proved by v i vasyunin

a sharper threshold for bootstrap percolation in two dimensions two dimensional bootstrap percolation is a cellular automaton in which sites become infected by contact with two or more already infected nearest neighbours we consider these dynamics which can be interpreted as a monotone version of the ising model on an n n square with sites initially infected independently with probability p the critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1 2 holroyd determined the sharp first order approximation pc 2 18 log n as n here we sharpen this result proving that the second term in the expansion is log n 3 2 o 1 and moreover determining it up to a poly log log n factor the exponent 3 2 corrects numerical predictions from the physics literature

asymptotic formulas for vasyunin cotangent sums abstract we study the vasyunin type cotangent sum c 0 h k m 1 k 1 m k cot h m k where h and k are positive coprime integers this sum is related to estermann zeta function by applying the euler maclaurin summation formula to a suitable function we improve a previous large k asymptotic approximation of c0 h k we also provide a procedure to compute an arbitrary number of terms of the approximation

existence results for n point boundary value problem of second order ordinary differential equations abstract this study concerns the existence of positive solutions to the boundary value problem u a t f u 0 t 0 1 u 0 i 1 n 2 b i u i u 1 i 1 n 2 a i u i where i 0 1 with 0 1 2 n 2 1 a i b i 0 with 0 i 1 n 2 a i 1 and i 1 n 2 b i 1 by applying the krasnoselskii s fixed point theorem in banach spaces some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n point boundary value problem

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

general formalism for inhomogeneous random graphs we present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models where vertices come in different types and the probability of realizing an edge depends on the types of its terminal vertices this approach provides a general framework for the analysis of a large class of models the generic phase structure is derived using generating function techniques and relations to other classes of models are pointed out

on mean values of mollifiers and l functions associated to primitive cusp forms we study the second moment of the l function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers this family is a natural extension of the mollifers considered by conrey and by bui conrey and young as an application we improve the current lower bound on critical zeros of holomorphic primitive cusp forms

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

bootstrap percolation on a graph with random and local connections let g n p 1 be a superposition of the random graph g n p and a one dimensional lattice the n vertices are set to be on a ring with fixed edges between the consecutive vertices and with random independent edges given with probability p between any pair of vertices bootstrap percolation on a random graph is a process of spread of activation on a given realization of the graph with a given number of initially active nodes at each step those vertices which have not been active but have at least r ge 2 active neighbours become active as well we study the size of the final active set in the limit when n rightarrow infty the parameters of the model are n the size a 0 a 0 n of the initially active set and the probability p p n of the edges in the graph the bootstrap percolation process on g n p was studied earlier here we show that the addition of n local connections to the graph g n p leads to a more narrow critical window for the phase transition preserving however the critical scaling of parameters known for the model on g n p we discover a range of parameters which yields percolation on g n p 1 but not on g n p

a sharp threshold for a modified bootstrap percolation with recovery bootstrap percolation is a type of cellular automaton on graphs introduced as a simple model of the dynamics of ferromagnetism vertices in a graph can be in one of two states healthy or infected and from an initial configuration of states healthy vertices become infected by local rules while the usual bootstrap processes are monotone in the sets of infected vertices in this paper a modification is examined in which infected vertices can return to a healthy state vertices are initially infected independently at random and the central question is whether all vertices eventually become infected the model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected

three point boundary value problems for second order ordinary differential equations in banach spaces in this paper by using the sadovskii fixed point theorem we study the existence of at least one solution for the second order three point boundary value problem u t f t u t u t email protected 0

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

bootstrap percolation in a polluted environment let a low densityp of sites on the lattice z2 be occupied remove a proportionq of them and call the remaining sites empty then update this configuration in discrete time by iteration of the following synchronous rule an empty site becomes occupied by contact with at least two occupied nearest neighbors while occupied and removed sites nerver change their states ifq p2 is large most sites remain unoccupied forever while ifq p2 is small this dynamics eventually makes most sites occupied this demonstrates how sensitive the usual bootstrap percolation rule theq 0 case is to the pollution of space

distribution of a cotangent sum related to the nyman beurling criterion for the riemann hypothesis abstract a certain category of cotangent sums has been proven of importance in the nyman beurling criterion for the riemann hypothesis in previous work 12 13 the authors proved the existence of a unique positive measure on r with respect to which certain normalized cotangent sums evaluated at rational numbers with fixed denominators are equidistributed the tools applied in this paper belong to various fields of mathematics for instance the relation between the equidistribution mod 1 of the multiples of a number and the diophantine approximation properties of that number in this paper we prove an analogous result for the case that the denominator of the rational numbers is a fixed prime number and that the numerator is also prime

