approximation of minimum weight spanners for sparse graphs a t spanner of a graph g is its spanning subgraph s such that the distance between every pair of vertices in s is at most t times their distance in g the sparsestt spanner problem asks to find for a given graph g and an integer t a t spanner of g with the minimum number of edges the problem is known to be np hard for all t 2 and even more it is np hard to approximate it with ratio o logn for every t 2 for t 5 the problem remains np hard for planar graphs and the approximability status of the problem on planar graphs was open we resolve this open issue by showing that the sparsestt spanner problem admits the efficient polynomial time approximation scheme eptas for every t 1 our result holds for a much wider class of graphs namely the class of apex minor free graphs which contains the classes of planar and bounded genus graphs moreover it is possible to extend our results to weighted apex minor free graphs when the maximum edge weight is bounded by some constant

normal forms and bifurcations of planar vector fields normal forms and bifurcations of planar vector fields by s n chow c li and d wang cambridge university press cambridge and new york 1994 viii 472 pp 69 95 isbn 0 521 37226 7

on the parametric description of log growth rates of cities sizes of four european countries and the usa abstract we have studied the parametric description of the distribution of the log growth rates of the sizes of cities of france germany italy spain and the usa we have considered several parametric distributions well known in the literature as well as some others recently introduced there are some models that provide similar excellent performance for all studied samples the normal distribution is not the one observed empirically

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

a note on pareto tails and lognormal body of us cities size distribution abstract luckstead and devadoss 2017 fitted a distribution made up of lower tail pareto lognormal body and upper tail pareto to city sizes in the us here we propose a class of distributions providing better fits the fits are assessed in terms of akaike information criterion bayesian information criterion and probability plots

perturbation of homoclinics and subharmonics in duffing s equation abstract for lambda mu and g f small the equation x x 2x cubed g x lambda x mu cos t f t is considered near the separatrix s of the unperturbed equation lambda mu 0 0 for g f in a neighborhood of the zero functions a complete description is given of the bifurcation curves in lambda mu space to homoclinic points the perturbation of subharmonics outside s is also considered in a symmetric and nonsymmetric case

from entropy maximization to equality maximization gauss laplace pareto and subbotin abstract the entropy maximization paradigm of statistical physics is well known to generate the omnipresent gauss law in this paper we establish an analogous socioeconomic model which maximizes social equality rather than physical disorder in the context of the distributions of income and wealth in human societies we show that on a logarithmic scale the laplace law is the socioeconomic equality maximizing counterpart of the physical entropy maximizing gauss law and that this law manifests an optimized balance between two opposing forces i the rich and powerful striving to amass ever more wealth and thus to increase social inequality and ii the masses struggling to form more egalitarian societies and thus to increase social equality our results lead from log gauss statistics to log laplace statistics yield paretian power law tails of income and wealth distributions and show how the emergence of a middle class depends on the underlying levels of socioeconomic inequality and variability also in the context of asset prices with laplace distributed returns our results imply that financial markets generate an optimized balance between risk and predictability

a langevin approach to the log gauss pareto composite statistical structure abstract the distribution of wealth in human populations displays a log gauss pareto composite statistical structure its density is log gauss in its central body and follows power law decay in its tails this composite statistical structure is further observed in other complex systems and on a logarithmic scale it displays a gauss exponential structure its density is gauss in its central body and follows exponential decay in its tails in this paper we establish an equilibrium langevin explanation for this statistical phenomenon and show that i the stationary distributions of langevin dynamics with sigmoidal force functions display a gauss exponential composite statistical structure ii the stationary distributions of geometric langevin dynamics with sigmoidal force functions display a log gauss pareto composite statistical structure this equilibrium langevin explanation is universal as it is invariant with respect to the specific details of the sigmoidal force functions applied and as it is invariant with respect to the specific statistics of the underlying noise

transverse homoclinic orbits in periodically perturbed parabolic equations on considere le systeme autonome x ax g x ou a est un operateur sectoriel sur un espace de banach x on suppose que l equation differentielle a une solution bornee t telle que son equation variationnelle a une dichotomie exponentielle sur des demi droites on utilise la methode de lyapunov schmidt pour obtenir la fonction de bifurcation pour des solutions bornees de l equation perturbee non autonome

homoclinic finger rings in rn abstract in this paper we investigate bifurcations of a degenerate homoclinic loop in r n we prove that a homoclinic finger ring an invariant manifold of a definite dimension textured with homoclinic orbits arises from the degenerate homoclinic orbit the size of the homoclinic finger ring is decided by not only its dimension of manifold but also its width for the rise of homoclinic finger rings of different dimensions we give conditions which are proved to form bifurcation manifolds in the parameter space we further estimate the width for the homoclinic finger ring and give a method to compute the bifurcation manifolds approximately

