fuzzy convex set based pattern classification for analysis of mammographic microcalcifications abstract there are many different criteria for the comparative analysis of pattern classifiers they include generalization ability computational complexity and understanding of the feature space in some applications such as the medical diagnostic systems it is crucial to use reliable tools whose behavior is always predictable so that the risk of misdiagnosis is minimized in such applications the use of the popular feedforward backpropagation bp neural network algorithm can be seen as questionable this is because it is not inherent for the backpropagation method to analyze the problem s feature space during training which can sometimes result in inadequate decision surfaces a novel convex set based neuro fuzzy algorithm for classification of difficult to diagnose instances of breast cancer is described in this paper with its structural approach to feature space the new method offers rational advantages over the backpropagation algorithm the classification performance computational and structural efficiencies are analyzed and compared with that of the bp network a 20 dimensional set of difficult to diagnose mammographic microcalcifications was used to evaluate the neuro fuzzy pattern classifier nfpc and the bp methods in order to evaluate the learning ability of both methods the relative size of training sets was varied from 40 to 90 the comparative results obtained using receiver operating characteristic roc analysis show that the ability of the convex set based method to infer knowledge was better than that of backpropagation in all of the tests performed making it more suitable for use in real diagnostic systems

differential geometry palatini gravity and reduction the present article deals with a formulation of the so called vacuum palatini gravity as a general variational principle in order to accomplish this goal some geometrical tools related to the geometry of the bundle of connections of the frame bundle lm are used a generalization of lagrange poincare reduction scheme to these types of variational problems allows us to relate it with the einstein hilbert variational problem relations with some other variational problems for gravity found in the literature are discussed

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

the lie poisson structure of integrable classical non linear sigma models the canonical structure of classical non linear sigma models on riemannian symmetric spaces which constitute the most general class of classical non linear sigma models known to be integrable is shown to be governed by a fundamental poisson bracket relation that fits into ther s matrix formalism for non ultralocal integrable models first discussed by maillet the matricesr ands are computed explicitly and being field dependent satisfy fundamental poisson bracket relations of their own which can be expressed in terms of a new numerical matrixc it is proposed that all these poisson brackets taken together are representation conditions for a new kind of algebra which for this class of models replaces the classical yang baxter algebra governing the canonical structure of ultralocal models the poisson brackets for the transition matrices are also computed and the notorious regularization problem associated with the definition of the poisson brackets for the monodromy matrices is discussed

hamiltonian formulation of the kdv equation we consider the canonical formulation of whitham s variational principle for the kdv equation this lagrangian is degenerate and we have found it necessary to use dirac s theory of constrained systems in constructing the hamiltonian earlier discussions of the hamiltonian structure of the kdv equation were based on various different decompositions of the field which is avoided by this new approach

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

quantum neural networks qnns inherently fuzzy feedforward neural networks this paper introduces quantum neural networks qnns a class of feedforward neural networks ffnns inherently capable of estimating the structure of a feature space in the form of fuzzy sets the hidden units of these networks develop quantized representations of the sample information provided by the training data set in various graded levels of certainty unlike other approaches attempting to merge fuzzy logic and neural networks qnns can be used in pattern classification problems without any restricting assumptions such as the availability of a priori knowledge or desired membership profile convexity of classes a limited number of classes etc experimental results presented here show that qnns are capable of recognizing structures in data a property that conventional ffnns with sigmoidal hidden units lack

the geometry of the bundle of connections a generalized symplectic structure on the bundle of connections p colon c p rightarrow m of an arbitrary principal g bundle pi colon p rightarrow m is defined by means of a p ast mathrm ad p valued differential 2 form omega 2 on c p which is related to the generalized contact structure on j 1 p the hamiltonian properties of omega 2 are also analyzed

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

euler poincare reduction for discrete field theories in this note we develop a theory of euler poincare reduction for discrete lagrangian field theories we introduce the concept of euler poincare equations for discrete field theories as well as a natural extension of the moser veselov scheme and show that both are equivalent the resulting discrete field equations are interpreted in terms of discrete differential geometry an application to the theory of discrete harmonic mappings is also briefly discussed

a fast and faithful collocation method with efficient matrix assembly for a two dimensional nonlocal diffusion model abstract the peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities which cannot be described properly by the classical theory of solid mechanics however the operator in the peridynamic theory is nonlocal so the resulting numerical methods generate dense or full coefficient matrices which require o n 2 of memory where n is the number of unknowns in the discretized system gaussian types of direct solvers which were traditionally used to solve these problems require o n 3 of operations furthermore due to the singularity of the kernel in the peridynamic model the evaluation and assembly of the coefficient matrix can be very expensive numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main computational cost the significantly increased computational work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications especially in multiple space dimensions we develop a fast and faithful collocation method for a two dimensional nonlocal diffusion model which can be viewed as a scalar valued version of a peridynamic model without using any lossy compression but rather by exploiting the structure of the coefficient matrix the new method reduces the evaluation and assembly of the coefficient matrix by o n reduces the computational work from o n 3 required by the traditional methods to o n log 2 n and the memory requirement from o n 2 to o n numerical results are presented to show the utility of the fast method

