dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

permitted and forbidden sets in discrete time linear threshold recurrent neural networks the concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks such concepts exhibit interesting dynamics in recurrent neural networks this paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete time recurrent neural networks the linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations networks in discrete time can directly provide algorithms for efficient implementation in digital hardware the main contribution of this paper is to establish foundations of permitted and forbidden sets necessary and sufficient conditions for the linear threshold discrete time recurrent neural networks are obtained for complete convergence existence of permitted and forbidden sets as well as conditionally multiattractivity respectively simulation studies explore some possible interesting practical applications

continuous time relaxation labeling processes abstract we study the dynamical properties of two new relaxation labeling schemes described in terms of differential equations and hence evolving in continuous time this contrasts with the customary approach to defining relaxation labeling algorithms which prefers discrete time continuous time dynamical systems are particularly attractive because they can be implemented directly in hardware circuitry and the study of their dynamical properties is simpler and more elegant they are also more plausible as models of biological visual computation we prove that the proposed models enjoy exactly the same dynamical properties as the classical relaxation labeling schemes and show how they are intimately related to hummel and zucker s now classical theory of constraint satisfaction in particular we prove that when a certain symmetry condition is met the dynamical systems behavior is governed by a liapunov function which turns out to be the negative of a well known consistency measure moreover we prove that the fundamental dynamical properties of the systems are retained when the symmetry restriction is relaxed we also analyze the properties of a simple discretization of the proposed dynamics which is useful in digital computer implementations simulation results are presented which show the practical behavior of the models

continuous attractors of lotka volterra recurrent neural networks with infinite neurons continuous attractors of lotka volterra recurrent neural networks lv rnns with infinite neurons are studied in this brief a continuous attractor is a collection of connected equilibria and it has been recognized as a suitable model for describing the encoding of continuous stimuli in neural networks the existence of the continuous attractors depends on many factors such as the connectivity and the external inputs of the network a continuous attractor can be stable or unstable it is shown in this brief that a lv rnn can possess multiple continuous attractors if the synaptic connections and the external inputs are gussian like in shape moreover both stable and unstable continuous attractors can coexist in a network explicit expressions of the continuous attractors are calculated simulations are employed to illustrate the theory

mode detection by relaxation a relaxation process is described and applied to the detection of the modes of the underlying probability density function for clustering multidimensional data when the process is iterated the label probabilities become very high in the modal region and very low in the valleys so that mode detection becomes trivial

upper bounds for some graph energies a general inequality for non negative real numbers is proven based on it upper bounds for ordinary graph energy minimum dominating energy minimum covering energy laplacian energy like invariant laplacian energy randic energy and incidence energy are obtained

labeling of ambiguous subvoxel fibre bundle configurations in high angular resolution diffusion mri whereas high angular resolution reconstruction methods for diffusion mri can estimate multiple dominant fibre orientations within a single imaging voxel they are fundamentally limited in certain cases of complex subvoxel fibre structures resulting in ambiguous local orientation distribution functions in this article we address the important problem of disambiguating such complex subvoxel fibre tract configurations with the purpose of improving the performance of fibre tractography we do so by extending a curve inference method to distinguish between the cases of curving and fanning fibre bundles using differential geometric estimates in a local neighbourhood the key benefit of this method is the inference of curves instead of only fibre orientations to model the underlying fibre bundles this in turn allows distinct fibre geometries that contain nearly identical sets of fibre orientations at a voxel to be distinguished from one another experimental results demonstrate the ability of the method to successfully label voxels into one of the above categories and improve the performance of a fibre tracking algorithm

texture segmentation using voronoi polygons textures are defined in terms of primitives called tokens a texture segmentation algorithm based on the voronoi tessellation is discussed the algorithm first builds the voronoi tessellation of the tokens that make up the textured image it then computes a feature vector for each voronoi polygon these feature vectors are used in a probabilistic relaxation labeling on the tokens to identify the interior and the border regions of the textures some experimental results are shown

