average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

the jacobi rosochatius problem on an ellipsoid the lax representations and billiards in this article we construct the lax representations of the geodesic flow the jacobi rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid we prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces also we consider billiards within an ellipsoid under the influence of the hook and rosochatius potentials between the impacts a geometric interpretation of the integrability analogous to the classical chasles and poncelet theorems is given

evolving apollonian networks with small world scale free topologies we propose two types of evolving networks evolutionary apollonian networks eans and general deterministic apollonian networks gdans established by simple iteration algorithms we investigate the two networks by both simulation and theoretical prediction analytical results show that both networks follow power law degree distributions with distribution exponents continuously tuned from 2 to 3 the accurate expression of clustering coefficient is also given for both networks moreover the investigation of the average path length of ean and the diameter of gdan reveals that these two types of networks possess small world feature in addition we study the collective synchronization behavior on some limitations of the ean

synchronizability of complex networks recent interest in the study of networks associated with complex systems has led to a better understanding of the factors and parameters relating the topology of the associated networks with the dynamics of the systems and in particular their synchronization in this paper by using known and new results from spectral graph theory we characterize relevant factors which affect the synchronization of complex networks

scaling of load in communications networks we show that the load at each node in a preferential attachment network scales as a power of the degree of the node for a network whose degree distribution is p k k we show that the load is l k k with 1 implying that the probability distribution for the load is p l 1 l 2 independent of the results are obtained through scaling arguments supported by finite size scaling studies they contradict earlier claims but are in agreement with the exact solution for the special case of tree graphs results are also presented for real communications networks at the ip layer using the latest available data our analysis of the data shows relatively poor power law degree distributions as compared to the scaling of the load versus degree this emphasizes the importance of the load in network analysis

hyperelliptic jacobians as billiard algebra of pencils of quadrics beyond poncelet porisms the thirty years old programme of griffiths and harris of understanding higher dimensional analogues of poncelet type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper starting with the observation of the billiard nature of some classical constructions and configurations we construct the billiard algebra that is a group structure on the set t of lines simultaneously tangent to d 1 quadrics from a given confocal family in the d dimensional euclidean space using this tool the related results of reid donagi and knorrer are further developed realized and simplified we derive a fundamental property of t any two lines from this set can be obtained from each other by at most d 1 billiard reflections at some quadrics from the confocal family we introduce two hierarchies of notions s skew lines in t and s weak poncelet trajectories s 1 0 d 2 the interrelations between billiard dynamics linear subspaces of intersections of quadrics and hyperelliptic jacobians developed in this paper enabled us to obtain higher dimensional and higher genera generalizations of several classical genus 1 results cayley s theorem weyr s theorem griffiths harris theorem and darboux theorem

range limited centrality measures in complex networks here we present a range limited approach to centrality measures in both nonweighted and weighted directed complex networks we introduce an efficient method that generates for every node and every edge its betweenness centrality based on shortest paths of lengths not longer than ensuremath ell 1 l in the case of nonweighted networks and for weighted networks the corresponding quantities based on minimum weight paths with path weights not larger than w ensuremath ell ensuremath ell ensuremath delta ensuremath ell 1 2 l r ensuremath delta these measures provide a systematic description on the positioning importance of a node edge with respect to its network neighborhoods one step out two steps out etc up to and including the whole network they are more informative than traditional centrality measures as network transport typically happens on all length scales from transport to nearest neighbors to the farthest reaches of the network we show that range limited centralities obey universal scaling laws for large nonweighted networks as the computation of traditional centrality measures is costly this scaling behavior can be exploited to efficiently estimate centralities of nodes and edges for all ranges including the traditional ones the scaling behavior can also be exploited to show that the ranking top list of nodes edges based on their range limited centralities quickly freezes as a function of the range and hence the diameter range top list can be efficiently predicted we also show how to estimate the typical largest node to node distance for a network of n nodes exploiting the afore mentioned scaling behavior these observations were made on model networks and on a large social network inferred from cell phone trace logs ensuremath sim 5 5 ifmmode times else texttimes fi 10 6 nodes and ensuremath sim 2 7 ifmmode times else texttimes fi 10 7 edges finally we apply these concepts to efficiently detect the vulnerability backbone of a network defined as the smallest percolating cluster of the highest betweenness nodes and edges and illustrate the importance of weight based centrality measures in weighted networks in detecting such backbones