on the emergence of a power law in the distribution of covid 19 cases the first confirmed case of coronavirus disease 2019 covid 19 in the us was reported on january 21 2020 by the end of march 2020 there were more than 180 000 confirmed cases in the us distributed across more than 2000 counties we find that the right tail of this distribution exhibits a power law with pareto exponent close to one we investigate whether a simple model of the growth of covid 19 cases involving gibrat s law can explain the emergence of this power law the model is calibrated to match i the growth rates of confirmed cases and ii the varying lengths of time during which covid 19 had been present within each county thus calibrated the model generates a power law with pareto exponent nearly exactly equal to the exponent estimated directly from the distribution of confirmed cases across counties at the end of march

the power law distribution for the income of poor households abstract this study proposes a reverse pareto model to describe the power law behavior for the lower tail data of income distribution and illustrates an application on malaysian and italian household income data a robust method based on probability integral transform statistic is used for estimating the shape parameter of the reverse pareto model to allow for the existence of outlying observations in the lower tail data besides that the optimal threshold of reverse pareto is determined by using kolmogorov smirnov statistic it is found that the fitted reverse pareto adequately describes the lower tail data of both datasets suggesting that the power law behavior is obeyed in addition the estimated optimal threshold of reverse pareto model can be utilized as an alternative measure for the relative poverty line based on the reverse pareto model the lorenz curve gini and theil coefficients are determined and it is found that low income inequality is observed for the period of the study the fitted lorenz curve shows that nearly 80 of the total household income is owned by the bottom 80 whereas the remaining total household income is owned by the top 20 finally comparison of the reverse pareto model with some alternative distributions such as shifted reverse exponential shifted reverse stretched exponential and shifted reverse lognormal in terms of model fitting for the lower tail data is also conducted the results show that the reverse pareto model outperform all the other models

on stable pareto laws in a hierarchical model of economy abstract this study considers a model of the income distribution of agents whose pairwise interaction is asymmetric and price invariant asymmetric transactions are typical for chain trading groups who arrange their business such that commodities move from senior to junior partners and money moves in the opposite direction the price invariance of transactions means that the probability of a pairwise interaction is a function of the ratio of incomes which is independent of the price scale or absolute income level these two features characterize the hierarchical model the income distribution in this class of models is a well defined double pareto function which possesses pareto tails for the upper and lower incomes for gross and net upper incomes the model predicts definite values of the pareto exponents a gross and a net which are stable with respect to quantitative variation of the pair interaction the pareto exponents are also stable with respect to the choice of a demand function within two classes of status dependent behavior of agents linear demand a gross 1 a net 2 and unlimited slowly varying demand a gross a net 1 for the sigmoidal demand that describes limited returns a gross a net 1 with some 0 satisfying a transcendental equation the low income distribution may be singular or vanishing in the neighborhood of the minimal income in any case it is l 1 integrable and its pareto exponent is given explicitly the theory used in the present study is based on a simple balance equation and new results from multiplicative markov chains and exponential moments of random geometric progressions

how to fit the degree distribution of the air network abstract we investigate three different approaches for fitting the degree distributions of china us and the composite china us air network in order to reveal the nature of such distributions and the potential theoretical background on which they are based our first approach is the fitting with q statistics probability distribution done separately in two regimes this yields acceptable outcomes but generates two sets of fitting parameters the second approach is an entire fitting to all data points with the formula proposed by tsallis et al so far this trial is not able to produce consistent results in the third approach we fit the data with two composite distributions which may lack theoretical support for the moment

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

collective additive tree spanners of bounded tree breadth graphs with generalizations and consequences in this paper we study collective additive tree spanners for families of graphs enjoying special robertson seymour s tree decompositions and demonstrate interesting consequences of obtained results we say that a graph g admits a system of m collective additive tree r spanners resp multiplicative tree t spanners if there is a system t g of at most m spanning trees of g such that for any two vertices x y of g a spanning tree t t g exists such that d t x y d g x y r resp d t x y t d g x y when m 1 one gets the notion of additive tree r spanner resp multiplicative tree t spanner it is known that if a graph g has a multiplicative tree t spanner then g admits a robertson seymour s tree decomposition with bags of radius at most t 2 in g we use this to demonstrate that there is a polynomial time algorithm that given an n vertex graph g admitting a multiplicative tree t spanner constructs a system of at most log 2n collective additive tree o tlogn spanners of g that is with a slight increase in the number of trees and in the stretch one can turn a multiplicative tree spanner into a small set of collective additive tree spanners we extend this result by showing that if a graph g admits a multiplicative t spanner with tree width k 1 then g admits a robertson seymour s tree decomposition each bag of which can be covered with at most k disks of g of radius at most t 2 each this is used to demonstrate that for every fixed k there is a polynomial time algorithm that given an n vertex graph g admitting a multiplicative t spanner with tree width k 1 constructs a system of at most k 1 log 2n collective additive tree o tlogn spanners of g

homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations nonautomonous ordinary differential equations depending on two parameters 1 and 2 are considered in rn it is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space rn or on the dimension of intersection of the stable and unstable manifolds by means of the method of lyapunov schmidt a bifurcation function h is constructed between two finite dimensional spaces where the zeros of h correspond to homoclinic solutions at nonzero parameter values the independent variables of h consist of scalars 1 2 and a vector where is a phase angle and corresponds to directions other than along the original homoclinic solution tangent to both the stable and unstable manifolds when is fixed the equation h 0 yields in general several bifurcation curves through the origin in the 1 2 plane along which there exists a homoclinic solution when is varied these become a number of wedge shaped regions the theory is applied to two examples one in r6 where the invariant manifolds meet in dimension three and a second in r4 where these manifolds agree