metastable behavior for bootstrap percolation on regular trees we examine bootstrap percolation on a regular b 1 ary tree with initial law given by bernoulli p the sites are updated according to the usual rule a vacant site becomes occupied if it has at least occupied neighbors occupied sites remain occupied forever it is known that when b 2 the limiting density q q p of occupied sites exhibits a jump at some p t p t b 0 1 from q t q p t p t we investigate the metastable behavior associated with this transition explicitly we pick p p t h with h 0 and show that as h 0 the system lingers around the critical state for time order h 1 2 and then passes to fully occupied state in time o 1 the law of the entire configuration observed when the occupation density is q q t 1 converges as h 0 to a well defined measure

sharp thresholds in bootstrap percolation in the standard bootstrap percolation on the d dimensional grid gnd in the initial position each of the nd sites is occupied with probability p and empty with probability 1 p independently of the state of every other site once a site is occupied it remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial configuration percolates by making use of a theorem of friedgut and kalai proc amer math soc 124 1996 2993 we shall show that the threshold function of the percolation is sharp we shall prove similar results for three other models of bootstrap percolation as well

bootstrap percolation in three dimensions by bootstrap percolation we mean the following deterministic process on a graph g given a set a of vertices infected at time 0 new vertices are subsequently infected at each time step if they have at least n previously infected neighbors when the set a is chosen at random the main aim is to determine the critical probability p c g r at which percolation infection of the entire graph becomes likely to occur this bootstrap process has been extensively studied on the d dimensional grid n d with 2 r d fixed it was proved by cerf and cirillo for d r 3 and by cerf and manzo in general that p c n d r 1 log r 1 n d r 1 where log r is an r times iterated logarithm however the exact threshold function is only known in the case d r 2 where it was shown by holroyd to be 1 o 1 2 18 log n in this paper we shall determine the exact threshold in the crucial case d r 3 and lay the groundwork for solving the problem for all fixed d and r

rates for the probability of large cubes being non internally spanned in modified bootstrap percolation we find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation it is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line

boundary value problems for second order differential equations on unbounded domains in a banach space in this paper the sadovskii fixed point theorem is used to investigate the existence of solutions of two point boundary value problems bvp for second order nonlinear differential equations on infinite intervals in a banach space

integrals partitions and cellular automata we prove that 1 0 log f x x dx 2 3ab where f x is the decreasing function that satisfies f a f b x a x b for 0 a b when a is an integer and b a 1 we deduce several combinatorial results these include an asymptotic formula for the number of integer partitions not having a consecutive parts and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation

the time of bootstrap percolation in two dimensions we study the distribution of the percolation time t of 2 neighbour bootstrap percolation on n 2 with initial set a sim text bin n 2 p we determine t up to a constant factor with high probability for all p above the critical probability for percolation and to within a 1 o 1 factor for a large range of p

geometry of dynamically available empty space is the key to near arrest dynamics received 6 october 2004 revised manuscript received 23 may 2005 published 3 august 2005we study several examples of kinetically constrained lattice models using dynamically accessible volume asan order parameter thereby we identify two distinct regimes exhibiting dynamical slowing with a sharpthreshold between them these regimes are identi ed both by a new response function in dynamically availablevolume as well as directly in the dynamics results for the self diffusion constant in terms of the connectedhole density are presented and some evidence is given for scaling in the limit of dynamical arrest doi 10 1103 physreve 72 021401 pacs number s 61 43 fs 64 60 ak

bootstrap percolation in power law random graphs a bootstrap percolation process on a graph is an infection process which evolves in rounds initially there is a subset of infected nodes and in each subsequent round each uninfected node which has at least infected neighbours becomes infected and remains so forever the parameter is fixed such processes have been used as models for the spread of ideas or trends within a network of individuals we analyse this process in the case where the underlying graph is an inhomogeneous random graph which exhibits a power law degree distribution and initially there are randomly infected nodes the main focus of this paper is the number of vertices that will have been infected by the end of the process the main result of this work is that if the degree sequence of the random graph follows a power law with exponent where then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph with high probability more specifically we determine explicitly a critical function such that with the following property assuming that is the number of vertices of the underlying random graph if then the process does not evolve at all with high probability as grows whereas if then there is a constant such that with high probability the final set of infected vertices has size at least this behaviour is in sharp contrast with the case where the underlying graph is a random graph with it follows from an observation of balogh and bollobas that in this case if the number of initially infected vertices is sublinear then there is lack of evolution of the process it turns out that when the maximum degree is then depends also on but when the maximum degree is then