reduction of polysymplectic manifolds the aim of this paper is to generalize the classical marsden weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure this generalization allows us to reduce polysymplectic hamiltonian systems with symmetries such as those appearing in certain kinds of classical field theories as an application of this technique an analogue to the kirillov kostant souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed our procedure corrects some mistakes and inaccuracies in previous papers gunther 1987 j differ geom 25 23 53 munteanu et al 2004 j math phys 45 1730 51 on this subject

incorporating fuzzy membership functions into the perceptron algorithm the perceptron algorithm one of the class of gradient descent techniques has been widely used in pattern recognition to determine linear decision boundaries while this algorithm is guaranteed to converge to a separating hyperplane if the data are linearly separable it exhibits erratic behavior if the data are not linearly separable fuzzy set theory is introduced into the perceptron algorithm to produce a fuzzy algorithm which ameliorates the convergence problem in the nonseparable case it is shown that the fuzzy perceptron like its crisp counterpart converges in the separable case a method of generating membership functions is developed and experimental results comparing the crisp to the fuzzy perceptron are presented

su n and so n invariant chiral fields one and two dimensional subspaces applying the harmonic map ansatz we find new classes of su n and so n chiral fields we write the corresponding one dimensional subgroups of su n and so n chiral fields in terms of one and two harmonic maps and the corresponding two dimensional subgroups in terms of su 2 and so 3 chiral fields in other words we reduce the su n and so n chiral equations to harmonic maps in one and two dimensional spaces

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

on the geometry of spontaneous symmetry breaking given a principal bundle p x with a structure group g and an associated higgs bundle with a standard fiber g h the case of a matter bundle e whose standard fiber admits action only of an exact symmetry subgroup h of g is examined in the presence of a fixed higgs field x matter fields are represented by sections of a matter bundle eh associated with the corresponding reduced subbundle ph of p the totality of matter fields and higgs fields is described by sections of the bundle e which is the composite bundle eh x where eh is the bundle associated with the principal h bundle p the bundle e fails to be associated with a principal bundle to construct a connection e j1e on e the canonical jet bundle morphism j1eh j1 j1e is used

reduced classical field theories k cosymplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k cosymplectic geometry we discuss the relation between lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied mainly on reduction of classical field theories

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

classical field theory on lie algebroids variational aspects the variational formalism for classical field theories is extended to the setting of lie algebroids given a lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to being morphisms of lie algebroids in addition to the standard case our formalism includes as particular examples the case of systems with symmetry covariant euler poincare and lagrange poincare cases sigma models or chern simons theories

neurofuzzy networks with nonlinear quantum learning nonlinear quantum processing allows the solution of an optimization problem by the exhaustive search on all its possible solutions hence it can replace advantageously the algorithms for learning from a training set in order to pursue this possibility in the case of neurofuzzy networks we propose in this paper to tailor their architectures to the requirements of quantum processing in particular superposition is introduced to pursue parallelism and entanglement to associate the network performance with each solution present in the superposition two aspects of the proposed method are considered in detail the binary structure of membership functions and fuzzy reasoning and the use of a particular nonlinear quantum algorithm for extracting the optimal neurofuzzy network by exhaustive search

reduction in principal bundles covariant lagrange poincare equations a general reduction theory for field theoretical lagrangians on principal fiber bundles is presented the reduced variational problem as well as the associated equations are formulated the link between the solutions of the original problem and that of the reduced problem is discussed and an obstruction to the reconstruction of the solutions is isolated the important case of semidirect products is discussed in detail and several concrete examples are presented