relaxation by hopfield network in stereo image matching abstract this paper outlines a relaxation approach using the hopfield neural network for solving the global stereovision matching problem the primitives used are edge segments the similarity smoothness and uniqueness constraints are transformed into the form of an energy function whose minimum value corresponds to the best solution of the problem we combine two methods a optimization relaxation 1 and b relaxation merit 2 with the above three constraints mapped in an energy function the main contribution is made 1 by applying a learning strategy in the similarity constraint and 2 by introducing specific conditions to overcome the violation of the smoothness constraint and to avoid the serious problem arising from the required fixation of a disparity limit so we improve the stereovision matching process a better performance of the proposed method is illustrated with a comparative analysis against a classical relaxation method

incremental rough set approach for hierarchical multicriteria classification abstract multicriteria classification refers to classify objects evaluated by a set of criteria to preference ordered decision classes dominance based rough set approach has been successfully introduced to express and reason inconsistencies with a dominance principle in multicriteria classification problems hierarchical attribute values exist extensively within many real world applications which provide a hierarchical form to organize view and analyze data from different perspectives for accommodating the preference variability in this study we consider an extension of dominance based rough set approach by applying an incremental learning technique for hierarchical multicriteria classification while attribute values dynamically vary across different levels of granulations we formalize the dynamic characteristics of knowledge granules with the cut refinement and coarsening through attribute value taxonomies in the hierarchical multicriteria decision systems in consequence incremental algorithms for computing dominance based rough approximations of preference ordered decision classes are developed by applying the resulted prior knowledge as the input and only recomputing those outputs which depend on the changed attribute values this paper presents the theoretical foundation of the proposed approach example analysis and experimental evaluation are also provided for illustration of the feasibility and efficiency

robust stability analysis for interval cohen grossberg neural networks with unknown time varying delays in this paper robust stability problems for interval cohen grossberg neural networks with unknown time varying delays are investigated using linear matrix inequality m matrix theory and halanay inequality techniques new sufficient conditions independent of time varying delays are derived to guarantee the uniqueness and the global robust stability of the equilibrium point of interval cohen grossberg neural networks with time varying delays all these results have no restriction on the rate of change of the time varying delays compared to some existing results these new criteria are less conservative and are more convenient to check two numerical examples are used to show the effectiveness of the present results

constraint satisfaction neural networks for image segmentation abstract constraint satisfaction neural networks csnns are proposed for image recognition which is the process to identify objects of interest in an image the image recognition problem is formulated as a constraint satisfaction problem csp the regions resulting from a signal based image segmentation algorithm can be considered as objects and their entities as labels in the csp the proposed neural network is multi layered and the function of each layer is to identify one physical entity label from the input regions objects through cascading these layers each label can be assigned to the corresponding region s attention is focused on recognizing x ray ct computed tomography images of human heads e g anatomic structure in a medical image the proposed csnn is capable of identifying the components of a brain in a preliminarily segmented image the major components in a human head include bone ventricle and gyrus therefore three layers are needed for this task each layer is constructed based on the principles proposed in kohonen s self organizing feature maps and the optimal linear associative memories and their weights are trained by the features of each entity in the prestored model the experimental results show that the proposed approach is very promising

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

ordering of oriented unicyclic graphs by skew energies ordering of the oriented unicyclic graphs with n vertices according to their minimal skew energies is investigated we derive the first 22 oriented unicyclic graphs for n 1784 and a series of oriented unicyclic graphs for 3 n 1783

stability of a complex dynamical network model abstract in this paper firstly we propose a complex dynamical network model which is always uniformly asymptotically stable about its equilibrium then we give theoretical analysis of its uniform asymptotical stability by using lyapunov stability theory finally we show a numerical example to verify our theoretical result

determinants of matrices of the conference type an n by n conference type matrix has 0 s on the main diagonal and 1 s elsewhere we investigate the largest possible determinant of such a matrix the literature is extensive for n even but for n odd the question has not been previously studied we determine the maxima up to n 11 for n 3 5 7 9 11 they are 2 22 394 8760 240786 it will require a new idea or a huge computation to go beyond n 11

