random walks on combs we develop techniques to obtain rigorous bounds on the behaviour of random walks on combs using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length we also calculate exactly the spectral dimension of some fixed non translationally invariant combs we relate the spectral dimension to the critical exponent of the mass of the two point function for random walks on random combs and compute mean displacements as a function of walk duration we prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension

periodic response of piecewise non linear oscillators under harmonic excitation the method of equivalent linearization has been extended to obtain periodic responses of harmonically excited piecewise non linear oscillators a dual representation of the solution is used to greatly enhance the algebraic simplicity the stability analysis of the solutions so obtained is carried out by the method of error propagation these different systems having piecewise non linearity are considered numerical results are compared with those obtained from the simple harmonic balance method and direct numerical integration the proposed method not only gives better results than harmonic balance but is also capable of including super and subharmonics

consistency of spectral clustering consistency is a key property of all statistical procedures analyzing randomly sampled data surprisingly despite decades of work little is known about consistency of most clustering algorithms in this paper we investigate consistency of the popular family of spectral clustering algorithms which clusters the data with the help of eigenvectors of graph laplacian matrices we develop new methods to establish that for increasing sample size those eigenvectors converge to the eigenvectors of certain limit operators as a result we can prove that one of the two major classes of spectral clustering normalized clustering converges under very general conditions while the other unnormalized clustering is only consistent under strong additional assumptions which are not always satisfied in real data we conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

the spectral dimension of random trees we present a simple yet rigorous approach to the determination of the spectral dimension of random trees based on the study of the massless limit of the gaussian model on such trees as a by product we obtain evidence in favour of a new scaling hypothesis for the gaussian model on generic bounded graphs and in favour of a previously conjectured exact relation between spectral and connectivity dimensions on more general tree like structures

symmetric submodular system contractions and gomory hu tree abstract let s v f be a symmetric submodular system for two distinct elements s and l of v let s l denote the set of all subsets of v which separate s from l by using any gomory hu tree of s we can obtain an element of s l which has minimum value among all the elements of s l this tree can be constructed iteratively by solving v 1 minimum s l separator problem pendant pairs of a symmetric submodular system play a key role in finding a minimizer of this system in this paper we obtain a gomory hu tree of a contraction of s with respect to some subsets of v only by using contraction in a gomory hu tree of s furthermore we obtain some pendant pairs of s and of its contractions by using a gomory hu tree of s

on the complexity of computing the k restricted edge connectivity of a graph the k restricted edge connectivity of a graph g denoted by k g is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices this graph invariant which can be seen as a generalization of a minimum edge cut has been extensively studied from a combinatorial point of view however very little is known about the complexity of computing k g very recently in the parameterized complexity community the notion of good edge separation of a graph has been defined which happens to be essentially the same as the k restricted edge connectivity motivated by the relevance of this invariant from both combinatorial and algorithmic points of view in this article we initiate a systematic study of its computational complexity with special emphasis on its parameterized complexity for several choices of the parameters we provide a number of np hardness and w 1 hardness results as well as fpt algorithms

scale free random branching trees in supercritical phase we study the size and the lifetime distributions of scale free random branching trees in which k branches are generated from a node at each time step with probability qk k in particular we focus on finite size trees in a supercritical phase where the mean branching number c k kqk is larger than 1 the tree size distribution p s exhibits a crossover behaviour when 2 3 a characteristic tree size sc exists such that for

on the growth of bounded trees bounded infinite graphs are defined on the basis of natural physical requirements when specialized to trees this definition leads to a natural conjecture that the average connectivity dimension of bounded trees cannot exceed two we verify that this bound is saturated by a class of random trees in which case we also derive explicit expressions for the growth probabilities

