on the complexity of algebraic numbers i expansions in integer bases let b 2 be an integer we prove that the b ary expansion of every irrational algebraic number cannot have low complexity furthermore we establish that irrational morphic numbers are transcendental for a wide class of morphisms in particular irrational automatic numbers are transcendental our main tool is a new combinatorial transcendence criterion

error threshold in optimal coding numerical criteria and classes of universalities for complexity the free energy of the random energy model at the transition point between the ferromagnetic and spin glass phases is calculated at this point equivalent to the decoding error threshold in optimal codes the free energy has finite size corrections proportional to the square root of the number of degrees the response of the magnetization to an external ferromagnetic phase is maximal at values of magnetization equal to one half we give several criteria of complexity and define different universality classes according to our classification at the lowest class of complexity are random graphs markov models and hidden markov models at the next level is the sherrington kirkpatrick spin glass connected to neuron network models on a higher level are critical theories the spin glass phase of the random energy model percolation and self organized criticality the top level class involves highly optimized tolerance design error thresholds in optimal coding language and maybe financial markets living systems are also related to the last class the concept of antiresonance is suggested for complex systems

reductions of hidden information sources in all but special circumstances measurements of time dependent processes reflect internal structures and correlations only indirectly building predictive models of such hidden information sources requires discovering in some way the internal states and mechanisms unfortunately there are often many possible models that are observationally equivalent here we show that the situation is not as arbitrary as one would think we show that generators of hidden stochastic processes can be reduced to a minimal form and compare this reduced representation to that provided by computational mechanics the e machine on the way to developing deeper measure theoretic foundations for the latter we introduce a new two step reduction process the first step internal event reduction produces the smallest observationally equivalent algebra and the second internal state reduction removes algebra components that are redundant for optimal prediction for several classes of stochastic dynamical systems these reductions produce representations that are equivalent to e machines

automated pattern detection an algorithm for constructing optimally synchronizing multi regular language filters in the computational mechanics structural analysis of one dimensional cellular automata the following automata theoretic analogue of the change point problem from time series analysis arises given a string and a collection di offinite automata identify the regions of that belong to each di and in particular the boundaries separating them we present two methods for solving this multi regular language filtering problem the first although providing the ideal solution requires a stack has a worst case compute time that grows quadratically in s length and conditions its output at any point on arbitrarily long windows of future input the second method is to algorithmically construct a finite transducer that approximates the first algorithm in contrast to the stack based algorithm however the transducer requires only a finite amount of memory runs in linear time and gives immediate output for each letter read it is moreover the best possible finite state approximation with these three features

nonregular languages in the kicked rotor a symbolic dynamics of the markov chain model for trajectories of the kicked rotor standard map in sticky regions near resonant orbits is developed two sets of trajectories are shown to correspond to a context free and a context sensitive language respectively

rigorous properties and uncertainty like relationships on product complexity measures application to atomic systems most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework valid for any arbitrary n dimensional distribution in the present work different product complexities constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated different results on product complexities such as rigorous bounds uncertainty like inequalities relationships among themselves and also statistical interpretations are given for the sake of completeness some of these analytical results on product complexities are numerically checked for the one particle densities of atomic systems in both conjugated spaces

learning hidden markov models using nonnegative matrix factorization the baum welch algorithm together with its derivatives and variations has been the main technique for learning hidden markov models hmms from observational data we present an hmm learning algorithm based on the nonnegative matrix factorization nmf of higher order markovian statistics that is structurally different from the baum welch and its associated approaches the described algorithm supports estimation of the number of recurrent states of an hmm and iterates the nmf algorithm to improve the learned hmm parameters numerical examples are provided as well

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

convolutions and multiplier transformations of convex bodies rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to minkowski and blaschke addition are investigated the main focus is on blaschke minkowski homomorphisms we show that such maps are represented by a spherical convolution operator an application of this representation is a complete classification of all even blaschke minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator among further applications is the following result if an even blaschke minkowski homomorphism maps a convex body to a polytope then it is a constant multiple of the projection body operator

