on the critical capacity of the hopfield model we estimate the critical capacity of the zero temperature hopfield model by using a novel and rigorous method the probability of having a stable fixed point is one when 0 113 for a large number of neurons this result is an advance on all rigorous results in the literature and the relationship between the capacity and retrieval errors obtained here for small coincides with replica calculation results

energy gap for yang mills connections ii arbitrary closed riemannian manifolds abstract we prove an l d 2 energy gap result for yang mills connections on principal g bundles p over arbitrary closed riemannian smooth manifolds of dimension d 2 we apply our version of the ojasiewicz simon gradient inequality 16 19 to remove a positivity constraint on a combination of the ricci and riemannian curvatures in a previous l d 2 energy gap result due to gerhardt 23 theorem 1 2 and a previous l energy gap result due to bourguignon lawson and simons 10 theorem c 11 theorem 5 3 as well as an l 2 energy gap result due to nakajima 42 corollary 1 2 for a yang mills connection over the sphere s d but with an arbitrary riemannian metric

embedded area constrained willmore tori of small area in riemannian three manifolds ii morse theory the synthesis of cationic rhodium and iridium complexes of a bis imidazol 2 thione functionalised calix 4 arene ligand and their surprising capacity for potassium binding is described in both cases uptake of the alkali metal into the calix 4 arene cavity occurs despite adverse electrostatic interactions associated with close proximity to the transition metal fragment rh k 3 715 1 a ir k 3 690 1 a the formation and constituent bonding of these unusual heterobimetallic adducts has been interrogated through extensive solution and solid state characterisation examination of the host guest chemistry of the ligand and its upper rim unfunctionalised calix 4 arene analogue and computationally using dft based energy decomposition analysis eda this is the second part of a series of two papers where we construct embedded willmore tori with small area constraint in riemannian three manifolds in both papers the construction relies on a lyapunov schmidt reduction the difficulty being the moebius degeneration of the tori in the first paper the construction was performed via minimization here by morse theory to this aim we establish new geometric expansions of the derivative of the willmore functional on small clifford tori in geodesic normal coordinates which degenerate to small geodesic spheres with a small handle under the action of the moebius group by using these sharp asymptotics we give sufficient conditions in terms of the ambient curvature tensors and morse inequalities for having existence multiplicity of embedded tori which are stationary for the willmore functional under the constraint of prescribed sufficiently small area

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we

the hopfield model on a sparse erdos renyi graph we analyze the storage capacity of the hopfield model on a sparse g n p random graph we show that it is proportional to pn in the entire regime where the corresponding random graph is asymptotically connected and for all value of c 0 03

variational principles for immersed surfaces with l 2 bounded second fundamental form in this work we present new fundamental tools for studying the variations of the willmore functional of immersed surfaces into r m this approach gives for instance a new proof of the existence of a willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension we explain how the same approach can solve constraint minimization problems for the willmore functional we show in particular that for a given closed surface and a given conformal class for this surface there is an immersion in r m away possibly from isolated branched points which minimizes the willmore energy among all possible lipschitz immersions in r m having an l 2 bounded second fundamental form and realizing this conformal class this branched immersion is either a smooth conformal willmore branched immersion or an isothermic branched immersion we show that branched points do not exist whenever the minimal energy in the conformal class is less than 8 and that these immersions extend to smooth conformal willmore embeddings or global isothermic embeddings of the surface in that case finally as a by product of our analysis we establish that inside a compact subspace of the moduli space the following holds weak limit of palais smale willmore sequences are conformal willmore that weak limits of palais smale sequences of conformal willmore are either conformal willmore or global isothermic and finally we observe also that weakly converging palais smale sequences of global isothermic immersions are global isothermic the analysis developped along the paper in particular these last results opens the door to the possibility of constructing new critical saddle points of the willmore functional without or with constraints using min max methods

adaptive estimation of fuzzy cognitive maps with proven stability and parameter convergence fuzzy cognitive maps fcms have been introduced by kosko to model complex behavioral systems in various scientific areas one issue that has not been adequately studied so far is the conditions under which they reach a certain equilibrium point after an initial perturbation this is equivalent to studying the existence and uniqueness of solutions for their concept values in this paper we study the existence of solutions of fcms equipped with continuous differentiable sigmoid functions having contractive or at least non expansive properties this is done by using an appropriately defined contraction mapping theorem and the non expansive mapping theorem it is proved that when the weight interconnections fulfill certain conditions the concept values will converge to a unique solution regardless of the exact values of the initial concept values perturbations or in some cases a solution exists that may not necessarily be unique otherwise the existence or the uniqueness of equilibrium cannot be assured based on these results an adaptive weight estimation algorithm is proposed that employs appropriate weight projection criteria to assure that the uniqueness of fcm solution is not compromised in view of these results recently proposed extensions of fcm which are the fuzzy cognitive networks fcn are invoked

on the free energy of the hopfield model the general theory of inhomogeneous mean field systems of raggio and werner provides a variational expression for the almost sure limiting free energy density of the hopfield model h n p xi s frac 1 2n sum limits i j 1 n sum limits mu 1 n xi i mu xi j mu s i s j for ising spinssi andp random patterns 1 2 n under the assumption that mathop lim limits n to gamma n 1 sum limits i 1 n delta xi i lambda xi i xi i 1 xi i 2 xi i p exists almost surely in the space of probability measures overp copies of 1 1 including an external field term p h i 1 n i si we give a number of general properties of the free energy density and compute it for a p 2 in general and b p arbitrary when is uniform and at most the two componentsh 1 andh 2 are nonzero obtaining the almost sure formula f beta h tfrac 1 2 f ew beta h mu 1 h mu 2 tfrac 1 2 f ew beta h mu 1 h mu 2 for the free energy wherefcw denotes the limiting free energy density of the curie weiss model with unit interaction constant in both cases we obtain explicit formulas for the limiting almost sure values of the so called overlap parameters m n mu beta h n 1 sum limits i 1 n xi i mu left langle s i right rangle in terms of the curie weiss magnetizations for the general i i d case with prob i 1 1 2 we obtain the lower bound 1 4 2 p 1 for the temperaturetc separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero this lower bound is exact forp 2 or e 0 or e 1 2 forp 2 we identify an intermediate temperature region between t 1 4 2 and tc 1 4 2 where the overlap vector is homogeneous i e all its components are equal and nonzero t marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct we conjecture that the homogeneous nonzero regime exists forp 3 and that t max 1 4 2 p 1 0

a lower bound for the memory capacity in the potts hopfield model we consider potts hopfield networks of sizen we prove the result c 0 such that for all 0 0 in such a way that whenn we can store n patterns all of them being sorrounded by energy barriers at distance

fuzzy causal networks general model inference and convergence in this paper we first propose a general framework for fuzzy causal networks fcns then we study the dynamics and convergence of such general fcns we prove that any general fcn with constant weight matrix converges to a limit cycle or a static state or the trajectory of the fcn is not repetitive we also prove that under certain conditions a discrete state general fcn converges to its limit cycle or static state in o n steps where n is the number of vertices of the fcn this is in striking contrast with the exponential running time 2 sup n which is accepted widely for classic fcns

on the temporal granularity in fuzzy cognitive maps the theory of fuzzy cognitive maps fcms is a powerful approach to modeling human knowledge that is based on causal reasoning taking advantage of fuzzy logic and cognitive map theories fcms enable system designers to model complex frameworks by defining degrees of causality between causal objects they can be used to model and represent the behavior of simple and complex systems by capturing and emulating the human being to describe and present systems in terms of tolerance imprecision and granulation of information however fcms lack the temporal concept that is crucial in many real world applications and they do not offer formal mechanisms to verify the behavior of systems being represented which limit conventional fcms in knowledge representation in this paper we present an extension to fcms by exploiting a theory from formal languages namely the timed automata which bridges the aforementioned inadequacies indeed the theory of timed automata enables fcms to effectively deal with a double layered temporal granularity extending the standard idea of b time that characterizes the iterative nature of a cognitive inference engine and offering model checking techniques to test the cognitive and dynamic comportment of the framework being designed

