curve shortening and the rendezvous problem for mobile autonomous robots if a smooth closed and embedded curve is deformed along its normal vector field at a rate proportional to its curvature it shrinks to a circular point this curve evolution is called euclidean curve shortening and the result is known as the gage hamilton grayson theorem motivated by the rendezvous problem for mobile autonomous robots we address the problem of creating a polygon shortening flow a linear scheme is proposed that exhibits several analogues to euclidean curve shortening the polygon shrinks to an elliptical point convex polygons remain convex and the perimeter of the polygon is monotonically decreasing

evolution by mean curvature flow of lagrangian spherical surfaces in complex euclidean plane abstract we describe the evolution under the mean curvature flow of embedded lagrangian spherical surfaces in the complex euclidean plane c 2 in particular we answer the question 4 7 addressed in 10 by a neves about finding out a condition on a starting lagrangian torus in c 2 such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the clifford torus

distributed formation control for fractional order systems dynamic interaction and absolute relative damping abstract this paper studies the distributed formation control problem for multiple fractional order systems under dynamic interaction and with absolute relative damping in the context of this paper formation control means that a group of systems reaches the desired state deviations via a local interaction we first study a formation control algorithm in the case of a directed dynamic network topology the convergence conditions on both the network topology and the fractional orders are presented when the fractional order satisfies 0 1 1 2 n sufficient conditions on the network topology are given to ensure the formation control we then propose fractional order formation control algorithms with absolute relative damping and study the conditions on the network topology and the control gains such that the formation control will be achieved under a directed fixed network topology the final equilibria are also given explicitly finally several simulation examples are presented as a proof of concept

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

a unified approach to some recurrence sequences via faa di bruno s formula abstract the well known faa di bruno formula for higher derivatives of a composite function plays an important role in combinatorics in this paper we obtain a unified approach to some recurrence sequences for the complete bell polynomials by using faa di bruno s formula furthermore we obtain an inverse relation of lah numbers of two types and by using this relation we obtain some new and interesting combinatorial identities

lp dual affine surface area abstract according to the notion of l p affine surface area by lutwak in this paper we introduce the concept of l p dual affine surface area further we establish the affine isoperimetric inequality and the blaschke santalo inequality for l p dual affine surface area besides the dual brunn minkowski inequality for l p dual affine surface area is presented

a geometric ginzburg landau problem for surfaces embedded in a three dimensional euclidean space consider a functional consisting of two terms a version of the willmore energy and an anisotropic area penalising the first component of the normal vector the latter weighted with the factor 1 epsilon 2 the asymptotic behaviour of such functionals as epsilon tends to 0 is studied in this paper the results include a lower and an upper bound on the minimal energy subject to suitable constraints moreover for embedded spheres a compactness result is obtained under appropriate energy bounds

on yau s problem of evolving one curve to another convex case abstract a revised yau s curvature difference flow is considered to deform one convex curve x 0 to another one x it is proved that this flow exists globally on time interval 0 and the evolving curve preserving its convexity and bounded area a converges to a fixed limiting curve x congruent to a a x as time tends to infinity where a is the area bounded by the target curve x

classification of limiting shapes for isotropic curve flows with a 5 0 and initial condition x p 0 xzo p this produces a family of curves yt x si t here n is the curvature and n is the outward pointing unit normal vector these equations are particularly natural in that they are isotropic equivariant under rotations in the plane and homogeneous equivariant under dilation of space if time is also scaled accordingly the main aim of this paper is to provide a complete description of the behaviour of embedded convex curves moving by equations of the form 1 1 in certain cases this description has already been provided if a 1 then equation 1 1 is the curve shortening flow for which the following holds