a trade off between space and efficiency for routing tables two conflicting goals play a crucial role in the design of routing schemes for communication networks a routing scheme should use paths that are as short as possible for routing messages in the network while keeping the routing information stored in the processors local memory as succinct as possible the efficiency of a routing scheme is measured in terms of its stretch factor the maximum ratio between the length of a route computed by the scheme and that of a shortest path connecting the same pair of vertices most previous work has concentrated on finding good routing schemes with a small fixed stretch factor for special classes of network topologies in this paper the problem for general networks is studied and the entire range of possible stretch factors is examined the results exhibit a trade off between the efficiency of a routing scheme and its space requirements almost tight upper and lower bounds for this trade off are presented specifically it is proved that any routing scheme for general n vertex networks that achieves a stretch factor k 1 must use a total of o n 1 1 2 k 4 bits of routing information in the networks this lower bound is complemented by a family k k of hierarchical routing schemes for every k l for unit cost general networks which guarantee a stretch factor of o k require storing a total of o k 3 n 1 1 h log n bits of routing information in the network name the vertices with o log 2 n bit names and use o log n bit headers

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

beyond lognormal inequality the lorenz flow structure abstract observed from a socioeconomic perspective the intrinsic inequality of the lognormal law happens to manifest a flow generated by an underlying ordinary differential equation in this paper we extend this feature of the lognormal law to a general lorenz flow structure of lorenz curves objects that quantify socioeconomic inequality the lorenz flow structure establishes a general framework of size distributions that span continuous spectra of socioeconomic states ranging from the pure communism extreme to the absolute monarchy extreme this study introduces and explores the lorenz flow structure analyzes its statistical properties and its inequality properties unveils the unique role of the lognormal law within this general structure and presents various examples of this general structure beyond the lognormal law the examples include the inverse pareto and pareto laws which often govern the tails of composite size distributions

the pareto law of incomes an explanation and an extension abstract a stochastic model for the generation of observed income distributions is used to provide an explanation for the pareto law of incomes analysis of the model also yields a prediction of paretian power law behaviour in the lower tail of the distribution and this is shown to occur in a number of empirical distributions a tractable four parameter distribution is derived and shown to fit extremely well to a number of different empirical income distributions

approximating k spanner problems for k 2 given a graph g v e a subgraph g v h h e is a k spanner of g if for any pair of vertices u w v it satisfies dh u w kdg u w the basic k spanner problem is to find a k spanner of a given graph g with the smallest possible number of edges this paper considers approximation algorithms for this and some related problems for k 2 known to be 2log1 n inapproximable the basic k spanner problem over undirected graphs with k 2 has been given a sublinear ratio approximation algorithm with ratio roughly o n2 k 1 but no such algorithms were known for other variants of the problem including the directed and the client server variants as well as for the related k dss problem we present the first approximation algorithms for these problems with sublinear approximation ratio the second contribution of this paper is in characterizing some wide families of graphs on which the problems do admit a logarithmic and a polylogarithmic approximation ratios these families are characterized as containing graphs that have optimal or near optimal spanners with certain desirable properties such as being a tree having low arboricity or having low girth all our results generalize to the directed and the client server variants of the problems as a simple corollary we present an algorithm that given a graph g builds a subgraph with o n edges and stretch bounded by the tree stretch of g namely the minimum maximal stretch of a spanning tree for g the analysis of our algorithms involves the novel notion of edge dominating systems developed in the paper the technique introduced in the paper reduces the studied algorithmic approximability questions on k spanners to purely graph theoretical questions concerning the existence of certain combinatorial objects in families of graphs

pareto tails and lognormal body of us cities size distribution abstract we consider a distribution which consists of lower tail pareto lognormal body and upper tail pareto to estimate the size distribution of all us cities this distribution fits the data more accurately than a distribution that comprises of only lognormal and the upper tail pareto

computation of bifurcation manifolds of linearly independent homoclinic orbits abstract regarding the small perturbation as a parameter in an appropriate space of functions we can discuss co existence of homoclinic orbits for non autonomous perturbations of an autonomous system in r n and describe conditions of parameters for such degenerate homoclinic bifurcations with some bifurcation manifolds of infinite dimension since those manifolds determine the relation among parameters for such bifurcations in this paper we give an algorithm to compute approximately those manifolds and concretely obtain their first order approximates

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

exponential dichotomies heteroclinic orbits and melnikov functions the interest in the behaviour of the solutions to perturbed autonomous systems with bounded perturbative term like

linearly independent homoclinic bifurcations parameterized by a small function abstract in this paper we discuss a small nonautonomous perturbation of an autonomous system on r n which has a homoclinic solution regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co existence of homoclinic orbits those conditions of various co existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations