w2 p estimates for the parabolic monge ampere equation when u is a solution to the equation ut det dx2u f with f positive continuous and ft satisfying certain growth conditions we establish estimates in l for ut and show that dx2u satisfies uniform interior estimates in lp for 0 p

modified curvature motion for image smoothing and enhancement we formulate a general modified mean curvature based equation for image smoothing and enhancement the key idea is to consider the image as a graph in some r sup 72 and apply a mean curvature type motion to the graph we consider some special cases relevant to grey scale and color images

the monge ampere equation and its link to optimal transportation we survey old and new regularity theory for the monge ampere equation show its connection to optimal transportation and describe the regularity properties of a general class of monge ampere type equations arising in that context

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

two disjoint cycle cover bipancyclicity of balanced hypercubes abstract a bipartite graph g is two disjoint cycle cover r1 r2 bipancyclic if for any even integer l satisfying r1 l r2 there exist two vertex disjoint cycles c1 and c2 in g such that v c 1 l and v c 2 v g l where v g denotes the number of vertices in g in this paper we study the two disjoint cycle cover bipancyclicity of the n dimensional balanced hypercube bhn which is a hypercube variant network and is superior to hypercube due to having a smaller diameter as a consequence we show that bhn is two disjoint cycle cover 4 2 2 n 1 bipancyclic for n 2

free boundary regularity on the focusing problem for the gauss curvature flow with flat sides we consider the motion of a compact weakly convex two dimensional surface of revolution sigma under the gauss curvature flow we assume that the initial surface has a flat side and as a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate at points where the curvature is zero expressing the strictly convex part of the surface near the interface as the graph of a function z f r t we show that if at ti me t 0 g sqrt f vanishes linearly at the flat side then g r t will become smooth up to the interface for t 0 and it will remain smooth up to the focusing time t of the flat side we also show that at the focusing time of the flat side the function g is of class c 1 beta for all beta and no better than c 1 2 over 5 this implies that at the focusing time the surface sigma is of class c 2 beta for all beta and no better than c 2 2 over 5 in the case of the evolution monge ampere equation we find the exact self similar profile of the function g at its focusing time

curvature flows in the sphere we consider contracting and expanding curvature flows in mathbb s n 1 when the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the gaus map the contracting hypersurfaces shrink to a point x 0 while the expanding hypersurfaces converge to the equator of the hemisphere mathcal h x 0 after rescaling by the same scale factor the rescaled hypersurfaces converge to the unit spheres with centers x 0 resp x 0 exponentially fast in c infty mathbb s n

two node disjoint paths in balanced hypercubes the balanced hypercube bhn proposed by wu and huang is a variation of the hypercube it has been proved that the balanced hypercube is a node transitive and bipartite graph assume that the nodes are divided into two bipartite node sets x and y u and x are two different nodes in x and v and y are two different nodes in y in this paper we prove that there exist two node disjoint paths p x y and r u v in bhn and v p x y v r u v v bhn where n 1 the hamiltonian laceability of bhn which was proved by xu et al is also obtained from the corollary of our result

path embedding in star graphs abstract the star graph interconnection network has been introduced as an attractive alternative to the hypercube network in this paper we consider the path embedding problem in star graphs assume that n 4 we prove that paths of all even lengths from d x y to n 2 can be embedded between two arbitrary vertices x and y from the same partite set in the n dimensional star graph in addition paths of all odd lengths from d x y to n 1 can be embedded between two arbitrary vertices x and y from different partite sets in the n dimensional star graph except that if x and y are adjacent there is no path of length 3 between them the result is optimal in the sense that paths of all possible lengths are found in star graphs

on adaptive control of nonlinearly parameterized nonlinear systems towards a constructive procedure abstract a new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized nonlinear systems was recently proposed by the authors the key step is the construction of a monotone mapping via a suitable selection of a controller tuning function which has to satisfy some integrability conditions this translates into the need to solve a partial differential equation pde in this paper this result is extended providing some answers to the questions of characterization of monotonizable systems and solvability of the pde first adding to the design a nonlinear dynamic scaling we obviate the need to solve the pde second for the case of factorizable nonlinearities the following results are established i it is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible ii directly verifiable involutivity conditions that ensure the solution of the pde are presented iii an explicit formula for the required tuning function is given provided the regressor matrix satisfies some rank conditions hence adding a dynamic scaling this yields a constructive solution to the problem

