an infinity norm bound for the inverse of dashnic zusmanovich type matrices with applications abstract an upper bound for the infinity norm for the inverse of dashnic zusmanovich type matrices is given it is proved that the upper bound is sharper than the well known varah s bound for strictly diagonally dominant matrices by introducing a new subclass of p matrices dashnic zusmanovich type b matrices dz type b matrices and using the proposed infinity norm bound an error bound is given for the linear complementarity problems of dz type b matrices we also give a new pseudospectra localization to measure the distance to instability

a bootstrap method for structure detection of narmax models many systems may be described by narmax models using only a few terms however depending on the order of the system the number of candidate terms can become very large selection of a subset of these candidate terms is necessary for an efficient system description this is an unresolved issue in system identification for over parameterized models therefore in this paper we develop a bootstrap structure detection bsd algorithm as a means of determining the structure of highly over parameterized models the performance of this bsd technique was evaluated by using it to estimate the structure of a 1 simple narmax model 2 moderately over parameterized narmax model and 3 highly over parameterized narmax model the results demonstrate that the bsd algorithm is a robust method for detecting the structure of narmax models this method provides accurate estimates of parameter statistics without relying on assumptions made by traditional procedures and yields a parsimonious description of the system

laplace transforms and valuations it is proved that the classical laplace transform is a continuous valuation which is positively gl n covariant and logarithmic translation covariant conversely these properties turn out to be sufficient to characterize this transform

on extendability by continuity of valuations on convex polytopes abstract there is a well known construction of weakly continuous valuations on convex compact polytopes in r n in this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the hausdorff metric to all convex compact subsets of r n it is shown that there is a necessary condition on the initial data for such an extension in the case of r 3 more explicit results are obtained

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

inequalities for mixed p affine surface area we prove new alexandrov fenchel type inequalities and new affine isoperimetric inequalities for mixed p affine surface areas we introduce a new class of bodies the illumination surface bodies and establish some of their properties we show for instance that they are not necessarily convex we give geometric interpretations of l p affine surface areas mixed p affine surface areas and other functionals via these bodies the surprising new element is that not necessarily convex bodies provide the tool for these interpretations

the even orlicz minkowski problem the classical minkowski problem leads to the lp minkowski problem and now to the orlicz minkowski problem existence is demonstrated for the even orlicz minkowski problem a byproduct is a new approach to the solution of the classical minkowski problem

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

sparse kernel regression modeling using combined locally regularized orthogonal least squares and d optimality experimental design the note proposes an efficient nonlinear identification algorithm by combining a locally regularized orthogonal least squares lrols model selection with a d optimality experimental design the proposed algorithm aims to achieve maximized model robustness and sparsity via two effective and complementary approaches the lrols method alone is capable of producing a very parsimonious model with excellent generalization performance the d optimality design criterion further enhances the model efficiency and robustness an added advantage is that the user only needs to specify a weighting for the d optimality cost in the combined model selecting criterion and the entire model construction procedure becomes automatic the value of this weighting does not influence the model selection procedure critically and it can be chosen with ease from a wide range of values

model construction rule reduction and robust compensation for generalized form of takagi sugeno fuzzy systems this paper presents a systematic procedure of fuzzy control system design that consists of fuzzy model construction rule reduction and robust compensation for nonlinear systems the model construction part replaces the nonlinear dynamics of a system with a generalized form of takagi sugeno fuzzy systems which is newly developed by us the generalized form has a decomposed structure for each element of a sub i and b sub i matrices in consequent parts the key feature of this structure is that it is suitable for constructing if then rules and reducing the number of if then rules the rule reduction part provides a successive procedure to reduce the number of if then rules furthermore we convert the reduction error between reduced fuzzy models and a system to model uncertainties of reduced fuzzy models the robust compensation part achieves the decay rate controller design guaranteeing robust stability for the model uncertainties finally two examples demonstrate the utility of the systematic procedure developed

fuzzy modeling of high dimensional systems complexity reduction and interpretability improvement fuzzy modeling of high dimensional systems is a challenging topic this paper proposes an effective approach to data based fuzzy modeling of high dimensional systems an initial fuzzy rule system is generated based on the conclusion that optimal fuzzy rules cover extrema redundant rules are removed based on a fuzzy similarity measure then the structure and parameters of the fuzzy system are optimized using a genetic algorithm and the gradient method during optimization rules that have a very low firing strength are deleted finally interpretability of the fuzzy system is improved by fine training the fuzzy rules with regularization the resulting fuzzy system generated by this method has the following distinct features 1 the fuzzy system is quite simplified 2 the fuzzy system is interpretable and 3 the dependencies between the inputs and the output are clearly shown this method has successfully been applied to a system that has 11 inputs and one output with 20 000 training data and 80 000 test data

the dual minkowski problem for symmetric convex bodies abstract the dual minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q th dual curvature measure of an origin symmetric convex body in r n a full solution to this is given when 1 q n the necessary and sufficient conditions turn out to be an explicit measure concentration condition to obtain the results a variational approach is used where the functional is the sum of a dual quermassintegral and an entropy integral the proof requires two crucial estimates the first is an estimate of the entropy integral which is obtained by using a spherical partition the second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body

centro affine tensor valuations abstract all measurable sl n covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors are completely classified it is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from l p surface area measures this generalizes and unifies earlier results for the scalar vector and matrix valued case

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

on the extendability by continuity of angular valuations on polytopes abstract a classical theorem of p mcmullen describes all valuations on polytopes that are invariant under translations and weakly continuous i e continuous with respect to parallel displacements of the facets of a polytope while it is typically not difficult to check that a valuation is weakly continuous it is not clear how to decide whether it admits a continuous extension to convex bodies in a special case of mcmullen s construction a simple necessary and sufficient condition on the initial data of such an extension is obtained

fractional pseudospectra and their localizations abstract motivated by the growing successful use of fractional differential equations in the modeling of different important phenomena in this paper we derive tools for practical analysis of the robust asymptotic stability of a incommensurate fractional order linear system first the concept of fractional pseudospectra is introduced second driven by the simplicity and usefulness of spectral localizations in the analysis of various matrix properties we introduce adequate localization techniques using the ideas that come from diagonally dominant matrices in order to localize the fractional pseudospectra in such way many theoretical and practical applications of pseudospectra robust stability transient behavior nonnormal dynamics etc in fractional order differential systems can be linked to the specificity of the matrix entries allowing one to understand certain phenomena in practice better third we consider the fractional distance to instability in l l 1 and l 2 norms and determine efficient lower bounds finally this novel approach is implemented on the realistic model of empirical food web to link the stability that incorporates hereditary dynamics of living organisms with the empirical data and their uncertainty limitations

structure of the unitary valuation algebra s alesker has shown that if g is a compact subgroup of o n acting transitively on the unit sphere sn 1 then the vector space valg of continuous translation invariant g invariant convex valuations on rn has the structure of a finite dimensional graded algebra over r satisfying poincare duality we show that the kinematic formulas for g are determined by the product pairing using this result we then show that the algebra valu n is isomorphic to r s t fn 1 fn 2 where s t have degrees 2 and 1 respectively and the polynomial fi is the degree i term of the power series log 1 s t

asymmetric affine lp sobolev inequalities abstract a new sharp affine l p sobolev inequality for functions on r n is established this inequality strengthens and implies the previously known affine l p sobolev inequality which in turn is stronger than the classical l p sobolev inequality

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

compact and transparent fuzzy models and classifiers through iterative complexity reduction in our previous work 2000 we showed that genetic algorithms gas provide a powerful tool to increase the accuracy of fuzzy models for both systems modeling and classification in addition to these results we explore the ga to find redundancy in the fuzzy model for the purpose of model reduction an aggregated similarity measure is applied to search for redundancy in the rule base description as a result we propose an iterative fuzzy identification technique starting with data based fuzzy clustering with an overestimated number of local models the ga is then applied to find redundancy among the local models with a criterion based on maximal accuracy and maximal set similarity after the reduction steps the ga is applied with another criterion searching for minimal set similarity and maximal accuracy this results in an automatic identification scheme with fuzzy clustering rule base simplification and constrained genetic optimization with low human intervention the proposed modeling approach is then demonstrated for a system identification and a classification problem

valuations and surface area measures we consider valuations defined on polytopes containing the origin which have measures on the sphere as values we show that the classical surface area measure is essentially the only such valuation which is sl n contravariant of degree one moreover for all real p an lp version of the above result is established for gl n contravariant valuations of degree p this provides a characterization of the lp surface area measures from the lp brunn minkowski theory mathematics subject classification 52b45 28a10

between gersgorin and minimal gersgorin sets the eigenvalues of a given matrix a can be localized by the well known gersgorin theorem they belong to the gersgorin set which is the union of the gersgorin disks each of them is a simple function of the matrix entries by applying the same theorem to a similar matrix x 1 ax a new inclusion set can be obtained taking the intersection over x being a positive diagonal matrix will lead us to the minimal gersgorin set defined by varga r s varga gersgorin and his circles springer series in computational mathematics vol 36 2004 but this set is not easy to calculate in this paper we will take the intersection over some special structured matrices x and show that this intersection can be expressed by the same formula as the eigenvalue inclusion set cs a in l j cvetkovic v kostic r varga a new gersgorin type eigenvalue inclusion set etna 18 2004 73 80

a fast identification algorithm for box cox transformation based radial basis function neural network in this letter a box cox transformation based radial basis function rbf neural network is introduced using the rbf neural network to represent the transformed system output initially a fixed and moderate sized rbf model base is derived based on a rank revealing orthogonal matrix triangularization qr decomposition then a new fast identification algorithm is introduced using gauss newton algorithm to derive the required box cox transformation based on a maximum likelihood estimator the main contribution of this letter is to explore the special structure of the proposed rbf neural network for computational efficiency by utilizing the inverse of matrix block decomposition lemma finally the box cox transformation based rbf neural network with good generalization and sparsity is identified based on the derived optimal box cox transformation and a d optimality based orthogonal forward regression algorithm the proposed algorithm and its efficacy are demonstrated with an illustrative example in comparison with support vector machine regression

fisher information and matrix valued valuations abstract all continuous and affinely contravariant matrix valued valuations on the sobolev space w 1 2 r n are completely classified it is shown that there is a unique such valuation this valuation turns out to be the fisher information matrix

on determining the structure of a non linear system abstract a method is presented for constructing a model of a non linear system from the input output data the structure of the system is assembled from a library of nonlinear terms that are possible suspects for inclusion in the model while each possible model structure is represented by a node in a tree it is shown that searching the tree for the optimal n term model structure can be done in n stages using an algorithm based on conjugate direction ideas the essential feature of the technique is that it generates a sequence of projection matrices such that the original vector valued functions become conjugate directions this leaves the original non linear terms intact and allows the structure for the optimal n term model to be assembled in n selections from the library examples are presented from regression analysis and non linear dynamic systems

p q mixed geominimal surface area and p q mixed affine surface area abstract in this paper the p q mixed geominimal surface area and the p q mixed affine surface area are introduced they are extensions of the l p geominimal surface area and the l p affine surface area respectively affine invariance under unimodular affine transformations and continuity for the p q mixed geominimal surface area and the p q mixed affine surface area are discussed the existence the uniqueness and the uniform boundness of the p q mixed petty bodies are obtained affine isoperimetric inequalities for the p q mixed geominimal surface area and the p q mixed affine surface area are established

the orlicz brunn minkowski theory a general framework additions and inequalities the orlicz brunn minkowski theory introduced by lutwak yang and zhang is a new extension of the classical brunn minkowski theory it represents a generalization of the lp brunn minkowski theory analogous to the way that orlicz spaces generalize lp spaces for appropriate convex functions 0 1 m 0 1 a new way of combining arbitrary sets in r n is introduced this operation called orlicz addition and denoted by has several desirable properties but is not associative unless it reduces to lp addition a general framework is introduced for the orlicz brunn minkowski theory that includes both the new addition and previously introduced concepts and makes clear for the rst time the relation to orlicz spaces and norms it is also shown that orlicz addition is intimately related to a natural and fundamental generalization of minkowski addition called m addition the results obtained show roughly speaking that the orlicz brunn minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the lp brunn minkowski theory inequalities of the brunn minkowski type are obtained both for m addition and orlicz addition the new orlicz brunn minkowski inequality implies the lp brunn minkowski in equality new orlicz minkowski inequalities are obtained that generalize the lp minkowski inequality one of these has connections with the conjectured log brunn minkowski inequal ity of lutwak yang and zhang and in fact these two inequalities together are shown to split the classical brunn minkowski inequality

a hybrid constructive algorithm for single layer feedforward networks learning single layer feedforward networks slfns have been proven to be a universal approximator when all the parameters are allowed to be adjustable it is widely used in classification and regression problems the slfn learning involves two tasks determining network size and training the parameters most current algorithms could not be satisfactory to both sides some algorithms focused on construction and only tuned part of the parameters which may not be able to achieve a compact network other gradient based optimization algorithms focused on parameters tuning while the network size has to be preset by the user therefore trial and error approach has to be used to search the optimal network size because results of each trial cannot be reused in another trial it costs much computation in this paper a hybrid constructive hc algorithm is proposed for slfn learning which can train all the parameters and determine the network size simultaneously at first by combining levenberg marquardt algorithm and least square method a hybrid algorithm is presented for training slfn with fixed network size then with the hybrid algorithm an incremental constructive scheme is proposed a new randomly initialized neuron is added each time when the training entrapped into local minima because the training continued on previous results after adding new neurons the proposed hc algorithm works efficiently several practical problems were given for comparison with other popular algorithms the experimental results demonstrated that the hc algorithm worked more efficiently than those optimization methods with trial and error and could achieve much more compact slfn than those construction algorithms

minkowski valuations under volume constraints abstract we provide a description of the space of continuous and translation invariant minkowski valuations k n k n for which there is an upper and a lower bound for the volume of k in terms of the volume of the convex body k itself although no invariance with respect to a group acting on the space of convex bodies is imposed we prove that only two types of operators appear a family of operators having only cylinders over n 1 dimensional convex bodies as images and a second family consisting essentially of 1 homogeneous operators using this description we give improvements of some known characterization results for the difference body

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

invariant valuations on star shaped sets the brunn minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in euclidean space among the most beautiful results of twentieth century convexity is hadwiger s characterization theorem for the elementary mixed volumes quermassintegrals see 3 5 9 hadwiger s characterization leads to effortless proofs of numerous results in geometric convexity including mean projection formulas for convex bodies 13 p 294 and various kinematic formulas 7 12 14 15 hadwiger s theorem also provides a connection between rigid motion invariant set functions and symmetric polynomials 1 7 recently advancements have been made in a theory introduced by lutwak 8 that is dual to the brunn minkowski theory a theory tailored for dealing with analogous questions involving star shaped sets and intersections with subspaces see also 2 4 6 in the dual theory convex bodies are replaced by star shaped sets and support functions are replaced by radial functions hadwiger s characterizaton theorem is of such fundamental importance that any candidate for a dual theory must possess a dual analogue however the dual theory in its original form was not sufficiently rich to be able to accommodate a dual of hadwiger s theorem in 6 it was shown that the natural setting for the dual theory is larger than that envisioned by previous investigators by defining the dual topology on star shaped sets in terms of the l topology on the space of n integrable functions on the unit sphere the author was able to extend the dual theory to a broad class of star shaped sets called l stars many new theorems can be proved within this larger framework including a hadwiger style classification theorem for continuous valuations on star shaped sets that are homogeneous with respect to dilation in the present paper we discard the stringent requirement of homogeneity and continue with classification theorems for continuous valuations on article no ai971601

smooth solutions to the l_p l p dual minkowski problem in this paper we consider the l p dual minkowski problem by geometric variational method using anisotropic gauss kronecker curvature flows we establish the existence of smooth solutions of the l p dual minkowski problem when pq ge 0 and the given data is even if f equiv 1 we show under some restrictions on p and q that the only even smooth uniformly convex solution is the unit ball

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

recovering map static nonlinearities from chaotic data using dynamical models abstract this paper is concerned with the estimation from chaotic data of maps with static nonlinearities a number of issues concerning model construction such as structure selection over parametrization and model validation are discussed in the light of the shape of the static non linearities reproduced by the estimated maps a new interpretation of term clusters and cluster coefficients of polynomial models is provided based on this approach the paper discusses model limitations and some useful principles to select the structure of nonlinear maps some of the ideas have been tested using several nonlinear systems including a boost voltage regulator map and a set of real data from a chaotic circuit

construction of neurofuzzy models for imbalanced data classification we propose a new class of neurofuzzy construction algorithms with the aim of maximizing generalization capability specifically for imbalanced data classification problems based on leave one out loo cross validation the algorithms are in two stages first an initial rule base is constructed based on estimating the gaussian mixture model with analysis of variance decomposition from input data the second stage carries out the joint weighted least squares parameter estimation and rule selection using orthogonal forward subspace selection ofss procedure we show how different loo based rule selection criteria can be incorporated with ofss and advocate either maximizing the leave one out area under curve of the receiver operating characteristics or maximizing the leave one out fmeasure if the data sets exhibit imbalanced class distribution extensive comparative simulations illustrate the effectiveness of the proposed algorithms

symmetric rbf classifier for nonlinear detection in multiple antenna aided systems in this paper we propose a powerful symmetric radial basis function rbf classifier for nonlinear detection in the so called overloaded multiple antenna aided communication systems by exploiting the inherent symmetry property of the optimal bayesian detector the proposed symmetric rbf classifier is capable of approaching the optimal classification performance using noisy training data the classifier construction process is robust to the choice of the rbf width and is computationally efficient the proposed solution is capable of providing a signal to noise ratio snr gain in excess of 8 db against the powerful linear minimum bit error rate ber benchmark when supporting four users with the aid of two receive antennas or seven users with four receive antenna elements

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

term and variable selection for non linear system identification the purpose of variable selection is to pre select a subset consisting of the significant variables or to eliminate the redundant variables from all the candidate variables of a system under study prior to model term detection it is required that the selected significant variables alone should sufficiently represent the system generally not all the model terms which are produced by combining different variables make an equal contribution to the system output and terms which make little contribution can be omitted a parsimonious representation which contains only the significant terms can often be obtained without the loss of representational accuracy by eliminating the redundant terms based on these observations a new variable and term selection algorithm is proposed in this paper the term detection algorithm can be applied to the general class of non linear modelling problems which can be expressed as a linear in the parameters form the variable selection procedure is based on locally linea

orthogonal least squares methods and their application to non linear system identification abstract identification algorithms based on the well known linear least squares methods of gaussian elimination cholesky decomposition classical gram schmidt modified gram schmidt householder transformation givens method and singular value decomposition are reviewed the classical gram schmidt modified gram schmidt and householder transformation algorithms are then extended to combine structure determination or which terms to include in the model and parameter estimation in a very simple and efficient manner for a class of multivariate discrete time non linear stochastic systems which are linear in the parameters

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

crofton measures and minkowski valuations a description of continuous rigid motion compatible minkowski valuations is established as an application we present a brunn minkowski type inequality for intrinsic volumes of these valuations

continuous rotation invariant valuations on convex sets the notion of valuation on convex sets can be considered as a generalization of the notion of measure which is deflned only on the class of convex compact sets it is well known that there are important and interesting examples of valuations on convex sets which are not measures in the usual sense as for example the mixed volumes basic deflnitions and some classical examples are discussed in section 2 of this paper for more detailed information we refer to the surveys mc sch and mc3 throughout this paper all the valuations are assumed to be continuous with respect to the hausdorfi metric note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry there exists an explicit description of translation invariant continuous valuations on 1 and 2 due to hadwiger h1 the case of 2 is nontrivial continuous rigid motion invariant valuations on d are completely classifled by the remarkable hadwiger theorem as linear combinations of the quermassintegrals cf h2 or for a simpler proof k

identification of partial differential equation models for a class of multiscale spatio temporal dynamical systems in this article the identification of a class of multiscale spatio temporal dynamical systems which incorporate multiple spatial scales from observations is studied the proposed approach is a combination of adams integration and an orthogonal least squares algorithm in which the multiscale operators are expanded using polynomials as basis functions and the spatial derivatives are estimated by finite difference methods the coefficients of the polynomials can vary with respect to the space domain to represent the feature of multiple scales involved in the system dynamics and are approximated using a b spline wavelet multi resolution analysis the resulting identified models of the spatio temporal evolution form a system of partial differential equations with different spatial scales examples are provided to demonstrate the efficiency of the proposed method

a neurofuzzy network knowledge extraction and extended gram schmidt algorithm for model subspace decomposition this paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets based on a takagi sugeno t s inference mechanism and a new extended gram schmidt orthogonal decomposition algorithm for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules the paper introduces a one to one mapping between a fuzzy rule base and a model matrix feature subspace hence rule based knowledge can be extracted to enhance model transparency model transparency is explored by the derivation of an equivalence between an a optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule the a optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule base an extended gram schmidt algorithm is then developed to estimate the parameter vector for each rule this new algorithm decomposes the model rule bases via an orthogonal subspace decomposition approach so as to enhance model transparency with the capability of interpreting the derived rule base energy level

improved structure optimization for fuzzy neural networks fuzzy neural network based inference systems are well known universal approximators which can produce linguistically interpretable results unfortunately their dimensionality can be extremely high due to an excessive number of inputs and rules which raises the need for overall structure optimization in the literature various input selection methods are available but they are applied separately from rule selection often without considering the fuzzy structure this paper proposes an integrated framework to optimize the number of inputs and the number of rules simultaneously first a method is developed to select the most significant rules along with a refinement stage to remove unnecessary correlations an improved information criterion is then proposed to find an appropriate number of inputs and rules to include in the model leading to a balanced tradeoff between interpretability and accuracy simulation results confirm the efficacy of the proposed method

nonlinear model structure detection using optimum experimental design and orthogonal least squares a very efficient learning algorithm for model subset selection is introduced based on a new composite cost function that simultaneously optimizes the model approximation ability and model adequacy the derived model parameters are estimated via forward orthogonal least squares but the subset selection cost function includes an a optimality design criterion to minimize the variance of the parameter estimates that ensures the adequacy and parsimony of the final model an illustrative example is included to demonstrate the effectiveness of the new approach

geometric measures in the dual brunn minkowski theory and their associated minkowski problems a longstanding question in the dual brunn minkowski theory is what are the dual analogues of federer s curvature measures for convex bodies the answer to this is provided this leads naturally to dual versions of minkowski type problems what are necessary and sufficient conditions for a borel measure to be a dual curvature measure of a convex body sufficient conditions involving measure concentration are established for the existence of solutions to these problems

identification of mimo non linear systems using a forward regression orthogonal estimator abstract an orthogonal least squares estimator which was originally derived for single input single output systems is extended to multi input multi output non linear systems the estimator can provide information about the structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner multivariable non linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included

from local similarities to global coding a framework for coding applications feature coding has received great attention in recent years as a building block of many image processing algorithms in particular the importance of the locality assumption in coding approaches has been studied in many previous works we review this assumption and claim that using the similarity of data points to a more global set of anchor points does not necessarily weaken the coding method as long as the underlying structure of the anchor points is considered we propose to capture the underlying structure by assuming a random walker over the anchor points we also show that our method is a fast approximation to the diffusion map kernel experiments on various data sets show that with a knowledge of the underlying structure of anchor points different state of the art coding algorithms may boost their performance in different learning tasks by utilizing the proposed method

the steiner formula for minkowski valuations a steiner type formula for continuous translation invariant minkowski valuations is established in combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations this new steiner type formula is used to obtain a family of brunn minkowski type inequalities for rigid motion intertwining minkowski valuations

new criteria for identifying h matrices in the recent paper of gan and huang linear algebra appl 374 2003 317 several simple criteria as well as a necessary condition for nonsingular h matrices have been obtained inspired by this work we will define several new subclasses of nonsingular h matrices and give necessary conditions for a matrix to be an h matrix finally as a result of numerical experiments we establish relations between defined and some already known subclasses

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

hermitian integral geometry we give in explicit form the principal kinematic formula for the action of the ane unitary group on c n together with a straightforward algebraic method for computing the full array of unitary kinematic formulas expressed in terms of certain convex valuations introduced essentially by h tasaki we introduce also several other canonical bases for the algebra of unitary invariant valuations explore their interrelations and characterize in these terms the cones of positive and monotone elements

moments and valuations all measurable and rm sl n covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified the moment vector is shown to be essentially the only such valuation

orthogonal parameter estimation algorithm for non linear stochastic systems an orthogonal parameter estimation algorithm is derived which allows each parameter in a linear in the parameters non linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model the algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model several simulated examples are included to demonstrate the effectiveness of the algorithm

a valuation theoretic approach to translative equidecomposability abstract all simple translation invariant valuations on polytopes are classified as a direct consequence the well known conditions for translative equidecomposability are recovered furthermore a simplified proof of the classification of continuous simple translation invariant valuations is presented

distributed dictionary learning for sparse representation in sensor networks this paper develops a distributed dictionary learning algorithm for sparse representation of the data distributed across nodes of sensor networks where the sensitive or private data are stored or there is no fusion center or there exists a big data application the main contributions of this paper are 1 we decouple the combined dictionary atom update and nonzero coefficient revision procedure into two stage operations to facilitate distributed computations first updating the dictionary atom in terms of the eigenvalue decomposition of the sum of the residual correlation matrices across the nodes then implementing a local projection operation to obtain the related representation coefficients for each node 2 we cast the aforementioned atom update problem as a set of decentralized optimization subproblems with consensus constraints then we simplify the multiplier update for the symmetry undirected graphs in sensor networks and minimize the separable subproblems to attain the consistent estimates iteratively and 3 dictionary atoms are typically constrained to be of unit norm in order to avoid the scaling ambiguity we efficiently solve the resultant hidden convex subproblems by determining the optimal lagrange multiplier some experiments are given to show that the proposed algorithm is an alternative distributed dictionary learning approach and is suitable for the sensor network environment

application of generalized diagonal dominance in wireless sensor network optimization problems abstract the recent application of the diagonal dominance in the development of the optimization algorithms in the wireless sensor networks design has been done by yuan and yu 2006 14 extended in yu et al 2006 9 and surveyed in machado and tekinay 11 in this paper we will use the concept of generalized diagonal dominance to improve the obtained results regarding the power control game in three directions we also discuss the applicability of such improvements

the lp minkowski problem for n p 1 abstract chou and wang s existence result for the l p minkowski problem on s n 1 for p n 1 and an absolutely continuous measure is discussed and extended to more general measures in particular we provide an almost optimal sufficient condition for the case p 0 1

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

prediction error estimation algorithm for non linear output affine systems a prediction error estimation algorithm is developed for non linear discrete time systems which can be represented by the output affine difference equation model the theory of hypothesis testing is employed to select a model with the correct structure problems relating to the use of output affine models in the identification of non linear systems are discussed and a comparison with the narmax nonlinear armax model is given

variable selection algorithm for the construction of mimo operating point dependent neurofuzzy networks an input variable selection procedure is introduced for the identification and construction of multi input multi output mimo neurofuzzy operating point dependent models the algorithm is an extension of a forward modified gram schmidt orthogonal least squares procedure for a linear model structure which is modified to accommodate nonlinear system modeling by incorporating piecewise locally linear model fitting the proposed input nodes selection procedure effectively tackles the problem of the curse of dimensionality associated with lattice based modeling algorithms such as radial basis function neurofuzzy networks enabling the resulting neurofuzzy operating point dependent model to be widely applied in control and estimation some numerical examples are given to demonstrate the effectiveness of the proposed construction algorithm

flag numbers and floating bodies abstract we investigate weighted floating bodies of polytopes we show that the weighted volume depends on the complete flags of the polytope this connection is obtained by introducing flag simplices which translate between the metric and combinatorial structure our results are applied in spherical and hyperbolic space this leads to new asymptotic results for polytopes in these spaces we also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in euclidean space

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1