slow droplet driven relaxation of stochastic ising models in the vicinity of the phase coexistence region we consider the stochastic ising models glauber dynamics corresponding to the infinite volume basic ising model in arbitrary dimensiond 2 with nearest neighbor interaction and under a positive external magnetic fieldh under minimal assumptions on the rates of flip so that all the common choices are included we obtain results which state that when the system is at low temperaturet the relaxation time when the evolution is started with all the spins down blows up whenh 0 as exp t hd 1 the precise results are lower and upper bounds of this form moreover after a time which does not scale withh and before a time which also grows as an exponential of a multiple of 1 hd 1 ash 0 the law of the state of the process stays whenh is small close to the minus phase of the same ising model without an external field these results may be considered as a partial vindication of a conjecture raised by aizenman and lebowitz in connection to the metastable behavior of these stochastic ising models

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

bootstrap percolation on homogeneous trees has 2 phase transitions we study the threshold bootstrap percolation model on the homogeneous tree with degree b 1 2 b and initial density p it is known that there exists a nontrivial critical value for p which we call pf such that a for p pf the final bootstrapped configuration is fully occupied for almost every initial configuration and b if p pc then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration moreover we show that 3 for p pc the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail 4 at p pc the expected occupied cluster size in the final bootstrapped configuration is infinite 5 the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on 0 pf and analytic on pc pf admitting an analytic continuation from the right at pc and only in the case b also from the left at pf

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

twisted second moments of the riemann zeta function and applications abstract in order to compute a twisted second moment of the riemann zeta function two different mollifiers each being a combination of two different dirichlet polynomials were introduced separately by bui conrey and young and by feng in this article we introduce a mollifier which is a combination of four dirichlet polynomials of different shapes we provide an asymptotic result for the twisted second moment of s for such choice of mollifier a small increment on the percentage of zeros of the riemann zeta function on the critical line is given as an application of our results

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

boundary value problems for impulsive integro differential equations on unbounded domains in a banach space in this paper the fixed point theory is used to investigate the existence and uniqueness of solutions of two point boundary value problems bvps for second order non linear impulsive integro differential equations on infinite intervals in a banach space

finite size scaling behavior of a biased majority rule cellular automaton abstract we consider the biased majority rule cellular automaton in which the sites take the majority of the states 0 or 1 of their neighbors to update their states with a bias towards 1 in case of ties we observe that in any dimension for any initial density of 1 s the probability of final complete occupancy by 1 s converges to 1 as the lattice grows in two dimensions the value p l of the initial density for which the probability of eventual complete occupancy of a lattice l l is 0 1 is shown to satisfy for fixed the bounds a log l b log l for large l where a and b are positive and finite constants

positive solutions of a nonlinear four point boundary value problems in this paper by using the krasnoselskii s theorem in a cone we study the existence of at least one or two positive solutions to the four point boundary value problemy t a t f y t 0 00 as an application we also give some examples to demonstrate our results

multiple solutions of two point boundary value problems of ordinary differential equations in banach spaces soit p un cone normal de l espace de banach reel e on note par n la constante normale de p on considere le probleme aux deux points limites x f t x t i x 0 x 1 ou i 0 1 et f une application continue de i p dans p on etudie l existence de solutions positives multiples

the twisted fourth moment of the riemann zeta function we compute the asymptotics of the fourth moment of the riemann zeta func tion times an arbitrary dirichlet polynomial of length t 1 11 the study of the moments of the riemann zeta function has a long and distinguished history starting with the work of hardy and littlewood in 1918 and continuing to the present day one motivation for understanding moments is that they yield information about the maximum size of the zeta function the lindelof hypothesis another application is to zero density estimates which in turn have consequences for primes in short intervals however they have become an interesting topic in their own right very few rigorous results are known just the second and fourth power moments indeed it is only recently that a believable conjecture for higher powers has been made the twisted moments that is moments of the riemann zeta function times an arbitrary dirichlet polynomial are important too for example levinson s method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment in a series of papers duke friedlander and iwaniec used estimates for amplified moments of a family of l functions in order to deduce a subconvexity bound for an individual member of the family of course there are far easier methods to give a subconvexity bound for zeta but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general in this paper we prove an asymptotic formula for the twisted fourth moment of the riemann zeta function where we may take a dirichlet polynomial of length up to t 1 11 e

