free energy and correlations of the number theoretical spin chain we analyze the number theoretical spin chain with partition function z 1 using the polymer model technique the finite grand canonical chains give bounds for the limit free energy and internal energy the correlation functions for inverse temperature 1 are products of two point functions a combinatorial result for general interval graphs is derived

constructing singular functions via farey fractions abstract to illustrate some points about continued fractions h minkowski in 1904 introduced the so called function this function and some generalizations of it are known to be singular i e strictly monotone with derivative 0 almost everywhere they can be characterized by systems of functional equations such as f biggl x over x 1 biggr tf x qquad f biggl 1 over x 1 biggr 1 1 t f x qquad hbox for all x in 0 1 eqno hbox f wheref 0 1 r is the unknown and r biggl x over x 1 biggr tr x qquad r biggl 1 over 2 x biggr t 1 t r x qquad hbox for all x in 0 1 eqno hbox r wherer 0 1 r is the unknown in both cases t 0 1 is a given parameter in the present note we give a general construction of singular functions based on the farey fractions and including as a special case the minkowski function and its generalizations in contrast to earlier proofs the methods presented here do not make explicit use of the theory of continued fractions

fluctuation theory of connectivities for subcritical random cluster models we develop a fluctuation theory of connectivities for subcritical random cluster models the theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one dimensional chains of irreducible objects statistics of local observables for example displacement over such chains obey classical limit laws and our construction leads to an effective random walk representation of percolation clusters the results include a derivation of a sharp ornstein zernike type asymptotic formula for two point functions a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling in two dimensions duality considerations enable a reformulation of these results for supercritical nearest neighbor random cluster measures in particular for nearest neighbor potts models in the phase transition regime accordingly we prove that in two dimensions potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive thus no roughening transition is possible in the whole regime where our results apply our results hold under an assumption of exponential decay of finite volume wired connectivities assumption 1 2 below in rectangular domains that is conjectured to hold in the whole subcritical regime the latter is known to be true in any dimensions when q 1 q 2 and when q is sufficiently large in two dimensions assumption 1 2 holds whenever there is an exponential decay of connectivities in the infinite volume measure by duality this includes all supercritical nearest neighbor potts models with positive surface tension between ordered phases

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

random cluster dynamics in mathrm mathbb z 2 the random cluster model has been widely studied as a unifying framework for random graphs spin systems and electrical networks but its dynamics have so far largely resisted analysis in this paper we analyze the glauber dynamics of the random cluster model in the canonical case where the underlying graph is an n times n box in the cartesian lattice mathrm mathbb z 2 our main result is a o n 2 log n upper bound for the mixing time at all values of the model parameter p except the critical point p p c q and for all values of the second model parameter q ge 1 we also provide a matching lower bound proving that our result is tight our analysis takes as its starting point the recent breakthrough by beffara and duminil copin on the location of the random cluster phase transition in mathrm mathbb z 2 it is reminiscent of similar results for spin systems such as the ising and potts models but requires the reworking of several standard tools in the context of the random cluster model which is not a spin system in the usual sense

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

convergence analysis of strang splitting for vlasov type equations a rigorous convergence analysis of the strang splitting algorithm for vlasov type equations in the setting of abstract evolution equations is provided it is shown that under suitable assumptions the convergence is of second order in the time step tau as an example it is verified that the vlasov poisson equations in 1 1 dimensions fit into the framework of this analysis further numerical experiments for the latter case are presented

power law corrections to exponential decay of connectivities and correlations in lattice models consider a translation invariant bond percolation model on the integer lattice which has exponential decay of connectivities that is the probability of a connection 0 x by a path of open bonds decreases like exp m x for some positive constant m which may depend on the direction x x in two and three dimensions it is shown that if the model has an appropriate mixing property and satisfies a special case of the fkg property then there is at most a power law correction to the exponential decay there exist a and c such that exp m x p 0 x a x c exp m x for all nonzero x in four or more dimensions a similar bound holds with x c replaced by exp c log x 2 in particular the power law lower bound holds for the fortuin kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially since the mixing property is known to hold in that case consequently a similar bound holds for correlations in the potts model at supercritical temperatures