symmetry reduction of variational bicomplexes and the principle of symmetric criticality consider a system of differential equations a 0 which is invariant under a lie group g of point transformations acting on the space e of independent and dependent variables by a method due to lie the g invariant solutions of these differential equations are found by solving a reduced system of differential equations a 0 on the space e of invariants of g in this paper we explore the relationship between the g invariant conservation laws and variational principles for the system of equations a 0 and the conservation laws and variational principles for the reduced equations a 0 this problem translates into one of constructing a certain cochain map qx between the g invariant variational bicomplex for the infinite jet space on e and the free variational bicomplex for e we prove that such a cochain map exists locally if and only if the relative lie algebra cohomology condition h f tooo 0 is satisfied where q is the orbit dimension of g r the lie algebra of vector fields on e which generate the infinitesimal action of g and toc o the linear isotropy subalgebra of t at eo as a simple consequence we prove that the vanishing of hq t tooo is the only local obstruction to palais principle of symmetric criticality 1 introduction when a system of partial differential equations is invari ant under a lie group g of point transformations one can construct a reduced system of partial or ordinary differential equations whose local solutions cor respond precisely to the local g invariant solutions of the original system of equations this simple and effective construction pioneered by lie himself is still one of the most powerful and extensively used techniques for finding exact solutions of many nonlinear partial differential equations 4 15 23 in this paper we explore the relationship between the g invariant conservation laws and variational principles for a system of partial differential equations and the con servation laws and variational principles of the reduced system of equations for the g invariant solutions since conservation laws and variational principles for differential equations are naturally represented by certain cohomology classes in the variational bicomplex associated to the given system of partial differential equations 1 22 our problem translates into that of constructing a cochain map between certain variational bicomplexes remarkably and contrary to the usual spirit of symmetry reduction these cochain maps do not always exist even lo cally we are able to completely classify the local obstructions to the existence of such cochain maps

the hamilton pontryagin principle and multi dirac structures for classical field theoriesa we introduce a variational principle for field theories referred to as the hamilton pontryagin principle and we show that the resulting field equations are the euler lagrange equations in implicit form second we introduce multi dirac structures as a graded analog of standard dirac structures and we show that the graph of a multisymplectic form determines a multi dirac structure we then discuss the role of multi dirac structures in field theory by showing that the implicit euler lagrange equations for fields obtained from the hamilton pontryagin principle can be described intrinsically using multi dirac structures finally we show a number of illustrative examples including time dependent mechanics nonlinear scalar fields maxwell s equations and elastostatics

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

possibility and necessity pattern classification using neural networks abstract we propose two learning algorithms of neural networks for two group discriminant problems from the view point of possibility and necessity one algorithm corresponds to the possibility analysis and the other to the necessity analysis the proposed algorithms are similar to the back propagation algorithm and the difference stems from a formulation of a cost function to be minimized in each algorithm each cost function of the proposed algorithms is the weighted sum of squared errors that is the sum of squared errors with different penalties when we discuss the possibility of group 1 the penalty for the squared errors relating to the patterns in group 1 is greater than that of group 2 this means that in the possibility analysis of group 1 we attach greater importance to the patterns in group 1 than to those in group 2 on the other hand when we discuss the necessity of group 1 the penalty for the squared errors relating to the patterns in group 2 is greater than that of group 1

quantum learning for neural associative memories abstract quantum information processing in neural structures results in an exponential increase of patterns storage capacity and can explain the extensive memorization and inferencing capabilities of humans an example can be found in neural associative memories if the synaptic weights are taken to be fuzzy variables in that case the weights update is carried out with the use of a fuzzy learning algorithm which satisfies basic postulates of quantum mechanics the resulting weight matrix can be decomposed into a superposition of associative memories thus the fundamental memory patterns attractors can be mapped into different vector spaces which are related to each other via unitary rotations quantum learning increases the storage capacity of associative memories by a factor of 2 n where n is the number of neurons

reformulated radial basis neural networks trained by gradient descent this paper presents an axiomatic approach for constructing radial basis function rbf neural networks this approach results in a broad variety of admissible rbf models including those employing gaussian rbfs the form of the rbfs is determined by a generator function new rbf models can be developed according to the proposed approach by selecting generator functions other than exponential ones which lead to gaussian rbfs this paper also proposes a supervised learning algorithm based on gradient descent for training reformulated rbf neural networks constructed using the proposed approach a sensitivity analysis of the proposed algorithm relates the properties of rbfs with the convergence of gradient descent learning experiments involving a variety of reformulated rbf networks generated by linear and exponential generator functions indicate that gradient descent learning is simple easily implementable and produces rbf networks that perform considerably better than conventional rbf models trained by existing algorithms

meshfree explicit local radial basis function collocation method for diffusion problems this paper formulates a simple explicit local version of the classical meshless radial basis function collocation kansa method the formulation copes with the diffusion equation applicable in the solution of a broad spectrum of scientific and engineering problems the method is structured on multiquadrics radial basis functions instead of global the collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node the computational effort thus grows roughly linearly with the number of the nodes the developed approach thus overcomes the principal large scale problem bottleneck of the original kansa method two test cases are elaborated the first is the boundary value problem nafems test associated with the steady temperature field with simultaneous involvement of the dirichlet neumann and robin boundary conditions on a rectangle the second is the initial value problem associated with the dirichlet jump problem on a square the accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes number of nodes in the domain of influence multiquadrics free parameter and timestep length on uniform and nonuniform node arrangements the developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the dirichlet jump where the approximation properties appear similar