fuzzy cognitive maps for stereovision matching this paper outlines a method for solving the stereovision matching problem using edge segments as the primitives in stereovision matching the following constraints are commonly used epipolar similarity smoothness ordering and uniqueness we propose a new matching strategy under a fuzzy context in which such constraints are mapped the fuzzy context integrates both fuzzy clustering and fuzzy cognitive maps with such purpose a network of concepts nodes is designed each concept represents a pair of primitives to be matched each concept has associated a fuzzy value which determines the degree of the correspondence the goal is to achieve high performance in terms of correct matches the main findings of this paper are reflected in the use of the fuzzy context that allows building the network of concepts where the matching constraints are mapped initially each concept value is loaded via the fuzzy clustering and then updated by the fuzzy cognitive maps framework this updating is achieved through the influence of the remainder neighboring concepts until a good global matching solution is achieved under this fuzzy approach we gain quantitative and qualitative matching correspondences this method works as a relaxation matching approach and its performance is illustrated by comparative analysis against some existing global matching methods

a fast and robust cluster update algorithm for image segmentation in spin lattice models without annealing visual latencies revisited image segmentation in spin lattice models relies on the fast and reliable assignment of correct labels to those groups of spins that represent the same object commonly used local spin update algorithms are slow because in each iteration only a single spin is flipped and a careful annealing schedule has to be designed in order to avoid local minima and correctly label larger areas updating of complete spin clusters is more efficient but often clusters that should represent different objects will be conjoined in this study we propose a cluster update algorithm that similar to most local update algorithms calculates an energy function and determines the probability for flipping a whole cluster of spins by the energy gain calculated for a neighborhood of the regarded cluster the novel algorithm called energy based cluster update ecu algorithm is compared to its predecessors a convergence proof is derived and it is shown that the algorithm outperforms local update algorithms by far in speed and re

boundary detection by constrained optimization a statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions the model is a joint probability distribution for the array of pixel gray levels and an array of labels in boundary finding the labels are binary zero or one representing the absence or presence of boundary elements in partitioning the label values are generic two labels are the same when the corresponding scene locations are considered to belong to the same region the distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels using the kolmogorov smirnov nonparametric measures of difference between the distributions of these features the number of model parameters is minimized by forbidding label configurations which are assigned probability zero the maximum a posteriori estimator of boundary placements and partitionings is examined the forbidden states introduce constraints into the calculation of these configurations stochastic relaxation methods are extended to accommodate constrained optimization

bicyclic digraphs with maximal energy if d is a digraph with n vertices then the energy of d is defined as e d k 1 n re z k where re z 1 re zn are the real parts of the eigenvalues z 1 z n of d in this paper we solve a problem proposed in khan et al 2015 we find the maximal value of the energy over the set of all bicyclic digraphs b n with n vertices

activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks this technical note proposes to study the activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks the concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons keeps invariant all the time conditions are obtained for locating activity invariant sets under some conditions it shows that an activity invariant set can have one equilibrium point which attracts exponentially all trajectories starting in the set since the attractors are located in activity invariant sets each attractor has binary pattern and also carries analog information such results can provide new perspective to apply attractor networks for applications such as group winner take all associative memory etc

subspace clustering using a low rank constrained autoencoder the performance of subspace clustering is affected by data representation data representation for subspace clustering maps data from the original space into another space with the property of better separability many data representation methods have been developed in recent years typical among them are low rank representation lrr and an autoencoder lrr is a linear representation method that captures the global structure of data with low rank constraint alternatively an autoencoder nonlinearly maps data into a latent space using a neural network by minimizing the difference between the reconstruction and input to combine the advantages of an lrr globality and autoencoder self supervision based locality we propose a novel data representation method for subspace clustering the proposed method called low rank constrained autoencoder lrae forces the latent representation of the neural network to be of low rank and the low rank constraint is computed as a prior from the input space one major advantage of the lrae is that the learned data representation not only maintains the local features of the data but also preserves the underlying low rank global structure extensive experiments on several datasets for subspace clustering were conducted they demonstrated that the proposed lrae substantially outperformed state of the art subspace clustering methods

multistability of recurrent neural networks with time varying delays and the piecewise linear activation function in this brief stability of multiple equilibria of recurrent neural networks with time varying delays and the piecewise linear activation function is studied a sufficient condition is obtained to ensure that n neuron recurrent neural networks can have 4k 1 n equilibrium points and 2k n of them are locally exponentially stable this condition improves and extends the existing stability results in the literature simulation results are also discussed in one illustrative example