c 1 α regularity of solutions to parabolic monge ampere equations we study interior c 1 alpha regularity of viscosity solutions of the parabolic monge amp ere equation u t b x t big det d 2 u big p with exponent p 0 and with coefficients b which are bounded and measurable we show that when p is less than the critical power 1 over n 2 then solutions become instantly c 1 alpha in the interior also we prove the same result for any power p0 at those points where either the solution separates from the initial data or where the initial data is c 1 beta

flow of nonconvex hypersurfaces into spheres the flow of surfaces by functions of their principal curvatures has been intensively studied it started with the work of brakke 1 who used the formalism of geometric measure theory a more classical approach had been chosen by huisken who looked at the so called inward flow in 5 the outward flow of surfaces this term will be explained in the sequel is scaling invariant and therefore more natural than the inward flow producing more general results so far huisken 6 and urbas 8 proved that the outward flow of convex surfaces by suitable functions of their principal curvatures converges to spheres the convexity assumption is essential in their work in this paper we would like to present a different method for proving the convergence of star shaped surfaces into spheres via the outward flow let be a symmetric positive function homogeneous of degree one being defined on an open cone of rn with vertex in the origin which contains the positive diagonal i e all tuples of the form

convexity estimates for a nonhomogeneous mean curvature flow we study the evolution of a closed immersed hypersurface whose speed is given by a function phi h h of the mean curvature asymptotic to h ln h for large h compared with other nonlinear functions of the curvatures this speed has some good properties which allow for an easier study of the formation of singularities in the nonconvex case we prove apriori estimates showing that any surface with positive mean curvature at the initial time becomes asymptotically convex near a singularity similar estimates also hold for the mean curvature flow for the flow considered here they admit a simpler proof based only on the maximum principle

conditional diagnosability of alternating group graphs let an be the alternating group of degree n with n 3 set s 1 2 i 1 i 2 3 i n the alternating group graph denoted by agn is defined as the cayley graph on an with respect to s jwo et al networks 23 1993 315 326 introduced alternating group graph agn as an interconnection network topology for computing systems conditional diagnosability a new measure of diagnosability introduced by lai et al ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of agn is 4 for n 4 and 6n 18 for each n 5

bipanconnectivity of balanced hypercubes the balanced hypercube proposed by wu and huang is a variant of the hypercube network in this paper paths of various lengths are embedded into balanced hypercubes a bipartite graph g is bipanconnected if for two arbitrary nodes x and y of g with distance d x y there exists a path of length l between x and y for every integer l with d x y l v g 1 and l d x y 0 mod 2 we prove that the n dimensional balanced hypercube bh n is bipanconnected for all n 1 this result is stronger than that obtained by xu et al which shows that the balanced hypercube is edge bipancyclic and hamiltonian laceable

edge pancyclicity and hamiltonian laceability of the balanced hypercubes abstract the balanced hypercube bh n is a variant of the hypercube q n huang and wu proved that bh n has better properties than q n with the same number of links and processors in particularly they showed that there exists a cycle of length 2 l in bh n for all l 2 l 2 n in this paper we improve this result by showing that bh n is edge pancyclic which means that for arbitrary edge e there exists a cycle of even length from 4 to 2 2 n containing e in bh n we also show that the balanced hypercubes are hamiltonian laceable

triangular surface mesh fairing via gaussian curvature flow surface mesh fairing by the mean curvature flow and its various modifications have become a popular topic however very few researches have been attempted on using the gaussian curvature flow in surface fairing the aim of this paper is to investigate such a problem we find that gaussian curvature flow can only be used to smooth convex meshes hence it cannot be used to smooth noisy surface meshes because a noisy surface mesh is not convex to overcome this difficulty we design a new diffusion equation whose evolution direction depends on the mean curvature normal and the magnitude is a properly defined function of the gaussian curvature experimental results show that the designed fairing scheme can effectively remove the noise and simultaneously preserve the sharp features such as corners and edges

