a two step matrix splitting iteration for computing pagerank the pagerank algorithm plays an important role in determining the importance of web pages the core of this algorithm involves using the classical power method to compute the pagerank vector which is the principal eigenvector of the matrix representing the web link graph nevertheless it is well known that the power method may perform poorly when the second largest eigenvalue is close to the dominant one in this article we present a new approach that is based on the two step splitting iteration framework the description and convergence of the new algorithm are discussed in detail numerical examples are given to illustrate the performance of this algorithm

amenability definable groups and automorphism groups abstract we prove several theorems relating amenability of groups in various categories discrete definable topological automor phism group to model theoretic invariants quotients by connected components lascar galois group g compactness among the main tools which is possibly of independent interest is the adaptation and generalization of theorem 12 of 27 to other settings including amenable topological groups whose topology is generated by a family of subgroups on the side of definable groups we prove that if g is definable in a model m and g is definably amenable then the connected components g m 00 and g m 000 coincide answering positively a question from 22 we also prove some natural counterparts for topological groups using our generalizations of 27 by finding a dictionary relating quotients by connected components and galois groups of categorical theories we conclude that if m is countable and categorical and aut m is amenable as a topological group then t th m is g compact i e the lascar galois group gal l t is compact hausdorff equivalently the natural epimorphism from gal l t to gal k p t is an isomorphism we also take the opportunity to further develop the model theoretic approach to topological dynamics obtaining for example some new invariants for topological groups as well as allowing a uniform approach to the theorems above and the various categories

accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

convergence of multi level iterative aggregation disaggregation methods this paper introduces an error propagation formula of a certain class of multi level iterative aggregation disaggregation iad methods for numerical solutions of stationary probability vectors of discrete finite markov chains the formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific markov chains numerical experiments indicate that the same type of the formula could be used for a wider class of the multi level iad methods using the formula we show that for given data there is no relation between convergence of two level and of multi level iad methods

a reordering for the pagerank problem we describe a reordering particularly suited to the pagerank problem which reduces the computation of the pagerank vector to that of solving a much smaller system and then using forward substitution to get the full solution vector we compare the theoretical rates of convergence of the original pagerank algorithm to that of the new reordered pagerank algorithm showing that the new algorithm can do no worse than the original algorithm we present results of an experimental comparison on five datasets which demonstrate that the reordered pagerank algorithm can provide a speedup of as much as a factor of 6 we also note potential additional benefits that result from the proposed reordering

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

an efficient elimination strategy for solving pagerank problems in web link structures similar link distributions often occur especially for pages from same hosts for pagerank problems similar in link distributions of pages result in similar row patterns of the transition matrix we demonstrate that common row patterns in the transition matrix determine identical sub rows thus the identical sub rows with a large proportion of nonzeros can be eliminated to decrease the density of pagerank problems we propose an elimination strategy that exploits such identical sub rows and generates an elimination operator for transferring the problem to an equivalent but more sparse problem numerical experiments are reported to illustrate the effectiveness of this strategy for decreasing the computational cost of solving pagerank problems

metric structures and probabilistic computation continuous first order logic is used to apply model theoretic analysis to analytic structures e g hilbert spaces banach spaces probability spaces etc classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first order logic the present paper shows that probabilistic computation sometimes called randomized computation and continuous logic stand in a similar close relationship the main result of this paper is an effective completeness theorem showing that every decidable continuous first order theory has a probabilistically decidable model we also show that probabilistically computable structures give rise to a model of aca 0in a natural way and describe a connection with complexity theory

subgroups close to normal subgroups abstract let g be a group and h a subgroup it is shown that the set of indices h h ghg 1 g g has a finite upper bound n if and only if there is a normal subgroup n g which is commensurable with h i e such that h n h and n n h are finite moreover the latter indices admit bounds depending only on n if the bounded index hypothesis is assumed only for g running over some subgroup k g the conclusion holds with normal weakened to normalized by k more detailed information is obtained under the assumption that h h ghg 1 g g 1 p for p prime in particular when p 2 there exists n g such that either h has index 2 in n or n has index 2 in h

sampling rare fluctuations of discrete time markov chains we describe a simple method that can be used to sample the rare fluctuations of discrete time markov chains we focus on the case of markov chains with well defined steady state measures and derive expressions for the large deviation rate functions and upper bounds on such functions for dynamical quantities extensive in the length of the markov chain we illustrate the method using a series of simple examples and use it to study the fluctuations of a lattice based model of active matter that can undergo motility induced phase separation

a quantitative version of the idempotent theorem in harmonic analysis suppose that g is a locally compact abelian group and write m g for the algebra of bounded regular complex valued measures under convolution a measure g m g is said to be idempotent if x i or alternatively if j5 takes only the values 0 and 1 the cohen helson rudin idempotent theorem states that a measure x is idempotent if and only if the set 7 g g i2 7 1 belongs to the coset ring of g that is to say we may write l i i