billiard systems on quadric surfaces and the poncelet theorem billiard systems constrained to move on a quadric surface with confocal quadric boundaries were studied the trajectory of a billiard is described as a collection of geodesic segments joining on the boundary at the joining points the trajectory obeys the law of reflection it was found that the geodesic segments are all tangent to a common confocal quadric curve a caustic on this quadric surface if a trajectory is closed then all trajectories sharing the same caustic quadric are all closed and have the same period and length thus a generalization of poncelet s theorem on a quadric surface is achieved the elliptical billiard systems on spheres and pseudospheres were studied and results were obtained that are similar to those on a plane the results can be extended to n dimensions

ellipsoidal billiards in pseudo euclidean spaces and relativistic quadrics abstract we study the geometry of confocal quadrics in pseudo euclidean spaces of arbitrary dimension d and any signature and related billiard dynamics the goal is to give a complete description of periodic billiard trajectories within ellipsoids the novelty of our approach is the introduction of a new discrete combinatorial geometric structure associated with a confocal pencil of quadrics a colouring in d colours this is used to decompose quadrics of d 1 geometric types of a pencil into new relativistic quadrics of d relativistic types a study of what we term discriminant sets of tropical lines and and their singularities provides insight into the related geometry and combinatorics this yields an analytic criterion describing all periodic billiard trajectories including light like trajectories as a case of special interest

erratum cayley type conditions for billiards within k quadrics in ℝd journal of physics a mathematical and general 2004 37 1269 1279 doi 10 1088 0305 4470 38 36 c01

betweenness centrality in a weighted network when transport in networks follows the shortest paths the union of all shortest path trees g union or logical sum spt can be regarded as the transport overlay network overlay networks such as peer to peer networks or virtual private networks can be considered as a subgraph of g union or logical sum spt the traffic through the network is examined by the betweenness bl of links in the overlay g union or logical sum spt the strength of disorder can be controlled by e g tuning the extreme value index alpha of the independent and identically distributed polynomial link weights in the strong disorder limit alpha 0 all transport flows over a critical backbone the minimum spanning tree mst we investigate the betweenness distributions of wide classes of trees such as the mst of those well known network models and of various real world complex networks all these trees with different degree distributions e g uniform exponential or power law are found to possess a power law betweenness distribution pr bl j approximately j c the exponent c seems to be positively correlated with the degree variance of the tree and to be insensitive of the size n of a network in the weak disorder regime transport in the network traverses many links we show that a link with smaller link weight tends to carry more traffic this negative correlation between link weight and betweenness depends on alpha and the structure of the underlying topology

schrodinger maps and anti ferromagnetic chains we study the problem of restricting schrodinger maps onto lagrangian submanifolds the restriction is imposed by an infinite constraining potential we show that in the limit solutions of the schrodinger map equations converge to solutions of generalized wave map equations this result is applied to the anti ferromagnetic systems where we prove rigorously that the dynamics is governed by wave map equations into open image in new window

towards a physics of internet traffic in a geographic network abstract a set of equations from a biased random walk are shown to describe the time based gaussian distributions of internet traffic relative to the earth s time zones the internet is an example of a more general physical problem dealing with motion near the speed of light relative to different time frames of reference the second order differential equation de takes the form of time diffusion near the speed of light or alternatively considered as a complex variable with real time and imaginary longitudinal components congestion waves are generated by peak global traffic from different time zones following the earth s revolution the de is divided into space and time operators for discussion and each component solution including constants is illustrated using data from a global network compiled by the stanford linear accelerator centre slac indices of global and regional phase congestion for the monitoring sites are calculated from standardised regressions from the earth s rotation there is also a j curve limit to transferring information by the internet and this is expressed as an inequality underpinned by the speed of light with examples from us and european traffic the research returns to an often little known theme of isaac newton s mixing physics with geography in our case the equations define trajectories of information packets travelling near the speed of light navigating within networks and between longitudes relative to the earth s rotation

erratum apollonian networks simultaneously scale free small world euclidean space filling and with matching graphs phys rev lett 94 018702 2005