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

a geometric model for active contours in image processing we propose a new model for active contours based on a geometric partial differential equation our model is intrinsec stable satisfies the maximum principle and permits a rigorous mathematical analysis it enables us to extract smooth shapes we cannot retrieve angles and it can be adapted to find several contours simultaneously moreover as a consequence of the stability we can design robust algorithms which can be engineed with no parameters in applications numerical experiments are presented

calibrations and spinors introduction i spinors in dimension 8m 1 1 spinors and pinors 1 2 constructing calibrations from spinors 1 3 triality and the octonians ii dimension 16 and es ii 1 a model for the clifford algebra cl 16 ii 2 the orbit structure of spin 16 on 8 16 ii 3 maximal isotropy subgroups of spin 16 ii 4 calibration on r 16 constructed by squaring spinors iii additional results in higher dimension iii 1 a differential system for each geometry iii 2 pure spinors and calibrations iii 3 spinors in dimensions 50 mod 8 references

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

asynchronous rendezvous analysis via set valued consensus theory this paper presents the design and analysis result for asynchronous rendezvous control of multiagent systems with continuous time dynamics and intermittent interactions the protocol designing strategies only impose weak restrictions on anticipated way point sets from which the way points are selected and path planning of each agent and can be applied to the networks of arbitrary dimensional subsystems explicitly the anticipated way point sets are in a polytope like form and the path between any two consecutive way points is required to be included within the minimum convex region covering the two associated anticipated way point sets under the assumption of directed and switching interaction topology and the assumption of intermittent and asynchronous interactions with time varying delays we perform the set valued consensus analysis on the evolution of anticipated way point sets with respect to update times and provide mild sufficient conditions for the solvability of the asynchronous rendezvous problem the proof techniques rely much on graph theory and nonnegative matrix theory the obtained result extends greatly the existing work in the literature and several examples demonstrate its broad potential applications particularly additional distributed control rules different from the circumcenter algorithm are devised for network connectivity maintenance

an introduction to potential theory in calibrated geometry in this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry these functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties moreover they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce a number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds this paper introduces and investigates questions of pseudo convexity in the context of a general calibrated manifold x phi analogues of totally real submanifolds are introduced and used to construct enormous families of strictly phi convex spaces with every topological type allowed by morse theory specific calibrations are used as examples throughout in a sequel the duality between phi pluri sub harmonic functions and phi positive currents is investigated this study involves boundaries generalized jensen measures and other geometric objects on a calibrated manifold

divided differences and a general explicit formula for sequences of mansour mulay shattuck divided differences is a very important tool often used both in theory and in practice for the approximation of functions in this paper we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying mansour mulay shattuck s two term recurrence with arbitrary b i i 0

convergent systems vs incremental stability two similar stability notions are considered one is the long established notion of convergent systems the other is the younger notion of incremental stability both notions require that any two solutions of a system converge to each other yet these stability concepts are different in the sense that none implies the other as is shown in this paper using two examples it is shown under what additional assumptions one property indeed implies the other furthermore this paper contains necessary and sufficient characterizations of both properties in terms of lyapunov functions

generalised g 2 manifolds we define new riemannian structures on 7 manifolds by a differential form of mixed degree which is the critical point of a possibly constrained variational problem over a fixed cohomology class the unconstrained critical points generalise the notion of a manifold of holonomy g 2 while the constrained ones give rise to a new geometry without a classical counterpart we characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type ii supergravity theory with bosonic background fields in particular this geometry can be described by two linear metric connections with skew symmetric torsion finally we construct explicit examples by introducing the device of t duality

parabolic equations related to curve motion abstract we study curve motions based on differential equations curve motion equations are classified into two types adaptive equations which depend on the choice of coordinate systems and non adaptive equations examples from both types of equations are studied and the global existence for these equations is proved based on integral estimates