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

stability and diversity in collective adaptation we derive a class of macroscopic differential equations that describe collective adaptation starting from a discrete time stochastic microscopic model the behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior we show that although individual agents interact with their environment and other agents in a purely self interested way macroscopic behavior can be interpreted as game dynamics application to several familiar explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors the simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation we also analyze the adaptation dynamics from an information theoretic viewpoint and discuss self organization induced by the dynamics of uncertainty giving a novel view of collective adaptation

exponential finite elements for diffusion advection problems a new finite element method for the solution of the diffusion advection equation is proposed the method uses non isoparametric exponentially varying interpolation functions based on exact one and two dimensional solutions of the laplace transformed differential equation two eight noded elements are developed and tested for convergence stability peclet number limit anisotropy material heterogeneity dirichlet and neumann boundary conditions and tolerance for mesh distortions their performance is compared to that of conventional eight and 12 noded polynomial elements the exponential element based on two dimensional analytical solutions fails basic tests of convergence the one based on one dimensional solutions performs particularly well it reduces by about 75 the number of elements and degrees of freedom required for convergence yielding an error that is one order of magnitude smaller than that of the eight noded polynomial element the exponential element is stable and robust under relatively high degrees of heterogeneity anisotropy and mesh distortions copyright 2005 john wiley sons ltd

complexity of dynamics as variability of predictability we construct a complexity measure from first principles as an average over the obstruction against prediction of some observable that can be chosen by the observer our measure evaluates the variability of the predictability for characteristic system behaviors which we extract by means of the thermodynamic formalism using theoretical and experimental applications we show that complex and chaotic are different notions of perception in comparison to other proposed measures of complexity our measure is easily computable non divergent for the classical 1 d dynamical systems and has properties of non overuniversality the measure can also be computed for higher dimensional and experimental systems including systems composed of different attractors moreover the results of the computations made for classical 1 d dynamical systems imply that it is not the nonhyperbolicity but the existence of a continuum of characteristic system length scales that is at the heart of complexity

on the complexity of algebraic numbers ii continued fractions the continued fraction expansion of an irrational number alpha is eventually periodic if and only if alpha is a quadratic irrationality however very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three because of some numerical evidence and a belief that these numbers behave like most numbers in this respect it is often conjectured that their partial quotients form an unbounded sequence more modestly we may expect that if the sequence of partial quotients of an irrational number alpha is in some sense simple then alpha is either quadratic or transcendental the term simple can of course lead to many interpretations it may denote real numbers whose continued fraction expansion has some regularity or can be produced by a simple algorithm by a simple turing machine for example or arises from a simple dynamical system the aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity complexity for the continued fraction expansion of algebraic real numbers of degree at least three

subword complexity and projection bodies abstract a polytope p 0 1 d and an 0 1 d induce a so called hartman sequence h p 0 1 z which is by definition 1 at the kth position if k mod 1 p and 0 otherwise k z we prove an asymptotic formula for the subword complexity of such a hartman sequence this result establishes a connection between symbolic dynamics and convex geometry if the polytope p is convex then the subword complexity of h p asymptotically equals the volume of the projection body p of p for almost all 0 1 d

complexity and fisher information the connection between an observer based theory of measurement and a hierarchical classification of complex systems in terms of topological exponents is discussed this leads to generalized definitions of complexity that capture different aspects of the structure of the trajectory space of complex systems