the minkowski question mark function explicit series for the dyadic period function and moments previously several natural integral transforms of the minkowski question mark function f x were introduced by the author each of them is uniquely characterized by certain regularity conditions and the functional equation thus encoding intrinsic information about f x one of them the dyadic period function g z was defined as a stieltjes transform in this paper we introduce a family of distributions f p x for rp 1 such that f 1 x is the question mark function and f 2 x is a discrete distribution with support on x 1 we prove that the generating function of moments of f p x satisfies the three term functional equation this has an independent interest though our main concern is the information it provides about f x this approach yields the following main result we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients

orthogonal polynomials for minkowski s question mark function hermann minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as minkowski s question mark function since minkowski used the notation x this function is a distribution function on 0 1 which defines a singular continuous measure with support 0 1 our interest is in the monic orthogonal polynomials p n n n for the minkowski measure and in particular in the behavior of the recurrence coefficients of the three term recurrence relation we will give some numerical experiments using the discretized stieltjes gautschi method with a discrete measure supported on the minkowski sequence we also explain how one can compute the moments of the minkowski measure and compute the recurrence coefficients using the chebyshev algorithm

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

moments of discrete measures with dense jumps induced by β expansions abstract let 1 through an appeal to expansions we define a strictly increasing and left continuous function on 0 1 then turns out to be a pure jump distribution in other words its associated lebesgue stieltjes measure is discrete i e a summation of point masses the present note studies the moment of this discrete measure and its asymptotics

a farey fraction spin chain we introduce a new number theoretic spin chain and explore its thermodynamics and connections with number theory the energy of each spin configuration is defined in a translation invariant manner in terms of the farey fractions and is also expressed using pauli matrices we prove that the free energy exists and a phase transition occurs for positive inverse temperature 2 the free energy is the same as that of related non translation invariant number theoretic spin chain using a number theoretic argument the low temperature 3 state is shown to be completely magnetized for long chains the number of states of energy e log n summed over chain length is expressed in terms of a restricted divisor problem we conjecture that its asymptotic form is n log n consistent with the phase transition at 2 and suggesting a possible connection with the riemann function the spin interaction coefficients include all even many body terms and are translation invariant computer results indicate that all the interaction coefficients except the constant term are ferromagnetic

on the error propagation of semi lagrange and fourier methods for advection problems in this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired the numerical methods considered are based on the fast fourier transform polynomial interpolation semi lagrangian methods using a lagrange or spline interpolation and a discontinuous galerkin semi lagrangian approach which is conservative and has to store more than a single value per cell we demonstrate by carrying out numerical experiments that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation based semi lagrangian methods for the discontinuous galerkin semi lagrangian method however we find that the characteristic property of semi lagrangian error estimates namely the fact that the error increases proportionally to the number of time steps is not observed we provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi lagrangian methods the method based on the fast fourier transform is exact but due to round off errors susceptible to a linear increase of the error in the number of time steps we show how to modify the cooley tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps finally we show for a simple model that our conclusions hold true if the advection solver is used as part of a splitting scheme

on the 2d ising wulff crystal near criticality we study the behavior of the two dimensional ising model in a finite box at temperatures that are below but very close to the critical temperature in a regime where the temperature approaches the critical point and simultaneously the size of the box grows fast enough we establish a large deviation principle that proves the appearance of a round wulff crystal

generalized farey trees transfer operators and phase transitions we consider a family of markov maps on the unit interval interpolating between the tent map and the farey map the latter is not uniformly expanding each map being composed of two fractional linear transformations the family generalizes many particular properties which for the case of the farey map have been successfully exploited in number theory we analyze the dynamics through the spectral properties of the generalized transfer operator application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function

fibonacci van der corput and riesz nagy abstract what do the three names in the title have in common the purpose of this paper is to relate them in a new and hopefully interesting way starting with the fibonacci numeration system also known as zeckendorff s system we will pose ourselves the problem of extending it in a natural way to represent all real numbers in 0 1 we will see that this natural extension leads to what is known as the system restricted to the unit interval the resulting complete system of numeration replicates the uniqueness of the binary system which in our opinion is responsible for the possibility of defining the van der corput sequence in 0 1 a very special sequence which besides being uniformly distributed has one of the lowest discrepancy a measure of the goodness of the uniformity lastly combining the fibonacci system and the binary in a very special way we will obtain a singular function more specifically the inverse of one of the family of riesz nagy

the derivative of minkowski s x function abstract minkowski s x function can be seen as the confrontation of two number systems regular continued fractions and the alternated dyadic system this way of looking at it enables us to prove that its derivative when it exists in a wide sense can only attain two values zero and infinity it is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k 5 31972 and x exists then x 0 in the same way if the same average is less than 2 log 2 where is the golden ratio then x finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions

the 2d ising model near criticality a fk percolation analysis we study the 2d ising model defined on finite boxes at temperatures that are below but very close from the critical point when the temperature approaches the critical point and the size of the box grows fast enough we establish large deviations estimates on fk percolation events that concern the phenomenon of phase coexistence

a performance comparison of semi lagrangian discontinuous galerkin and spline based vlasov solvers in four dimensions abstract the purpose of the present paper is to compare two semi lagrangian methods in the context of the four dimensional vlasov poisson equation more specifically our goal is to compare the performance of the more recently developed semi lagrangian discontinuous galerkin scheme to cubic spline interpolation which is widely used in eulerian vlasov simulation to that end we perform simulations for nonlinear landau damping and a two stream instability and provide benchmarks for the selalib and sldg codes both on a workstation and using mpi on a cluster in addition we will present results for the graphic processing unit gpu implementation contained in sldg we find that the semi lagrangian discontinuous galerkin scheme shows a moderate improvement in run time for nonlinear landau damping and a substantial improvement for the two stream instability it should be emphasized that these results are markedly different from results obtained in the asymptotic regime which favor spline interpolation thus we conclude that the traditional approach of evaluating numerical methods is misleading even for short time simulations in addition the absence of any global communication in the semi lagrangian discontinuous galerkin method gives it a decisive advantage for scaling to more than 256 cores

accuracy of the discrete fourier transform and the fast fourier transform fast fourier transform fft based computations can be far more accurate than the slow transforms suggest discrete fourier transforms computed through the fft are far more accurate than slow transforms and convolutions computed via fft are far more accurate than the direct results however these results depend critically on the accuracy of the fft software employed which should generally be considered suspect popular recursions for fast computation of the sine cosine table or twiddle factors are inaccurate due to inherent instability some analyses of these recursions that have appeared heretofore in print suggesting stability are incorrect even in higher dimensions the fft is remarkably stable

pattern theorems ratio limit theorems and gumbel maximal clusters for random fields we study occurrences of patterns on clusters of size n in random fields on d we prove that for a given pattern there is a constant a 0 such that the probability that this pattern occurs at most na times on a cluster of size n is exponentially small moreover for random fields obeying a certain markov property we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value this leads to an elegant and simple proof of the ratio limit theorem for these random fields which states that the ratio of the probabilities that the cluster of the origin has sizes n 1 and n converges as n implications for the maximal cluster in a finite box are discussed

the number theoretical spin chain and the riemann zeroes it is an empirical observation that the riemann zeta function can be well approximated in its critical strip using the number theoretical spin chain a proof of this would imply the riemann hypothesis here we relate that question to the one of spectral radii of a family of markov chains this in turn leads to the question whether certain graphs are ramanujan

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

a singular function with a non zero finite derivative on a dense set with hausdorff dimension one abstract this article closes a trilogy on the existence of singular functions with non zero finite derivatives in two previous papers the authors had exhibited a continuous strictly increasing singular function from 0 1 into 0 1 with a derivative that takes non zero finite values at two different zero measure sets first at the points of an uncountable set then at the points of a dense set in 0 1 in the present paper the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval a b has hausdorff dimension one another feature of this third article is the construction of the required function using the most paradigmatic of the singular functions the cantor lebesgue one

towards a complete determination of the spectrum of a transfer operator associated with intermittency the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point for an example of such an intermittent map the farey map we give a simple proof that the transfer operator is self adjoint on a suitably defined hilbert space and characterize its spectrum using a suitable first return map we present a highly efficient numerical method for the determination of all the eigenvalues including those embedded in the continuous spectrum

a splitting algorithm for vlasov simulation with filamentation filtration a fourier fourier transformed version of the splitting algorithm for simulating solutions of the vlasov poisson system of equations is introduced it is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm it is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle in cell counterpart it is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results

the asymmetric random cluster model and comparison of ising and potts models we introduce the asymmetric random cluster or arc model which is a graphical representation of the potts lattice gas and establish its basic properties the arc model allows a rich variety of comparisons in the fkg sense between models with different parameter values we give for example values h for which the 0 s configuration in the potts lattice gas is dominated by the configuration of the h ising model the potts model with possibly an external field applied to one of the spins is a special case of the potts lattice gas which allows our comparisons to yield rigorous bounds on the critical temperatures of potts models for example we obtain 0 571 1 exp c 0 600 for the 9 state potts model on the hexagonal lattice another comparison bounds the movement of the critical line when a small potts interaction is added to a lattice gas which otherwise has only interparticle attraction arc models can also be compared to related models such as the partial fk model obtained by deleting a fraction of the nonsingleton clusters from a realization of the fortuin kasteleyn random cluster model this comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the potts model

