generalized suq 1 2 gaudin model in this paper we propose the hamiltonians of the generalized suq 1 2 gaudin model corresponding to the periodic generalized t j model with the help of the well defined graded quantum determinant we obtain the eigenstate and eigenvalues of the generating function and the hamiltonians of the gaudin model in the fermionic background in the framework of the graded quantum inverse scattering method the bethe ansatz equations are also obtained

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

exact results for the critical behaviour of a nienhuis o n model on the square lattice exact results are obtained for a square lattice fifteen vertex model and related o n model recently proposed by nienhuis 1990 the partition function per site and finite size corrections are obtained along with the critical exponents via the temperley lieb equivalent six vertex model the exact exponents nu 4 7 and gamma 6 7 follow in the limit n 0 for even system sizes odd system sizes are also discussed the fifteen vertex model is also the vertex formulation of the anisotropic or q deformed spin 1 biquadratic model

on the clebsch gordan coefficients for the two parameter quantum algebra su 2 p q we show that the clebsch gordan coefficients for the su 2 p q algebra depend on a single parameter q square root pq contrary to the explicit calculation of smirnov and wehrhahn 1992

optimal storage capacity of neural networks at finite temperatures gardner s analysis 1989 of the optimal storage capacity of neural networks is extended to study finite temperature effects the typical volume of the space of interactions is calculated for strongly diluted networks as a function of the storage ratio alpha temperature t and the tolerance parameter m from which the optimal storage capacity alpha c is obtained as a function of t and m at zero temperature it is found that alpha c 2 regardless of m while alpha c in general increases with the tolerance at finite temperatures the authors show how the best performance for given alpha and t is obtained which reveals a first order transition from high quality performance to a low quality one at low temperatures an approximate criterion for recalling which is valid near m 1 is also discussed

a critical exponent for shortest path scaling in continuum percolation we carry out monte carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation these studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation we use splitting a technique that has not yet been fully exploited in the physics literature to increase the speed of our simulations this technique can also be applied to other models where clusters are grown sequentially

exact solution of the deformed biquadratic spin 1 chain we solve for the sped of the deformed biquadratic spin l chain by a generalization of the coordinate bethe ansatz the model is invariant under the quantum group u sl z an6 depending on the deformation parameter q it s hamiltonian may not be hermitian we consider several boundary conditions and find in particular that for periodic ones the complete specmm of the spin 4 xxz model is always contained in the spec of the biquadratic model for free boundary conditions the spectrum is identicd to the xxz model speci only the multiplicities are of course different bethe states are highest weight states of the quantum group except possibly for zerwnergy states

a retinex model based on absorbing markov chains the retinex algorithm developed by land and mccann provides an abstract model of the mechanism of color sensation in the human vision system at the basis of model lies the fact that the color appearance of a point does not depend only on its color value but rather on the comparison among itself and other pixels according to the model separately for each chromatic channel an image pixel receives suitably filtered information about the brightness of other image regions based on which its own brightness is eventually re scaled the original formulation land and mccann 1971 uses a path based sampling approach the information is transported by memoryless random walks starting from randomly chosen points along the path the information is filtered based on the brightness of the travelled regions by a specific path function computed through chains of ratios of pixel intensities such a function is path dependent and retains the value of the brightest point found along the path the overall correction to a pixel depends on the specific realizations of two sampling processes the starting point sampling process and the path sampling process as a consequence of the sampling this algorithm is known to be intrinsically noisy this draw back can be overcome by passing from the path sampling algorithm to the probabilistic representation of the corresponding diffusion process in this paper we start from the random path simulative model of retinex we respell the standard path based sampling process representation of the retinex model as formalized in provenzi et al 2005 and we show that despite the overall path dependence the model can be given a representation in terms of absorbing markov chains by means of the embedding into a suitable state space we derive the corresponding analytic model accounting for the combined effects of path function path sampling process and starting point sampling process finally we provide a numerical algorithm for working out its solution using such a model the output brightness of a pixel can be computed based on the solution of a simple sparse linear system we show that the output of the random walk sampling algorithm and the markov chain based algorithm agree to an extent that can be controlled by few model parameters we have found also that the markov chain based algorithm is more efficient than the basic random path sampling in obtaining noise free images those analytic probabilistic models and simulative models can be used as complementary tools for studying the retinex mechanism and for identifying and comparing variants

