discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

first order phase transitions in lattice and continuous systems extension of pirogov sinai theory we generalize the notion of ground states in the pirogov sinai theory of first order phase transitions at low temperatures applicable to lattice systems with a finite number of periodic ground states to that of restricted ensembles with equal free energies a restricted ensemble is a gibbs ensemble i e equilibrium probability measure on a restricted set of configurations in the phase space of the system when a restricted ensemble contains only one configuration it coincides with a ground state in the more general case the entropy is also important

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

uniqueness of gibbs measures and absorption probabilities 1 introduction in studying uniqueness of gibbs measures it may be useful to make a comparison with a stochastic process we present a method connecting absorption probabilities of a simple markov chain at 0 with uniqueness our specific result theorem 1 1 is used for one dimensional systems it includes a well known result on uniqueness in ruelle 1968 and succeeds in section 6 to find a natural improvement of other results that needed refined estimates in our method the key role is played by a certain inequality comparison of processes is quite well known already in the literature duality see liggett 1985 is a nice technique based on an equality we hope that our comparison method using absorption probabilities is useful in more problems let us now describe our problem setting let s be a countable set and x a finite or countable set for a c s and a e xs write ga gj j ea we define gibbs measures on xs in terms of a given energy difference function ah a a for the energy difference between a and a in xs this function has to be properly defined only in case aa a o for the complement a of any finite set a assume that for any finite a c s this energy difference has the form

complex networks in confined comminution the physical process of confined comminution is investigated within the framework of complex networks we first characterize the topology of the unweighted contact networks as generated by the confined comminution process we find this process gives rise to an ultimate contact network which exhibits a scale free degree distribution and small world properties in particular if viewed in the context of networks through which information travels along shortest paths we find that the global average of the node vulnerability decreases as the comminution process continues with individual node vulnerability correlating with grain size a possible application to the design of synthetic networks e g sensor networks is highlighted next we turn our attention to the physics of the granular comminution process and examine force transmission with respect to the weighted contact networks where each link is weighted by the inverse magnitude of the normal force acting at the associated contact we find that the strong forces i e force chains are transmitted along pathways in the network which are mainly following shortest path routing protocols as typically found for example in communication systems motivated by our earlier studies of the building blocks for self organization in dense granular systems we also explore the properties of the minimal contact cycles the distribution of the contact strain energy intensity of 4 cycle motifs in the ultimate state of the confined comminution process is shown to be consistent with a scale free distribution with infinite variance thereby suggesting that 4 cycle arrangements of grains are capable of storing vast amounts of energy in their contacts without breaking

the relationship between phase transitions and percolation in models with ground state degeneracy a criterion for the uniqueness of limiting gibbs states in classical models with unique ground states is formulated various applications of this criterion formulated in the terminology of percolation theory are discussed

packing limited growth of irregular objects we study growth limited by packing for irregular objects in two dimensions we generate packings by seeding objects randomly in time and space and allowing each object to grow until it collides with another object the objects we consider allow us to investigate the separate effects of anisotropy and nonunit aspect ratio by means of a connection to the decay of pore space volume we measure power law exponents for the object size distribution we carry out a scaling analysis showing that it provides an upper bound for the size distribution exponent we find that while the details of the growth mechanism are irrelevant the exponent is strongly shape dependent potential applications lie in ecological and biological environments where sessile organisms compete for limited space as they grow

topological evolution in dense granular materials a complex networks perspective abstract one of the great challenges in the science of complex materials materials capable of emergent behavior such as self organized pattern formation is deciphering their inherent structural design principles as they deform in response to external loads we have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to i uncover such design principles and ii identify suitable metrics that quantify the evolution of structure during deformation herein we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level i e a particle and its immediate neighbors as well as the macroscopic level we explore network motifs and other topological properties at these multiple length scales in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation spatially and with respect to strain key processes or rearrangement events that cause loss of connectivity within the material domain e g microbanding and force chain buckling are investigated network statistics of these processes previously shown to be major sources of energy dissipation and nonaffine deformation are then tied to corresponding trends observed in the evolving macroscopic network it is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity

on thue s equation and more precisely for the number of primitive solutions to 1 1 that is solutions in coprime integers x y the first important results on the number of solutions of thue s equation were obtained by c l siegel in the case r 3 and in the binomial case f x y a x r b y r in view of his results and comments we should attribute to siegel the problem to decide whether the number of primitive solutions to 1 1 for irreducible f of degree r 3 could be bounded by a function depending only on r and h but otherwise independent of f in 1983 siegel s question was answered in the affirmative by j h evertse 2 as a special case of his results he also treats equations in number fields evertse obtains the bound

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

cycles and clustering in bipartite networks we investigate the clustering coefficient in bipartite networks where cycles of size three are absent and therefore the standard definition of clustering coefficient cannot be used instead we use another coefficient given by the fraction of cycles with size four showing that both coefficients yield the same clustering properties the new coefficient is computed for two networks of sexual contacts one bipartite and another where no distinction between the nodes is made monopartite in both cases the clustering coefficient is similar furthermore combining both clustering coefficients we deduce an expression for estimating cycles of larger size which improves previous estimations and is suitable for either monopartite and multipartite networks and discuss the applicability of such analytical estimations

on the number of solutions of polynomial congruences and thue equations has only finitely many solutions in integers x and yv in the first part of this paper we shall establish upper bounds for the number of solutions of 1 in coprime integers x and y under the assumption that the discriminant d f of f is nonzero for most integers h these bounds improve upon those obtained by bombieri and schmidt in 5 in the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of nagell 30 ore 32 sandor 33 and huxley 19 in fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is in general best possible in the second part we shall address the problem of finding forms f for which 1 has many solutions for arbitrarily large integers h finally we shall obtain upper bounds for the number of solutions of certain thue mahler and ramanujan nagell equations by appealing to estimates of evertse gy6ry stewart and tijdeman 17 for the number of solutions of s unit equations

precise determination of the fractal dimensions of apollonian packing and space filling bearings the authors present high precision numerical values for the fractal dimensions of apollonian packing and various space filling bearings which might have application for mechanical gearworks for turbulence or for tectonic motion they find for the simplest apollonian packing df 1 305 684 or 0 000 010

a block effect of external field in the one dimensional ferromagnetical ising model with long range interaction we consider the one dimensional ferromagnetical ising model with long range interaction under external field blocks of equal length with alternating signs and investigate the low temperature phase diagram of this model it turns out that when the absolute value of the external field is sufficiently small the set of gibbs states substantially depends on block size at small block sizes there are at least two gibbs states and at large block sizes there is a unique gibbs state

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

intensity and coherence of motifs in weighted complex networks the local structure of unweighted networks can be characterized by the number of times a subgraph appears in the network the clustering coefficient reflecting the local configuration of triangles can be seen as a special case of this approach in this paper we generalize this method for weighted networks we introduce subgraph intensity as the geometric mean of its link weights and coherence as the ratio of the geometric to the corresponding arithmetic mean using these measures motif scores and clustering coefficient can be generalized to weighted networks to demonstrate these concepts we apply them to financial and metabolic networks and find that inclusion of weights may considerably modify the conclusions obtained from the study of unweighted characteristics

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

the number of solutions of norm form equations a norm form is a form f xi xn with rational coefficients which factors into linear forms over c but is irreducible or a power of an irreducible form over q it is known that a nondegenerate norm form equation f xl x m has only finitely many solutions xl x e zn we derive explicit bounds for the number of solutions when f has coefficients in z these bounds depend only on n m and the degree of f but are independent of the size of the coefficients of f