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

turbulence amalgamation and generic automorphisms of homogeneous structures we study topological properties of conjugacy classes in polish groups with emphasis on automorphism groups of homogeneous countable structures we first consider the existence of dense conjugacy classes the topological rokhlin property we then characterize when an automorphism group admits a comeager conjugacy class answering a question of truss and apply this to show that the homeomorphism group of the cantor space has a comeager conjugacy class answering a question of akin hurley and kennedy finally we study polish groups that admit comeager conjugacy classes in any dimension in which case the groups are said to admit ample generics we show that polish groups with ample generics have the small index property generalizing results of hodges hodkinson lascar and shelah and arbitrary homomorphisms from such groups into separable groups are automatically continuous moreover in the case of oligomorphic permutation groups they have uncountable cofinality and the bergman property these results in particular apply to automorphism groups of many stable 0 categorical structures and of the random graph in this connection we also show that the infinite symmetric group s has a unique non trivial separable group topology for several interesting groups we also establish serre s properties fh and fa

a general omitting types theorem in mathematical fuzzy logic this paper is a contribution to the theoretical study of weighted structures in fuzzy logic we consider an important item from classical model theory the construction of models that do not have any collection satisfying certain prescribed properties that is an omitting types theorem we generalize the work done by cintula and diaconescu omitting types theorem for fuzzy logics ieee transactions on fuzzy systems 27 2 273 277 2019 who solved the problem for standard one sided types instead we introduce types for fuzzy structures as pairs of sets of formulas with free variables expressing respectively properties to be satisfied and those to be avoided and prove the corresponding omitting types theorem in the framework of uninorm based logics

hilbert s fifth problem for local groups we solve hilbert s fifth problem for local groups every locally euclidean local group is locally isomorphic to a lie group jacoby claimed a proof of this in 1957 but this proof is seriously flawed we use methods from nonstandard analysis and model our solution after a treatment of hilbert s fifth problem for global groups by hirschfeld

uncoupling the perron eigenvector problem abstract for a nonnegative irreducible matrix a with spectral radius this paper is concerned with the determination of the unique normalized perron vector which satisfies a 62 0 j j 1 it is explained how to uncouple a large matrix a into two or more smaller matrices say p 11 p 22 p kk such that this sequence of smaller matrices has the following properties 1 each p ii is also nonnegative and irreducible so that each pii has a unique perron vector i 2 each pii has the same spectral radius as a 3 it is possible to determine the i s completely independently of each other so that one can execute the computation of the i s parallel 4 it is easy to couple the smaller perron vectors i back together in order to produce the perron vector for the original matrix a

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

non equivalence of dynamical ensembles and emergent non ergodicity dynamical ensembles have been introduced to study constrained stochastic processes in the microcanonical ensemble the value of a dynamical observable is constrained to a given value in the canonical ensemble a bias is introduced in the process to move the mean value of this observable the equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result in this paper we study the physical conditions associated with ensemble equivalence and the consequences of non equivalence for continuous time markov jump processes we show that ergodicity guarantees ensemble equivalence for non ergodic systems or systems with emergent ergodicity breaking we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition we illustrate our results on the infinite range ising model by characterizing the fluctuations of magnetization and activity we discuss the emergence of non ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins

computation of large deviation statistics via iterative measurement and feedback procedure we propose a computational method for large deviation statistics of time averaged quantities in general markov processes in our proposed method we repeat a response measurement against external forces where the forces are determined by the previous measurement as feedback consequently we obtain a set of stationary states corresponding to an exponential family of distributions each of which shows rare events in the original system as the typical behavior as a demonstration of our method we study large deviation statistics of one dimensional lattice gas models

on computing pagerank via lumping the google matrix computing google s pagerank via lumping the google matrix was recently analyzed in i c f ipsen t m selee pagerank computation with special attention to dangling nodes siam j matrix anal appl 29 2007 1281 1296 it was shown that all of the dangling nodes can be lumped into a single node and the pagerank could be obtained by applying the power method to the reduced matrix furthermore the stochastic reduced matrix had the same nonzero eigenvalues as the full google matrix and the power method applied to the reduced matrix had the same convergence rate as that of the power method applied to the full matrix therefore a large amount of operations could be saved for computing the full pagerank vector in this note we show that the reduced matrix obtained by lumping the dangling nodes can be further reduced by lumping a class of nondangling nodes called weakly nondangling nodes to another single node and the further reduced matrix is also stochastic with the same nonzero eigenvalues as the google matrix

