Function Ofn Research Articles

Lower Bounds for Maximal Matchings and Maximal Independent Sets

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of n .

Stein's Method for Mean Field Approximations in Light and Heavy Traffic Regimes

Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ= limN→ ∞ρ(N), where ρ(N)is the load of the stochastic system of sizeN.Now if ρ(N)→ 1asN→ ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions ofN.This paper first outlines an analytical method to bound the approximation error based on Stein's method and the perturbation theory. We further present two examples: theM/M/Nqueueing system and the supermarket model under the power-of-two-choices algorithm. For both applications, the method enables us to characterize the system performance under a broad range of traffic loads. For the supermarket model, this is the first paper that rigorously quantifies the steady-state performance of the-power-of-two-choices in the heavy-traffic regime. These results in the heavy-traffic regime cannot be obtained using the traditional mean-field analysis and the interchange of the limits.

Amplifiers for the Moran Process

The Moran process, as studied by Lieberman, Hauert, and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or nonmutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected, then the algorithm will eventually reach fixation , in which all individuals are mutants, or extinction , in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the article. Strong amplification is a rather surprising property—it means that in such graphs, the fixation probability of a uniformly placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n ). The name “strongly amplifying” comes from the fact that this selective advantage is “amplified.” Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly amplifying families—superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this article, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family “megastars.” When the algorithm is run on an n -vertex graph in this family, starting with a uniformly chosen mutant, the extinction probability is roughly n − 1/2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n ). Finally, we prove that our analysis of megastars is fairly tight—there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al.

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

AbstractLetn ≥2 be an integer and denote byθnthe real root in (0, 1) of the trinomialGn(X) = −1 +X+Xn. The sequence of Perron numbers(θn−1)n≥2$(\theta _n^{ - 1} )_{n \ge 2} $tends to 1. We prove that the Conjecture of Lehmer is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the rootsθn,zj,n, ofGn(X) lying in|z|< 1, as a function ofn, jonly. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measuresM(Gn)=M(θn)=M(θn−1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$of the trinomialsGnas a function ofnonly, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house, and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots ofGn, near the unit circle, and limit equidistribution of the conjugates, forntending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

Expanders Are Universal for the Class of All Spanning Trees

A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

Probabilities of Competing Binomial Random Variables

Suppose that both you and your friend toss an unfair coinntimes, for which the probability of heads is equal to α. What is the probability that you obtain at leastdmore heads than your friend if you makeradditional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function ofn, and demonstrate surprising phase transition phenomenon as the parametersd,r, and α vary. Our main tools are integral representations based on Fourier analysis.

The role of H2O in controlling bond topology:I. The[6]Mg(SO4)(H2O)n(n= 0–11) structures

AbstractHere, we examine the role of (H2O) in controlling polymerization of structural units in simple hydrated oxysalt compounds of the form Mg(SO4)(H2O)n. As the number of (H2O) groups in the structure increases, the number ofM—M(M= Mg) linkages decreases in a 1 : 1 ratio with the (increasing) number of (H2O) groups until noM—Mlinkages are left, and then the number ofM—T(T= S) linkages decreases with the (increasing) number of (H2O) groups until noM—Tlinkages are left (atn= 6). The change in bond topology is monotonic as a function of (H2O) content except forn= 5 where, instead of replacing anM—Tlinkage, the fifth (H2O) group is held in the structure by hydrogen bonds only. The valence-sum rule of bond-valence theory constrains the anion-coordination numbers to [2], [3] and [4] in these structures. The handshaking lemma of graph theory allows us to derive an equation relating the coordination numbers of the cations (Mg = [6], S6+= [4], H = [2]) to the coordination numbers of the anions. There are four integer solutions to this equation, and the Mg(SO4)(H2O)nstructures correspond to two of these solutions. Structures are known forn= 0–2.5, 4–7, 11; considering the variations in connectivity and coordination number as a function ofn, a possible structural arrangement is proposed for Mg(SO4)(H2O)3.

