一个模拟相关数据的新方法

介绍了一种产生相关数据的模拟方法.区别于以往的利用函数关系进行模拟的方法,本文主要从变量之间因子结构的相似性出发产生满足一定相关性的模拟数据.最后,通过一个例子说明了这个方法的应用.     

A note on the existence of fractional f-factors in random graphs

Let G=Gn,p be a binomial random graph with n vertices and edge probability p=p(n),and f be a nonnegative integer-valued function defined on V(G) such that 0a≤f(x)≤bnp-2np ㏒n for every x ∈V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0,1] so that for each vertex x,we have dh G(x)=f(x),where dh G(x) = x∈e h(e) is the fractional degree of x in G. Set Eh = {e:e ∈E(G) and h(e)=0}.If Gh is a spanning subgraph of G such that E(Gh)=Eh,then Gh is called an fractional f-factor of G. In this paper,we prove that for any binomial random graph Gn,p with p≥n-23,almost surely Gn,p contains an fractional f-factor.     

Global optimization by random perturbation of the gradient method with a fixed parameter

The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.     

A self normalized law of the iterated logarithm for random walk in random scenery

LetX,X _i ,i1, be a sequence of i.i.d. random vectors in ^d . LetS _o=0 and, forn1, letS _n =X ₁+...+X _n . LetY,Y(), ^d , be i.i.d. -valued random variables which are independent of theX _i . LetZ _n =Y(S _o )+...+Y(S _n ). We will callZ _n arandom walk in random scenery.In this work, we consider the law of the iterated logarithm for random walk in random sceneries. Under fairly general conditions, we obtain arandomly normalized law of the iterated logarithm.Supported in part by NSF Grants DMS-85-21586 and DMS-90-24961.     

An upper bound for the solvability probability of a random stable roommates instance

It is well-known that not all instances of the stable roommates problem admit a stable matching. Here we establish the first nontrivial upper bound on the limiting behavior of P_n, the probability that a random roommates instance of size n has a stable matching, namely, lim _n→∞ Pn? e^1/2/2 (=0.8244…). © 1994 John Wiley & Sons, Inc.     

The ζ(2) limit in the random assignment problem

The random assignment (or bipartite matching) problem asks about A_n=min_π ∑c(i, π(i)), where (c(i, j)) is a n×n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. Mézard and Parisi (1987) used the replica method from statistical physics to argue nonrigorously that EA_n→ζ(2)=π²/6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge‐costs and their rank‐orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost‐optimal matching coincides with the optimal matching except on a small proportion of edges. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 381–418, 2001     

N值随机序列的随机选择的强极限定理

将赌博系统的随机选择理论扩展到N值随机序列,利用似然比概念及分析技术,得到一个随机选择下有序数偶相对频率的强极限定理     

On the complexity of branch‐and‐bound search for random trees

Let T_n be a b‐ary tree of height n, which has independent, non‐negative, identically distributed random variables associated with each of its edges, a model previously considered by Karp, Pearl, McDiarmid, and Provan. The value of a node is the sum of all the edge values on its path to the root. Consider the problem of finding the minimum leaf value of T_n. Assume that the edge random variable X is nondegenerate, has E {X^θ}<∞ for some θ>2, and satisfies bP{X=c}<1 where c is the leftmost point of the support of X. We analyze the performance of the standard branch‐and‐bound algorithm for this problem and prove that the number of nodes visited is in probability (β+o(1))ⁿ, where β∈(1, b) is a constant depending only on the distribution of the edge random variables. Explicit expressions for β are derived. We also show that any search algorithm must visit (β+o(1))ⁿ nodes with probability tending to 1, so branch‐and‐bound is asymptotically optimal where first‐order asymptotics are concerned. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 309–327, 1999     

The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

A set A of non‐negative integers is called a Sidon set if all the sums , with and a₁, , are distinct. A well‐known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erd?s, Singer and Turán from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non‐differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erd?s (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015     

A nonstandard finite difference method for solving a mathematical model of HIV-TB co-infection

We design and analyze an unconditionally convergent nonstandard finite-difference method to study transmission dynamics of a mathematical model of HIV-TB co-infection. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves positivity of the solution which is one of the essential requirements when modelling epidemic diseases. Furthermore, we show that the numerical method is unconditionally stable. Competitive numerical results confirming theoretical investigations are provided. Comparisons are also made with other conventional approaches that are routinely used to solve these types of problems.     

A limit theorem for the eigenvalues of product of two random matrices

The existence of limit spectral distribution of the product of two independent random matrices is proved when the number of variables tends to infinity. One of the above matrices is the Wishart matrix and the other is a symmetric nonnegative definite matrix.     

A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any .     

Iterative adaptive RBF methods for detection of edges in two-dimensional functions

Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimension-by-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results.     

A threshold for the size of random caps to cover a sphere

Consider a unit sphere on which are placed N random spherical caps of area 4p(N). We prove that if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% qGSbGaaeyAaiaab2gaaaWaaeWaaeaacaWGWbWaaeWaaeaacaWGobaa% caGLOaGaayzkaaGaai4Taiaad6eacaGGVaGaaeiBaiaab+gacaqGNb% Gaaeiiaiaad6eaaiaawIcacaGLPaaacqGH8aapcaaIXaaaaa!454E!\[\overline {{\rm{lim}}} \left( {p\left( N \right)\cdotN/{\rm{log }}N} \right) < 1\], then the probability that the sphere is completely covered by N caps tends to 0 as N , and if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca% qGSbGaaeyAaiaab2gaaaWaaeWaaeaacaWGWbWaaeWaaeaacaWGobaa% caGLOaGaayzkaaGaai4Taiaad6eacaGGVaGaaeiBaiaab+gacaqGNb% Gaaeiiaiaad6eaaiaawIcacaGLPaaacqGH+aGpcaaIXaaaaa!4551!\[\underline {{\rm{lim}}} \left( {p\left( N \right)\cdotN/{\rm{log }}N} \right) > 1\], then for any integer n>0 the probability that each point of the sphere is covered more than n times tends to 1 as N .     

A linearly convergent derivative-free descent method for the second-order cone complementarity problem

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the Fischer–Burmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function ψ_FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan P-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104(2005), pp. 293–327), which confirm the theoretical results and the effectiveness of the algorithm.     

A class of heuristics for the constrained forest problem

The constrained forest problem seeks a minimum-weight spanning forest in an undirected edge-weighted graph such that each tree spans at least a specified number of vertices. We present a structured class of greedy heuristics for this NP-hard problem, and identify the best heuristic.     

On the convergence rate for a penalty function method of exponential type

Recently, Kort and Bertsekas (Ref. 1) and Hartman (Ref. 2) presented independently a new penalty function algorithm of exponential type for solving inequality-constrained minimization problems. The main purpose of this work is to give a proof on the rate of convergence of a modification of the exponential penalty method proposed by these authors. We show that the sequence of points generated by the modified algorithm converges to the solution of the original nonconvex problem linearly and that the sequence of estimates of the optimal Lagrange multiplier converges to this multiplier superlinearly. The question of convergence of the modified method is discussed. The present paper hinges on ideas of Mangasarian (Ref. 3), but the case considered here is not covered by Mangasarian's theory.     

Modelling and system identification of an experimental apparatus for anomaly detection in mechanical systems

This paper presents design, modelling and system identification of a laboratory test apparatus that has been constructed to experimentally validate the concepts of anomaly detection in complex mechanical systems. The test apparatus is designed to be complex in itself due to partially correlated interactions amongst its individual components and functional modules. The experiments are conducted on the test apparatus to represent operations of mechanical systems where both dynamic performance and structural durability are critical.