volume of mixed bodies by using inequalities obtained for the volume of mixed bodies and the petty projection inequality sharp isoperimetric inequalities are derived for the projection measures quermassintegrale of a convex body these projection measure inequalities which involve mixed projection bodies zonoids are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body the inequality obtained for the volume of mixed bodies is also used to derive a form of the brunn minkowski inequality involving mixed bodies as an application inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies o introduction in 24 it was shown that the petty projection inequality 26 will yield extensions of the known isoperimetric inequalities between the volume of a convex body and its projection measures quermassintegrale in this article we show that the petty projection inequality can be used to obtain sharp isoperimetric inequalities corollary 5 20 which are strengthened forms of the classical inequalities between the projection measures of a convex body this is accomplished by using the so called mixed convex bodies introduced by firey 17 geometric inequalities are derived for the volume of mixed bodies theorems 4 1 and 4 3 which when combined with the petty projection inequality yield inequalities involving projection bodies zonoids which are related to the inequalities between the projection measures of a convex body the inequalities for mixed bodies are also used to obtain a version of the brunn minkowski inequality theorem 4 2 involving mixed bodies also derived are inequalities between the volume of convex bodies and their associated mixed projection integral theorems 6 4 and 6 4 that are extensions and improvements of some previously established inequalities since some of our results concern projection bodies a topic of interest to workers outside the area of geometric convexity we have restated a few basic definitions and standard results from the field of geometric inequalities in 1 we collect for quick reference some well known results concerning mixed volumes and mixed area measures which will be required later references for material presented in this section are bonnesen and fenchel 7 for mixed volumes busemann 9 and leichtweiss 20 the original works of aleksandrov 1 4 and fenchel and jessen 15 supplemented by the survey of schneider 28 are recommended references for mixed area measures 2 contains introductory material regarding blaschke addition received by the editors january 8 1985 1980 mcithematics subject classihecition primary 52a40 secondary 53a15 46b20

reexamination of an information geometric construction of entropic indicators of complexity abstract information geometry and inductive inference methods can be used to model dynamical systems in terms of their probabilistic description on curved statistical manifolds in this article we present a formal conceptual reexamination of the information geometric construction of entropic indicators of complexity for statistical models specifically we present conceptual advances in the interpretation of the information geometric entropy ige a statistical indicator of temporal complexity chaoticity defined on curved statistical manifolds underlying the probabilistic dynamics of physical systems

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

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

context tree modeling of observed symbolic dynamics modern techniques invented for data compression provide efficient automated algorithms for the modeling of the observed symbolic dynamics we demonstrate the relationship between coding and modeling motivating the well known minimum description length mdl principle and give concrete demonstrations of the context tree weighting and context tree maximizing algorithms the predictive modeling technique obviates many of the technical difficulties traditionally associated with the correct mdl analyses these symbolic models representing the symbol generating process as a finite state automaton with probabilistic emission probabilities provide excellent and reliable entropy estimations the resimulations of estimated tree models satisfying the mdl model selection criterion are faithful to the original in a number of measures the modeling suggests that the automated context tree model construction could replace fixed order word lengths in many traditional forms of empirical symbolic analysis of the data we provide an explicit pseudocode for implementation of the context tree weighting and maximizing algorithms as well as for the conversion to an equivalent markov chain

information accessibility and cryptic processes we give a systematic expansion of the crypticity a recently introduced measure of the inaccessibility of a stationary process s internal state information this leads to a hierarchy of k cryptic processes and allows us to identify finite state processes that have infinite cryptic order the internal state information is present across arbitrarily long observed sequences the crypticity expansion is exact in both the finite and infinite order cases it turns out that k crypticity is complementary to the markovian finite order property that describes state information in processes one application of these results is an efficient expansion of the excess entropy the mutual information between a process s infinite past and infinite future that is finite and exact for finite order cryptic processes

affine inequalities and radial mean bodies we introduce for p 1 the radial pth mean body rpk of a convex body k in en the distance from the origin to the boundary of rpk in a given direction is the pth mean of the distances from points inside k to the boundary of k in the same direction the bodies rpk form a spectrum linking the difference body of k and the polar projection body of k which correspond to p oo and p 1 respectively we prove that rpk is convex when p 0 we also establish a strong and sharp affine inequality relating the volume of rpk to that of rqk when 1 1 and q n it becomes a reverse form of the petty projection inequality proved previously by the second author 0 introduction two important objects in convex geometry are the differ ence body and the polar projection body of a convex body the difference body was studied by minkowski and is ubiquitous in geometry and elsewhere as the vector sum of a set and its reflection in the origin the operation that forms the difference body is essentially that known as central or radial symmetrization and as such finds many applications in mathematical physics and partial differential equations see for example the books of bandle ba and kawohl k projection bodies also originated in the work of minkowski and have found application in the theory of vector valued measures liapounov s theorem the local theory of banach spaces stochastic geometry random determinants hubert s fourth prob lem mathematical economics and other areas see the articles of bourgain and lindenstrauss bl goodey and weil gw and lutwak li the polar pro jection body the polar body of the projection body appears explicitly but very frequently in the more recent literature its behavior under linear transformations often renders it more natural than the projection body itself both the difference body and the polar projection body appear in known affine inequalities the first is an ingredient in the famous rogers shephard inequality corollary 5 8 below an affine inequality relating the volume of a convex body to that of its difference body the second appears in another affine inequality corollary 5 9 first proved by the second author relating the volume of a convex body to that of its polar projection body the latter is a sharp reverse form of the well known petty projection inequality see for example g theorem 9 2 9

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

comment i on simple measure for complexity we critique the measure of complexity introduced by shiner davison and landsberg phys rev e 59 1459 1999 in particular we point out that it is over universal in the sense that it has the same dependence on disorder for structurally distinct systems we then give counterexamples to the claim that complexity is synonymous with being out of equilibrium equilibrium systems can be structurally complex and nonequilibrium systems can be structurally simple we also correct a misinterpretation of a result given by two of the present authors j p crutchfield and d p feldman phys rev e 55 r1239 1997

a higher order discontinuous enrichment method for the solution of high peclet advection diffusion problems on unstructured meshes a higher order discontinuous enrichment method dem with lagrange multipliers is proposed for the efficient finite element solution on unstructured meshes of the advection diffusion equation in the high peclet number regime following the basic dem methodology the usual galerkin polynomial approximation is enriched with free space solutions of the governing homogeneous partial differential equation pde in this case these are exponential functions that exhibit a steep gradient in a specific flow direction exponential lagrange multipliers are introduced at the element interfaces to weakly enforce the continuity of the solution the construction of several higher order dem elements fitting this paradigm is discussed in detail numerical tests performed for several two dimensional benchmark problems demonstrate their computational superiority over stabilized galerkin counterparts especially for high peclet numbers copyright 2009 john wiley sons ltd

the laplace transform galerkin technique a time continuous finite element theory and application to mass transport in groundwater a time continuous numerical technique referred to as the laplace transform galerkin ltg method is developed and applied to the problem of solute transport in porous media after application of galerkin s procedure and subdivision of the domain into finite elements the method involves a simple application of the laplace transformation to eliminate the temporal derivatives appearing in the space discretized set of ordinary differential equations then by solving the resulting transformed system of algebraic equations in laplace p space numerical inversion of the laplace transformed nodal concentration is performed using the robust and accurate crump 1976 algorithm the crump algorithm permits the concentration to be evaluated from a range of time values from a single set of laplace p space solutions because each of the needed p space solutions are independent the algorithm is well suited for execution on multiprocessor parallel computers it is demonstrated by means of a series of examples that the ltg scheme is capable of providing highly accurate solutions essentially devoid of numerical dispersion for grid peclet numbers in excess of 30 examination of the complex valued laplace domain concentration profiles reveal that they are generally smooth well behaved oscillatory functions compared to the profiles in the time domain thus permitting the use of a coarse finite element grid because of the nature of the laplace transformation the ltg method is particularly well suited to the problem of transient groundwater flow and solute transport in fractured porous media or multiple aquifer aquitard systems based on the dual porosity integrodifferential equation approach

