information theoretic thresholds from the cavity method abstract vindicating a sophisticated but non rigorous physics approach called the cavity method we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models we work out several concrete applications of these general results for instance we pinpoint the exact condensation phase transition in the potts antiferromagnet on the random graph thereby improving prior approximate results contucci et al 2013 34 further we prove the conjecture from krzakala et al 2007 55 about the condensation phase transition in the random graph coloring problem for any number q 3 of colors moreover we prove the conjecture on the information theoretic threshold in the disassortative stochastic block model decelle et al 2011 35 additionally our general result implies the conjectured formula for the mutual information in low density generator matrix codes montanari 2005 73

universality of the mean field for the potts model we consider the potts model with q colors on a sequence of weighted graphs with adjacency matrices a n allowing for both positive and negative weights under a mild regularity condition on a n we show that the mean field prediction for the log partition function is asymptotically correct whenever mathrm tr a n 2 o n in particular our results are applicable for the ising and the potts models on any sequence of graphs with average degree going to infty using this we establish the universality of the limiting log partition function of the ferromagnetic potts model for a sequence of asymptotically regular graphs and that of the ising model for bi regular bipartite graphs in both ferromagnetic and anti ferromagnetic domain we also derive a large deviation principle for the empirical measure of the colors for the potts model on asymptotically regular graphs

how to avoid a compact set abstract a first order expansion of the r vector space structure on r does not define every compact subset of every r n if and only if topological and hausdorff dimension coincide on all closed definable sets equivalently if a r k is closed and the hausdorff dimension of a exceeds the topological dimension of a then every compact subset of every r n can be constructed from a using finitely many boolean operations cartesian products and linear operations the same statement fails when hausdorff dimension is replaced by packing dimension

the equivalence of multi axis spine systems recommended stiffness limits using a standardized testing protocol abstract the complexity of multi axis spine testing often makes it challenging to compare results from different studies the aim of this work was to develop and implement a standardized testing protocol across three six axis spine systems compare them and provide stiffness and phase angle limits against which other test systems can be compared standardized synthetic lumbar specimens n 5 comprising three springs embedded in polymer at each end were tested on each system using pure moments in flexion extension lateral bending and axial rotation tests were performed using sine and triangle waves with an amplitude of 8 nm a frequency of 0 1 hz and with axial preloads of 0 and 500 n the stiffness phase angle and r2 value of the moment against rotation in the principal axis were calculated at the center of each specimen the tracking error was adopted as a measure of each test system to minimize non principal loads defined as the root mean squared difference between actual and target loads all three test systems demonstrated similar stiffnesses with small

nonlinear large deviations abstract we present a general technique for computing large deviations of nonlinear functions of independent bernoulli random variables the method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs previous technology based on szemeredi s regularity lemma works only for dense graphs applications are also made to exponential random graphs and three term arithmetic progressions in random sets of integers

characterization of the l4 l5 s1 motion segment using the stepwise reduction method the two aims of this study were to generate data for a more accurate calibration of finite element models including the l5 s1 segment and to find mechanical differences between the l4 l5 and l5 s1 segments then the range of motion rom and facet forces for the l4 s1 segment were measured using the stepwise reduction method this consists of sequentially testing and reducing each segment in nine stages by cutting the ligaments facet capsules and removing the nucleus five l4 s1 human segments median 65 years range 53 84 years sd 11 0 years were loaded under a maximum pure moment of 8nm the rom was measured using stereo photogrammetry via tracking of three markers and the facet contact forces cf were measured using a tekscan system the rom for the l4 l5 segment and all stages showed good agreement with published data the major differences in rom between the l4 l5 and l5 s1 segments were found for lateral bending and all stages for which the l4 l5 rom was about 1 5 3 times higher than that of the l5 s1 segment consistent with l5 s1 facet cf about 1 3 to 4 times higher than those measured for the l4 l5 segment for the other movements and few stages the l4 l5 rom was significantly lower that of the l5 s1 segment rom and cf provide important baseline data for more accurate calibration of fe models and to understand the role that their structures play in lower lumbar spine mechanics

on the decidability of termination of query evaluation in transitive closure logics for polynomial constraint databases the formalism of constraint databases in which possibly infinite data sets are described by boolean combinations of polynomial inequality and equality constraints has its main application area in spatial databases the standard query language for polynomial constraint databases is first order logic over the reals because of the limited expressive power of this logic with respect to queries that are important in spatial data base applications various extensions have been introduced we study extensions of first order logic with different types of transitive closure operators and we are in particular interested in deciding the termination of the evaluation of queries expressible in these transitive closure logics it turns out that termination is undecidable in general however we show that the termination of the transitive closure of a continuous function graph in the two dimensional plane viewed as a binary relation over the reals is decidable and even expressible in first order logic over the reals based on this result we identify a particular transitive closure logic for which termination of query evaluation is decidable and which is more expressive than first order logic over the reals furthermore we can define a guarded fragment in which exactly the terminating queries of this language are expressible

definable sets in ordered structures on introduit la notion de theorie o minimale des structures ordonnees une telle theorie etant telle que les sous ensembles definissables de ses modeles soient particulierement simples

biomechanical properties of human lumbar spine ligaments biomechanical properties of the six major lumbar spine ligaments were determined from 38 fresh human cadaveric subjects for direct incorporation into mathematical and finite element models anterior and posterior longitudinal ligaments joint capsules ligamentum flavum interspinous and supraspinous ligaments were evaluated using the results from in situ isolation tests individual force deflection responses from 132 samples were transformed with a normalization procedure into mean force deflection properties to describe the nonlinear characteristics ligament responses based on the mechanical characteristics as well as anatomical considerations were grouped into t12 l2 l2 l4 and l4 s1 levels maintaining individuality and nonlinearity a total of 18 data curves are presented geometrical measurements of original length and cross sectional area for these six major ligaments were determined using cryomicrotomy techniques derived parameters including failure stress and strain were computed using the strength and geometry information these properties for the lumbar spinal ligaments which are based on identical definitions used in mechanical testing and geometrical assay will permit more realistic and consistent inputs for analytical models

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

taming correlations through entropy efficient measure decompositions with applications to mean field approximation the analysis of various models in statistical physics relies on the existence of decompositions of measures into mixtures of product like components where the goal is to attain a decomposition into measures whose entropy is close to that of the original measure yet with small correlations between coordinates we prove a related general result for every measure mu on mathbb r n and every varepsilon 0 there exists a decomposition mu int mu theta d m theta such that h mu mathbb e theta sim m h mu theta le mathrm tr mathrm cov mu varepsilon and mathbb e theta sim m mathrm cov mu theta preceq mathrm id varepsilon as an application we derive a general bound for the mean field approximation of ising and potts models which is in a sense dimension free in both continuous and discrete settings in particular for an ising model on pm 1 n or on 1 1 n we show that the deficit between the mean field approximation and the free energy is at most c frac 1 p p left n vert j vert s p right frac p 1 p for all p 0 where vert j vert s p denotes the schatten p norm of the interaction matrix for the case p 2 this recovers the result of jain et al mean field approximation convex hierarchies and the optimality of correlation rounding a unified perspective arxiv 1808 07226 2018 but for an optimal choice of p it often allows to get almost dimension free bounds

an estimate for the siciak extremal function subanalytic geometry approach we study the ojasiewicz siciak inequality in cn this concerns the estimating from below the siciak extremal function which is very much related to the green function in c as an application we present some results describing how fast holomorphic functions defined in a neighbourhood of a compact holomorphic polyhedron can be approximated uniformly on this polyhedron by complex polynomials

relational expressive power of constraint query languages the expressive power of first order query languages with several classes of equality and inequality constraints is studied in this paper we settle the conjecture that recursive queries such as parity test and transitive closure cannot be expressed in the relational calculus augmented with polynomial inequality constraints over the reals furthermore noting that relational queries exhibit several forms of genericity we establish a number of collapse results of the following form the class of generic boolean queries expressible in the relational calculus augmented with a given class of constraints coincides with the class of queries expressible in the relational calculus with or without an order relation we prove such results for both the natural and active domain semantics as a consequence the relational calculus augmented with polynomial inequalities expresses the same classes of generic boolean queries under both the natural and active domain semantics in the course of proving these results for the active domin semantics we establish ramsey type theorems saying that any query involving certain kinds of constraints coincides with a constraint free query on databases whose elements come from a certain infinite subset of the domain to prove the collapse results for the natural semantics we make use of techniques from nonstandard analysis and from the model theory of ordered structures

factor models on locally tree like graphs we consider homogeneous factor models on uniformly sparse graph sequences converging locally to a unimodular random tree t and study the existence of the free energy density the limit of the log partition function divided by the number of vertices n as n tends to infinity we provide a new interpolation scheme and use it to prove existence of and to explicitly compute the quantity subject to uniqueness of a relevant gibbs measure for the factor model on t by way of example we compute for the independent set or hard core model at low fugacity for the ferromagnetic ising model at all parameter values and for the ferromagnetic potts model with both weak enough and strong enough interactions even beyond uniqueness regimes our interpolation provides useful explicit bounds on in the regimes in which we establish existence of the limit we show that it coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation bethe recursions on t in the special case that t has a galton watson law this formula coincides with the nonrigorous bethe prediction obtained by statistical physicists using the replica or cavity methods thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees we also provide a variational characterization for the bethe prediction in this general setting which is of independent interest

inclusion dynamics hybrid automata hybrid systems are dynamical systems with the ability to describe mixed discrete continuous evolution of a wide range of systems consequently at first glance hybrid systems appear powerful but recalcitrant neither yielding to analysis and reasoning through a purely continuous time modeling as with systems of differential equations nor open to inferential processes commonly used for discrete state transition systems such as finite state automata a convenient and popular model called hybrid automata was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting intuitively a hybrid automaton is simply a finite state automaton with each state augmented by continuous variables which evolve according to a set of well defined continuous laws each specified separately for each state this article investigates both the notion of hybrid automaton and the model checking problem over such a structure in particular it relates first order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions x f x t to a decidability problem for first order formulaeover the reals furthermore the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first order formulaeover the reals despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite we show that our techniques permit effective model checking for a nontrivial fragment of ctl

moment rotation responses of the human lumbosacral spinal column abstract the objective of this study was to test the hypothesis that the human lumbosacral joint behaves differently from l1 l5 joints and provides primary moment rotation responses under pure moment flexion and extension and left and right lateral bending on a level by level basis in addition range of motion rom and stiffness data were extracted from the moment rotation responses ten t12 s1 column specimens with ages ranging from 27 to 68 years mean 50 6 13 2 were tested at a load level of 4 0 n m nonlinear flexion and extension and left and right lateral bending moment rotation responses at each spinal level are reported in the form of a logarithmic function the mean rom was the greatest at the l5 s1 level under flexion 7 37 3 69 and extension 4 62 2 56 and at the l3 l4 level under lateral bending 4 04 1 11 the mean rom was the least at the l1 l2 level under flexion 2 42 0 90 l2 l3 level under extension 1 58 0 63 and l1 l2 level under lateral bending 2 50 0 75 the present study proved the hypothesis that l5 s1 motions are significantly greater than l1 l5 motions under flexion and extension loadings but the hypothesis was found to be untrue under the lateral bending mode these experimental data are useful in the improved validation of fe models which will increase the confidence of stress analysis and other modeling applications

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

structures having o minimal open core the open core of an expansion of a dense linear order is its reduct in the sense of definability generated by the collection of all of its open definable sets in this paper expansions of dense linear orders that have o minimal open core are investigated with emphasis on expansions of densely ordered groups the first main result establishes conditions under which an expansion of a densely ordered group has an o minimal open core specifically the following is proved let r be an expansion of a densely ordered group r that is definably complete and satisfies the uniform finiteness property then the open core of r is o minimal two examples of classes of structures that are not o minimal yet have o minimal open core are discussed dense pairs of o minimal expansions of ordered groups and expansions of o minimal structures by generic predicates in particular such structures have open core interdefinable with the original o minimal structure these examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane a phenomenon that can occur in dense pairs but not in expansions by generic predicates the property of having no dense graphs is examined and related to uniform finiteness definable completeness and having o minimal open core