retrieval phase diagrams for attractor neural networks with optimal interactions the authors consider the retrieval properties of attractor neural networks whose synaptic matrices have been constructed to maximise the number of patterns which can be stored in a perceptron satisfying certain constraints retrieval is studied in the absence as well as in the presence of fast noise temperature the discussion is restricted to dilute networks for which dynamical equations for the overlaps are available when the patterns are stored with a prescribed lower limit on the stability parameter on every site the full stability of the perceptron ensures the existence of an attractor with perfect retrieval it is found that the curve of critical storage capacity alpha against temperature t as a line of first order transitions for high values of alpha and becomes second order for low alpha at a point of a tricritical nature the phase diagram is compared with the dilute hopfield model it is found that at high synaptic noise levels the diluted hopfield net stores more effectively than the network trained for optimal perceptron storage when a given fraction of sites is allowed to violate the stability bound the solution of the perceptron learning problem does not ensure the existence of an attractor of finite overlap even in the absence of noise this case is studied separately for t 0 and for finite t

issues about retinex theory and contrast enhancement we present an interpretation of land s retinex theory that we show to be consistent with the original formulation the proposed model relies on the computation of the expectation value of a suitable random variable weighted with a kernel function thus the name kernel based retinex kbr for the corresponding algorithm kbr shares the same intrinsic characteristics of the original retinex it can reduce the effect of a color cast and enhance details in low key images but since it can only increase pixel intensities it is not able to enhance over exposed pictures comparing the analytical structure of kbr with that of a recent variational model of color image enhancement we are able to perform an analysis of the action of kbr on contrast showing the need to anti symmetrize its equation in order to produce a two sided contrast modification able to enhance both under and over exposed pictures the anti symmetrized kbr equations show clear correspondences with other existing color correction models in particular ace whose relationship with retinex has always been difficult to clarify finally from an image processing point of view we mention that both kbr and its antisymmetric version are free from the chromatic noise due to the use of paths in the original retinex implementation and that they can be suitably approximated in order to reduce their computational complexity from mathcal o n 2 to mathcal o n log n being n the number of input pixels

vanishing gaps in 1d bandstructures the criterion for vanishing or zero energy gaps in the spectrum of 1d electronic bandstructures is obtained from perturbation theory it is in agreement with examples of periodic potentials which never exhibit zero gaps as well as with those which may including certain smooth periodic potentials the results have relevance to recent work in the field of semiconductor layered heterostructures or superlattices the analysis should also have implications for other problems that involve hill s determinant

random surfaces from polymer membranes to strings i review the state of knowledge about random surface ensembles in continuous embedding spaces and their possible role in defining strings in arbitrary dimensions the application of rigorous statistical mechanics approximate calculation and numerical simulation is described

isotropy of semisimple group actions on manifolds with geometric structure we prove results concerning isotropy of noncompact semisimple lie group actions that preserve pseudo riemannian or affine structures on compact or finite volume manifolds for example if a lie group g acts locally faithfully on a connected compact complete affine manifold m and preserves the affine structure then the connected component of the identity of each isotropy subgroup is a compact extension of a solvable group this is a consequence of the result that nontrivial affine actions of connected groups locally isomorphic to sl 2 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i on such manifolds cannot have fixed points we also prove that if g is connected semisimple without compact factors and acts isometrically and topologically transitively on a connected finite volume pseudo riemannian m then the action must be everywhere locally free this answers affirmatively a special case of a conjecture by g d ambra and m gromov with only the assumption that no factor of g acts trivially local freeness is also assured when m is compact and complete

the matrix theory of automorphisms for twisted affine kac moody algebras for and for 1 the application to the two series of classical twisted affine kac moody algebras for and for of the matrix formulation of automorphisms developed earlier for untwisted affine kac moody algebras is investigated the conjugacy classes of involutive automorphisms and the corresponding real forms are deduced explicitly for the special case of

involutive automorphisms of the class of affine kac moody algebras bl 1 all the conjugacy classes of involutive automorphisms of the affine kac moody algebras bl 1 for l or 1 are determined using the matrix formulation of automorphisms of an affine kac moody algebra

from statistical mechanics to neural networks and back abstract the neural network models which have attracted so much attention in the physics community consist of a large assembly of two state elements highly interconnected by ferromagnetic and antiferromagnetic interactions methods of statistical mechanics have been applied to these models bringing new discoveries and surprises to neural network science at the same time neural network models have posed new problems and incentives to statistical mechanics one of the basic problems is that of learning which in the context of these models means a choice of the interactions for which the network dynamics leads to specific attractors the main emphasis in this paper is on the essential developments and results on this problem derived from methods of statistical mechanics