fault tolerant cycle embedding in alternating group graphs abstract cycle embedding is an important issue in evaluating the efficiency of interconnection networks and is also an extension of the theoretical research on hamiltonicity in this paper we study the fault tolerant hamiltonicity of alternating group graphs which were recently proposed as interconnection topologies for parallel and distributed systems a cycle of length l is said to be an l cycle let f be a set of faulty vertices in an n dimensional alternating group graph ag n and let m be the number of vertices in ag n f a previous result in j m chang j s yang y l wang y cheng panconnectivity fault tolerant hamiltonicity and hamiltonian connectivity in alternating group graphs networks 44 2004 302 310 showed that for n 4 ag n f is hamiltonian i e ag n f possesses a hamiltonian cycle if f n 2 and ag n f is hamiltonian connected i e every two vertices of ag n f are connected by a hamiltonian path if f n 3 in this paper we show the following results i ag n f is pancyclic i e ag n f contains an l cycle for each l with 3 l m if f n 2 ii ag n f is vertex pancyclic i e every vertex of ag n f is contained in an l cycle for each l with 3 l m if f n 3 and iii ag n f is edge 4 pancyclic i e every edge of ag n f is contained in an l cycle for each l with 4 l m if f n 4 the first result is an improvement over the previous one and the last two results are new characterization of fault tolerant pancyclicity on ag n all the results we achieved are the best possible in the sense that the number of faulty vertices cannot be increased

the balanced hypercube a cube based system for fault tolerant applications in this paper we present an interconnection structure called the balanced hypercube which is a variation of the standard hypercube with desirable properties of strong connectivity regularity and symmetry the proposed structure is a special type of load balanced graph designed to tolerate processor failure in balanced hypercubes each processor has a backup matching processor that shares the same set of neighboring nodes therefore tasks that run on a faulty processor can be reactivated in the backup processor to provide efficient system reconfiguration other properties of balanced hypercubes are examined it is also shown that odd dimensional balanced hypercubes have smaller diameters than that of standard hypercubes as an application of balanced hypercubes we show a fault tolerant embedding of rings in balanced hypercubes

nonlinear pi control of uncertain systems an alternative to parameter adaptation a novel approach to stabilization and trajectory tracking for nonlinear systems with unknown parameters and uncertain disturbances is developed we take a drastic departure from the classical adaptive control approach consisting of a parameterized feedback law and an identifier which tries to minimize a tracking or prediction error instead we propose a simple nonlinear pi structure that generates a stable error equation with a perturbation function that exhibits at least one root trajectories are forced to converge to this root by suitably adjusting the nonlinear pi gains we consider the two basic problems of i matched uncertainties when the uncertain terms are in the image of the input matrix and ii unknown control directions when the control signal is multiplied by a gain of unknown sign we show that without knowing the system parameters and with only basic information on the uncertainties we can achieve global asymptotic stability and global tracking without injecting high gains into the loop interestingly we prove that we can take as our nonlinear pi structure an activation function reminiscent of that used in neural networks although most of the results are derived assuming full state measurement we also present an observer based solution for a chain of integrators with unknown control direction the procedure is shown to provide simple solutions to the classical problems of neural network function approximation as well as eccentricity control and friction compensation of mechanical systems

conditional fault tolerant edge bipancyclicity of hypercubes with faulty vertices and edges let f be a faulty set in an n dimensional hypercube q n such that in q n f each vertex is incident to at least two edges and let f v f e be the numbers of faulty vertices and faulty edges in f respectively in this paper we consider the fault tolerant edge bipancyclicity of hypercubes it is shown that each edge in q n f for n 3 lies on a fault free cycle of any even length from 6 to 2 n 2 f v if f v f e 2 n 5 this gives an answer for a problem proposed by yang et al 2016 33 consider hypercubes with mixed faulty edges and faulty vertices under conditional fault model improve the previous results about the fault tolerant edge bipancyclicity of hypercubes give an affirmative answer for a problem proposed by yang et al 2016

fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs the arrangement graph a sub n k is a generalization of the star graph there are some results concerning fault hamiltonicity and fault hamiltonian connectivity of the arrangement graph however these results are restricted in some particular cases and thus are less completed we improve these results and obtain a stronger and simpler statement let n k spl ges 2 and f spl sube v a sub n k spl cup e a sub n k we prove that a sub n k f is hamiltonian if f spl les k n k 2 and a sub n k f is hamiltonian connected if f spl les k n k 3 these results are optimal

hamiltonian cycle embedding for fault tolerance in balanced hypercubes abstract the balanced hypercube bh n defined by wu and huang is a variant of the hypercube network yang proposed that fault tolerance of balanced hypercube bh n is an important issue in parallel computing which needs further study yang 2010 24 in this paper we prove that there exists a fault free hamiltonian path between any two adjacent vertices in bh n with 2 n 2 faulty edges as a corollary we derive that for any fault free edge e there exists a fault free hamiltonian cycle containing e in bh n with 2 n 2 faulty edges which is optimal in the sense of the number of faulty edges

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

pinching estimates and motion of hypersurfaces by curvature functions second derivative pinching estimates are proved for a class of elliptic and parabolic equations including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature the estimates imply convergence of convex hypersurfaces to spheres under these flows improving earlier results of b chow and the author the result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives

symmetric property and reliability of balanced hypercube huang and wu in ieee transactions on computers 46 1997 484 490 introduced the balanced hypercube bh n as an interconnection network topology for computing systems and they proved that bh n is vertex transitive however some other symmetric properties say edge transitivity and arc transitivity of bh n remained unknown in this paper we solve this problem and prove that bh n is an arc transitive cayley graph using this we also investigate some reliability measures including super connectivity cyclic connectivity etc in bh n first we prove that every minimum edge cut of bh n n ge 2 isolates a vertex and every minimum vertex cut of bh n n ge 3 isolates a vertex this is stronger than that obtained by wu and huang which shows the connectivity and edge connectivity of bh n are 2n second yang applied mathematics and computation 219 2012 970 975 proved that for n ge 2 the super connectivity of bh n is 4n 4 and the super edge connectivity of bh n is 4n 2 in this paper we proved that bh n n ge 2 is super lambda prime but not super kappa prime that is every minimum super edge cut of bh n n ge 2 isolates an edge but the minimum super vertex cut of bh n n ge 2 does not isolate an edge third we also obtain that for n ge 2 the cyclic connectivity of bh n is 4n 4 and the cyclic edge connectivity of bh n is 4 2n 2 that is to become a disconnected graph which has at least two components containing cycles we need to remove at least 4n 4 vertices resp 4 4n 2 edges from bh n n ge 2

evolution of convex hypersurfaces by powers of the mean curvature we study the evolution of a closed convex hypersurface in n 1 in direction of its normal vector where the speed equals a positive power k of the mean curvature we show that the flow exists on a maximal finite time interval and that approaching the final time the surfaces contract to a point

edge fault tolerant bipanconnectivity of hypercubes this paper shows that for any two distinct vertices u and v with distance d in the hypercube q n n 3 with at most 2n 5 faulty edges and each vertex incident with least two fault free edges there exist fault free uv paths of length in q n for every with d 4 2 n 1 and d 0 mod2 this result improves some known results on edge fault bipanconnectivity of hypercubes the proof is based on the recursive structure of q n

α gauss curvature flows with flat sides abstract in this paper we study the deformation of the 2 dimensional convex surfaces in r 3 whose speed at a point on the surface is proportional to power of positive part of gauss curvature first for 1 2 1 we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with c 1 1 estimate before the collapsing time if the initial surface is only convex moreover we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while we also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 t t 0 under certain necessary regularity and non degeneracy initial conditions where t 0 is the vanishing time of the flat side

immersion and invariance adaptive control of nonlinearly parameterized nonlinear systems a new framework to design adaptive controllers for nonlinearly parameterized systems is proposed in this technical note the key step is the construction of a monotone mapping which explicitly depends on some of the estimator tuning parameters convexity that is related to monotonicity has been explored by several authors the property is assumed a priori and is useful for the design only in some region of state space which is completed switching the controller in our approach monotonicity can be used in the whole state space and moreover is enforced by the designer effectively becoming a synthesis tool one consequence of these features is that the controller does not rely on state dependent switching in order to dispose of degrees of freedom to render the function monotone we depart from standard adaptive control and adopt instead the recently introduced immersion and invariance approach

worn stones with flat sides all time regularity of the interface we study the all time regularity of the free boundary problem associated to the deformation of a compact weakly convex surface in 3 with a flat side by its gaussian curvature we show that under certain necessary regularity and non degeneracy initial conditions the interface separating the flat from the strictly convex side remains smooth on 0 t tc up to the vanishing time tc of the flat side