beyond graph energy norms of graphs and matrices abstract in 1978 gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues and ever since then graph energy has been intensively studied since graph energy is the trace norm of the adjacency matrix matrix norms provide a natural background for its study thus this paper surveys research on matrix norms that aims to expand and advance the study of graph energy the focus is exclusively on the ky fan and the schatten norms both generalizing and enriching the trace norm as it turns out the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics the survey brings to the fore the exceptional role of hadamard matrices conference matrices and conference graphs in matrix norms in addition a vast new matrix class is studied a relaxation of symmetric hadamard matrices the survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics thus open problems and questions are raised to outline topics for further investigation

analysis of stochastic automata algorithm for relaxation labelling a parallel stochastic algorithm for relaxation labeling is analyzed for the case of symmetric compatibility functions it is proved that time algorithm will always converge to a consistent labeling

multistability of discrete time recurrent neural networks with unsaturating piecewise linear activation functions this paper studies the multistability of a class of discrete time recurrent neural networks with unsaturating piecewise linear activation functions it addresses the nondivergence global attractivity and complete stability of the networks using the local inhibition conditions for nondivergence are derived which not only guarantee nondivergence but also allow for the existence of multiequilibrium points under these nondivergence conditions global attractive compact sets are obtained complete stability is studied via constructing novel energy functions and using the well known cauchy convergence principle examples and simulation results are used to illustrate the theory

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

on the normalized laplacian energy and general randic index r 1 of graphs in this paper we consider the energy of a simple graph with respect to its normalized laplacian eigenvalues which we call the l energy over graphs of order n that contain no isolated vertices we characterize the graphs with minimal l energy of 2 and maximal l energy of 2 n 2 we provide upper and lower bounds for l energy based on its general randic index r 1 g we highlight known results for r 1 g most of which assume g is a tree we extend an upper bound of r 1 g known for trees to connected graphs we provide bounds on the l energy in terms of other parameters one of which is the energy with respect to the adjacency matrix finally we discuss the maximum change of l energy and r 1 g upon edge deletion

the maximum determinant of 1 matrices we give a new proof for the bound on the value of the determinant of a 1 matrix of dimension n 1 mod 4 first given by barba adapting a construction of a e brouwer we give examples to show that the bound is sharp for infinitely many values of n this in turn gives an infinite family of examples which attain the bound given by h ehlich and by m wojtas for the determinant of a 1 matrix of dimension n 2 mod 4 for n 3 mod4 we construct an infinite family of 1 examples which attain slightly more than of the bound given by ehlich elsevier science inc 1997

genetic algorithms for ambiguous labelling problems abstract consistent labelling problems frequently have more than one solution most work in the field has aimed at disambiguating early in the interpretation process using only local evidence this paper starts with a review of the literature on labelling problems and ambiguity based on this review we propose a strategy for simultaneously extracting multiple related solutions to the consistent labelling problem in a preliminary experimental study we show that an appropriately modified genetic algorithm is a robust tool for finding multiple solutions to the consistent labelling problem these solutions are related by common labellings of the most strongly constrained junctions we have proposed three run time measures of algorithm performance the maximum fitness of the genetic algorithm s population its shannon entropy and the total hamming distance between its distinct members the results to date indicate that when the shannon entropy falls below a certain threshold new solutions are unlikely to emerge and that most of the diversity in the population disappears within the first few generations

dynamic stability conditions for lotka volterra recurrent neural networks with delays the lotka volterra model of neural networks derived from the membrane dynamics of competing neurons have found successful applications in many winner take all types of problems this paper studies the dynamic stability properties of general lotka volterra recurrent neural networks with delays conditions for nondivergence of the neural networks are derived these conditions are based on local inhibition of networks thereby allowing these networks to possess a multistability property multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process under these nondivergence conditions a compact set that globally attracts all the trajectories of a network can be computed explicitly if the connection weight matrix of a network is symmetric in some sense and the delays of the network are in l2 space we can prove that the network will have the property of complete stability