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

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

unified cooperative control of multiple agents on a sphere for different spherical patterns in this technical note we consider the cooperative control of multiple agents on a sphere provide the appropriate attraction repulsion interactions between agents with introduction of the state dependent repulsive coefficients and design the unified control laws for agents to achieve three fundamental spherical configurations or patterns rendezvous uniform deployment and formation using both the first and second order models we present the analysis of the stability scaling and geometric patterns of the models on the sphere under different attraction repulsion parameters with illustration of the geometrical patterns and the geometrical isomerism by simulations the closed loop dynamics of agents are invariant with respect to group rotation actions on the sphere

lp mixed affine surface area abstract according to the three notions of mixed affine surface area l p affine surface area and l p mixed affine surface area proposed by lutwak in this article we give the concept of ith l p mixed affine surface area such that the first and second notions of lutwak are its special cases further some lutwak s results are extended associated with this concept besides applying this concept we establish an inequality for the volumes and dual quermassintegrals of a class of star bodies

two chain rules for divided differences and faa di bruno s formula in this paper we derive two formulas for divided differences of a function of a function both formulas lead to other divided difference formulas such as reciprocal and quotient rules the two formulas can also be used to derive faa di bruno s formula and other formulas for higher derivatives of composite functions we also derive a divided difference version of faa di bruno s determinant formula

calibrations and t duality for a subclass of hitchin s generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold under a suitable integrability condition on the ambient geometry these generalised calibrated submanifolds minimise a functional occurring as d brane energy in type ii string theories further we investigate the behaviour of calibrated cycles under t duality and construct non trivial examples

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

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

anisotropic flows for convex plane curves where v and k are respectively the normal velocity and curvature of the interface equation 1 is sometimes called the curve shortening problem because it is the negative l2 gradient flow of the length of the interface by drawing pictures one can be easily convinced that a simple closed curve stays simple and smooth along 1 and shrinks to a point in finite time with the limiting shape of a circle however a mathematical treatment of 1 turned out to be rather delicate in fact a rigorous study did not exist until the early 1980s when differential geometers considered 1 as a tool in the search for simple closed geodesics on surfaces it was also regarded as a model case for more general curvature flows which are believed to be important in the topological classification of low dimensional manifolds as a first attempt gage 10 proved that the isoperimetric ratio decreases along convex curves then in gage and hamilton 12 it was shown that a convex curve stays convex and shrinks to a point in finite time moreover if one normalizes the flow by dilating it so that the enclosing area is constant the normalized flow converges smoothly to a circle finally grayson 14 completed this line of investigation by showing that a simple curve evolves into a convex one before shrinks to a point for an alternative approach to this result one may consult hamilton 18 recently mullins s theory was generalized by gurtin 15 16 and by angenent and gurtin 4 5 see also the monograph of gurtin 17 to include anisotropy and the possibility of a difference in bulk energies between phases anisotropy is indispensable in dealing with crystalline materials for perfect conductors the temperatures in both phases are constant the equation becomes

kahler reduction of metrics with holonomy g2 a torsion free g 2 structure admitting an infinitesimal isometry such that the quotient is a kahler manifold is shown to give rise to a 4 manifold equipped with a complex symplectic structure and a 1 parameter family of functions and 2 forms linked by second order equations reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to g 2

mean curvature flow of monotone lagrangian submanifolds we use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone lagrangian submanifolds in mathbb c n

combinatorial formulae for the derivatives of lame functions lame functions play a central role in the theory of ellipsoidal harmonics and have many varied applications in mathematical physics in this article a generalized approach to the computation of lame function derivatives of arbitrary order is derived and it is demonstrated how these derivatives can be expressed recursively in terms of the original lame functions by making use of combinatorial formulae discovered by di bruno girard and waring

a steiner formula in the lp brunn minkowski theory abstract we prove an analogue of the classical steiner formula for the l p affine surface area of a minkowski outer parallel body for any real parameter p we show that the classical steiner formula and the steiner formula of lutwak s dual brunn minkowski theory are special cases of this new steiner formula this new steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature they have the intrinsic volumes of the classical brunn minkowski theory as well as the dual quermassintegrals of the dual brunn minkowski theory as special cases properties of these new quantities are investigated a connection to information theory among them a steiner formula for the s th mixed l p affine surface area of a minkowski outer parallel body for any real parameters p and s is also given