Entanglement entropy and the complex plane of replicas

The entanglement entropy of a subsystemA of aquantum system is expressed, in the replica method, through analytic continuation with respect ton of the trace of thenth power of the reduceddensity matrix trρAn. We study the analytic properties of this quantity as a function ofn in some quantum critical Ising-like models in1+1 and2+1 dimensions. Although we find no true singularities forn>0, there is athreshold value of n close to 2 which separates two very different ‘phases’. The region with largern ischaracterized by rapidly convergent Taylor expansions and is very smooth. The region with smallern has a very rich and varied structure in the complexn plane and is characterized by Taylor coefficients which instead of being monotonedecreasing, have a maximum growing with the size of the subsystem. Finite truncations ofthe Taylor expansion in this region lead to increasingly poor approximations oftrρAn. The computation of the entanglement entropy from the knowledge oftrρAn for positiveinteger n becomes extremely difficult, particularly in spatial dimensions larger than 1, whereone cannot use conformal field theory as a guidance in the extrapolations ton = 1.

Force analysis of jamming with disks of different sizes in a two-dimensional hopper

We calculate using force analysis the jamming probability of mixed disks that movedownwards under gravity in a two-dimensional hopper. Jamming happens when disks forman arc across the hopper opening and we determine the stability of the forcesacting at different arc points. To test the accuracy of our technique, we performedhopper experiments (side-wall separation: 0.2 cm) using disks (average thickness:0.198 ± 0.02 cm) at variousd/R values whered is the hopper openingdimension and R isthe disk radius. For N identical disks, the jamming probability is a sigmoidal function ofN. We measured thejamming probability at d/R = 2.97, 3.48, 4.0, 4.3 and 4.97, and verified the presence of three possible flow regimes: (1) jam-free continuous flow forN less thanthe onset Nd, (2) transitory flow where the probability increases withN forNd≤N≤Nu and (3) anN-independent jammedstate for N greater thanthe saturation threshold Nu. Good agreement is found between theory and experiments. The existence of three flow regimes is confirmedfor all d/R valueswith the Nu valueincreasing with d/R—the 2D hopper is more robust against jamming when relatively smallhomogeneous disks are used. We also experimented with mixtures ofN largeand n small disks and confirmed that jamming becomes more likely with increasingN for aconstant n (and vice versa) until the maximum jamming probability is attained.

Birkhoff normalization and superintegrability of Hamiltonian systems

AbstractWe study Birkhoff normalization in connection with superintegrability of ann-degree-of-freedom Hamiltonian systemXHwith holomorphic HamiltonianH. Without assuming any Poisson commuting relation among integrals, we prove that, if the system XHhasn+qholomorphic integrals near an equilibrium point of resonance degreeq≥0, there exists a holomorphic Birkhoff transformation φ such thatH∘φ becomes a holomorphic function ofn−qvariables and thatXH∘φcan be solved explicitly. Furthermore, the Birkhoff normal formH∘φ is determined uniquely, independently of the choice of φ, as convergent power series. We also show that the systemXHis superintegrable in the sense of Mischenko–Fomenko as well as Liouville-integrable near the equilibrium point.

Electrical conduction in alkali borate glasses; a unique dependence on the concentration ofmodifier ions

The electrical conduction of Li2O–B2O3, Na2O–B2O3 and K2O–B2O3 glasses seems, at first sight, to be dominated by the activation energy.Regardless of the size of the alkali ion, there is a unique dependence ofconductivity, at a certain temperature, on the alkali–alkali distance and thus onN (the number of ionsper cm3). The lineardependence of logσ onN−3/2 for all types of alkaliions reveals that N is the basic parameter that determines the conductivity at a certain temperature. Aderived semi-empirical relation can be used to calculate the conductivity as a function ofN and temperature.

Edge-disjoint paths revisited

The approximability of the maximum edge-disjoint paths problem (EDP) in directed graphs was seemingly settled by an Ω( m 1/2 - ϵ)-hardness result of Guruswami et al. [2003], and an O (√ m ) approximation achievable via a natural multicommodity-flow-based LP relaxation as well as a greedy algorithm. Here m is the number of edges in the graph. We observe that the Ω( m 1/2 - ϵ)-hardness of approximation applies to sparse graphs, and hence when expressed as a function of n , that is, the number of vertices, only an Ω( n 1/2 - ϵ)-hardness follows. On the other hand, O (√ m )-approximation algorithms do not guarantee a sublinear (in terms of n ) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the flow-based LP relaxation: an Ω(√ n ) lower bound and O (√ m ) upper bound. Motivated by this discrepancy in the upper and lower bounds, we study algorithms for EDP in directed and undirected graphs and obtain improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O (min( n 2/3 , √ m )) in undirected graphs and a ratio of O (min( n 4/5 , √ m )) in directed graphs. For acyclic graphs we give an O (√ n ln n ) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. The results also extend to EDP with weights and to the uniform-capacity unsplittable flow problem (UCUFP).