use of transformations with the higher order method of multiple scales to determine the steady state periodic response of harmonically excited non linear oscillators part i transformation of derivative abstract use of the higher order method of multiple scales with reconstitution to determine the periodic steady state response of harmonically excited non linear oscillators is considered in this method frequency detuning can be introduced in several different ways the main objective of this work is to compare the second order results determined by introducing detuning in the square of the excitation frequency against the second order results determined by introducing detuning in the excitation frequency itself a modified reconstitution procedure is used to uncover the relative approximations used in these second order solutions the well known primary resonance in the duffing oscillator is used as a prototype example and it is shown that the combined effects of using transformed time and introducing a detuning parameter in the square of the excitation frequency can lead to a non uniform expansion within the excitation frequency band in which large amplitude primary response is excited

hierarchical inference of unicast network topologies based on end to end measurements in this paper we address the problem of topology discovery in unicast logical tree networks using end to end measurements without any cooperation from the internal routers topology estimation can be formulated as hierarchical clustering of the leaf nodes based on pairwise correlations as similarity metrics unlike previous work that first assumes the network topology is a binary tree and then tries to generalize to a nonbinary tree we provide a framework that directly deals with general logical tree topologies a hierarchical algorithm to estimate the topology is developed in a recursive manner by finding the best partitions of the leaf nodes level by level our simulations show that the algorithm is more robust than binary tree based methods

a classical perturbation technique which is valid for large parameters there are various perturbation techniques for constructing analytical approximations to the oscillatory solutions of second order non linear differential equations 1 2 but many of them apply to weakly non linear cases only to overcome the limitations many novel techniques have been proposed in recent years for example cheung et al 3 proposed a modified lindstedt poincare method and lim et al 4 presented a modified mickens procedure for certain non linear oscillators recently he 5 proposed a perturbation technique which is valid for large parameters but unfortunately o1 and b in ref 5 are the functions of z which is unknown the main purpose of this paper is to point out that there already exists a classical perturbation technique which is valid for large parameters in this paper the following well known duffing equation is considered

i o efficient algorithms for the minimum cut problem on unweighted undirected graphs the problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model first a lower bound of e v sort v on the number of i os is shown for the problem where v is the number of vertices and e is the number of edges then the following are presented for m b 2 1 a minimum cut algorithm that uses o c log e rm msf v e frac v b rm sort v i os here msf v e is the number of i os needed to compute a minimum spanning tree of the graph and c is the value of the minimum cut the algorithm performs better on dense graphs than the algorithm of 7 which requires o e c 2 v log v c i os when executed on the external memory model for a fat graph for 0 the maximum tree packing of the graph is at least 1 c 2 our algorithm computes a minimum cut in o c loge msf v e sort e i os 2 a randomized algorithm that computes minimum cut with high probability in o c log e cdot rm msf v e rm sort e log 2 v frac v b rm sort v log v i os 3 a 2 e minimum cut algorithm that requires o e v msf v e i os and performs better on sparse graphs than our exact minimum cut algorithm

weight reduction problems with certain bottleneck objectives abstract this paper is concerned with bottleneck weight reduction problems wrps stated as follows we are given a finite set e a class f of nonempty subsets of e a weight w e r and a cost c e r for each e e c e stands for the cost of reducing weight w e by one unit for each subset f f the bottleneck weight of f is w f min e f w e the weight of the family f is the maximum of w f for all f in f the problem is to determine new weights x e w e such that the weight of f is minimized under the constraint that the overall reduction cost does not exceed a given budget b similarly to capacity expansion problems wrps include np hard problems a wrp can be formulated as a parametric optimization problem over all transversal sets t of the class f this leads to strongly polynomial solution procedures for special systems f in particular we outline a polynomial algorithm in the case when f is the class of all spanning trees in an undirected graph

analytical approximation of the primary resonance response of a periodically excited piecewise non linear linear oscillator an analytical approximate solution is constructed for the primary resonance response of a periodically excited non linear oscillator which is characterized by a combination of a weakly non linear and a linear differential equation without eliminating the secular terms a valid asymptotic expansion solution for the weakly non linear equation is analytically determined for the case of primary resonances then a symmetric periodic solution for the overall system is obtained by imposing continuity and matching conditions the stability characteristic of the symmetric periodic solution is investigated by examining the asymptotic behaviour of perturbations to the steady state solution the validity of the developed analysis is highlighted by comparing the first order approximate solutions with the results of numerical integration of the original equations