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

stabilization of nonlinear systems nonlinearly depending on fast time varying parameters an immersion and invariance approach the problem of stabilization of nonlinear systems which depend nonlinearly on fast time varying parameters is considered in the technical note it is assumed that if the plant parameters were known a static state feedback controller that achieves the stabilization objective with a lyapunov like function that is independent of the parameters is known a constructive procedure to update the unknown parameters of the controller based on the immersion and invariance approach is proposed the main contribution of the paper is to show that the proposed controller guarantees global convergence to zero of the systems state for arbitrary time variations of the plant parameters provided the controller parameters are bounded to ensure the latter condition an assumption that in the single parameter case is strictly weaker than the monotonicity condition invoked in previous studies is imposed stabilization is achieved via a rather unique combination of gradient like parameter estimation and the construction of a monotonic signal that counters the deleterious effect of the parameter variations several simulation examples illustrate the applicability of the suggested method

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

edge fault tolerant edge bipancyclicity of balanced hypercubes abstract the balanced hypercube bh n is a variant of hypercube q n hao et al 2014 showed that there exists a fault free hamiltonian path between any two adjacent vertices in bh n with 2 n 2 faulty edges cheng et al 2015 proved that bh n is 6 edge bipancyclic after 2 n 3 faulty edges occur for all n 2 in this paper we improve these two results by demonstrating that bh n is 6 edge bipancyclic even when there exist 2 n 2 faulty edges for all n 2 our result is optimal with respect to the maximum number of tolerated edge faults

hamiltonian paths passing through prescribed edges in balanced hypercubes abstract the balanced hypercube was proposed as a novel interconnection network for large scale parallel systems it is known that the balanced hypercube is edge bipancyclic given a set of edges p with p n 1 and two vertices x and y in different bipartite sets we show that there exists a hamiltonian path from x to y passing through p in b h n if and only if p is a linear forest and neither x nor y is an internal vertex of p and x and y are not end vertices of a component of p simultaneously consequently the balanced hypercube contains a hamiltonian cycle passing through a set p of at most n prescribed edges if and only if p is a linear forest moreover our result improves some known results

anisotropic flow of convex hypersurfaces by the square root of the scalar curvature abstract we show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor when the strictly convex initial hypersurface in euclidean space is suitably pinched we also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere for a general case in dimension two it is shown that with a volume preserving rescaling the limit profile satisfies a soliton equation

matching preclusion for balanced hypercubes matching preclusion is a measure of robustness in the event of edge failure in interconnection networks the matching preclusion number of a graph g is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching and the conditional matching preclusion number of g is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or an almost perfect matching in this paper we consider balanced hypercubes we obtain that an n dimension balanced hypercube bh n has the matching preclusion number 2n and mainly prove that for the balanced hypercube bh n each matching preclusion set of cardinality 2n is trivial and the conditional matching preclusion number of balanced hypercube is 4n 2 whenever n 2

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

ancient solutions of the affine normal flow we construct noncompact solutions to the affine normal flow of hypersurfaces and show that all ancient solutions must be either ellipsoids shrinking solitons or paraboloids translating solitons we also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines which is originally due to cheng yau the main techniques are local second derivative estimates for a parabolic

global bifurcation and convex solutions for the monge ampere equation abstract we study the following monge ampere equation det d 2 u 1 n f u in u 0 on by bifurcation technique we establish some results about the existence nonexistence uniqueness and multiplicity of convex solutions for this problem our results generalize and improve many important known results from previous literature

parameter convergence in nonlinearly parameterized systems a large class of problems in parameter estimation concerns nonlinearly parametrized systems over the past few years a stability framework for estimation and control of such systems has been established we address the issue of parameter convergence in such systems in this paper systems with both convex concave and general parameterizations are considered in the former case sufficient conditions are derived under which parameter estimates converge to their true values using a min max algorithm in the latter case to achieve parameter convergence a hierarchical min max algorithm is proposed where the lower level consists of a min max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie using this hierarchical algorithm a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization in both cases the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions a general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper

embedding of cycles in arrangement graphs arrangement graphs have been proposed as an attractive interconnection topology for large multiprocessor systems the authors study these graphs by proving the existence of hamiltonian cycles in any arrangement graph they also prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph they show that an arrangement graph can be decomposed into node disjoint cycles in many different ways

what is the best causal scale space for three dimensional images we study the unique affine invariant morphological scale space in three dimensions we discuss its properties and show that it improves the dynamic shape model we explain the algorithms and display the first numerical experiments

convexity estimates for surfaces moving by curvature functions we consider the evolution of compact surfaces by fully nonlinear parabolic curvature ows for which the normal speed is given by a smooth degree one homogeneous function of the principal curvatures of the evolving surface under no further restrictions on the speed function we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow this generalises the corresponding result 26 of huisken and sinestrari for the mean curvature ow to the largest possible class of degree one homogeneous surface flows

on a prioric 1 α andw 2 p estimates for a parabolic monge ampere equation in the gauss curvature flows this paper establishes holder estimates of du and l p estimates of d 2 u for solutions u to the parabolic monge ampere equation au t det d 2 u 1 n

adaptive control of a class of non linearly parametrized systems using convexification in this paper we deal with the problem of adaptive control of a class of non linearly parametrized plants the main contribution is to show that through a well chosen reparametrization it is possible to transform the non convex nonlinear parametrization into a convex one a property that is used to design a globally stable adaptive controller

volume preserving flow by powers of symmetric polynomials in the principal curvatures we study a volume preserving curvature flow of convex hypersurfaces driven by a power of the k th elementary symmetric polynomial in the principal curvatures unlike most of the previous works on related problems we do not require assumptions on the curvature pinching of the initial datum we prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm in the case of the volume preserving scalar curvature flow we can prove that the hypersurfaces converge smoothly to a round sphere

flow by powers of the gauss curvature abstract we prove that convex hypersurfaces in r n 1 contracting under the flow by any power 1 n 2 of the gauss curvature converge after rescaling to fixed volume to a limit which is a smooth uniformly convex self similar contracting solution of the flow under additional central symmetry of the initial body we prove that the limit is the round sphere for 1

convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature we consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity we prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero we apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one for smooth strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time recently several papers have considered the flow of convex hypersurfaces by speeds which are homogeneous functions of the principal curvatures of degree 1 under suitable pinching conditions on the curvature of the initial hypersurface the aim is to prove that solutions become spherical as they contract to points this behaviour has been established for a wide range of flows where the speed is homogeneous of degree 1 in the principal curvatures including the mean curvature flow h1 the flow by nth root of gauss curvature ch1 square root of scalar curvature ch2 and a large family of other speeds a1 a3 a4 the first such result with degree of homogeneity higher than 1 was due to ben chow ch1 and concerned flow by powers of the gauss curvature he proved that flow by k with 1 n produces a spherical limiting shape provided the initial hypersurface is sufficiently pinched i n the sense that hij c hgij later such results were proved by schulze s3 for powers of the mean curvature by alessandroni and sinestrari for powers of the scalar curvature a as and by cabezas rivas and sinestrari crs for normalized flows by powers of elementary symmetric functions a feature of the results mentioned above is that the flows all have some divergence structure a point which is used crucially in deriving sufficient regularity of solutions to deduce the existence of a smooth limiting hypersurface in ch1 the divergence structure was used to deduce pinching of the principal curvatures using integral estimates in a manner similar to h1 in s3 a and as the curvature pinching was proved using maximum principle arguments but the divergence structure was still needed because higher regularity of solutions was established using results for divergence form porous medium equations to date we are not aware of any work which provides holder continuity for solutions of porous medium equations in non divergence form without assuming regularity of the coefficients in this paper we provide a geometric estimate which circumvents this difficulty we prove in section 3 that if the ratios of principal curvatures at each point are bounded in terms of the maximum principal curvature by a function which approaches one at infinity then the ratio of circumradius to inradius is as close to one as desired if the circumradius is sufficiently small using