Shannon's monotonicity problem for free and classical entropy

We give a short unified proof of the following theorem, valid in the context of both classical probability theory and Voiculescu's free probability theory: let (X(j)((1)), ..., X(j)((n))) be independent (resp., freely independent) n-tuples of random variables. Let Z(N)((p)) = N(-1/2)(X(1)((p)) + ... + X(N)((p))) be their central limit sums. Then the entropy (resp., free entropy) of the n-tuple (Z(N)((1)), ..., Z(N)((n))) is a monotone function of N. The classical case (for n = 1) is a celebrated result of Artstein, Ball, Barthe, and Naor, and our proof is an adaptation and simplification of their argument.

Curvature corrections in DGP brane cosmology

We consider a DGP-inspired brane scenario where the action on the brane is augmented bya function of the Ricci scalar, . The cosmological implications that such a scenario entails are examined forRn and shown to be consistent with a universe expanding with power-law acceleration. Itis shown that two classes of solutions exist for the usual FRW metric andsmall Hubble radii. When the Hubble radius becomes larger, we either have atransition to a fully 5D regime or to a self-inflationary solution which producesa late accelerated expansion such that the radius becomes a function ofn.

System size dependence of the transport coefficients and Stokes–Einstein relationship ofhard sphere and Weeks–Chandler–Andersen fluids

Molecular dynamics simulations have been used to calculate the self-diffusion coefficient,D, and other transport coefficients of the hard sphere and Weeks–Chandler–Andersen (WCA)fluids over a wide density range. Simulations were carried out with different numbers of particles,N, in the range between 500 and 273 375 for the WCA and up to 10 976 for the hard spherefluid. These data were fitted to the relationship , where the parameters , A and α were all allowed to be density dependent. The self-diffusion coefficient in the thermodynamiclimit was obtained for both fluids. The Stokes–Einstein (SE) relationship stick–slip parameter,c = kBT/πDηs, wherekB is Boltzmann’sconstant, T is thetemperature and ηs is the shear viscosity, was calculated for the two fluids at each state point as a function ofN. Because of therelatively strong N dependence of D, theparameter c is alsoshown to be sensitive to N. It is shown that data taken for a few hundred particles can significantly overestimate the value ofc. At liquid-like densities, with increasing system size,c tends towards the slip value of 2. The same trend is observed for hard spheresand WCA particles. Therefore for any study of the SE stick–slip parameter it isimportant to perform several simulations for different system sizes and extrapolate theself-diffusion coefficient to the thermodynamic limit, and it is this value which shouldbe used to compare with theory. At the same packing fraction the self-diffusioncoefficient of the WCA fluid is larger than the value for the hard sphere fluidin the thermodynamic limit by, for example, 10% at a packing fraction of0.3 and 60% at a packing fraction of 0.49. The trend for the shear viscosity is the reverse, bothof which could be attributed to the softness of the potential in the WCA case and itseffect in inducing more cooperative interparticle trajectories than for the hardsphere.

Vorticity and magnetic shielding in a type-II superconductor

We study in detail, solving the Bogoliubov–de Gennes equations, the magnetic field,supercurrent and order parameter profiles originated by a solenoid or magnetic whiskerinserted in a type-II superconductor. We consider solutions of different vorticities,n, in the various cases. The results confirm the connection between the vorticity, the internalcurrents and the boundstates in a self-consistent way. The number of boundstates is givenby the vorticity of the phase of the gap function as in the case with no external solenoid. Inthe limiting case of an infinitely thin solenoid, like a Dirac string, the solution isqualitatively different. The quasiparticle spectrum and wavefunctions are a function ofn−next,where next is the vorticity of the solenoid. The flux is in all cases determined by the vorticity of thegap function.