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

flexible construction of hierarchical scale free networks with general exponent extensive studies have been done to understand the principles behind architectures of real networks recently evidence for hierarchical organization in many real networks has also been reported here we present a hierarchical model that reproduces the main experimental properties observed in real networks scale free of degree distribution p k frequency of the nodes that are connected to k other nodes decays as a power law p k ensuremath sim k ensuremath ensuremath gamma and power law scaling of the clustering coefficient c k ensuremath sim k ensuremath 1 the major points of our model can be summarized as follows a the model generates networks with scale free distribution for the degree of nodes with general exponent ensuremath gamma g2 and arbitrarily close to any specified value being able to reproduce most of the observed hierarchical scale free topologies in contrast previous models cannot obtain values of ensuremath gamma g2 58 b our model has structural flexibility because i it can incorporate various types of basic building blocks e g triangles tetrahedrons and in general fully connected clusters of n nodes and ii it allows a large variety of configurations i e the model can use more than n ensuremath 1 copies of basic blocks of n nodes the structural features of our proposed model might lead to a better understanding of architectures of biological and nonbiological networks

unraveling the structure of treelike networks from first passage times of lazy random walkers we study the problem of random search in finite networks with a tree topology where it is expected that the distribution of the first passage time f t decays exponentially we show that the slope of the exponential tail is independent of the initial conditions of entering the tree in general and scales exponentially or as a power law with the extent of the tree l depending on the tendency p to jump toward the target node it is unfeasible to uniquely determine l and p from measuring the tail slope or the mean first passage time mfpt of an ordinary diffusion along the tree to unravel the structure we consider lazy random walkers that take steps with probability m when jumping on the nodes and return with probability q from the leaves by deriving an exact analytical expression for the mfpt of the intermittent random walk we verify that the structural information of the tree can be uniquely extracted by measuring the mfpt for two randomly chosen types of tracer particles with distinct experimental parameters m and q we also address the applicability of our approach in the presence of disorder in the structure of the tree or statistical uncertainty in the experimental parameters

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

periodic ellipsoidal billiard trajectories and extremal polynomials a comprehensive study of periodic trajectories of billiards within ellipsoids in d dimensional euclidean space is presented the novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line by leveraging deep but yet not widely known results of the krein levin nudelman theory of generalized chebyshev polynomials fundamental properties of billiard dynamics are proven for any d viz that the sequences of winding numbers are monotonic by employing the potential theory we prove the injectivity of the frequency map as a byproduct for d 2 a new proof of the monotonicity of the rotation number is obtained the case study of trajectories of small periods t d le t le 2d is given in particular it is proven that all d periodic trajectories are contained in a coordinate hyperplane and that for a given ellipsoid there is a unique set of caustics which generates d 1 periodic trajectories a complete catalog of billiard trajectories with small periods is provided for d 3

linking legendrian linking and causality the set n of all null geodesics of a globally hyperbolic d 1 dimensional spacetime m g is naturally a smooth 2d 1 dimensional contact manifold the sky of an event x in m is the subset x of n consisting of all null geodesics through x and is an embedded legendrian submanifold of n diffeomorphic to s d 1 it was conjectured by low that for d 2 two events x and y are causally related if and only if x and y are linked in an appropriate sense we use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases and suggest that for d 3 smooth linking should be replaced with legendrian linking

record length requirement of long range dependent teletraffic abstract this article contributes the highlights mainly in two folds on the one hand it presents a formula to compute the upper bound of the variance of the correlation periodogram measurement of teletraffic traffic for short with long range dependence lrd for a given record length t and a given value of the hurst parameter h theorems 1 and 2 on the other hand it proposes two formulas for the computation of the variance upper bound of the correlation periodogram measurement of traffic of fractional gaussian noise fgn type and the generalized cauchy gc type respectively corollaries 1 and 2 they may constitute a reference guideline of record length requirement of traffic with lrd in addition record length requirement for the correlation periodogram measurement of traffic with either the schuster type or the bartlett one is studied and the present results about it show that both types of periodograms may be used for the correlation measurement of traffic with a pre desired variance bound of correlation estimation moreover real traffic in the internet archive by the special interest group on data communication under the association for computing machinery of us acm sigcomm is analyzed in the case study in this topic

pseudo riemannian geodesics and billiards abstract in pseudo riemannian geometry the spaces of space like and time like geodesics on a pseudo riemannian manifold have natural symplectic structures just like in the riemannian case while the space of light like geodesics has a natural contact structure furthermore the space of all geodesics has a structure of a jacobi manifold we describe the geometry of these structures and their generalizations we also introduce and study pseudo euclidean billiards emphasizing their distinction from euclidean ones we present a pseudo euclidean version of the clairaut theorem on geodesics on surfaces of revolution we prove pseudo euclidean analogs of the jacobi chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo euclidean space

memory effect in growing trees abstract we show that the structure of a growing tree preserves an information on the shape of an initial graph for the exponential trees evidence of this kind of memory is provided by means of the iterative equations derived for the moments of the node node distance distribution numerical calculations confirm the result and allow to extend the conclusion to the barabasi albert scale free trees the memory effect almost disappears if subsequent nodes are connected to the network with more than one link