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

the number of metastable states of a simple perceptron with gradient descent learning algorithm the number of metastable states of a simple perceptron with gradient descent learning algorithm has been calculated as a function of the storage capacity alpha and the gain parameter beta for alpha not 0 and beta 0 exponential large number of local minima similar to the spin glasses and analogue attractor networks has been found

on lorentz dynamics from group actions to warped products via homogeneous spaces we show a geometric rigidity of isometric actions of non compact semisimple lie groups on lorentz manifolds namely we show that the manifold has a warped product structure of a lorentz manifold with constant curvature by a riemannian manifold

large systems of random linear equations with nonnegative solutions characterizing the solvable and the unsolvable phase large systems of linear equations are ubiquitous in science quite often e g when considering population dynamics or chemical networks the solutions must be nonnegative recently it has been shown that large systems of random linear equations exhibit a sharp transition from a phase where a nonnegative solution exists with probability one to one where typically no such solution may be found the critical line separating the two phases was determined by combining farkas lemma with the replica method here we show that the same methods remain viable to characterize the two phases away from criticality to this end we analytically determine the residual norm of the system in the unsolvable phase and a suitable measure of robustness of solutions in the solvable one our results are in very good agreement with numerical simulations

pair connectedness and shortest path scaling in critical percolation we present high statistics data on the distribution of shortest path lengths between two near by points on the same cluster at the percolation threshold our data are based on a new and very efficient algorithm for d 2 they clearly disprove a recent conjecture by m porto et al 1998 phys rev e 58 r5205 our data also provide upper bounds on the probability that two near by points are on different infinite clusters

higher order corrections to the effective potential close to the jamming transition in the perceptron model in view of the results achieved in a previously related work a altieri s franz and g parisi j stat mech 2016 093301 10 1088 1742 5468 2016 09 093301 regarding a plefka like expansion of the free energy up to the second order in the perceptron model we improve the computation here focusing on the role of third order corrections the perceptron model is a simple example of constraint satisfaction problem falling in the same universality class as hard spheres near jamming and hence allowing us to get exact results in high dimensions for more complex settings our method enables to define an effective potential or thouless anderson palmer free energy namely a coarse grained functional which depends on the generalized forces and the effective gaps between particles the analysis of the third order corrections to the effective potential reveals that albeit irrelevant in a mean field framework in the thermodynamic limit they might instead play a fundamental role in considering finite size effects we also study the typical behavior of generalized forces and we show that two kinds of corrections can occur the first contribution arises since the system is analyzed at a finite distance from jamming while the second one is due to finite size corrections we nevertheless show that third order corrections in the perturbative expansion vanish in the jamming limit both for the potential and the generalized forces in agreement with the isostaticity argument proposed by wyart and coworkers finally we analyze the relevant scaling solutions emerging close to the jamming line which define a crossover regime connecting the control parameters of the model to an effective temperature

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

a perceptually inspired variational framework for color enhancement basic phenomenology of human color vision has been widely taken as an inspiration to devise explicit color correction algorithms the behavior of these models in terms of significative image features such as e g contrast and dispersion can be difficult to characterize to cope with this we propose to use a variational formulation of color contrast enhancement that is inspired by the basic phenomenology of color perception in particular we devise a set of basic requirements to be fulfilled by an energy to be considered as perceptually inspired showing that there is an explicit class of functionals satisfying all of them we single out three explicit functionals that we consider of basic interest showing similarities and differences with existing models the minima of such functionals is computed using a gradient descent approach we also present a general methodology to reduce the computational cost of the algorithms under analysis from o n2 to o n logn being n the number of pixels of the input image