a perturbation technique that works even when the non linearity is not small abstract we extend the lindstedt poincare small non linearity based perturbation scheme to strongly non linear systems the extended technique starts from a physically non existent neighboring linear system this system is created by adding an optimal linear spring term to the system side of each linear differential equation starting at the zeroth level and balancing that term by adding an equal one to the forcing side one level down unlike the ritz method which also works for strongly non linear systems results can be obtained beyond the first level without resorting to numerical techniques thus the extended method keeps the intuition guiding straightforwardness of the perturbation method while giving accurate beyond first level hand derivable numerical results examples are presented for free vibrations with cubic and antisymmetric quadratic non linearities and for harmonic solutions of the undamped duffing equation the forms of the solutions allow the regions of validity of the expansions to be estimated an internal method of estimating accuracy of the perturbation solutions is developed and checked with the free vibration examples the exact solutions of which are available this internal method is then used to estimate the accuracy of the new solutions to the duffing equation these new duffing equation solutions give accurate results in all regions where stable harmonic solutions are known to exist in particular they give accurate results at and near the vertical tangent to the resonance curve a region where hayashi s amplitude expansion based perturbation solution necessarily breaks down

graph theory applied to noise and vibration control in statistical energy analysis models a fundamental aspect of noise and vibration control in statistical energy analysis sea models consists in first identifying and then reducing the energy flow paths between subsystems in this work it is proposed to make use of some results from graph theory to address both issues on the one hand linear and path algebras applied to adjacency matrices of sea graphs are used to determine the existence of any order paths between subsystems counting and labeling them finding extremal paths or determining the power flow contributions from groups of paths on the other hand a strategy is presented that makes use of graph cut algorithms to reduce the energy flow from a source subsystem to a receiver one modifying as few internal and coupling loss factors as possible

dynamic stability of weakly damped oscillators with elastic impacts and wear the dynamics of non linear oscillators comprising of a single degree of freedom system and beams with elastic two sided amplitude constraints subject to harmonic loads is analyzed the beams are clamped at one end and constrained against unilateral contact sites near the other end the structures are modelled by a bernoulli type beam supported by springs using the finite element method rayleigh damping is assumed symmetric and elastic double impact motions both harmonic and sub harmonic are studied by way of a poincare mapping that relates the states at subsequent impacts stability and bifurcation analyses are performed for these motions and domains of instability are delineated impact work rate which is the rate of energy dissipation to the impacting surfaces is evaluated and discussed in addition an experiment conducted by moon and shaw on the vibration of a cantilevered beam with one sided amplitude constraining stop is modelled bifurcation observed in the experiment could be captured

periodic forced vibration of unsymmetrical piecewise linear systems by incremental harmonic balance method abstract a single degree of freedom non linear system subjected to periodic excitation is considered the non linearity is in the restoring force and is represented by unsymmetrical piecewise linear stiffness the steady state response are analysed by an extended incremental harmonic balance ihb method all possible subharmonic harmonic and superharmonic resonances and bifurcations are sought the results reveal some interesting details of responses of unsymmetrical piecewise linear systems comparisons with available published findings are also made

spectral dimension of fractal trees we present in detail a calculation of the spectral dimension for a class of fractal trees called mathrm nt mathit d i e nice trees of dimension d defined as trees whose branches are splitting in r every time the distance from the origin is doubled where r is an integer greater than 1 which presents nonanomalous diffusion this is performed by an analytical technique related to an exact rescaling of the time variable which can be extended to more general geometrical structures