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

the appell hypergeometric functions and classical separable mechanical systems a relationship between two old mathematical subjects is observed the theory of hypergeometric functions and the separability in classical mechanics separable potential perturbations of integrable billiard systems and the jacobi problem for geodesics on an ellipsoid are expressed through the appell hypergeometric functions f4 of two variables even when the number of degrees of freedom increases if an ellipsoid is symmetric the number of variables in the hypergeometric functions does not increase wider classes of separable potentials are given by the obtained new formulae automatically

on schrodinger maps we study the question of well posedness of the cauchy problem for schrodinger maps from 2 to the sphere 2 or to 2 the hyperbolic space the idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear schrodinger system of equations and then study this modified schrodinger map system msm we then prove local well posedness of the cauchy problem for the msm with minimal regularity assumptions on the data and outline a method to derive well posedness of the schrodinger map itself from it in proving well posedness of the msm the heart of the matter is resolved by considering truly quatrilinear forms of weighted l2 functions 2002 wiley periodicals inc

stochastically evolving networks we discuss a class of models for the evolution of networks in which new nodes are recruited into the network at random times and links between existing nodes that are not yet directly connected may also form at random times the class contains both models that produce small world networks and less tightly linked models we produce both trees appropriate in certain biological applications and networks in which closed loops can appear which model communication networks and networks of human sexual interactions one of our models is closely related to random recursive trees and some exact results known in that context can be exploited the other models are more subtle and difficult to analyze our analysis includes a number of exact results for moments correlations and distributions of coordination number and network size we report simulations and also discuss some mean field approximations if the system has evolved for a long time and the state of a random node which thus has a random age is observed power law distributions for properties of the system arise in some of these models

on periodical trajectories of the billiard systems within an ellipsoid in rd and generalized cayley s condition the analytical condition for a trajectory of the billiard system within an ellipsoid of any dimension is given an interesting property is obtained every periodical trajectory with the period less than the dimension of the space is placed in a coordinate hyperplane

trajectory tracking for a particle in elliptical billiards an infinitely rigid unitary mass particle is considered moving on a planar region delimited by a rigid elliptical barrier elliptical billiards under the action of proper control forces a class of periodic trajectories involving an infinite sequence of non smooth impacts between the mass and the barrier at fixed times is found by using an lmis based procedure the jumps in the velocities at the impact times render difficult if not impossible to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour hence the tracking control problem is properly stated using notions similar to the quasi stability concept in v lakshmikantham d d bainov and p s simeonov theory of impulsive differential equations 6 world scientific 1989 a controller whose state is subject to discontinuities based on the internal model principle is shown to solve the proposed tracking problem giving rise to control forces that are piecewise continuous fun

elliptical billiards and poncelet s theorem two and three dimensional billiard systems with elliptical and ellipsoidal boundaries respectively are studied it is known that the trajectory of such a two dimensional system generates a caustic conic curve many properties of the elliptical billiard system in the language of projective geometry can be described one of these properties is that if a trajectory is closed after p bounces then all trajectories sharing the same caustic conic are also closed after p bounces in projective geometry this is known as poncelet s theorem many of the two dimensional results are extended to three dimensions in particular it is shown that all straight segments of a trajectory in a three dimensional ellipsoidal billiard system are always tangent to two caustic quadrics if a trajectory is closed after p bounces then all trajectories sharing the same two caustic quadrics are also closed after p bounces a generalized poncelet s theorem in three dimensions is thus established on the basis of numerical studies it is conjectured that this generalized poncelet s theorem is also valid for an arbitrary finite number field the implications of the results are discussed and their extension to n dimensions is outlined

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

cayley type conditions for billiards within k quadrics in bbb rd in section 4 of this paper the authors misstated theorems 3 and 4 please see pdf for corrected theorems

a general geometric growth model for pseudofractal scale free web abstract we propose a general geometric growth model for pseudofractal scale free web psw which is controlled by two tunable parameters we derive exactly the main characteristics of the networks degree distribution second moment of degree distribution degree correlations distribution of clustering coefficient as well as the diameter which are partially determined by the parameters analytical results show that the resulting networks are disassortative and follow power law degree distributions with a more general degree exponent tuned from 2 to 1 ln 3 ln 2 the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order the diameter grows logarithmically with the number of network nodes all these reveal that the networks described by our model have small world effect and scale free topology

shortest paths and load scaling in scale free trees the average node to node distance of scale free graphs depends logarithmically on n the number of nodes while the probability distribution function of the distances may take various forms here we analyze these by considering mean field arguments and by mapping the m 1 case of the barabasi albert model into a tree with a depth dependent branching ratio this shows the origins of the average distance scaling and allows one to demonstrate why the distribution approaches a gaussian in the limit of n large the load the number of the shortest distance paths passing through any node is discussed in the tree presentation

elliptical billiards and hyperelliptic functions the geometrical properties of the elliptical billiard system are related to poncelet s theorem this theorem states that if a polygon is inscribed in a conic and circumscribed about a second conic every point of the former conic is a vertex of a polygon with the same number of sides and the same perimeter chang and friedberg have extended this theorem to three and higher dimensions they have shown that the geometrical properties of the hyperelliptic billiard system are related to the algebraic character of a poincare map in the phase space the geometrical and algebraic properties of the system can be understood in terms of the analytical structure of the equations of motion these equations form a complete system of abelian integrals the integrability of the physical system is reflected by the topology of the riemann surfaces associated to these integrals the algebraic properties are connected with the existence of addition formulas for hyperelliptic functions the main purpose of this study is to establish such a connection and to provide an algebraic proof of poncelet s theorem in three and higher dimensions

marginally trapped submanifolds in lorentzian space forms and in the lorentzian product of a space form by the real line we give local explicit representation formulas for n dimensional spacelike submanifolds which are marginally trapped in the minkowski space r1n 2 the de sitter space d x1d54a n 2 the anti de sitter space ad x1d54a n 2 and the lorentzian products x1d54a n 1 and n 1 of the sphere and the hyperbolic space by the real line

mixed type surfaces with bounded gaussian curvature in three dimensional lorentzian manifolds abstract a mixed type surface is a connected regular surface in a lorentzian 3 manifold with non empty spacelike and timelike point sets the induced metric of a mixed type surface is a signature changing metric and their lightlike points may be regarded as singular points of such metrics in this paper we investigate the behavior of gaussian curvature at a non degenerate lightlike point of a mixed type surface to characterize the boundedness of gaussian curvature at a non degenerate lightlike points we introduce several fundamental invariants along non degenerate lightlike points such as the lightlike singular curvature and the lightlike normal curvature moreover using the results by pelletier and steller we obtain the gauss bonnet type formula for mixed type surfaces with bounded gaussian curvature

quantitatively investigating the locally weak stationarity of modified multifractional gaussian noise a b s t r a c t we suggest that there exists a critical point h 0 70 of the local holder exponent h t for describing the weak stationary stationary for short property of the modified multifractional gaussian noise mmgn from the point of view of engineering more precisely when h t 0 70 for t 0 the stationarity of mmgn is conditional relying on the variation ranges of h t when h t 0 70 on the other side mmgn is unconditionally stationary yielding a consequence that short memory mmgn is stationary in addition for h t 0 70 we introduce the concept of stationary range denoted by hmin hmax it means that corr r h t1 r h t2 0 70 if h t1 h t2 hmin hmax where r h t1 and r h t2 are the autocorrelation functions of mmgn with h t1 and h t2 for t1 t2 respectively and corr r h t1 r h t2 is the correlation coefficient between r h t1 and r h t2 we present a set of stationary ranges which may be used for a quantitative description of the local stationarity of mmgn a case study is demonstrated for applying the present method to testing the stationarity of a real traffic trace

symplectic flow for the square root of the negative laplacian abstract the paper is concerned with symplectic flow for the square root of the negative laplacian which is motivated when studying a one dimensional model about ferromagnetic thin film without gilbert damping term only gyromagnetic term remained derived from the stationary model of a desimone r v kohn s muller and f otto existence of weak solution to the model is proved by viscosity approximation and commutator estimate

elliptical billiard systems and the full poncelet s theorem in n dimensions in this work is presented a generalization of poncelet s theorem to n dimensions which is refered to as the full poncelet s theorem the theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon then there exists an n 1 parameter family of polygons having the same property a physical realization and a projective geometrical proof of this theorem are given if all the reflecting quadrics coincide the above theorem reduces to the n dimensional poncelet s theorem presented by chang and friedberg the geometrical proof is a finite construction based on a preliminary theorem which extends hart s lemma the full poncelet s theorem may thus be extended to projective geometries over most fields including discrete ones

