Perfect difference families,perfect difference matrices,and related combinatorial structures

The existence problems of perfect difference families with block size k, k=4,5, and additive sequences of permutations of length n, n=3,4, are two outstanding open problems in combinatorial design theory for more than 30 years. In this article, we mainly investigate perfect difference families with block size k=4 and additive sequences of permutations of length n=3. The necessary condition for the existence of a perfect difference family with block size 4 and order v, or briefly (v, 4,1)‐PDF, is v≡1(mod12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP (3, m), is m≡1(mod2). So far, (12t+1,4,1)‐PDFs with t<50 are known only for t=1,4−36,41,46 with two definiteexceptions of t=2,3, and ASP (3, m)'s with odd 3<m<200 are known only for m=5,7,13−29,35,45,49,65,75,85,91,95,105,115,119,121,125,133,135,145,147,161,169,175,189,195 with two definite exceptions of m=9,11. In this article, we show that a (12t+1,4,1)‐PDF exists for any t⩽1,000 except for t=2,3, and an ASP (3, m) exists for any odd 3<m<200 except for m=9,11 and possibly for m=59. The main idea of this article is to use perfect difference families and additive sequences of permutations with “holes”. We first introduce the concepts of an incomplete perfect difference matrix with a regular hole and a perfect difference packing with a regular difference leave, respectively. We show that an additive sequence of permutations is in fact equivalent to a perfect difference matrix, then describe an important recursive construction for perfect difference matrices via perfect difference packings with a regular difference leave. Plenty of perfect difference packings with a desirable difference leave are constructed directly. We also provide a general recursive construction for perfect difference packings, and as its applications, we obtain extensive recursive constructions for perfect difference families, some via incomplete perfect difference matrices with a regular hole. Examples of perfect difference packings directly constructed are used as ingredients in these recursive constructions to produce vast numbers of perfect difference families with block size 4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 415–449, 2010     

Two constructions of (v, (v − 1)/2, (v − 3)/2) difference families

In this article, two constructions of (v, (v ? 1)/2, (v ? 3)/2) difference families are presented. The first construction produces both cyclic and noncyclic difference families, while the second one gives only cyclic difference families. The parameters of the second construction are new. The difference families presented in this article can be used to construct Hadamard matrices. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 164–171, 2008     

Lagrangian chaos and multiphase processes in vortex flows

We discuss experimental and numerical studies of the effects of Lagrangian chaos (chaotic advection) on the stretching of a drop of an immiscible impurity in a flow. We argue that the standard capillary number used to describe this process is inadequate since it does not account for advection of a drop between regions of the flow with varying velocity gradient. Consequently, we propose a Lagrangian-generalized capillary number C_L number based on finite-time Lyapunov exponents. We present preliminary tests of this formalism for the stretching of a single drop of oil in an oscillating vortex flow, which has been shown previously to exhibit Lagrangian chaos. Probability distribution functions (PDFs) of the stretching of this drop have features that are similar to PDFs of C_L. We also discuss on-going experiments that we have begun on drop stretching in a blinking vortex flow.     

Interpretations of some parameter dependent generalizations of classical matrix ensembles

Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Andersons and Dixons work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.Supported by the Australian Research Council     

Cyclic (v; r, s; ��) Difference Families with Two Base Blocks and v �� 50

We construct many new cyclic (v; r, s; λ) difference families with v ≥ 2r ≥ 2s ≥ 4 and v ≤ 50. In particular, we construct the difference families with parameters (45; 18, 10; 9), (45; 22, 22; 21), (47; 21, 12; 12), (47; 19, 15; 12), (47; 22, 14; 14), (48; 20, 10; 10), (48; 24, 4; 12), (50; 25, 20; 20), for which the existence question was an open problem. We point out that the (45; 22, 22; 21) difference family gives a balanced incomplete block design (BIBD) with parameters v = 45, b = 90, r = 44, k = 22, and λ = 21, and that the one with parameters (50; 25, 20; 20) gives a pair of binary sequences of length 50 with zero periodic autocorrelation function (the periodic analog of a Golay pair). The new SDSs include nine new D-optimal designs. A normal form for cyclic difference families (with base blocks of arbitrary sizes) is proposed and used effectively in compiling our selective listings in Tables 3–6 of known and new difference families in the above range.     