Polymer in a double well: dynamics of translocation of short chains over a barrier

We consider the dynamics of a short chain polymer crossing over a free energy barrier inspace. Adopting the continuum version of the Rouse model, we find exact expressions forthe activation energy and the rate of crossing. For this model, the analysis of barriercrossing is analogous to semiclassical treatment of quantum tunnelling. Finding the saddlepoint for the process requires solving a Newton-like equation of motion for a fictitiousparticle. The analysis shows that short chains would cross the barrier as a globule. Theactivation free energy for this would increase linearly with the number of unitsN in the polymer. The saddle point for longer chains is an extendedconformation, in which the chain is stretched out. The stretching outlowers the energy and hence the activation free energy is no longer linear inN. The rates in both the cases are calculated using a multidimensional approach andanalytical expressions are derived, using a new formula for evaluating the infinite products.However, due to the harmonic approximation made in the derivation, the rates are found todiverge at the point where the saddle point changes over from the globule to the stretchedout conformation. The reason for this is identified to be the bifurcation of the saddle togive two new saddles, and a correction formula is derived for the rate in the vicinity ofthis point. Numerical results using the formulae are presented. As a function ofN, it is possible for the rate to have a minimum. This is due to confinement effects in theinitial state.

The off-resonant aspects of decoherence and a critique of the two-level approximation

Conditions in favour of a realistic multilevelled description of a decohering quantum systemare examined. In this regard the first crucial observation is that the thermal effects,contrary to the conventional belief, play a minor role at low temperatures in thedecoherence properties. The system–environment coupling and the environmentalenergy spectrum dominantly affect the decoherence. In particular, zero temperaturequantum fluctuations or non-equilibrium sources can be present and influentialon the decoherence rates in a wide energy range allowed by the spectrumof the environment. A crucial observation against the validity of the two-levelapproximation is that the decoherence rates are found to be dominated not by thelong time resonant but the short time off-resonant processes. This observationis demonstrated in two stages. Firstly, our zero temperature numerical resultsreveal that the calculated short time decoherence rates are Gaussian-like (thetime dependence of the density matrix is led by the second time derivative att = 0). Exact analytical results are also permitted in the short time limit, which, consistent withour numerical results, reveal that this specific Gaussian-like behaviour is a property of thenon-Markovian correlations in the environment. These Gaussian-like rates have no dependenceon any spectral parameter (position and the width of the spectrum) except, in totality, thespectral area itself. The dependence on the spectral area is a power law. Furthermore, theGaussian-like character at short times is independent of the number of levels(N), but the numerical value of the decoherence rates is a monotonic function ofN. In this context, we demonstrate that leakage, as a characteristic multilevel effect, isdominated by the non-resonant processes.The long time behaviour of decoherence is also examined. Since our spectral model allowsMarkovian environmental correlations at long times, the decoherence rates in this regimebecome exponential independently from the number of levels. The latter and the couplingstrengths play the major role in the quantitative values of the rates and the rates areindependent of the other spectral parameters.The validity of the presented results is restricted only by their reliance on theBorn–Oppenheimer approximation. This approximation is strongly dependent on theexternal observational time and its reliability depends on an additional timescale. In therest of the work, the crossover between the short and the long time behaviour ofthe density matrix of the multilevelled system is examined using an intuitiveargument. It is shown that the Born approximation weakens as the resonantcouplings become more effective at long times. This implies that, in calculations madewith this approximation in the long time regime, a need for a justification arisesfor the reliability of the results. This justification is made for the present work.

Population moments of sampling distributions

We describe a simple technique for computing population moments of a large class of sample statistics. We assume these to be symmetric functions ofn independent observations, wheren is relatively large. This enables us to expand the results in powers of $\frac{1}{n}$ The algorithm is demonstrated using Mathematica.

A growth gap for diffeomorphisms of the interval

Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of itsn-th iteration. We get a function ofn called the growth sequence. Its asymptotic behaviour is an interesting invariant, which naturally appears both in geometry of the diffeomorphism groups and in smooth dynamics. Our main result is the following Gap Theorem: the growth rate of this sequence is either exponential or at most quadratic withn. Further, we construct diffeomorphisms whose growth sequence has quite irregular behaviour. This construction easily extends to arbitrary manifolds.