hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

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

output feedback stabilization control design for boolean control networks this paper investigates the output feedback stabilization of boolean control networks bcns by using the semi tensor product method and presents a number of new results first based on the algebraic expression of bcns a necessary and sufficient condition is presented for the existence of output feedback stabilizers second a constructive procedure is proposed to design output feedback stabilization controllers for bcns the study of an illustrative example shows that the new results obtained in this paper are very effective in designing output feedback stabilizers for bcns

synchronization in an array of output coupled boolean networks with time delay this brief presents an analytical study of synchronization in an array of coupled deterministic boolean networks bns with time delay two kinds of models are considered in one model the outputs contain time delay while in another one the outputs do not one restriction in this brief is that the state delay and output delay are restricted to be equal by referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices some necessary and sufficient conditions are derived for the synchronization of delay coupled bns examples including a practical epigenetic example are given for illustration

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

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

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

brief paper controllability of boolean control networks with time delays in states this paper investigates the controllability of boolean networks with time invariant delays in states after a brief introduction on converting the logic dynamics to discrete time delay systems the controllability via two kinds of controls is studied one kind of control is generated by boolean control networks another kind of control is free boolean sequences in both cases necessary and sufficient conditions of the controllability of boolean control networks are proved finally examples are given to illustrate the efficiency of the obtained results

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

a characterization of affine surface area abstract we show that every upper semicontinuous and equi affine invariant valuation on the space of d dimensional convex bodies is a linear combination of affine surface area volume and the euler characteristic

on the periodic trajectories of boolean control networks abstract in this note we first characterize the periodic trajectories or equivalently the limit cycles of a boolean network and their global attractiveness we then investigate under which conditions all the trajectories of a boolean control network may be forced to converge to the same periodic trajectory if every trajectory can be driven to such a periodic trajectory this is possible by means of a feedback control law

fast time stability of temporal boolean networks in real systems most of the biological functionalities come from the fact that the connections are not active all the time based on the fact temporal boolean networks tbns are proposed in this paper and the fast time stability is analyzed via semi tensor product stp of matrices and incidence matrices first the algebraic form of a tbn is obtained based on the stp method and one necessary and sufficient condition for global fast time stability is presented moreover incidence matrices are used to obtain several sufficient conditions which reduce the computational complexity from mathcal o n2 n exponential type to mathcal o n 4 polynomial type compared with the stp method in addition the global fast time stabilization of tbns is considered and pinning controllers are designed based on the neighbors of controlled nodes rather than all the nodes finally the local fast time stability of tbns is considered based on the incidence matrices as well several examples are provided to illustrate the effectiveness of the obtained results

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

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

sl m c equivariant and translation covariant continuous tensor valuations abstract the space of continuous sl m c equivariant m 2 and translation covariant valuations taking values in the space of real symmetric tensors on c m r 2 m of rank r 0 is completely described the classification involves the moment tensor valuation for r 1 and is analogous to the known classification of the corresponding tensor valuations that are sl 2 m r equivariant although the method of proof cannot be adapted

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

moderate deviations for the overlap parameter in the hopfield model we derive moderate deviation principles for the overlap parameter in the hopfield model of spin glasses and neural networks if the inverse temperature is different from the critical inverse temperature c 1 and the number of patterns m n satisfies m n n 0 the overlap parameter multiplied by n 1 2 1 obeys a moderate deviation principle with speed n1 2 and a quadratic rate function i e the gaussian limit for 1 2 remains visible on the moderate deviation scale at the critical temperature we need to multiply the overlap parameter by n 1 4 1 if then m n satisfies m n 6 log n m n 2n4 log n n 0 the rescaled overlap parameter obeys a moderate deviation principle with speed n1 4 and a rate function that is basically a fourth power the random term occurring in the central limit theorem for the overlap at c 1 is no longer present on a moderate deviation scale if the scaling is even closer to n1 4 e g if we multiply the overlap parameter by n1 4 log log n the moderate deviation principle breaks down the case of variable temperature converging to one is also considered if n converges to c fast enough i e faster than open image in new window the non gaussian rate function persists whereas for n converging to one slower than open image in new window the moderate deviations principle is given by the gaussian rate at the borderline the moderate deviation rate function is the one at criticality plus an additional gaussian term

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

lp minkowski valuations on lq spaces abstract all continuous sl n covariant l p minkowski valuations on l q are classified the only non trivial operators appearing in this context are the asymmetric l p moment operators

the discrete planar l0 minkowski problem abstract in the discrete setting the l 0 minkowski problem extends the question posed and answered by the classical minkowski s existence theorem for polytopes in particular the planar extension which we address in this paper concerns the existence of a convex polygonal body which contains the origin whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

valuations on convex sets non commutative determinants and pluripotential theory abstract a new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space h n is presented in particular new examples of sp n sp 1 invariant translation invariant continuous valuations are constructed this method is based on the theory of plurisubharmonic functions of quaternionic variables developed by the author in two previous papers bull sci math 127 1 2003 1 j geom anal 13 2 2003 183

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

state feedback stabilization for boolean control networks state feedback stabilization for boolean control networks is investigated in this technical note based on the algebraic representation of logical dynamics in terms of the semi tensor product of matrices a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller and a general control design approach is proposed when global stabilization is feasible via state feedback instead of designing the logical form of a stabilizing feedback law directly we first construct its algebraic representation and then convert the algebraic representation back to the logical form an example is worked out to illustrate the proposed design procedure

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

brief paper controllability of boolean control networks via the perron frobenius theory boolean control networks bcns are recently attracting considerable interest as computational models for genetic and cellular networks addressing control theoretic problems in bcns may lead to a better understanding of the intrinsic control in biological systems as well as to developing suitable protocols for manipulating biological systems using exogenous inputs we introduce two definitions for controllability of a bcn and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive respectively our analysis is based on a result that may be of independent interest namely a simple algebraic formula for the number of different control sequences that steer a bcn between given initial and final states in a given number of time steps while avoiding a set of forbidden states

information theoretic inequalities for contoured probability distributions we show that for a special class of probability distributions that we call contoured distributions information theoretic invariants and inequalities are equivalent to geometric invariants and inequalities of bodies in euclidean space associated with the distributions using this we obtain characterizations of contoured distributions with extremal shannon and renyi entropy we also obtain a new reverse information theoretic inequality for contoured distributions

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

observability reconstructibility and state observers of boolean control networks the aim of this paper is to introduce and characterize observability and reconstructibility properties for boolean networks and boolean control networks described according to the algebraic approach proposed by d cheng and co authors in the series of papers 3 6 7 and in the recent monography a complete characterization of these properties based both on the boolean matrices involved in the network description and on the corresponding digraphs is provided finally the problem of state observer design for reconstructible bns and bcns is addressed and two different solutions are proposed

on a model of associative memory with huge storage capacity in krotov et al in lee eds advances in neural information processing systems curran associates inc red hook 2016 krotov and hopfield suggest a generalized version of the well known hopfield model of associative memory in their version they consider a polynomial interaction function and claim that this increases the storage capacity of the model we prove this claim and take the limit as the degree of the polynomial becomes infinite i e an exponential interaction function with this interaction we prove that model has an exponential storage capacity in the number of neurons yet the basins of attraction are almost as large as in the standard hopfield model

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

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

controllability of probabilistic boolean control networks based on transition probability matrices in this paper we propose a new approach to investigate the controllability and reachability of probabilistic boolean control networks pbcns with forbidden states we first give a simple algebraic formula for the transition probability between two states in a given number of time step while avoiding a set of forbidden states then we construct the controllability matrix based on a new operator and some necessary and sufficient conditions are obtained for the controllability and reachability of pbcns a numerical example is given to illustrate the efficiency of the obtained results

on the hopfield model at the critical temperature we study the hopfield model at temperature 1 when thenumber m n of patterns grows a bit slower than n we reach a goodunderstanding of the model whenever m n n log n 11 for example we show that if m n for two typical configurations 1 2 i n 1 i 2 i 2 is close to nm n

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

replica symmetry breaking and exponential inequalities for the sherrington kirkpatrick model we provide an extremely accurate picture of the sherrington kirkpatrick model in three cases for high temperature for large external field and for any temperature greater than or equal to 1 and sufficiently small external field we describe the system at the level of the central limit theorem or as physicists would say at the level of fluctuations around the mean field we also obtain much more detailed information in the form of exponential inequalities that express a uniform control over higher order moments we give a complete rigorous proof that at the generic point of the predicted low temperature region there is replica symmetry breaking in the sense that the system is unstable with respect to an infinitesimal coupling between two replicas

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

cycles of periodically time variant boolean networks this paper investigates cycles of periodically time variant boolean networks ptvbns some properties of cycles of ptvbns and the relationship between ptvbns and time invariant boolean networks tibns are revealed it is shown that each cycle of a ptvbn corresponds to a common cycle of some tibns and all the cycles of a ptvbn can be constructed by those of the corresponding tibns moreover some results on lengths of cycles of ptvbns are derived finally some examples are given to illustrate the obtained theoretical results

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

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

output tracking control of boolean control networks via state feedback this paper investigates the state feedback based output tracking control of boolean control networks bcns with a constant reference signal by using the semi tensor product method based on the algebraic expression of bcns and by constructing a series of reachable sets a general procedure is proposed for the design of the state feedback laws for bcns to track a constant reference signal the study of an illustrative example shows that the obtained new results are effective in designing state feedback based output tracking controllers for bcns to track a constant reference signal

a homogeneous decomposition theorem for valuations on convex functions abstract the existence of a homogeneous decomposition for continuous and epi translation invariant valuations on super coercive functions is established continuous and epi translation invariant valuations that are epi homogeneous of degree n are classified by duality corresponding results are obtained for valuations on finite valued convex functions

metastates in disordered mean field models ii the superstates we continue to investigate the size dependence of disordered mean field models with finite local spin space in more detail illustrating the concept of superstates as recently proposed by bovier and gayrard we discuss various notions of convergence for the behavior of the paths t tn t 0 1 in the thermodynamic limit n here n is the gibbs measure in the finite volume 1 n and is the disorder variable in particular we prove refined convergence statements in our concrete examples the hopfield model with finitely many patterns having continuous paths and the curie weiss random field ising model having singular paths

ordered overlaps in disordered mean field models let m n be a sequence of integers with m as n and m o n for bounded i i d r v i k and bounded i i d r v i we study the large deviation of the family of ordered scalar products x k n 1 i 1 n i i k k m under the distribution conditioned on the i k s to get a full large deviation principle it is necessary to specify also the total norm k m x k 2 1 2 which turns to be associated with some extra gaussian distribution our results apply to disordered mean field systems including generalized hopfield models in the regime of a sublinear number of patterns we build also a class of examples where this norm is the crucial order parameter

intersecting random half spaces toward the gardner derrida formula gardner and derrida have introduced a natural version of the problem of the capacity of the binary perceptron with temperature and they proposed based on physical methods remarkable formulas for this model we give a complete rigorous proof that these formulas are correct at sufficiently high temperature for a much larger class of models

valuations with crofton formula and finsler geometry valuations admitting a smooth crofton formula are studied using geometric measure theory and rumin s cohomology of contact manifolds the main technical result is a current representation of a valuation with a smooth crofton formula a geometric interpretation of alesker s product is given for such valuations as a first application in finsler geometry a short proof of the theorem of gelfand smirnov that crofton densities are projective is derived the holmes thompson volumes in a projective finsler space are studied it is shown that they induce in a natural way valuations and that the alesker product of the k dimensional and the l dimensional holmes thompson valuation is the k l dimensional holmes thompson valuation

volume polar volume and euler characteristic for convex functions abstract functional analogs of the euler characteristic and volume together with a new analog of the polar volume are characterized as non negative continuous sl n and translation invariant valuations on the space of finite convex and coercive functions on r n

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

contravariant minkowski valuations a complete classification of all continuous gl n contravariant minkowski valuations is established as an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical petty projection inequality

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

star valuations and dual mixed volumes since its creation by brunn and minkowski what has become known as the brunn minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data the machinery of the brunn minkowski theory includes mixed volumes of minkowski symmetrization techniques such as those of steiner and blaschke isoperimetric inequalities such as the brunn minkowski minkowski and aleksandrov fenchel inequalities integral transforms such as the cosine transform and important auxiliary bodies associated with these transforms such as minkowski s projection bodies schneider s recent book 22 on the brunn minkowski theory is the best available introduction to the subject while the brunn minkowski theory has proven to be of enormous value in answering inverse questions regarding projections of convex bodies onto subspaces the theory has been of little value in answering inverse questions with data regarding intersections with subspaces however recent advancements have been made in the development of a dual brunn minkowski theory 3 4 5 6 8 9 11 13 16 23 26 27 28 29 which has been tailored specifically for dealing with such questions in contrast to the brunn minkowski theory in the dual theory convex bodies are replaced by star shaped sets and projections onto subspaces are replaced by intersections with subspaces the machinery of the dual theory includes dual mixed volumes introduced by lutwak 15 16 dual isoperimetric inequalities 16 and important auxiliary bodies known as intersection bodies first introduced by busemann for special centered convex bodies in 1 later defined for star shaped sets by lutwak 16 see also 9 a comprehensive introduction to geometric tomography including the dual brunn minkowski theory may be found in gardner s book 17 unfortunately there still remain fundamental and foundational problems with the dual brunn minkowski theory one of the most beautiful and article no 0048

the lp busemann petty centroid inequality abstract the ratio between the volume of the p centroid body of a convex body k in r n and the volume of k attains its minimum value if and only if k is an origin symmetric ellipsoid this result the lp busemann petty centroid inequality was recently proved by lutwak yang and zhang in this paper we show that all the intrinsic volumes of the p centroid body of k are convex functions of a time like parameter when k is moved by shifting all the chords parallel to a fixed direction the lp busemann petty centroid inequality is a consequence of this general fact