the effect of frequency and clearance variations on single degree of freedom impact oscillators abstract a simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro impacting systems the presence of impacts makes the system discontinuous and non linear and to study it effectively the theory of dynamical systems is used by a combination of analytical and numerical techniques many phenomena are identified which are characteristic of non linear dynamical systems as the system parameters are varied in particular these include chaotic regimes and multiply coexisting stable states the domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states in addition phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems chatter when infinitely many impacts occur in a finite time and grazing when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters

constrained clustering by constraint programming cluster analysis is an important task in data mining with hundreds of different approaches in the literature since the last decade the cluster analysis has been extended to constrained clustering also called semi supervised clustering so as to integrate previous knowledge on data to clustering algorithms in this dissertation we explore constraint programming cp for solving the task of constrained clustering the main principles in cp are 1 users specify declaratively the problem in a constraint satisfaction problem 2 solvers search for solutions by constraint propagation and search relying on cp has two main advantages the declarativity which enables to easily add new constraints and the ability to find an optimal solution satisfying all the constraints when there exists one we propose two models based on cp to address constrained clustering tasks the models are flexible and general and supports instance level constraints and different cluster level constraints it also allows the users to choose among different optimization criteria in order to improve the efficiency different aspects have been studied in the dissertation experiments on various classical datasets show that our models are competitive with other exact approaches we show that our models can easily be embedded in a more general process and we illustrate this on the problem of finding the pareto front of a bi criterion optimization process

periodic and chaotic responses of an sdf system with piecewise linear stiffness subjected to combined harmonic and flow induced excitations the response of a single degree of freedom sdf vibrating system with unsymmetrical piecewise linear stiffness subjected to combined harmonic and flow induced excitations is investigated motion limiting stops different tension and compression behavior etc may introduce an unsymmetrical piecewise linear stiffness characteristic a multi harmonic balance cum newton raphson procedure in conjunction with an fft algorithm is adopted to determine the stable and unstable periodic solutions the stability of the periodic solutions is investigated by using floquet theory digital simulation results reveal periodic quasi periodic and chaotic motions of the system in a range of flow velocities mode locked oscillations with period 5 motions are found to occur in certain range of flow velocities bifurcation diagrams and lyapunov exponents are also presented

diffuse interface models on graphs for classification of high dimensional data this paper is a republication of an mms paper a l bertozzi and a flenner multiscale model simul 10 2012 pp 1090 1118 describing a new class of algorithms for classification of high dimensional data these methods combine ideas from spectral methods on graphs with nonlinear edge region detection methods traditionally used in the pde based imaging community the algorithms use the ginzburg landau functional extended to a graphical framework which has classical pde connections to total variation minimization convex splitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graph theoretic problems we review the diverse computational examples presented in the original paper involving both basic clustering and semisupervised learning for different applications case studies include feature identification in images segmentation in social networks and segmentation of shapes in high dimensional datasets since the pape

exact solution of stable periodic one contact per n cycles motion of a damped linear oscillator contacting a unilateral elastic stop abstract an exact closed form solution is developed for a damped single degree of freedom oscillator contacting a unilateral stop once per n cycles of external sinusoidal force it also includes the effect of a bias force this solution was obtained by using two different displacement and velocity expressions for a two region piecewise linear oscillator continuity of displacement and velocity at the boundary of the two regions and periodicity conditions contact duration is assumed as known and four simultaneous equations in four unknowns are solved exactly the four unknowns are the amplitude of the sinusoidal force phase angle of the assumed first contact entry velocity and exit velocity of mass the stability of motion is checked using the closed form expressions of elements of a 2 2 stability governing matrix these elements are obtained by considering only first order perturbations in periodic motion theoretical predictions based on exact solution agree with previous results and with results obtained using a numerical simulation approach the relationships are studied in detail between input parameters such as amplitude of external force frequency of external force bias force gap spring stiffness and damping constants and output parameters such as contact duration phase angle at contact entry velocity exit velocity maximum displacement and minimum displacement

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs we consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded rank hypergraphs with multiple edge cost functions for a fixed number of edge cost functions we show that the total number of supported non dominated snd cuts is bounded by a polynomial in the numbers of nodes and edges i e is strongly polynomial this bound also applies to the combinatorial facet complexity of the problem i e the maximum number of facets linear pieces of the parametric curve for the parametrized linear combination objective over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive we sharpen this bound in the case of two objectives the bicriteria problem for which we also derive a strongly polynomial upper bound on the total number of non dominated pareto optimal cuts in particular the bicriteria global minimum cut problem in an n node graph admits o n 3 log n o n3logn snd cuts and o n 5 log n o n5logn pareto optimal cuts these results significantly improve on earlier graph cut results by mulmuley siam j comput 28 4 1460 1509 1999 and armon and zwick algorithmica 46 1 15 26 2006 they also imply that the parametric curve and all snd cuts and for the bicriteria problems all pareto optimal cuts can be computed in strongly polynomial time when the number of objectives is fixed

sparse spectral clustering method based on the incomplete cholesky decomposition a novel sparse spectral clustering method using linear algebra techniques is proposed spectral clustering methods solve an eigenvalue problem containing a graph laplacian the proposed method exploits the structure of the laplacian to construct an approximation not in terms of a low rank approximation but in terms of capturing the structure of the matrix with this approximation the size of the eigenvalue problem can be reduced to obtain the indicator vectors from the eigenvectors the method proposed by zha et al 2002 26 which computes a pivoted lq factorization of the eigenvector matrix is adapted this formulation also gives the possibility to extend the method to out of sample points

stability and dynamics of a harmonically excited elastic perfectly plastic oscillator this paper deals with the stability and the dynamics of a harmonically excited elastic perfectly plastic oscillator the hysteretical system is written as a non smooth forced autonomous system it is shown that the dimension of the phase space can be reduced using adapted variables free vibrations of such a system are then considered for the damped system the extended direct method of liapounov is applied to this non smooth mechanical system and the asymptotic stability of the origin is proven in the new phase space the forced vibration of such an oscillator is treated by numerical approach by using the time locating techniques the stability of the limit cycles is analytically investigated with a perturbation approach the boundary between elastoplastic shakedown and alternating plasticity is given in closed form it is shown that this boundary corresponds to a bifurcation boundary for the undamped system period doubling bifurcation finally the equivalent damping of this hysteretic system is characterized from dynamical properties

dirac delta representation by exact parametric equations application to impulsive vibration systems a pair of closed parametric equations are proposed to represent the heaviside unit step function differentiating the step equations results in two additional parametric equations that are also hereby proposed to represent the dirac delta function these equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus the proposed delta representation complies exactly with the values of the definition it complies also with the sifting property and the requisite unit area and its laplace transform coincides with the most general form given in the tables furthermore it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems two example solutions are presented

deterministic edge connectivity in near linear time we present a deterministic algorithm that computes the edge connectivity of a graph in near linear time this is for a simple undirected unweighted graph g with n vertices and m edges this is the first o mn time deterministic algorithm for the problem our algorithm is easily extended to find a concrete minimum edge cut in fact we can construct the classic cactus representation of all minimum cuts in near linear time the previous fastest deterministic algorithm by gabow from stoc 91 took o m 2 n where is the edge connectivity but can be as big as n 1 karger presented a randomized near linear time monte carlo algorithm for the minimum cut problem at stoc 96 but the returned cut is only minimum with high probability our main technical contribution is a near linear time algorithm that contracts vertex sets of a simple input graph g with minimum degree producing a multigraph with o m edges which preserves all minimum cuts of g with at least two vertices on each side in our deterministic near linear time algorithm we will decompose the problem via low conductance cuts found using pagerank a la brin and page 1998 as analyzed by andersson chung and lang at focs 06 normally such algorithms for low conductance cuts are randomized monte carlo algorithms because they rely on guessing a good start vertex however in our case we have so much structure that no guessing is needed

on the approximate solution of a piecewise nonlinear oscillator under super harmonic resonance abstract an approximate solution for the super harmonic resonance response of a periodically excited nonlinear oscillator with a piecewise nonlinear linear characteristic is constructed using both a matching method and a modified averaging method the validity of the developed analysis is confirmed by comparing the approximate solutions with the results of direct numerical integration of the original equation