harmonic maps from degenerating riemann surfaces we study harmonic maps from degenerating riemann surfaces with uniformly bounded energy and show the so called generalized energy identity we find conditions that are both necessary and sufficient for the compactness in w 1 2 and c 0 modulo bubbles of sequences of such maps

multi group binary choice with social interaction and a random communication structure a random graph approach abstract we construct and analyze a random graph model for discrete choice with social interaction and several groups of equal size we concentrate on the case of two groups of equal sizes and we allow the interaction strength within a group to differ from the interaction strength between the two groups given that the resulting graph is sufficiently dense we show that with probability 1 the average decision in each of the two groups is the same as in the fully connected model in particular we show that there is a phase transition if the interaction among a group and between the groups is strong enough the average decision per group will either be positive or negative and the decision of the two groups will be correlated we also compute the free energy per particle in our model

rigorous bounds on the storage capacity of the dilute hopfield model we study a neural network model consisting ofn neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1 p for the hopfield hamiltonian on such a network we prove that ifp c lnn n 1 2 the model can store at leastm cpn patterns where c 0 027 ifc 3 and decreases proportional to 1 lnc forc small this generalizes the results of newman for the standard hopfield model

lower bounds on the restitution error in the hopfield model this paper contains a first rigorous results at the direction of the converse to newman s bounds 11 on the storage capacity for the hopfield model of associative memory let m denote the number of stored patterns and n that of neurons assuming that m n to alpha as n to infty we show that for alpha 0 the memory necessarily commits a positive fraction of errors on memorized patterns that becomes superior to some threshold value approximately 0 05 as alpha to infty

self averaging in a class of generalized hopfield models we prove the almost sure convergence to zero of the fluctuations of the free energy in a class of disordered mean field spin systems that generalize the hopfield model in two ways i multi spin interactions are permitted and ii the random variables xi imu describing the patterns can have arbitrary distributions with mean zero and finite 4 epsilon th moments the number of patterns m is allowed to be an arbitrary multiple of the system size this generalizes a previous result of bovier gayrard and picco for the standard hopfield model and improves a result of feng and tirozzi that required m to be a finite constant note that the convergence of the mean of the free energy is not proven

an extension to fuzzy cognitive maps for classification and prediction fuzzy cognitive maps fcms as an illustrative causative representation of modeling and manipulation of complex systems can be used to model the dynamic behavior of the investigated systems however due to defects in expression and architecture the traditional fcms and most of their relevant extensions are not applicable to classification problems to solve this problem this paper presents an approach that directly extends the model by translating the reasoning mechanism of traditional fcms to a set of fuzzy if then rules moreover the proposed approach fully considers the contribution of the inputs to the activation of the fuzzy rules and quantifies the causalities using mutual subsethood which works in conjunction with volume defuzziflcation in a gradient descent learning framework in this manner our approach enhances the capability of the conventional fcms to automatically identify membership functions and quantify causalities despite the increase in the number of tunable parameters experimental results show that the proposed approach efficiently extends the application of the traditional fcms into classification problems while keeping the ability for prediction and approximation

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

willmore surfaces with nonremovable singularities and number of critical levels in this paper we show that for every prescribed genus p in mathbb n and n in mathbb n with n ge 3 there exists a closed and orientable willmore surface in mathbb r n of genus p with at least two nonremovable point singularities furthermore we prove an energy gap theorem for smooth closed and orientable willmore surfaces in mathbb r n of prescribed genus p in mathbb n with n in mathbb n and n ge 3 moreover we show that the willmore functional as a map on the set of smooth closed and orientable surfaces in mathbb r n of genus p in mathbb n with n in 3 4 has only a finite number of critical levels strictly below a douglas condition on the willmore energy in order to exclude topological splitting

the retrieval phase of the hopfield model a rigorous analysis of the overlap distribution standard large deviation estimates or the use of the hubbard stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function n on m in this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns we denote by m the modulus of the spontaneous magnetization in the curie weiss model and by the ratio between the number of the stored patterns and the system size we show that there exist strictly positive numbers 0 a c such that 1 if a m 2 then the absolute minima of are located within small balls around the points m e where e denotes the th unit vector while 2 if c m 2 at least a local minimum surrounded by extensive energy barriers exists near these points the random location of these minima is given within precise bounds these are used to prove sharp estimates on the support of the gibbs measures

rigorous results for the hopfield model with many patterns we perform a thorough investigation of the main aspects of the hopfield model with many patterns advances are made toward the validity of the replica symmetric solution strong evidence of the validity of this solution is given over the entire domain where this validity is conjectured complete proof is given in a subregion that contains strictly the ergodic region