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

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

the equivalence issue of two kinds of controllers in boolean control networks abstract in this paper the equivalence issue between state feedback controller and free sequence controller in boolean control network bcn is investigated based on the algebraic representation of boolean networks we prove that a boolean control network can be stabilized to a cycle or a fixed point via free sequence controller if and only if there is a state feedback controller which can derive the boolean control network to the cycle or the fixed point then two algorithms are provided to find the state feedback controller after that we prove that a bcn can be globally controllable by free sequence controller while the bcn is not necessarily globally controllable by state feedback controller at last an example is given to illustrate the results

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

fourier transform and valuations abstract it is proved that the classical fourier transform is a continuous positively gl n covariant and logarithmic translation covariant valuation moreover the fourier transform is shown to be the only such valuation on complex valued functions

valuations on manifolds and rumin cohomology smooth valuations on manifolds are studied by establishing a link with the rumin de rham complex of the co sphere bundle several operations on differential forms induce operations on smooth valuations signature operator rumin laplace operator euler verdier involution and derivation operator as an application alesker s hard lefschetz theorem for even translation invariant valuations on a finite dimensional euclidean space is generalized to all translation invariant valuations the proof uses kahler identities the rumin de rham complex and spectral geometry

on controllability of delayed boolean control networks this paper is devoted to studying the trajectory and state controllability of boolean control networks bcns with time delay in contrast to bcns without time delay the dynamics of delayed bcns are determined by a sequence of initial states named here trajectories trajectory controllability means that there exists a control signal steering a system from an initial trajectory to a desired trajectory while state controllability means that there exists a control signal steering an initial state to a given state here both trajectory controllability and state controllability will be studied it should be noted that in this paper trajectory controllability does not mean tracking or following a given trajectory in fact it means to control bcns to a destination trajectory of length mu at the k th step using the semi tensor product of matrices the delayed bcns are first converted into an equivalent algebraic description and then some necessary and sufficient conditions are derived for the trajecto

exponential inequalities and convergence of moments in the replica symmetric regime of the hopfield model in a previous work we proved the validity of the replica symmetric solution for the hopfield model in a nontrivial domain of parameters this was done at the accuracy of the lln in a somewhat larger domain we obtain a description at the level of the clt or in the terminology of physics we calculate the fluctuations around the mean field this is obtained by refining the tools we developed for a rigorous use of the cavity method and proving new a priori estimates about the localization of the gibbs measure

controllability analysis and control design for switched boolean networks with state and input constraints this paper investigates the controllability analysis and the control design for switched boolean networks sbns with state and input constraints by using the semi tensor product method and presents a number of new results on their controllability optimal control and stabilization first the constrained incidence matrix is proposed for sbns based on which several necessary and sufficient conditions are obtained for the controllability second using the results on the controllability two algorithms are established to design proper switching sequence and controls that minimize the cost functional in a fixed designed shortest termination time third the constrained sbn is converted to an equivalent unconstrained one based on which some necessary and sufficient conditions are presented for the stabilization of the constrained sbn with open loop and closed loop controls respectively finally a practical example of the apoptosis network is studied by using the new results obtained in this paper the s

theory of valuations on manifolds iii multiplicative structure in the general case this is the third part of a series of articles where the theory of valuations on manifolds is constructed in the second part of this series the notion of a smooth valuation on a manifold was introduced the goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds thus extending some of the affine constructions from the first author s 2004 paper and from the first part of this series

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

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

theory of valuations on manifolds ii abstract this article is the second part in the series of articles where we are developing theory of valuations on manifolds roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions the goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it

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

strategy optimization for static games based on stp method abstract in this paper we consider the choice behavior of players in a static game and discuss solutions to static games of complete information problem via semi tensor product stp method some properties of the dominant behaviors are obtained based on which choosing the dominant strategies deleting the dominated strategies are used to optimize the strategy sets so as to obtain solutions to the considered game examples are given to show the effectiveness of the results

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

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

set stability and set stabilization of boolean control networks based on invariant subsets this study addresses the set stability of boolean networks bns and set stabilization of boolean control networks bcns set stability determines whether a bn converges to a given subset whereas set stabilizability addresses the issue of whether a bcn can be stabilized to a given subset many problems can be viewed as special cases of set stability and set stabilization including synchronization partial stability and partial stabilization problems the concepts of invariant subset and control invariant subset are introduced then algorithms for the largest invariant subset and the largest control invariant subset contained in a given subset are proposed based on the invariant subsets obtained the necessary and sufficient conditions for set stability and set stabilizability are established and formulae are provided to calculate the shortest transient periods for respective initial states a design procedure is proposed for finding all the time optimal set stabilizers finally an example is used to show the application of the proposed results to the synchronization problem of bns