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

exact single equation closed form solution of vibrating systems with piecewise linear springs abstract a method is proposed to obtain in a single equation the exact closed form displacement response of both free undamped vibrations and the steady state of forced damped vibrations of systems with piecewise linear springs with continuous or discontinuous force deflection relations since the displacement equations are exact the velocity and acceleration responses are obtained by differentiation the solutions are complete they apply for any value of time from zero to infinity the method is designed to be handled in a computer either in conjunction with a symbolic mathematics package like mathematica or maple or without it however for certain relatively simple problems the phase plane solution may be obtained even in a hand held calculator two examples are presented

fractality in complex networks critical and supercritical skeletons fractal scaling char22 a power law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box char22 is studied we introduce a box covering algorithm that is a modified version of the original algorithm introduced by song et al nature london 433 392 2005 this algorithm enables easy implementation fractal networks are viewed as comprising a skeleton and shortcuts the skeleton embedded underneath the original network is a special type of spanning tree based on the edge betweenness centrality it provides a scaffold for the fractality of the network when the skeleton is regarded as a branching tree it exhibits a plateau in the mean branching number as a function of the distance from a root for nonfractal networks on the other hand the mean branching number decays to zero without forming a plateau based on these observations we construct a fractal network model by combining a random branching tree and local shortcuts the scaffold branching tree can be either critical or supercritical depending on the small worldness of a given network for the network constructed from the critical supercritical branching tree the average number of vertices within a given box grows with the lateral size of the box according to a power law an exponential form in the cluster growing method the critical and supercritical skeletons are observed in protein interaction networks and the world wide web respectively the distribution of box masses i e the number of vertices within each box follows a power law p m m ensuremath sim m ensuremath ensuremath eta the exponent ensuremath eta depends on the box lateral size ensuremath ell b for small values of ensuremath ell b ensuremath eta is equal to the degree exponent ensuremath gamma of a given scale free network whereas ensuremath eta approaches the exponent ensuremath tau ensuremath gamma ensuremath gamma ensuremath 1 as ensuremath ell b increases which is the exponent of the cluster size distribution of the random branching tree finally we study the perimeter h ensuremath alpha of a given box ensuremath alpha i e the number of edges connected to different boxes from a given box ensuremath alpha as a function of the box mass m b ensuremath alpha it is obtained that the average perimeter over the boxes with box mass m b is likely to scale as h m b ensuremath sim m b irrespective of the box size ensuremath ell b

an investigation of stability in the large behaviour of a control surface with structural non linearities in supersonic flow it is well known that the presence of non linearities may significantly affect the aeroelastic response of an aerospace vehicle structure in this paper the aeroelastic behaviour at high mach numbers of an all moving control surface with a non linearity in the root support is investigated very often a stable equilibrium point corresponding to zero displacement of the structure together with an unstable limit cycle arising from a sub critical hopf bifurcation results from the presence of the non linearity the stable equilibrium point will then possess a domain of attraction in this paper this situation is investigated by first applying the averaging method to obtain a new set of aeroelastic equations in which the limit cycle is replaced by an unstable equilibrium point a fourth order power series approximation to the stable manifold in the neighbourhood of this equilibrium point is then determined from the stable manifold predictions of the domain of attraction of the stable equilibrium point may then be made the method is applied to two examples in which the non linearity in the root support was due to either a cubic hardening restoring moment or the presence of freeplay the approximation to the stable manifold was sufficient to enable significant information about the domain of attraction of the stable equilibrium point of the control surface to be obtained agreement with predictions from numerical integration of the aeroelastic equations in the time domain was shown to be generally good in the cases considered though outside the region of validity of the stable manifold expansion discrepancies will occur the averaging method was shown to be sufficiently accurate for this analysis even when the non linearities could not be considered as weak