largest j simplices in d cubes some relatives of the hadamard maximum determinant problem abstract this paper studies the computationally difficult problem of finding a largest j dimensional simplex in a given d dimensional cube the case in which j d is of special interest for it is equivalent to the hadamard maximum determinant problem it has been solved for infinitely many values of d but not for d 14 the subcase in which j d 3 mod 4 subsumes the famous problem on the existence of hadamard matrices the known results for the case j d are here summarized and used but the main focus is on fixed small values of j when j 1 the problem is trivial and when j 2 or j 3 it is here solved completely i e for all d beyond that the results are fragmentary but numerous and they lead to several attractive conjectures some other problems involving simplices in cubes are mentioned and the relationship of largest simplices to d optimal weighing designs is discussed

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

lower bounds for the energy of digraphs abstract the energy of a digraph d is defined as e d i 1 n re z i where z 1 z n are the eigenvalues of d in this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2 extending in this way the result obtained by caporossi et al g caporossi d cvetkovic i gutman p hansen variable neighborhood search for extremal graphs 2 finding graphs with extremal energy j chem inf comput sci 39 1999 984 996 2 m e g for all graphs g with m edges also we study digraphs with three eigenvalues

a markov random field model based approach to image interpretation an image is segmented into a collection of disjoint regions that form the nodes of an adjacency graph and image interpretation is achieved through assigning object labels or interpretations to the segmented regions or nodes using domain knowledge extracted feature measurements and spatial relationships between the various regions the interpretation labels are modeled as a markov random field mrf on the corresponding adjacency graph and the image interpretation problem is then formulated as a maximum a posteriori map estimation rule given domain knowledge and region based measurements simulated annealing is used to find this best realization or optimal map interpretation this approach also provides a systematic method for organizing and representing domain knowledge through appropriate design of the clique functions describing the gibbs distribution representing the pdf of the underlying mrf a general methodology is provided for the design of the clique functions results of image interpretation experiments on synthetic and real world images are described

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

improving consistency and reducing ambiguity in stochastic labeling an optimization approach we approach the problem of labeling a set of objects from a quantitative standpoint we define a world model in terms of transition probabilities and propose a definition of a class of global criteria that combine both ambiguity and consistency a projected gradient algorithm is developed to minimize the criterion we show that the minimization procedure can be implemented in a highly parallel manner results are shown on several examples and comparisons are made with relaxation labeling techniques

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

exponential stabilization of memristive neural networks with time delays in this paper a general class of memristive neural networks with time delays is formulated and studied some sufficient conditions in terms of linear matrix inequalities are obtained in order to achieve exponential stabilization the result can be applied to the closed loop control of memristive systems in particular several succinct criteria are given to ascertain the exponential stabilization of memristive cellular neural networks in addition a simplified and effective algorithm is considered for design of the optimal controller these conditions are the improvement and extension of the existing results in the literature two numerical examples are given to illustrate the theoretical results via computer simulations

energy of matrices abstract let m n c denote the space of n n matrices with entries in c we define the energy of a m n c as 1 e a k 1 n k t r a n where 1 n are the eigenvalues of a tr a is the trace of a and z denotes the modulus of z c if a is the adjacency matrix of a graph g then e a is precisely the energy of the graph g introduced by gutman in 1978 in this paper we compare the energy e with other well known energies defined over matrices then we find upper and lower bounds of e which extend well known results for the energies of graphs and digraphs also we obtain new results on energies defined over the adjacency laplacian and signless laplacian matrices of digraphs

upper bound for the trace norm of the laplacian matrix of a digraph and normally regular digraphs abstract the trace norm of m m n c is defined as m k 1 n k where 1 2 n 0 are the singular values of m i e the square roots of the eigenvalues of m m we are particularly interested in the trace norm l d a n i n where l d is the laplacian matrix of a digraph d with n vertices and a arcs and i n is the n n identity matrix when d g is a graph with n vertices and m edges then l d a n i n l g 2 m n i n l e g the laplacian energy of g introduced by gutman and zhou in 2006 we show that for a digraph d with n vertices and a arcs l d a n i n n a a 2 n i 1 n d i 2 where d 1 d n are the outer degrees of the vertices of d moreover the digraphs where this bound is attained are special classes of normally regular digraphs studied by jorgensen in 2015 6 finally we construct normally regular digraphs where the equality is attained

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

selectively grouping neurons in recurrent networks of lateral inhibition winner take all networks have been proposed to underlie many of the brain s fundamental computational abilities however not much is known about how to extend the grouping of potential winners in these networks beyond single neuron or uniformly arranged groups of neurons we show that competition between arbitrary groups of neurons can be realized by organizing lateral inhibition in linear threshold networks given a collection of potentially overlapping groups with the exception of some degenerate cases the lateral inhibition results in network dynamics such that any permitted set of neurons that can be coactivated by some input at a stable steady state is contained in one of the groups the information about the input is preserved in this operation the activity level of a neuron in a permitted set corresponds to its stimulus strength amplified by some constant sets of neurons that are not part of a group cannot be coactivated by any input at a stable steady state we analyze the storage capacity of such a network for random groups the number of random groups the network can store as permitted sets without creating too many spurious ones in this framework we calculate the optimal sparsity of the groups maximizing group entropy we find that for dense inputs the optimal sparsity is unphysiologically small however when the inputs and the groups are equally sparse we derive a more plausible optimal sparsity we believe our results are the first steps toward attractor theories in hybrid analog digital networks

new measurements and corner guidance for curve matching with probabilistic relaxation reliable curve matching is a difficult yet important problem in many vision based applications including image based modeling we describe in this paper two aspects of our research in this area a new algorithm for curve matching including lines within a probabilistic relaxation framework and an approach of incorporating previously matched points corners to guide curve matching we propose similarity invariant unary and binary measurements suitable for curves and introduce an additional measurement to model the uncertainty of the binary measurements the uncertainty measure is proven to be very important in computing the matching support from neighboring matches we also show how to use a set of previously matched points corners to guide the curve matching the role of the corner guidance is explicitly modeled by a set of unary measurements and a similarity function under the same relaxation framework preprocessing techniques contributing to the success of our curve matching techniques are also developed and discussed experiments with complex real scenes show that the rate of correct matching is higher than 98

bounds for the signless laplacian energy the energy of a graph g is the sum of the absolute values of the eigenvalues of the adjacency matrix of g the laplacian respectively the signless laplacian energy of g is the sum of the absolute values of the differences between the eigenvalues of the laplacian respectively signless laplacian matrix and the arithmetic mean of the vertex degrees of the graph in this paper among some results which relate these energies we point out some bounds to them using the energy of the line graph of g most of these bounds are valid for both energies laplacian and signless laplacian however we present two new upper bounds on the signless laplacian which are not upper bounds for the laplacian energy

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

foundations of implementing the competitive layer model by lotka volterra recurrent neural networks the competitive layer model clm can be described by an optimization problem the problem can be further formulated by an energy function called the clm energy function in the subspace of nonnegative orthant the set of minimum points of the clm energy function forms the set of solutions of the clm problem solving the clm problem means to find out such solutions recurrent neural networks rnns can be used to implement the clm to solve the clm problem the key point is to make the set of minimum points of the clm energy function just correspond to the set of stable attractors of the recurrent neural networks this paper proposes to use lotka volterra rnns lv rnns to implement the clm the contribution of this paper is to establish foundations of implementing the clm by lv rnns the contribution mainly contains three parts the first part is on the clm energy function necessary and sufficient conditions for minimum points of the clm energy function are established by detailed study the second part is on the convergence of the proposed model of the lv rnns it is proven that interesting trajectories are convergent the third part is the most important it proves that the set of stable attractors of the proposed lv rnn just equals the set of minimum points of the clm energy function in the nonnegative orthant thus the lv rnns can be used to solve the problem of the clm it is believed that by establishing such basic rigorous theories more and interesting applications of the clm can be found

applications of a theorem by ky fan in the theory of graph energy abstract the energy of a graph g is equal to the sum of the absolute values of the eigenvalues of g which in turn is equal to the sum of the singular values of the adjacency matrix of g let x y and z be matrices such that x y z the ky fan theorem establishes an inequality between the sum of the singular values of z and the sum of the sum of the singular values of x and y this theorem is applied in the theory of graph energy resulting in several new inequalities as well as new proofs of some earlier known inequalities