Inference in hybrid Bayesian networks using mixtures of polynomials

The main goal of this paper is to describe inference in hybrid Bayesian networks (BNs) using mixture of polynomials (MOP) approximations of probability density functions (PDFs). Hybrid BNs contain a mix of discrete, continuous, and conditionally deterministic random variables. The conditionals for continuous variables are typically described by conditional PDFs. A major hurdle in making inference in hybrid BNs is marginalization of continuous variables, which involves integrating combinations of conditional PDFs. In this paper, we suggest the use of MOP approximations of PDFs, which are similar in spirit to using mixtures of truncated exponentials (MTEs) approximations. MOP functions can be easily integrated, and are closed under combination and marginalization. This enables us to propagate MOP potentials in the extended Shenoy-Shafer architecture for inference in hybrid BNs that can include deterministic variables. MOP approximations have several advantages over MTE approximations of PDFs. They are easier to find, even for multi-dimensional conditional PDFs, and are applicable for a larger class of deterministic functions in hybrid BNs.     

Multiscale analysis and reconstruction of time series of stochastic cascade processes

We propose a new method to generate synthetical time series of hierarchical stochastic processes. Based on the statistics of n–scale joint PDFs, the stochastic properties of a time series are modeled simultaneously on many scales. The application to a data set of turbulent velocities is demonstrated, showing the ability of the approach to reproduce the correct statistics of the original time series on all considered scales. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two issues in using mixtures of polynomials for inference in hybrid Bayesian networks

We discuss two issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. First, in defining MOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z = X + Y, etc. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. Second, Shenoy and West suggest using the Taylor series expansion of differentiable functions for finding MOP approximations of PDFs. In this paper, we describe a new method for finding MOP approximations based on Lagrange interpolating polynomials (LIP) with Chebyshev points. We describe how the LIP method can be used to find efficient MOP approximations of PDFs. We illustrate our methods using conditional linear Gaussian PDFs in one, two, and three dimensions, and conditional log-normal PDFs in one and two dimensions. We compare the efficiencies of the hyper-rhombus condition with the hypercube condition. Also, we compare the LIP method with the Taylor series method.     

A powerful method for constructing difference families and optimal optical orthogonal codes

In this paper we proceed in the way indicated by R. M. Wilson for obtaining simple difference families from finite fields [28]. We present a theorem which includes as corollaries all the known direct techniques based on Galois fields, and provides a very effective method for constructing a lot of new difference families and also new optimal optical orthogonal codes.By means of our construction—just to give an idea of its power—it has been established that the only primesp<10⁵ for which the existence of a cyclicS(2, 9,p) design is undecided are 433 and 1009. Moreover we have considerably improved the lower bound on the minimumv for which anS(2, 15,v) design exists.     

Frequency domain criteria for robust stability of a family of linear difference equations 1

We provide necessary and sufficient conditions for stability of solutions to linear difference equations with coefficients in a prescribed set. The conditions encompass classes of uncertainty such as interval familiesl^p families, affine families.     

Calculation of probability density functions for nonlinear vibration systems

Technical systems are subjected to a variety of excitations that cannot generally be described in deterministic ways. Random excitations such as road roughness, wind gusts or loads on marine structures are commonly described by stochastic differential equations (SDEs). Given a set of SDEs, the main task is in finding probability density functions (PDFs), which yield statistical information about the system states. Monte-Carlo simulations represent a general way of generating PDFs, however, reliable integration methods can be time-consuming for complex systems. An alternative way of finding PDFs lies in the analysis of the Fokker-Planck equation, a partial differential equation of the PDF. Linear problems under Gaussian excitation can be solved analytically, which is the case only for a small class of nonlinear problems. As a result, there are a number of methods of approximating PDFs for general problems. Most of these are restricted to comparably low dimensions, such as d=4 ("curse of dimensionality"), limiting the relevance to technical applications. This paper presents solutions to problems of dimensions up to d=10, applying a Galerkin-method that expands approximate solutions into orthogonal polynomials. Problems include polynomial nonlinearities in damping and restoring terms, such as classical Duffing-elements, as well as other types of nonlinearities, demonstrated on a typical problem in vehicle dynamics. All results are compared with results from Monte-Carlo simulations or exact solutions, where available. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

(C_k \oplus G,k,\lambda ) Difference Families

Let G be an additive group and C _k be the additive group of the ring Z _k of residues modulo k. If there exist a (G, k, ) difference family and a (G, k, ) perfect Mendelsohn difference family, then there also exists a difference family. If the (G, k, ) difference family and the (G, k, ) perfect Mendelsohn difference family are further compatible, then the resultant difference family is elementary resolvable. By first constructing several series of perfect Mendelsohn difference families, many difference families and elementary resolvable difference families are thus obtained.     