on the diameter of permutation groups given a finite group g and a set a of generators the diameter diam g g a of the cayley graph g g a is the smallest l such that every element of g can be expressed as a word of length at most l in a a 1 we are concerned with bounding diam g max a diam g g a it has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n but the best previously known upper bound was exponential in nlogn v we give a quasipolynomial upper bound namely diam g exp o logn 4 loglogn exp loglog g o 1 for g sym n or g alt n where the implied constants are absolute this addresses a key open case of babai s conjecture on diameters of simple groups by a result of babai and seress 1992 our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

semialgebraic expansions of c we prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers

growth in solvable subgroups of mathrm gl _r mathbb z p mathbb z let k mathbb z p mathbb z and let a be a subset of mathrm gl r k such that langle a rangle is solvable we reduce the study of the growth of a under the group operation to the nilpotent setting fix a positive number c ge 1 we prove that either a grows meaning a 3 ge c a or else there are groups u r and s with u r unlhd s unlhd langle a rangle such that s u r is nilpotent a k cap s is large and u r subseteq a k where k depends only on the rank r of mathrm gl r k here a k x 1 x 2 cdots x k x i in a cup a 1 cup 1 when combined with recent work by pyber and szabo the main result of this paper implies that it is possible to draw the same conclusions without supposing that langle a rangle is solvable

continuous first order logic and local stability we develop continuous first order logic a variant of the logic described by chang and keisler 1966 we show that this logic has the same power of expression as the framework of open hausdorff cats and as such extends henson s logic for banach space structures we conclude with the development of local stability for which this logic is particularly well suited

growth in finite simple groups of lie type we prove that if l is a finite simple group of lie type and a a set of generators of l then either a grows i e vert a 3 vert vert a vert 1 varepsilon where varepsilon depends only on the lie rank of l or a 3 l this implies that for simple groups of lie type of bounded rank a well known conjecture of babai holds i e the diameter of any cayley graph is polylogarithmic we also obtain new families of expanders a generalization of our proof yields the following let a be a finite subset of sl n mathbb f mathbb f an arbitrary field satisfying big vert a 3 big vert le mathcal k vert a vert then a can be covered by mathcal k m i e polynomially many cosets of a virtually soluble subgroup of sl n mathbb f which is normalized by a where m depends on n see more at http www ams org journals jams 0000 000 00 s0894 0347 2014 00821 3 home html sthash lp65mzge dpuf

an arnoldi extrapolation algorithm for computing pagerank the arnoldi type algorithm proposed by golub and greif g golub c greif an arnoldi type algorithm for computing pagerank bit 46 2006 759 771 is a restarted krylov subspace method for computing pagerank however this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small in this paper we first develop an extrapolation method based on ritz values we then consider how to periodically knit this extrapolation method together with the arnoldi type algorithm the resulting algorithm is the arnoldi extrapolation algorithm the convergence of the new algorithm is analyzed numerical experiments demonstrate the numerical behavior of this algorithm

groups measures and the nip we discuss measures invariant measures on definable groups and genericity often in an nip failure of the independence property environment we complete the proof of the third author s conjectures relating definably compact groups g in saturated o minimal structures to compact lie groups we also prove some other structural results about such g for example the existence of a left invariant finitely additive probability measure on definable subsets of g we finally introduce a new notion compact domination domination of a definable set by a compact space and raise some new conjectures in the o minimal case

stochastic complementation uncoupling markov chains and the theory of nearly reducible systems a concept called stochastic complementation is an idea which occurs naturally although not always explicitly in the theory and application of finite markov chains this paper brings this idea to the forefront with an explicit definition and a development of some of its properties applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of markov chains furthermore the role of stochastic complementation in the development of the classical simon ando theory of nearly reducible system is presented

explicit helfgott type growth in free products and in limit groups abstract we adapt safin s result on powers of sets in free groups to obtain helfgott type growth in free products if a is any finite subset of a free product of two arbitrary groups then either a is conjugate into one of the factors or the triple product a 3 of a satisfies a 3 1 7776 a 2 or a generates an infinite cyclic or infinite dihedral group we also point out that if a is any finite subset of a limit group then a 3 satisfies the above inequality unless a generates a free abelian group this gives rise to many infinite groups g where there exist c 0 and 1 such that any finite subset a of g either satisfies a 3 c a 1 or generates a virtually nilpotent group

approximate subgroups of residually nilpotent groups we show that a k approximate subgroup a of a residually nilpotent group g is contained in boundedly many cosets of a finite by nilpotent subgroup the nilpotent factor of which is of bounded step combined with an earlier result of the author this implies that a is contained in boundedly many translates of a coset nilprogression of bounded rank and step the bounds are effective and depend only on k in particular if g is nilpotent they do not depend on the step of g as an application we show that there is some absolute constant c such that if g is a residually nilpotent group and if there is an integer n 1 such that the ball of radius n in some cayley graph of g has cardinality bounded by n c log log n then g is virtually log n step nilpotent