periodic solutions of a multi dof beam system with impact the steady state behaviour is analyzed of a periodically driven multi dof beam system which has an elastic stop at its middle the elastic stop is modelled in a continuous way by using the contact law of hertz the beam is modelled by using finite elements and subsequently reduced by using a component mode synthesis method the steady state behaviour of the system reduced to one two and four dof s is investigated by calculating periodic solutions at varying excitation frequency periodic solutions are calculated by solving two point boundary value problems by the multiple shooting method in combination with a path following technique it is shown that models with more than one dof are required for a good assessment of the long term behaviour of the system

perturbation analysis of a clearance type nonlinear system abstract this study applies the method of multiple scales to obtain periodic solutions of a two pulley belt system with clearance type nonlinearity the purpose is to explain the published numerical results and clarify how design parameters affect the system dynamics the validity of the perturbation method for such strong nonlinearity is evaluated the closed form frequency response relation is determined at the first order and an implicit expression is obtained for the second order approximation the preload applied to the accessory determines the softening level of the nonlinearity larger preload leads to less disengagement and less softening for a considerable range of practical parameter values the analytical solutions well approximate the numerical results from harmonic balance

application of non smooth modelling techniques to the dynamics of a flexible impacting beam non smooth modelling techniques have been successfully applied to lumped mass type structures for modelling phenomena such as vibro impact and friction oscillators in this paper the application of these techniques to continuous elements using the example of a cantilever beam is considered employing a galerkin reduction to form an n degree of freedom modal model a technique for modelling impact phenomena using a non smooth dynamics approach is demonstrated numerical simulations computed using the non smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side a method for dealing with sticking motions when numerically simulating the beam motion is presented in addition choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

the statistical geometry of scale free random trees the properties of random trees galton watson trees with scale free power like probability distribution of coordinations are investigated in the thermodynamic limit the scaling form of volume probability is found and the connectivity dimensions are determined and compared with other exponents which describe the growth the local spectral dimension is also determined through the study of the massless limit of the gaussian model on such trees

clustering by minimum cut hyperplanes minimum normalised graph cuts are highly effective ways of partitioning unlabeled data having been made popular by the success of spectral clustering this work presents a novel method for learning hyperplane separators which minimise this graph cut objective when data are embedded in euclidean space the optimisation problem associated with the proposed method can be formulated as a sequence of univariate subproblems in which the optimal hyperplane orthogonal to a given vector is determined these subproblems can be solved in log linear time by exploiting the trivial factorisation of the exponential function experimentation suggests that the empirical runtime of the overall algorithm is also log linear in the number of data asymptotic properties of the minimum cut hyperplane both for a finite sample and for an increasing sample assumed to arise from an underlying probability distribution are discussed in the finite sample case the minimum cut hyperplane converges to the maximum margin hyperplane as the scaling parameter is reduced to zero applying the proposed methodology both for fixed scaling and the large margin asymptotes is shown to produce high quality clustering models in comparison with state of the art clustering algorithms in experiments using a large collection of benchmark datasets

consensus and products of random stochastic matrices exact rate for convergence in probability we find the exact rate for convergence in probability of products of independent identically distributed symmetric stochastic matrices it is well known that if the matrices have positive diagonals almost surely and the support graph of the mean or expected value of the random matrices is connected the products of the matrices converge almost surely to the average consensus matrix and thus in probability in this paper we show that the convergence in probability is exponentially fast and we explicitly characterize the exponential rate of this convergence our analysis reveals that the exponential rate of convergence in probability depends only on the statistics of the support graphs of the random matrices further we show how to compute this rate for commonly used random models gossip and link failure with these models the rate is found by solving a min cut problem and hence it is easily computable finally as an illustration we apply our results to solving power allocation among networked sensors in a consensus innovations distributed detection problem