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

perceptual color correction through variational techniques in this paper we present a discussion about perceptual based color correction of digital images in the framework of variational techniques we propose a novel image functional whose minimization produces a perceptually inspired color enhanced version of the original the variational formulation permits a more flexible local control of contrast adjustment and attachment to data we show that a numerical implementation of the gradient descent technique applied to this energy functional coincides with the equation of automatic color enhancement ace a particular perceptual based model of color enhancement moreover we prove that a numerical approximation of the euler lagrange equation reduces the computational complexity of ace from o n2 to o nlogn where n is the total number of pixels in the image

facial expression recognition in perceptual color space this paper introduces a tensor perceptual color framework tpcf for facial expression recognition fer which is based on information contained in color facial images the tpcf enables multilinear image analysis in different color spaces and demonstrates that color components provide additional information for robust fer using this framework the components in either rgb ycbcr cielab or cieluv space of color images are unfolded to 2 d tensors based on multilinear algebra and tensor concepts from which the features are extracted by log gabor filters the mutual information quotient method is employed for feature selection these features are classified using a multiclass linear discriminant analysis classifier the effectiveness of color information on fer using low resolution and facial expression images with illumination variations is assessed for performance evaluation experimental results demonstrate that color information has significant potential to improve emotion recognition performance due to the complementary characteristics of image textures furthermore the perceptual color spaces cielab and cieluv are better overall for fer than other color spaces by providing more efficient and robust performance for fer using facial images with illumination variation

statistical mechanics of lossy data compression using a nonmonotonic perceptron the performance of a lossy data compression scheme for uniformly biased boolean messages is investigated via methods of statistical mechanics inspired by a formal similarity to the storage capacity problem in neural network research we utilize a perceptron of which the transfer function is appropriately designed in order to compress and decode the messages employing the replica method we analytically show that our scheme can achieve the optimal performance known in the framework of lossy compression in most cases when the code length becomes infinite the validity of the obtained results is numerically confirmed

information capacity of a perceptron the authors study the information storage capacity of a simple perceptron in the error regime for random unbiased patterns the geometrical analysis gives a logarithmic dependence for the information content in the asymptotic limit in this case the statistical physics approach when used at the simplest level of replica theory does not give satisfactory results however for perceptrons with finite stability the information content can be simply calculated with statistical physics methods in a region above the critical storage level for biased as well as for unbiased patterns

a precise characterization of three dimensional percolating backbones the backbones of three dimensional critical percolation clusters are extracted and the fractal dimension is accurately calculated using various mass scaling and box counting techniques to be dfb 1 855 or 0 015 the eigenvalue spectrum of the brownian transition probability matrix is studied numerically which results in the estimates dwb 3 13 or 0 03 for the walk dimension and dsb 1 18 or 0 01 for the spectral dimension of the backbone independent calculation of all three quantities allows the conductivity exponent mu to be determined and also allows the validity of the scaling relation ds 2df dw proposed by alexander and orbach 1982 to be tested on the backbone

random cellular froths in spaces of any dimension and curvature froth is a random partition of a d dimensional space by cells this assembly of cells obeys two fundamental laws euler s relation and the condition of maximum vertex figure imposed by geometry and by topological stability respectively these two conditions generate a set of relations between the variables that fully characterize the system topologically the number of degrees of freedom of the system and a set of useful independent variables the even valences have been found the influence of the space dimension and curvature on the range of variability of these valences is discussed and up to d 5 the regions in valence space corresponding to differently curved froths are calculated explicitly

noncompact simple automorphism groups of lorentz manifolds and other geometric manifolds when such a g does act zimmer has proven that the full automorphism group of the lorentz manifold must be locally isomorphic to sl 2 r x k where k is a compact lie group m gromov has shown 2 that the universal cover m must then be lorentz isometric to a warped product sl 2 r f x s where sl 2 ir is the universal cover of sl 2 ii with the cartan killing metric s is a riemannian manifold and f s 0 o oo is a warping function this paper is mostly concerned with the following situation a noncompact simple r algebraic group g acting on a manifold m preserving a lorentz metric we do not assume m compact or finite volume or even complete in contrast the theorems above require compactness and use it in an essential way in their proofs but as we show strong analogues of these theorems hold even for m noncompact our first result is the following

an analytic estimate of the maximum lyapunov exponent in products of tridiagonal random matrices in this work we present a theoretical and numerical study of the behaviour of the maximum lyapunov exponent in products of random tridiagonal matrices in the limit of small coupling and small fluctuations such a problem is directly motivated by the investigation of coupled map lattices in a regime where the chaotic properties are quite robust and yet a complete understanding has still not been reached we derive some approximate analytic expressions by introducing a suitable continuous time formulation of the evolution equation as a first result we show that the perturbation of the lyapunov exponent due to the coupling depends only on a single scaling parameter which in the case of strictly positive multipliers is the ratio of the coupling strength with the variance of local multipliers an explicit expression for the lyapunov exponent is obtained by mapping the original problem onto a chain of nonlinear langevin equations which are eventually reduced to a single stochastic equation the probability distribution of this dynamical equation provides an excellent description for the behaviour of the lyapunov exponent finally multipliers with random signs are also considered finding that the lyapunov exponent again depends on a single scaling parameter which however has a different expression