global solutions of a dynamical equation in ferrimagnet abstract we consider global solutions of a dynamical equation in ferrimagnet we show that it admits a global weak solution by using the penalty method by the energy estimates method we show there exists a unique global smooth solution finally we establish the relationship between this equation and wave maps

numerical analysis of an explicit approximation scheme for the landau lifshitz gilbert equation the landau lifshitz gilbert equation describes magnetic behavior in ferromagnetic materials construction of numerical strategies to approximate weak solutions for this equation is made difficult by its top order nonlinearity and nonconvex constraint in this paper we discuss necessary scaling of numerical parameters and provide a refined convergence result for the scheme first proposed by alouges and jaisson 2006 as an application we numerically study discrete finite time blowup in two dimensions

traveling wave solutions of the schrodinger map equation we first construct traveling wave solutions for the schrodinger map in r 2 m t m m m 3 e 3 in 2 of the form m x 1 x 2 t where m has exactly two vortices at approximately 1 2 0 r 2 of degree 1 we use a perturbative approach that gives a complete characterization of the asymptotic behavior of the solutions with a few modifications a similar construction yields traveling wave solutions of schrodinger map equations in higher dimensions 2010 wiley periodicals inc

random walks and diameter of finite scale free networks abstract dynamical scalings for the end to end distance r e e and the number of distinct visited nodes n v of random walks rws on finite scale free networks sfns are studied numerically r e e shows the dynamical scaling behavior r e e l t l n g t l z where l is the average minimum distance between all possible pairs of nodes in the network n is the number of nodes is the degree exponent of the sfn and t is the step number of rws especially r e e l t in the limit t satisfies the relation r e e l d where d is the diameter of network with d l ln n for 3 or d l ln ln n for 3 based on the scaling relation r e e we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using rws

conditions of cayley s type for ellipsoidal billiard the billiard system within an ellipsoid in three dimensional space is considered we describe periodical trajectories of that mechanical system deriving analytical condition of cayley s type for generalized poncelet s theorem using methods of algebrogeometric integration

global weak solutions of the fractional landau lifshitz maxwell equation abstract the fractional landau lifshitz maxwell equation is considered in this paper we show the global existence of suitable weak solutions by galerkin method and the vanishing viscosity method the main difficulties in this study are due to the loss of compactness of this system and the fact that the nonlinear term is nonlocal and of the same order of the equation to overcome these difficulties we introduce the commutator estimates and some cancellation properties to this equation which prove to be proper tools in the study of landau lifshitz type equations

on integrable potential perturbations of the billiard system within an ellipsoid generalizing kozlov s method of integrable potential perturbations for systems with three degrees of freedom new integrable billiard systems within ellipsoid in r3 are constructed two countable families of the basic solutions in the class of the laurent polynomials are obtained explicit formulas are given

learning by natural gradient on noncompact matrix type pseudo riemannian manifolds this paper deals with learning by natural gradient optimization on noncompact manifolds in a riemannian manifold the calculation of entities such as the closed form of geodesic curves over noncompact manifolds might be infeasible for this reason it is interesting to study the problem of learning by optimization over noncompact manifolds endowed with pseudo riemannian metrics which may give rise to tractable calculations a general theory for natural gradient based learning on noncompact manifolds as well as specific cases of interest of learning are discussed

geometrization and generalization of the kowalevski top a new view on the kowalevski top and the kowalevski integration procedure is presented for more than a century the kowalevski 1889 case has attracted full attention of a wide community as the highlight of the classical theory of integrable systems despite hundreds of papers on the subject the kowalevski integration is still understood as a magic recipe an unbelievable sequence of skillful tricks unexpected identities and smart changes of variables the novelty of our present approach is based on our four observations the first one is that the so called fundamental kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the kowalevski variables w x 1 x 2 as the pencil parameter and the darboux coordinates respectively the second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials all steps of the kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials the third observation connects the kowalevski integration and the pencil equation with the theory of multi valued groups the kowalevski change of variables is now recognized as an example of a two valued group operation and its action the final observation is surprising equivalence of the associativity of the two valued group operation and its action to the n 3 case of the great poncelet theorem for pencils of conics