On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

A Family of Estimators of Finite-Population Distribution Function Using Auxiliary Information

This paper considers the problem of estimating the finite-population distribution function and quantiles with the use of auxiliary information at the estimation stage of a survey. We propose the families of estimators of the distribution function of the study variate y using the knowledge of the distribution function of the auxiliary variate x. In addition to ratio, product and difference type estimators, many other estimators are identified as members of the proposed families. For these families the approximate variances are derived, and in addition, the optimum estimator is identified along with its approximate variance. Estimators based on the estimated optimum values of the unknown parameters used to minimize the variance are also given with their properties. Further, the family of estimators of a finite-population distribution function using two-phase sampling is given, and its properties are investigated.      

Level of nodes in increasing trees revisited

Simply generated families of trees are described by the equation T(z) = ϕ(T(z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ { 1,…,n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways. This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j,n → ∞ such that n − j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud, and others. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦⁿW)⁽ⁿ⁾W^-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.     

Equianalytic and equisingular families of curves on surfaces

Summary We consider flat families of reduced curves on a smooth surfaceS such that each memberC has the same number of singularities and each singularity has a fixed singularity type (up to analytic resp. topological equivalence). We show that these families are represented by a schemeH and give sufficient conditions for the smoothness ofH (atC). Our results improve previously known criteria for families with fixed analytic singularity type and seem to be quite sharp for curves in ℙ² of small degree. Moreover, for families with fixed topological type this paper seems to be the first in which arbitrary singularities are treated. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

The Kolmogorov–Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.     

Unchained polygons and the N-body problem

We study both theoretically and numerically the Lyapunov families which bifurcate in the vertical direction from a horizontal relative equilibrium in ℝ³. As explained in [1], very symmetric relative equilibria thus give rise to some recently studied classes of periodic solutions. We discuss the possibility of continuing these families globally as action minimizers in a rotating frame where they become periodic solutions with particular symmetries. A first step is to give estimates on intervals of the frame rotation frequency over which the relative equilibrium is the sole absolute action minimizer: this is done by generalizing to an arbitrary relative equilibrium the method used in [2] by V. Batutello and S. Terracini. In the second part, we focus on the relative equilibrium of the equal-mass regular N-gon. The proof of the local existence of the vertical Lyapunov families relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We compute the symmetry groups G _r/s (N, k, η) of the vertical Lyapunov families observed in appropriate rotating frames, and use them for continuing the families globally. The paradigmatic examples are the “Eight” families for an odd number of bodies and the “Hip- Hop” families for an even number. The first ones generalize Marchal’s P ₁₂ family for 3 bodies, which starts with the equilateral triangle and ends with the Eight [1, 3–6]; the second ones generalize the Hip-Hop family for 4 bodies, which starts from the square and ends with the Hip-Hop [1, 7, 8]. We argue that it is precisely for these two families that global minimization may be used. In the other cases, obstructions to the method come from isomorphisms between the symmetries of different families; this is the case for the so-called “chain” choreographies (see [6]), where only a local minimization property is true (except for N = 3). Another interesting feature of these chains is the deciding role played by the parity, in particular through the value of the angular momentum. For the Lyapunov families bifurcating from the regular N-gon whith N ≤ 6 we check in an appendix that locally the torsion is not zero, which justifies taking the rotation of the frame as a parameter. To the memory of J. Moser, with admiration     

Strong difference families,difference covers,and their applications for relative difference families

In (M. Buratti, J Combin Des 7:406–425, 1999), Buratti pointed out the lack of systematic treatments of constructions for relative difference families. The concept of strong difference families was introduced to cover such a problem. However, unfortunately, only a few papers consciously using the useful concept have appeared in the literature in the past 10 years. In this paper, strong difference families, difference covers and their connections with relative difference families and optical orthogonal codes are discussed.