complex network view of evolving manifolds we study complex networks formed by triangulations and higher dimensional simplicial complexes representing closed evolving manifolds in particular for triangulations the set of possible transformations of these networks is restricted by the condition that at each step all the faces must be triangles stochastic application of these operations leads to random networks with different architectures we perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties this characterization includes the hausdorff and spectral dimensions of the resulting networks their degree distributions and various structural correlations our results reveal a rich zoo of architectures and geometries of these networks some of which appear to be small worlds while others are finite dimensional with hausdorff dimension equal or higher than the original dimensionality of their simplices the range of spectral dimensions of the evolving triangulations turns out to be from about 1 4 to infinity our models include simplicial complexes representing manifolds with evolving topologies for example an h holed torus with a progressively growing number of holes this evolving graph demonstrates features of a small world network and has a particularly heavy tailed degree distribution

a pde formalization of retinex theory in 1964 edwin h land formulated the retinex theory the first attempt to simulate and explain how the human visual system perceives color his theory and an extension the reset retinex were further formalized by land and mccann several retinex algorithms have been developed ever since these color constancy algorithms modify the rgb values at each pixel to give an estimate of the color sensation without a priori information on the illumination unfortunately the retinex land mccann original algorithm is both complex and not fully specified indeed this algorithm computes at each pixel an average of a very large set of paths on the image for this reason retinex has received several interpretations and implementations which among other aims attempt to tune down its excessive complexity in this paper it is proved that if the paths are assumed to be symmetric random walks the retinex solutions satisfy a discrete poisson equation this formalization yields an exact and fast implementation using only two ffts several experiments on color images illustrate the effectiveness of the retinex original theory

dynamical partitions of space in any dimension topologically stable cellular partitions of d dimensional spaces are studied a complete statistical description of the average structural properties of such partitions is given in terms of a sequence of or variables for d even or odd these variables are the average coordination numbers of the 2k dimensional polytopes which make up the cellular structure a procedure to produce d dimensional space partitions through cell division and cell coalescence transformations is presented classes of structures which are invariant under these transformations are found and the average properties of such structures are illustrated homogeneous partitions are constructed and compared with the known structures obtained by voronoi partitions and sphere packings in high dimensions

storage capacity of ultrametric committee machines the problem of computing the storage capacity of a feed forward network with l hidden layers n inputs and k units in the first hidden layer is analyzed using techniques from statistical mechanics we found that the storage capacity strongly depends on the network architecture c log k 1 1 2l and that the number of units k limits the number of possible hidden layers l through the relationship 2l 1 2log k

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

dynamics of learning and generalization in perceptrons with constraints abstract depending on the kind of constraints imposed on the weights of a perceptron learning can be a combinatorially hard problem as an example of this type i discuss a perception with binary weights comparing results obtained from replica theory dynamic mean field theory and simulated annealing contrary to the replica calculation dynamics yields information about the performance of a polynomial algorithm in a situation where the best solution cannot be found in polynomial time i also discuss improved learning algorithms and results for finite size perceptrons

a variational framework for retinex retinex theory addresses the problem of separating the illumination from the reflectance in a given image and thereby compensating for non uniform lighting this is in general an ill posed problem in this paper we propose a variational model for the retinex problem that unifies previous methods similar to previous algorithms it assumes spatial smoothness of the illumination field in addition knowledge of the limited dynamic range of the reflectance is used as a constraint in the recovery process a penalty term is also included exploiting a priori knowledge of the nature of the reflectance image the proposed formulation adopts a bayesian view point of the estimation problem which leads to an algebraic regularization term that contributes to better conditioning of the reconstruction problem based on the proposed variational model we show that the illumination estimation problem can be formulated as a quadratic programming optimization problem an efficient multi resolution algorithm is proposed it exploits the spatial correlation in the reflectance and illumination images applications of the algorithm to various color images yield promising results

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

what geometric visual hallucinations tell us about the visual cortex many observers see geometric visual hallucinations after taking hallucinogens such as lsd cannabis mescaline or psilocybin on viewing bright flickering lights on waking up or falling asleep in near death experiences and in many other syndromes kluver organized the images into four groups called form constants i tunnels and funnels ii spirals iii lattices including honeycombs and triangles and iv cobwebs in most cases the images are seen in both eyes and move with them we interpret this to mean that they are generated in the brain here we summarize a theory of their origin in visual cortex area v1 based on the assumption that the form of the retino cortical map and the architecture of v1 determine their geometry a much longer and more detailed mathematical version has been published in philosophical transactions of the royal society b 356 2001 we model v1 as the continuum limit of a lattice of interconnected hypercolumns each comprising a number of interconnected iso orientation columns based on anatomical evidence we assume that the lateral connectivity between hypercolumns exhibits symmetries rendering it invariant under the action of the euclidean group e 2 composed of reflections and translations in the plane and a novel shift twist action using this symmetry we show that the various patterns of activity that spontaneously emerge when v1 s spatially uniform resting state becomes unstable correspond to the form constants when transformed to the visual field using the retino cortical map the results are sensitive to the detailed specification of the lateral connectivity and suggest that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges contours surfaces and textures

products and symmetrized powers of irreducible representations of and their associates the calculation of kronecker products and plethysms of the infinite dimensional harmonic series unitary irreducible representations of the non compact group is considered the complementarity of and o k is used to define associate irreducible representations of this leads to simple relationships between kronecker products and plethysms of irreducible representations of and those of their corresponding associate irreducible representations in the process of proving the validity of these previously conjectured relationships several new identities are found for plethysms involving infinite series of schur functions in addition a general formula for plethysms of arbitrary irreducible representations of is derived and its implementation is illustrated with a detailed example a remarkable analogy is then observed between plethysms of the basic harmonic irreducible representations of and those of the basic spin irreducible representations of so 2n

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

statistical mechanics of neural networks what are the differences between wide and narrow basins abstract we consider training noise in neural networks as a means of tuning the structure of retrieval basins and study how learning and retrieving properties depend on it the stability of the replica symmetric solution and the correlation in the weight space indicate that neural networks can be roughly classified into hebbian like and msn like msn meaning the maximally stable network re entrant retrieval noise robustness selectivity damage spreading and activity distribution all illustrate the differences in retrieval behaviours arising from the different basin structures

compact lorentz manifolds with local symmetry we prove a structure theorem for compact aspherical lorentz manifolds with abundant local symmetry if m is a compact aspherical real analytic complete lorentz manifold such that the isometry group of the universal cover has semisimple identity component then the local isometry orbits in m are roughly fibers of a fiber bundle a corollary is that if m has an open dense locally homogeneous subset then m is locally homogeneous acknowledgements i thank my dissertation advisor benson farb for his guidance and encouragement i also thank abdelghani zeghib with whom i had helpful conversations while working on this project in loving memory of laura sue jung kang

asymmetric anharmonic oscillators in the hill determinant picture a new version of the hill determinant construction of bound states is proposed it is based on the matching of two suitable power series ansatze for the wave function for the asymmetric well potential v x ax bx2 cx3 dx4 the convergence of the approximants to the physical solutions is proved for b bminimal the taylor coefficients for this anharmonic oscillator s wave function are specified by four term recurrences

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

actions of semisimple lie groups preserving a degenerate riemannian metric we prove a rigidity of the lightcone in minkowski space it is essentially the unique space endowed with a lightlike metric and supporting an isometric nonproper action of a semisimple lie group

a wavelet perspective on variational perceptually inspired color enhancement the issue of perceptually inspired correction of color and contrast in digital images has been recently analyzed with the help of variational principles these techniques allowed building a general framework in which the action of many already existing algorithms can be more easily understood and compared in terms of intensification of local contrast and control of dispersion around the average intensity value in this paper we analyze this issue from the dual perspective of wavelet theory showing that it is possible to build energy functionals of wavelet coefficients that lead to a multilevel perceptually inspired color correction by computing the euler lagrange equations associated to the wavelet based functionals we were able to find an analytical formula for the modification of wavelet detail coefficients that overcomes the problem of an ad hoc selection based on empirical considerations besides these theoretical results the wavelet perspective provides the computational advantage of generating much faster algorithms in comparison with the spatial variational framework

a replica analysis of the minimum weight solution to the linear equations problem we investigate through statistical mechanics techniques the problem of finding n tuples j ji j2 jn with a fixed fraction k of non vanishing entries that solve a set of p alpha n linear equations in particular we determine the region in the plane alpha k where these solutions exist with a probability of 1 in the limit of large n we also obtain at insight into the microscopic structure of these solutions by calculating the distribution of the probability of an arbitrary non vanishing entry ji moreover we evaluate analytically the performances of several easy to implement heuristic algorithms and compare them with the optimal solution a task that is suited very well to the statistical mechanics approach