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

approximate solution of three point boundary value problems for second order ordinary differential equations with variable coefficients three point boundary value problem is investigated and solved numerically a novel simple and effective numerical method is proposed the convergence and error estimate of approximate solution are given numerical results are carried out to verify the proposed methods some three point boundary value problems for a second order ordinary differential equation with variable coefficients are investigated in the present paper by using the integration method the second order three point boundary value problems are transformed into a fredholm integral equation of the second kind the solutions and green s functions for some special cases of the second order three point boundary value problems can be determined easily the existence and uniqueness of the solutions of the given fredholm integral equations are considered by using the fixed point theorem in banach spaces a new numerical method is further proposed to solve the second kind fredholm integral equation and an approximate solution is made the convergence and error estimate of the obtained approximate solution are further analyzed numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures

exact solutions of a class of second order nonlocal boundary value problems and applications in this paper solutions of a class of second order differential equations with some multi point boundary conditions are studied we give exact expressions of the solutions for the linear m point boundary problems by the green s functions as applications we study uniqueness and iteration of the positive solutions for a nonlinear singular second order m point boundary value problem

finite size scaling in three dimensional bootstrap percolation we consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior bootstrap percolation is an example of cellular automata defined on the d dimensional lattice 1 2 l d in which each site can be empty or occupied by a single particle in the starting configuration each site is occupied with probability p occupied sites remain occupied forever while empty sites are occupied by a particle if at least among their 2d nearest neighbor sites are occupied when d is fixed the most interesting case is the one d this is a sort of threshold in the sense that the critical probability p c for the dynamics on the infinite lattice z d switches from zero to one when this limit is crossed finite size effects in the three dimensional case are already known in the cases 2 in this paper we discuss the case 3 and we show that the finite size scaling function for this problem is of the form f l const in in l we prove a conjecture proposed by a c d van enter

bootstrap percolation on the hypercube in the bootstrap percolation on the n dimensional hypercube in the initial position each of the 2n sites is occupied with probability p and empty with probability 1 p independently of the state of the other sites every occupied site remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial position spans the hypercube we shall show that there are constants c1 c2 0 such that for open image in new window the probability of spanning tends to 1 as n while for open image in new window the probability tends to 0 furthermore we shall show that for each n the transition has a sharp threshold function

finite size effects for some bootstrap percolation models the consequences of schonmann s new proof that the critical threshold is unity for certain bootstrap percolation models are explored it is shown that this proof provides an upper bound for the finite size scaling in these systems comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics we show that the threshold for a finite system of sizel scales aso ln lnl for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled

universality for critical kcm finite number of stable directions kinetically constrained models kcm are reversible interacting particle systems on mathbb z d with continuous time constrained glauber dynamics they are a natural non monotone stochastic version of the family of cellular automata with random initial state known as mathscr u bootstrap percolation kcm have an interest in their own right owing to their use for modelling the liquid glass transition in condensed matter physics in two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes here we study in full generality the class termed critical together with the companion paper by hartarsky et al universality for critical kcm finite number of stable directions arxiv e prints arxiv 1910 06782 2019 we establish the universality classes of critical kcm and determine within each class the critical exponent of the infection time as well as of the spectral gap in this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart this is due to the occurrence of energy barriers which determine the dominant behaviour for these kcm but which do not matter for the monotone bootstrap dynamics our result confirms the conjecture of martinelli et al commun math phys 369 2 761 809 https doi org 10 1007 s00220 018 3280 z 2019 who proved a matching upper bound

the twisted mean square and critical zeros of dirichlet l functions in this work we obtain an asymptotic formula for the twisted mean square of a dirichlet l function with a longer mollifier whose coefficients are also more general than before as an application we obtain that for every dirichlet l function more than 41 72 of zeros are on the critical line and more than 40 74 of zeros are simple and on the critical line these proportions also improve previous results which were proved only for the riemann zeta function