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

thermodynamic limit of the q state potts hopfield model with infinitely many patterns we prove the almost sure convergence of the free energy and of the overlap order parameters in aq state version of the hopfield neural network model we compute explicitly these limits for all temperatures different from some critical value the number of stored patterns is allowed to grow with the size of the systemn like lnq lnn we study the limiting behavior of the extremal states of the model that are the measures induced on the gibbs measures by the overlap parameters

large deviation principles for the hopfield model and the kac hopfield model we study the kac version of the hopfield model and prove a lebowitz penrose theorem for the distribution of the overlap parameters at the same time we prove a large deviation principle for the standard hopfield model with infinitely many patterns

on causal inference in fuzzy cognitive maps fuzzy cognitive maps fcm is a powerful paradigm for representing human knowledge and causal inference this paper formally analyzes the causal inference mechanism of fcm we focus on binary concept states it is known that given initial conditions fcm is able to reach only certain states in its state space we prove that the problem of finding whether a state is reachable in the fcm is nondeterministic polynomial np hard that we can divide fuzzy cognitive maps containing circles into basic fcm modules the inference patterns in these basic modules can be studied individually in a hierarchical fashion this paper also presents a recursive formula for computing fcm s inference patterns in terms of key vertices the theoretical results presented in this paper provide a feasible and effective framework for the analysis and design of fuzzy cognitive maps in real world large scale applications

intuitionistic fuzzy cognitive maps uncertainty and imprecision characterize human cognitive and reasoning processes fuzzy cognitive maps fcms are computationally simple yet effective structures to approximately model and simulate such processes a limitation of current fcms is that they are unable to model the hesitancy introduced into a complex system due to imperfect facts missing information and indecision to cope with this issue we propose a novel extension of the fcm model which is based on the theory of intuitionistic fuzzy sets this intuitionistic fcm ifcm model which is denoted as ifcm ii inherently exploits the mathematical framework of intuitionistic fuzzy sets for the definition of the concepts constituting the cognitive map and their interrelations as well as for reasoning furthermore unlike the previous ifcm model which is denoted as ifcm i it enables an intuitionistic estimation of hesitancy at the output concepts thus offering a natural mechanism to assess the quality of its output the advantages of the proposed ifcm model over the current fcm and ifcm models are demonstrated with reproducible numeric examples for process control and decision support applications

some rigorous results on the hopfield neural network model we analyze the thermal equilibrium distribution of 2p mean field variables for the hopfield model withp stored patterns in the case where 2p is small compared to the number of spins in particular we give a full description of the free energy density in the thermodynamic limit and of the so called symmetric solutions for the mean field equations

estimation of the conformal factor under bounded willmore energy we prove for closed orientable surfaces in mathbb r 3 with willmore energy less that 8 pi delta and whose conformal structures are compactly contained in moduli space that after applying appropriate mobius transformations the conformal factors between the induced metrics and conformal metrics of constant curvature are uniformly bounded by constants depending only on delta 0 the genus of the surfaces and the compact subset of the moduli space secondly for a given sequence of closed orientable surfaces as above we prove that the conformal factor remains bounded without applying mobius transformations if and only if no topology is lost similar estimates hold in higher codimension

quotient fcms a decomposition theory for fuzzy cognitive maps in this paper we introduce a decomposition theory for fuzzy cognitive maps fcm first we partition the set of vertices of an fcm into blocks according to an equivalence relation and by regarding these blocks as vertices we construct a quotient fcm second each block induces a natural sectional fcm of the original fcm which inherits the topological structure as well as the inference from the original fcm in this way we decompose the original fcm into a quotient fcm and some sectional fcm as a result the analysis of the original fcm is reduced to the analysis of the quotient and sectional fcm which are often much smaller in size and complexity such a reduction is important in analyzing large scale fcm we also propose a causal algebra in the quotient fcm which indicates that the effect that one vertex influences another in the quotient depends on the weights and states of the vertices along directed paths from the former to the latter to illustrate the process involved we apply our decomposition theory to university management networks finally we discuss possible approaches to partitioning an fcm and major concerns in constructing quotient fcm the results represented in this paper provide an effective framework for calculating and simplifying causal inference patterns in complicated real world applications

energy quantization for willmore surfaces and applications we prove a bubble neck decomposition together with an energy quantization result for sequences of willmore surfaces into r m with uniformly bounded energy and nondegenerating conformal type we deduce the strong compactness of willmore closed surfaces of a given genus modulo

eigenvalue distribution of large dilute random matrices we study the eigenvalue distribution of dilute n n random matrices hn that in the pure undiluted case describe the hopfield model we prove that for the fixed dilution parameter a the normalized counting function ncf of hn converges as n to a unique a we find the moments of this distribution explicitly analyze the 1 a correction and study the asymptotic properties of a for large we prove that a converges as a to the wigner semicircle distribution scd we show that the scd is the limit of the ncf of other ensembles of dilute random matrices this could be regarded as evidence of stability of the scd to dilution or more generally to random modulations of large random matrices

dynamic domination in fuzzy causal networks this paper presents a dynamic domination theory for fuzzy causal networks fcn there are three major contributions first we propose a new inference procedure based on dominating sets second we introduce the concepts of dynamic and minimal dynamic dominating sets dds and mdds in an fcn to reflect changes of dominance with time we also introduce the concept of a dynamic dominating process ddp that has significant implications in many real world problems we pay a special attention to the minimal dynamic dominating process mddp and develop rules for generating ddp and mddp third we investigate dynamic dominating sets with extended feedback which we call effective dynamic dominating sets edds and related effective dynamic dominating process eddp this study unveils a very important phenomenon in fcn at any time t either an edds exists or there is a dramatic change of the states of vertices in the latter case we also identify the special structure of the sub fcn induced by active vertices

graph degree sequence solely determines the expected hopfield network pattern stability we analyze the effect of network topology on the pattern stability of the hopfield neural network in the case of general graphs the patterns are randomly selected from a uniform distribution we start the hopfield procedure from some pattern v an error in an entry e of v is the situation where if the procedure is started at e the value of e flips such an entry is an instability point note that we disregard the value at e by the end of the procedure as well as what happens if we start the procedure from another pattern or another entry of v we measure the instability of the system by the expected total number of instability points of all the patterns our main result is that the instability of the system does not depend on the exact topology of the underlying graph but rather only on its degree sequence moreover for a large number of nodes the instability can be approximated by where is the standard normal distribution function and are the degrees of the nodes

bubble tree convergence for harmonic maps let be a compact riemann surface any sequence fn m of harmonic maps with bounded energy has a bubble tree limit consisting of a harmonic map o m and a tree of bubbles fk s2 m we give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class and that the images of the fn converge pointwise we then give explicit counterexamples showing that bubble tree convergence fails i for harmonic maps fn when the conformal structure of varies with n and ii when the conformal structure is fixed and n is a palais smale sequence for the harmonic map energy

rigorous results on the thermodynamics of the dilute hopfield model we study the hopfield model of an autoassociative memory on a random graph onn vertices where the probability of two vertices being joined by a link isp n assuming thatp n goes to zero more slowly thano 1 n we prove the following results 1 if the number of stored patternsm n is small enough such thatm n np n 0 asn then the free energy of this model converges upon proper rescaling to that of the standard curie weiss model for almost all choices of the random graph and the random patterns 2 if in additionm n ln n ln 2 we prove that there exists fort 1 a gibbs measure associated to each original pattern whereas for higher temperatures the gibbs measure is unique the basic technical result in the proofs is a uniform bound on the difference between the hamiltonian on a random graph and its mean value