naturalness preserved enhancement algorithm for non uniform illumination images image enhancement plays an important role in image processing and analysis among various enhancement algorithms retinex based algorithms can efficiently enhance details and have been widely adopted since retinex based algorithms regard illumination removal as a default preference and fail to limit the range of reflectance the naturalness of non uniform illumination images cannot be effectively preserved however naturalness is essential for image enhancement to achieve pleasing perceptual quality in order to preserve naturalness while enhancing details we propose an enhancement algorithm for non uniform illumination images in general this paper makes the following three major contributions first a lightness order error measure is proposed to access naturalness preservation objectively second a bright pass filter is proposed to decompose an image into reflectance and illumination which respectively determine the details and the naturalness of the image third we propose a bi log transformation which is utilized to map the illumination to make a balance between details and naturalness experimental results demonstrate that the proposed algorithm can not only enhance the details but also preserve the naturalness for non uniform illumination images

emergence of chaotic behaviour in linearly stable systems strong nonlinear effects combined with diffusive coupling may give rise to unpredictable evolution in spatially extended deterministic dynamical systems even in the presence of a fully negative spectrum of lyapunov exponents this regime denoted as stable chaos has been so far mainly characterized by means of numerical studies in this paper we investigate the mechanisms that are at the basis of this form of unpredictable evolution generated by a nonlinear information flow through the boundaries in order to clarify how linear stability can coexist with nonlinear instability we construct a suitable stochastic model in the absence of spatial coupling the model does not reveal the existence of any self sustained chaotic phase nevertheless already this simple regime reveals peculiar differences between the behaviour of finite size and that of infinitesimal perturbations a mean field analysis of the spatially extended case provides a semi quantitative description of the onset of irregular behaviour possible relations with directed percolation as a synchronization transition are also briefly discussed

a probabilistic method for image enhancement with simultaneous illumination and reflectance estimation in this paper a new probabilistic method for image enhancement is presented based on a simultaneous estimation of illumination and reflectance in the linear domain we show that the linear domain model can better represent prior information for better estimation of reflectance and illumination than the logarithmic domain a maximum a posteriori map formulation is employed with priors of both illumination and reflectance to estimate illumination and reflectance effectively an alternating direction method of multipliers is adopted to solve the map problem the experimental results show the satisfactory performance of the proposed method to obtain reflectance and illumination with visually pleasing enhanced results and a promising convergence rate compared with other testing methods the proposed method yields comparable or better results on both subjective and objective assessments

isometries of lorentz surfaces and convergence groups we study the isometry group of a globally hyperbolic spatially compact lorentz surface such a group acts on the circle and we show that when the isometry group acts non properly the subgroups of mathrm diff mathbb s 1 obtained are semi conjugate to subgroups of finite covers of mathrm psl 2 mathbb r by using convergence groups under an assumption on the conformal boundary we show that we have a conjugacy in mathrm homeo mathbb s 1

size distribution of percolating clusters on cubic lattices we present monte carlo simulations of site percolation on cubic lattices with size l the cut off function of the size distribution at the characteristic size s can be described by a gaussian for site occupations p lower than the critical value for percolation pc while for p pc it is better described by a stretched exponential finite lattice size effects depend on both e pc p pc and l and cannot be eliminated by normalization with the distribution at e 0 we give a general expression of the cut off function valid for any e sufficiently small and l sufficiently large using this general expression we calculate the finite lattice size effects on the weight average sw and z average sz sizes which are in good agreement with values directly obtained from the simulations

satake diagrams of affine kac moody algebras satake diagrams of affine kac moody algebras untwisted and twisted are obtained from their dynkin diagrams these diagrams give a classification of restricted root systems associated with these algebras in the case of simple lie algebras these root systems and satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems quantum transport problems random matrix theories etc we hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine kac moody algebras if they exist

dilution in boolean perceptrons that learn from noisy examples we investigate the effect of dilution after learning on the generalization ability of single layer boolean perceptrons that learn from noisy examples we present a thorough comparison between the relative performances of several well known learning rules in particular we show that the effect of dilution is always deleterious and that the bayes algorithm always gives the best generalization performance