a discontinuous enrichment method for variable coefficient advection diffusion at high peclet number a discontinuous galerkin method with lagrange multipliers is presented for the solution of variable coefficient advection diffusion problems at high peclet number in this method the standard finite element polynomial approximation is enriched within each element with free space solutions of a local constant coefficient homogeneous counterpart of the governing partial differential equation hence in the two dimensional case the enrichment functions are exponentials each exhibiting a sharp gradient in a carefully chosen flow direction the continuity of the enriched approximation across the element interfaces is enforced weakly by the aforementioned lagrange multipliers numerical results obtained for two benchmark problems demonstrate that elements based on the proposed discretization method are far more competitive for variable coefficient advection diffusion analysis in the high peclet number regime than their standard galerkin and stabilized finite element comparables copyright 2010 john wiley sons ltd

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

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

the stability of the rogers shephard inequality and of some related inequalities abstract an essentially optimal stability version of the rogers shephard inequality and stability versions of some other related affine invariant inequalities are verified the characteristic feature of all these inequalities is that simplices provide the equality case

finite statistical complexity for sofic systems we propose a measure of complexity for symbolic sequences which is based on conditional probabilities and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata dfa moreover if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures causal states

stabilized element residual method serm a posteriori error estimation for the advection diffusion equation residual based a posteriori error estimation techniques have been developed for linear elliptic symmetric positive definite problems one asymptotically exact error estimator for the elliptic laplacian operator relies on solving local neumann problems in each element this technique is extended to the unsymmetric and positive semi definite advection diffusion ad operator here a novel approach the stabilized element residual method serm is presented in this method the unsymmetric advection terms are retained in the formulation of the local error problem through the use of stabilized methods the selection of the optimum stabilization parameter is discussed

a new affine invariant for polytopes and schneider s projection problem new affine invariant functionals for convex polytopes are introduced some sharp affine isoperimetric inequalities are established for the new functionals these new inequalities lead to fairly strong volume estimates for projection bodies two of the new affine isoperimetric inequalities are extensions of ball s reverse isoperimetric inequalities if k is a convex body i e a compact convex subset with nonempty interior in euclidean n space rn then on the unit sphere sn l its support function h k sn 1 r is defined for u e sn l by h k u max u y y e k where u y denotes the standard inner product of u and y the projection body hk of k can be defined as the convex body whose support function for u e sn 1 is given by h ihk u vol n1 klu1 where voln l denotes n 1 dimensional volume and klul denotes the image of the orthogonal projection of k onto the codimension 1 subspace orthogonal to u an important unsolved problem regarding projection bodies is schneider s projection problem what is the least upper bound as k ranges over the class of origin symmetric convex bodies in r of the affine invariant ratio v hnk v k n1 1 n1 where v is used to abbreviate vol see s1 s2 sw and le schneider s1 conjectured that this ratio is maximized by parallelotopes in s1 schneider also presented applications of such results in stochastic geometry however a counterexample was produced in br to show that this is not the case we will present a modified version of schneider s conjecture that has an affirmative answer in addition we will obtain an inequality that gives an upper bound for the affine ratio while our upper bound is not sharp for any n nevertheless it is asymptotically optimal to be more specific in this paper we introduce a new centro affine functional u defined on the class of polytopes which is closely related to the volume functional v while in general u k v k if k is a random polytope with many faces then u k is very close to v k we shall prove the following variation of schneider s projection conjecture received by the editors february 26 2000 1991 mathematics subject classification primary 52a40

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

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

extremum problems for the cone volume functional of convex polytopes abstract lutwak yang and zhang defined the cone volume functional u over convex polytopes in r n containing the origin in their interiors and conjectured that the greatest lower bound on the ratio of this centro affine invariant u to volume v is attained by parallelotopes in this paper we give affirmative answers to the conjecture in r 2 and r 3 some new sharp inequalities characterizing parallelotopes in r n are established moreover a simplified proof for the conjecture restricted to the class of origin symmetric convex polytopes in r n is provided

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100