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

multidimensional bell polynomials of higher order we develop some extensions of the classical bell polynomials previously obtained by introducing a further class of these polynomials called multidimensional bell polynomials of higher order they arise considering the derivatives of functions f in several variables f i i 1 2 m where f i are composite functions of different orders i e f i t i 1 i 2 i r i t i 1 2 m we show that these new polynomials are always expressible in terms of the ordinary bell polynomials by means of suitable recurrence relations or formal multinomial expansions moreover we give a recurrence relation for their computation

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

a curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area motivated by a problem from fluid mechanics we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate using formal asymptotic and numerical techniques we derive possible extinction shapes as the curve contracts to a point dependent on the rate of decreasing area we find there is a wider class of extinction shapes than for standard curve shortening for which initially simple closed curves are always asymptotically circular we also provide numerical evidence that self intersection is possible for non convex initial conditions distinguishing between pinch off and coalescence of the curve interior

symmetries and invariant solutions for the geometric heat flows we study lie symmetries and invariant solutions of the geometric heat flows the basic similarity reductions for the ghe are performed reduced equations and exact solutions associated with the symmetries are obtained group invariant solutions and reductions for the affine case are also discussed in a special case

the elastica problem under area constraint we show that the elastic energy e gamma of a closed curve gamma has a minimizer among all plane simple regular closed curves of given enclosed area a gamma and that the minimum is attained for a circle the proof is of a geometric nature and deforms parts of gamma in a finite number of steps to construct some related convex sets with smaller energy

some extensions of faa di bruno s formula with divided differences the well known formula of faa di bruno s for higher derivatives of a composite function has played an important role in combinatorics in this paper we generalize the divided difference form of faa di bruno s formula and give an explicit formula for the n th divided difference of a multicomposite function more generally we establish the relationships of the bell polynomials with respect to multicomposite functions applying these to multicomposite functions we obtain some extensions of faa di bruno s formula

mixed affine surface area various means over the unit sphere of powers of the curvature function of a convex body arise in geometric convexity for example both the surface area and the affine surface area of sufficiently smooth bodies can be expressed as such means the survey of gruber 111 on approximation of convex bodies contains a number of examples of means of powers of the curvature function arising in approximation problems see pp 137 141 143 and 1466148 of 11 i these means arise surprisingly in questions related to approximating convex bodies by polytopes see e g mcclure and vitale 18 schneider 24 and schneider and wieacker 27 an interesting conjecture of firey 9 p s also 26 p 2571 which was recently proved by gage lo for plane convex bodies is that a certain weighted mean of the curvature function lies between the arithmetic and harmonic means of the curvature function it turns out that there is also a close connection between various means of the curvature function of a convex body and some mixed volumes involving the body in this note we investigate means of the curvature function of a convex body which arise as coefficients in the expansion of the afftne surface area of outer and inner blaschke parallel bodies for a variety of reasons we will take a general approach to the study of these means an approach similar to that taken in 14 see also 4 pp 164 1671 in presenting dual mixed volumes let x denote the set of convex bodies compact convex sets with interior points in euclidean n space r we use h k to denote the support function of a convex body k i e

an extension of the bell polynomials abstract after recalling the most important properties and applications of the bell polynomials we introduce an extension of this special class of functions more precisely we consider the case of multicomposite functions and we show connections with the ordinary bell polynomials

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

some identities involving exponential functions and stirling numbers and applications guo and qi 2013 posed a problem asking to determine the coefficients a k i 1 for 1 i k such that 1 1 e t k 1 i 1 k a k i 1 1 e t 1 i 1 the authors answer this question alternatively by faa di bruno s formula unify the eight identities due to guo and qi to two identities involving two parameters and apply them to obtain an explicit expression for the apostol bernoulli numbers and the fubini numbers respectively

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

a note on the isoperimetric deficit abstract in this study for a strictly convex domain k enclosed by a c 2 curve with perimeter l and area a we find several lower and upper bounds of the isoperimetric deficit l 2 4 a of k