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

convergence analysis of a discontinuous galerkin strang splitting approximation for the vlasov poisson equations a rigorous convergence analysis of the strang splitting algorithm with a discontinuous galerkin approximation in space for the vlasov poisson equations is provided it is shown that under suitable assumptions the error is of order mathcal o tau 2 h q h q tau where tau is the size of a time step h is the cell size and q the order of the discontinuous galerkin approximation in order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed

riesz nagy singular functions revisited abstract in 1952 f riesz and sz nagy published an example of a monotonic continuous function whose derivative is zero almost everywhere that is to say a singular function besides the function was strictly increasing their example was built as the limit of a sequence of deformations of the identity function as an easy consequence of the definition the derivative when it existed and was finite was found to be zero in this paper we revisit the riesz nagy family of functions and we relate it to a system for real number representation which we call 1 expansions with the help of these real number expansions we generalize the family the singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity

a singular function with a non zero finite derivative abstract this paper exhibits for the first time in the literature a continuous strictly increasing singular function with a derivative that takes non zero finite values at some points for all the known classic singular functions cantor s hellinger s minkowski s and the riesz nagy one including its generalizations and variants the derivative when it existed and was finite had to be zero as a result there arose a strong suspicion almost a conjecture that this had to be the case for any singular function we present here a singular function constructed as a patchwork of known classic singular functions with derivative 1 on a subset of the cantor set

a thermodynamic formalism approach to the selberg zeta function for hecke triangle surfaces of infinite area we provide an explicit construction of a cross section for the geodesic flow on infinite area hecke triangle surfaces which allows us to conduct a transfer operator approach to the selberg zeta function further we construct closely related cross sections for the billiard flow on the associated triangle surfaces and endow the arising discrete dynamical systems and transfer operator families with two weight functions which presumably encode dirichlet respectively neumann boundary conditions the fredholm determinants of these transfer operator families constitute dynamical zeta functions which provide a factorization of the selberg zeta function of the hecke triangle surfaces

on a ferromagnetic spin chain ii thermodynamic limit the existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the riemann zeta function is proven

a singular function with a non zero finite derivative on a dense set abstract the authors had exhibited in a previous paper a continuous strictly increasing singular function from 0 1 into 0 1 with a derivative that takes non zero finite values at the points of an uncountable set in this article the construction is improved to encompass a dense set

differentiability of the minkowski question mark function abstract we get necessary and sufficient conditions for the derivative of the minkowski question mark function x to be equal to zero or infinity these conditions are formulated in terms of sums s x t a 1 a t of partial quotients of continued fraction expansion to x 0 a 1 a t in particular we prove that if there exists c such that s x t 1 t log t log 2 c with 1 2 log 1 5 2 log 2 1 38 8 then x exists and x another result is as follows assume that there exists a constant c such that s x t 2 t c with 2 4 log 5 29 2 5 log 2 5 log 5 29 2 log 2 5 log 2 4 40 1 then x exists and x 0 we show that our conditions on the sum s x t are the best possible our results improve upon earlier theorems by paradis viader bibiloni and dushistova moshchevitin

representation and poly time approximation for pressure of mathbb z 2 lattice models in the non uniqueness region we develop a new pressure representation theorem for nearest neighbour gibbs interactions and apply this to obtain the existence of efficient algorithms for approximating the pressure in the 2 dimensional ferromagnetic potts multi type widom rowlinson and hard core models for potts model our results apply to every inverse temperature but the critical for widom rowlinson and hard core models they apply to certain subsets of both the subcritical and supercritical regions the main novelty of our work is in the latter

a splitting approach for the kadomtsev petviashvili equation we consider a splitting approach for the kadomtsev petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented the error made by this numerical scheme is compared to exponential integrators which have been shown in klein and roidot 2011 2 to perform best for stiff solutions of the kadomtsev petviashvili equation since many classic high order splitting methods do not perform well we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four in addition the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated