metric spaces with linear extensions preserving lipschitz condition we study a new bi lipschitz invariant m of a metric space m its finiteness means that lipschitz functions on an arbitrary subset of m can be linearly extended to functions on m whose lipschitz constants are expanded by a factor controlled by m we prove that m is finite for several important classes of metric spaces these include metric trees of arbitrary cardinality groups of polynomial growth gromov hyperbolic groups certain classes of riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects on the other hand we construct an example of a two dimensional riemannian manifold m of bounded geometry for which m 8

phase diagrams of random field ising systems we show that the random field ising model may be tuned to obtain two distinct scenarios of phase diagram topology explicit evidence for this is presented in a numerically exact analysis in three dimensions and on a bethe lattice which allows us to investigate the effects of temperature coordination number and asymmetry in the field distribution

smooth extensions of functions on separable banach spaces let x be a banach space with a separable dual x let y subset x be a closed subspace and f y rightarrow mathbb r a c1 smooth function then we show there is a c1 extension of f to x

a variable structure gradient adaptive algorithm for a class of dynamical systems abstract a standard assumption in adaptive control is that the parameters being estimated are either constant or vary slowly as a function of time this paper investigates the adaptive control of a class of systems in which the parameters vary as a specified function of state the dynamic structure of the systems may be either linear or nonlinear for this class of systems the state space is separated into distinct subsets the parameters are then required to remain constant or be slowly time varying within the subsets given a controller for the system an analysis of the output error dynamics and the parameter error dynamics leads to a parameter adaptation algorithm with a variable structure the stability and convergence of both the parameter error and the output tracking error are investigated an analysis of siso linear systems with full state information is used to motivate and illustrate the treatment of siso feedback linearizable systems

fitting a cm smooth function to data iii suppose we are given a finite subset e rn and a function f e r how to extend f to a cm function f rn r with cm norm of the smallest possible order of magnitude in this paper and in 20 we tackle this question from the perspective of theoretical computer science we exhibit algorithms for constructing such an extension function f and for computing the order of magnitude of its cm norm the running time of our algorithms is never more than cn log n where n is the cardinality of e and c is a constant depending only on m and n

synchronization in the random field kuramoto model on complex networks we study the impact of random pinning fields on the emergence of synchrony in the kuramoto model on complete graphs and uncorrelated random complex networks we consider random fields with uniformly distributed directions and homogeneous and heterogeneous gaussian field magnitude distribution in our analysis we apply the ott antonsen method and the annealed network approximation to find the critical behavior of the order parameter in the case of homogeneous fields we find a tricritical point above which a second order phase transition gives place to a first order phase transition when the network is either fully connected or scale free with the degree exponent ensuremath gamma g5 interestingly for scale free networks with 2l ensuremath gamma ensuremath le 5 the phase transition is of second order at any field magnitude except for degree distributions with ensuremath gamma 3 when the transition is of infinite order at k c 0 independent of the random fields contrary to the ising model even strong gaussian random fields do not suppress the second order phase transition in both complete graphs and scale free networks although the fields increase the critical coupling for ensuremath gamma g3 our simulations support these analytical results

whitney extension theorems for convex functions of the classes c1 and c1 ω let c be a subset of n not necessarily convex f c r be a function and g c n be a uniformly continuous function with modulus of continuity we provide a necessary and sufficient condition on f g for the existence of a convex function f cc1 n such that f f on c and f g on c with a good control of the modulus of continuity of f in terms of that of g on the other hand assuming that c is compact we also solve a similar problem for the class of c1 convex functions on n with a good control of the lipschitz constants of the extensions namely lip f g finally we give a geometrical application concerning interpolation of compact subsets k of n by boundaries of c1 or c1 1 convex bodies with prescribed outer normals on k

explicit formulas for c 1 1 and c 1 omega _ textrm conv extensions of 1 jets in hilbert and superreflexive spaces abstract given x a hilbert space a modulus of continuity e an arbitrary subset of x and functions f e r g e x we provide necessary and sufficient conditions for the jet f g to admit an extension f f with f x r convex and of class c 1 x by means of a simple explicit formula as a consequence of this result if is linear we show that a variant of this formula provides explicit c 1 1 extensions of general not necessarily convex 1 jets satisfying the usual whitney extension condition with best possible lipschitz constants of the gradients of the extensions finally if x is a superreflexive banach space we establish similar results for the classes c conv 1 x

improved belief propagation algorithm finds many bethe states in the random field ising model on random graphs we first present an empirical study of the belief propagation bp algorithm when run on the random field ising model defined on random regular graphs in the zero temperature limit we introduce the notion of extremal solutions for the bp equations and we use them to fix a fraction of spins in their ground state configuration at the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size this in turn makes the associated optimization problem highly non trivial in the critical region using the bounds on the bp messages provided by the extremal solutions we design a new and very easy to implement bp scheme which is able to output a large number of stable fixed points on one hand this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time on the other hand we found that the number of fixed points of the bp algorithm grows with the system size in the critical region this unexpected feature poses new relevant questions about the physics of this class of models

higher order tangents and fefferman s paper on whitney s extension problem whitney w2 proved that a function defined on a closed subset of r is the restriction of a cm function if the limiting values of all mth divided differences form a continuous function we show that fefferman s solution of whitney s problem for rn f th 1 is equivalent to a variant of our conjecture in bmp2 giving a criterion for cm extension in terms of iterated limits of finite differences

sobolev lp2 functions on closed subsets of r2 abstract for each p 2 we give intrinsic characterizations of the restriction of the homogeneous sobolev space l p 2 r 2 to an arbitrary finite subset e of r 2 the trace criterion is expressed in terms of certain weighted oscillations of the second order with respect to a measure generated by the menger curvature of triangles with vertices in e

an extension theorem for convex functions of class c1 1 on hilbert spaces abstract let h be a hilbert space e h be an arbitrary subset and f e r g e h be two functions we give a necessary and sufficient condition on the pair f g for the existence of a convex function f c 1 1 h such that f f and f g on e we also show that if this condition is met f can be taken so that lip f lip g we give a geometrical application of this result concerning interpolation of sets by boundaries of c 1 1 convex bodies in h finally we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous

sobolev extension by linear operators let l r be the sobolev space of functions with mth derivatives lying in l r assume that n p for e r let l e denote the space of restrictions to e of functions in l r we show that there exists a bounded linear map t lm p e l r such that for any f l e we have tf f on e we also give a formula for the order of magnitude of f lm p e for a given f e r when e is finite

barycentric selectors and a steiner type point of a convex body in a banach space let f be a mapping from a metric space m into the family of all m dimensional affine subsets of a banach space x we present a helly type criterion for the existence of a lipschitz selection f of the set valued mapping f i e a lipschitz continuous mapping f m x satisfying f x f x x m the proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a lipschitz with respect to the hausdorff distance selector sx m defined on the family km x of all convex compacts in x of dimension at most m if x is a hilbert space then the classical steiner point of a convex body provides such a selector but in the non hilbert case there is no known way of constructing such a point we prove the existence of a lipschitz continuous selector sx m km x x for an arbitrary banach space x the proof is based on a new result about lipschitz properties of the center of mass of a convex set

cascades on clique based graphs we present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of highly clustered random graphs introduced by gleeson j p gleeson phys rev e 80 036107 2009 a condition for the existence of global cascades is also derived applications of this approach include analyses of percolation and watts s model we show how our techniques can be used to study the effects of in group bias in cascades on social networks

zero temperature hysteresis in an anti ferromagnetic ising chain with quenched random fields abstract we discuss the non equilibrium dynamics of an anti ferromagnetic ising chain in a quenched random field our discussion is mainly in the context of magnetic hysteresis but its relation to the problem if random sequential adsorption is also noted

one dimensional random field ising model gibbs states and structure of ground states we consider the random gibbs field formalism for the ferromagnetic id dichotomous random field ising model as the simplest example of a quenched disordered system we prove that for nonzero temperatures the gibb state is unique for any realization of the external field then we prove that ast 0 the gibbs state converges to a limit a ground state for almost all realizations of the external field the ground state turns out to be a probability measure concentrated on an infinite set of configurations and we give a constructive description of this measure

on the smoothness of convex envelopes we examine differentiability properties of the convex envelope conv e of a given function e rn cc cc in terms of properties of e it is shown that c1 as well as optimal c1 a regularity results 0 a 1 can be obtained under general conditions

bethe lattice spin glass the effects of a ferromagnetic bias and external fields i bifurcation analysis we present a rigorous analysis of the j ising spin glass model on the bethe lattice with fixed uncorrelated boundary conditions phase diagrams are derived as a function of temperature vs concentration of ferromagnetic bonds and for a symmetric distribution of bonds external field vs temperature in this part we characterize the bulk ordered phases using bifurcation theory we prove the existence of a distribution of single site magnetizations far inside the lattice which is stable with respect to changes in the boundary conditions

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

host parasite models on graphs the behavior of two interacting populations hosts and parasites is investigated on cayley trees and scale free networks in the former case analytical and numerical arguments elucidate a phase diagram for the susceptible infected susceptible model whose most interesting feature is the absence of a tricritical point as a function of the two independent spreading parameters for scale free graphs the parasite population can be described effectively by its dynamics in a host background this is shown both by considering the appropriate dynamical equations and by numerical simulations on barab asi albert networks with the major implication that in the thermodynamic limit the critical parasite spreading parameter vanishes some implications and generalizations are discussed

cascades on a class of clustered random networks we present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of random networks with arbitrary degree distribution and nonzero clustering introduced previously in m e j newman phys rev lett 103 058701 2009 a condition for the existence of global cascades is derived as well as a general criterion that determines whether increasing the level of clustering will increase or decrease the expected cascade size applications examples of which are provided include site percolation bond percolation and watts threshold model in all cases analytical results give excellent agreement with numerical simulations

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

low temperature behaviour of the random field ising model a relationship is established between the ising model in a random positive and negative magnetic field on a lattice with p 1 2 at t 0 each percolation cluster gives rise to a first order transition at a different value of the field there is a significant difference in behaviour between lattices for which pc 1 2 which do not by considering the bethe lattice and allowing the coordination number to become large the results of the mean field approximation are reproduced the above considerations do not apply to a gaussian distribution of fields and the absence of a first order transition can be understood in this case since large clusters overturn for small fields there are clear indications of metastable behaviour

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

on the sum of a sobolev space and a weighted lp space abstract let p n and let l p 1 r n be a homogeneous sobolev space for an arbitrary borel measure on r n we give a constructive characterization of the space l p 1 r n l p r n we express the norm of this space in terms of certain local oscillations with respect to the measure in particular we prove that if is a measure supported on a finite set s r n then there exist a sequence i of positive numbers and a family x i y i of two point subsets of s i 1 m such that m c n s and for every function f on r n f p i 1 m i f x i f y i p we also obtain another equivalent expression for the norm of the space which enables us to describe the k functional for the couple l p r n l p 1 r n in terms of p oscillations of functions and to prove that this couple is quasi linearizable

a bounded linear extension operator for l2 p r2 let l2 p r2 be the sobolev space of real valued functions on the plane whose hessian belongs to lp for any finite subset e r2 and p 2 let l2 p r2 e be the space of real valued functions on e equipped with the trace seminorm in this paper we construct a bounded linear extension operator t l2 p r2 e l2 p r2 we also provide an explicit formula that approximates the l2 p r2 e trace seminorm

controlling nonlinear time varying systems via euler approximations abstract a discrete time model reference control algorithm for continuous time systems represented by essentially unknown nonlinear time varying ordinary differential equations is presented and analysed provided the reference model and the inverse dynamics of the plant are bounded input bounded output stable it is shown that a discrete control using state measurements based on euler approximations for the systems response can achieve arbitrary tracking accuracy the effect of unmodelled dynamics and disturbances is analysed

sup inf explicit formulas for minimal lipschitz extensions for 1 fields on r n abstract we study the relationship between the lipschitz constant of 1 field introduced in 12 and the lipschitz constant of the gradient canonically associated with this 1 field moreover we produce two explicit formulas which are two extremal minimal lipschitz extensions for 1 fields as a consequence of the previous results for the problem of minimal extension by lipschitz continuous functions from r m to r n we produce explicit formulas similar to those of bauschke and wang see 7 finally we show that wells s extensions see 24 of 1 fields are absolutely minimal lipschitz extensions when the domain of 1 field to expand is finite we provide a counter example showing that this result is false in general

potts models on feynman diagrams we investigate numerically and analytically potts models on thin random graphs generic feynman diagrams using the idea that such models may be expressed as the limit of a matrix model the thin random graphs in this limit are locally tree like in distinction to the fat random graphs that appear in the planar feynman diagram limit more familiar from discretized models of two dimensional gravity the interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree like structures such as the bethe lattice q state potts models display a first order transition in the mean field for q 2 so the thin graph potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation such discontinuous transitions also appear in multiple ising models on thin graphs and may have implications for the use of the replica trick in spin glass models on random graphs

random field p spin glass model on regular random graphs we investigate in detail the phase diagrams of the p body j ising model with and without random fields on random graphs with fixed connectivity one of our most interesting findings is that a thermodynamic spin glass phase is present in the three body purely ferromagnetic model in random fields unlike for the canonical two body interaction random field ising model we also discuss the location of the phase boundary between the paramagnetic and spin glass phases that does not depend on the change of the ferromagnetic bias the independence of the ferromagnetic bias has also been observed from the perspective of the distribution of the partition function zeros on the complex field plane this behavior is explained by a gauge transformation which shows that gauge invariant properties generically do not depend on the strength of the ferromagnetic bias for the j ising model on regular random graphs

differentiable functions defined in closed sets a problem of whitney in 1934 whitney raised the question of how to recognize whether a function f defined on a closed subset x of n is the restriction of a function of class p a necessary and sufficient criterion was given in the case n 1 by whitney using limits of finite differences and in the case p 1 by glaeser 1958 using limits of secants we introduce a necessary geometric criterion for general n and p involving limits of finite differences that we conjecture is sufficient at least if x has a tame topology we prove that if x is a compact subanalytic set then there exists q q x p such that the criterion of order q implies that f is p the result gives a new approach to higher order tangent bundles or bundles of differential operators on singular spaces

whitney type extension theorems for jets generated by sobolev functions abstract let l p m r n 1 p be the homogeneous sobolev space and let e r n be a closed set for each p n and each non negative integer m we give an intrinsic characterization of the restrictions d f e m to e of m jets generated by functions f l p m 1 r n our trace criterion is expressed in terms of variations of corresponding taylor remainders of m jets evaluated on a certain family of well separated two point subsets of e for p this result coincides with the classical whitney glaeser extension theorem for m jets our approach is based on a representation of the sobolev space l p m 1 r n p n as a union of c m d r n spaces where d belongs to a family of metrics on r n with certain nice properties here c m d r n is the space of c m functions on r n whose partial derivatives of order m are lipschitz functions with respect to d this enables us to show that for every non negative integer m and every p n the very same classical linear whitney extension operator as in 31 provides an almost optimal extension of m jets generated by l p m 1 functions

analytical results for bond percolation and k core sizes on clustered networks an analytical approach to calculating bond percolation thresholds sizes of k cores and sizes of giant connected components on structured random networks with nonzero clustering is presented the networks are generated using a generalization of trapman s p trapman theor popul biol 71 160 2007 model of cliques embedded in treelike random graphs the resulting networks have arbitrary degree distributions and tunable degree dependent clustering the effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature for very high levels of clustering the percolation threshold in these generalized trapman networks is increased above the value it takes in a randomly wired unclustered network of the same degree distribution in assortative scale free networks where the variance of the degree distribution is infinite this clustering effect can lead to a nonzero percolation epidemic threshold

whitney s extension problem for c m let f be a real valued function on a compact set in rn and let m be a positive integer we show how to decide whether f extends to a cm function on rn

nonnegative c2 r2 interpolation abstract in this paper we prove two improved versions of the finiteness principle for nonnegative c 2 r 2 interpolation previously proven by fefferman israel and luli the first version sharpens the finiteness constant to 64 and the second version carries better computational practicality along the way we also provide a detailed construction of nonnegative c 2 interpolants in one dimension and prove the nonexistence of a bounded linear c 2 extension operator that preserves nonnegativity

sobolev w1p spaces on closed subsets of rn abstract for each p n we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the sobolev space w p 1 r n to an arbitrary closed subset of r n

sobolev inequalities in arbitrary domains abstract a theory of sobolev inequalities in arbitrary open sets in r n is established boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order the relevant sobolev inequalities exhibit the same critical exponents as in the classical framework moreover they involve constants independent of the geometry of the domain and hence yield genuinely new results even in the case when just smooth domains are considered our approach relies upon new representation formulas for sobolev functions and on ensuing pointwise estimates which hold in any open set

on smooth extensions of vector valued functions defined on closed subsets of banach spaces let x and z be banach spaces a a closed subset of x and a mapping f a rightarrow z we give necessary and sufficient conditions to obtain a c 1 smooth mapping f x rightarrow z such that f mid a f when either i x and z are hilbert spaces and x is separable or ii x is separable and z is an absolute lipschitz retract or iii x l 2 and z l p with 1 p 2 or iv x l p and z l 2 with 2 p infty where l p is any separable banach space l p s sigma mu with s sigma mu a sigma finite measure space

smooth extension theorems for one variable maps abstract we characterize the real valued functions f defined on perfect subsets p of r which admit n times differentiable extensions f r r in this characterization no continuity of f n is imposed in particular it generalizes jarnik s extension theorem according to which f admits differentiable extension f r r if and only if f is differentiable the new characterization is also closely related to the whitney s extension theorem which characterizes partial maps f admitting n times differentiable extensions f r r with continuous nth derivative f n we also provide an elegant description of a linear extension operator t n c p c r such that t n f d n r for every d n r extendable f c p and t n f c n r whenever f c p is c n r extendable

a hybridizable discontinuous galerkin method for elliptic interface problems in the formulation of boundary integral equations abstract a novel numerical method is presented to solve the general elliptic interface problem with piecewise constant diffusion coefficients to overcome the difficulties caused by piecewise constant diffusion coefficients a combination of the kernel free boundary integral kfbi method and the unfitted mesh based hybridizable discontinuous galerkin hdg method is considered to this end we first reformulate the elliptic interface problem into a fredholm boundary integral equation of the second kind and further solve the well conditioned discrete integral equation iteratively with the gmres method in each iteration evaluation of boundary and volume integrals is made by interpolating numerical solutions to equivalent but simple interface problems which is efficiently calculated with a hdg method on unfitted meshes the advantage of using the unfitted hdg method to treat the equivalent but simple interface problem is that the interpolation along boundary integrals on the interface can be easily evaluated via a linear lagrange polynomial element by element which is an important improvement from ying and henriquez 2007 numerical experiments are given to verify the effectiveness of the novel method which has second order accuracy even in the case of high contrast diffusion coefficients

the free energy at t 0 of the spin glass in the case of the continuous distribution of the effective field on the bethe lattice for z 4 5 6 abstract we evaluated the free energy at t 0 of the j random ising model on the bethe lattice of the coordination number z 4 5 6 in the case where the distribution function of the effective field has a continuous distribution by using the method of the integral equation of the effective field

hard rigid rods on a bethe like lattice we study a system of long rigid rods of fixed length k with only excluded volume interaction we show that contrary to the general expectation the self consistent field equations of the bethe approximation do not give the exact solution of the problem on the bethe lattice in this case we construct a new lattice called the random locally treelike layered lattice which allows a dense packing of rods and we show that the bethe self consistent equations are exact for this lattice for a four coordinated lattice k mers with k 4 undergo a continuous isotropic nematic phase transition for even coordination number q 6 the transition exists only for k k min q and is discontinuous

c2 r2 nonnegative extension by bounded depth operators abstract in this paper we prove the existence of a nonnegative parameter dependent nonlinear c 2 r 2 extension operator with bounded depth

cm omega extension by bounded depth linear operators abstract let c m r n be the space of functions on r n whose m th derivatives have modulus of continuity and c m e the space of restrictions to e r n of functions in c m r n we show that for any closed set e r n there exists a linear extension operator t c m e c m r n that assumes a particularly simple form namely t has depth d depending only on m and n

the whitney extension problem and lipschitz selections of set valued mappings in jet spaces we study a variant of the whitney extension problem 1934 for the space c k r n we identify c k r n with a space of lipschitz mappings from r n into the space p k x r of polynomial fields on r n equipped with a certain metric this identification allows us to reformulate the whitney problem for c k r n as a lipschitz selection problem for set valued mappings into a certain family of subsets of p k x r we prove a helly type criterion for the existence of lipschitz selections for such set valued mappings defined on finite sets with the help of this criterion we improve estimates for finiteness numbers in finiteness theorems for c k r n due to c fefferman

a decomposition of multidimensional point sets with applications to k nearest neighbors and n body potential fields we define the notion of a well separated pair decomposition of points in d dimensional space we then develop efficient sequential and parallel algorithms for computing such a decomposition we apply the resulting decomposition to the efficient computation of k nearest neighbors and n body potential fields

random field ising model on complete graphs and trees we present exact results for the critical behaviour of the random field ising model on complete graphs and trees both at equilibrium and away from equilibrium i e models for hysteresis and barkhausen noise we show that for stretched exponential and power law distributions of random fields the behaviour on complete graphs is non universal while the behaviour on cayley trees is universal even in the limit of large co ordination

on planar sobolev lpm extension domains abstract for each m 1 and p 2 we characterize bounded simply connected sobolev l p m extension domains r 2 our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in an important geometrical ingredient for obtaining these results is a new square separation theorem it states that under certain natural assumptions on the relative positions of a point x and a square s there exists a similar square q which touches s and has the property that x and s belong to distinct connected components of q

hysteresis in the antiferromagnetic random field ising model at zero temperature we study hysteresis in antiferromagnetic random field ising model at zero temperature the external field is cycled adiabatically between ensuremath ensuremath infty and ensuremath infty two different distributions of the random field are considered i a uniform distribution of width 2 ensuremath delta centered at the origin and ii a gaussian distribution with average value zero and standard deviation ensuremath sigma in each case the hysteresis loop is determined exactly in one dimension and compared with numerical simulations of the model

extension operators for smooth functions on compact subsets of the reals we introduce sufficient as well as necessary conditions for a compact set k such that there is a continuous linear extension operator from the space of restrictions c infty k f k f in c infty mathbb r to c infty mathbb r this allows us to deal with examples of the form k a n n in mathbb n cup 0 for a n rightarrow 0 previously considered by fefferman and ricci as well as vogt

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

equations of motion approach to the spin 1 2 ising model on the bethe lattice we exactly solve the ferromagnetic spin 1 2 ising model on the bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the green s function formalism in particular such an approach is applied to an isomorphic model of localized fermi particles interacting via an intersite coulomb interaction a complete set of eigenoperators is found together with the corresponding eigenvalues the green s functions and the correlation functions are written in terms of a finite set of parameters to be self consistently determined a procedure is developed that allows us to exactly fix the unknown parameters in the case of a bethe lattice with any coordination number z nonlocal correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities

whitney s extension problems and interpolation of data given a function f e r with e rn we explain how to decide whether f extends to a cm function f on rn if e is finite then one can efficiently compute an f as above whose cm norm has the least possible order of magnitude joint work with b klartag

whitney s problem on extendability of functions and an intrinsic metric abstract we consider the space of functions with bounded k 1 th derivatives in a general domain in r n is every such function extendible to a function of the same class defined on the whole r n h whitney showed that the equivalence of the intrinsic geodesic metric in this domain to the euclidean one is sufficient for such extendability there was an old conjecture going back to h whitney that this equivalence is also necessary for extendability we disprove this conjecture and construct examples of domains in r 2 such that the above extendability holds but the analogous property for smaller k fails our study is based on a duality approach

bond percolation on a class of clustered random networks analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering the network s degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real world networks

a sharp form of whitney s extension theorem 3 order relations involving multi indices 4 statement of two main lemmas 5 plan of the proof 6 starting the main induction 7 nonmonotonic sets 8 a consequence of the main inductive assumption 9 setup for the main induction 10 applying helly s theorem on convex sets 11 a calder6n zygmund decomposition 12 controlling auxiliary polynomials i 13 controlling auxiliary polynomials ii 14 controlling the main polynomials 15 proof of lemmas 9 1 and 5 2 16 a rescaling lemma 17 proof of lemma 5 3 18 proofs of the theorems 19 a bound for k references