The challenges of statistical patterns of language: The case of Menzerath's law in genomes

The importance of statistical patterns of language has been debated over decades. Although Zipf's law is perhaps the most popular case, recently, Menzerath's law has begun to be involved. Menzerath's law manifests in language, music and genomes as a tendency of the mean size of the parts to decrease as the number of parts increases in many situations. This statistical regularity emerges also in the context of genomes, for instance, as a tendency of species with more chromosomes to have a smaller mean chromosome size. It has been argued that the instantiation of this law in genomes is not indicative of any parallel between language and genomes because (a) the law is inevitable and (b) noncoding DNA dominates genomes. Here mathematical, statistical, and conceptual challenges of these criticisms are discussed. Two major conclusions are drawn: the law is not inevitable and languages also have a correlate of noncoding DNA. However, the wide range of manifestations of the law in and outside genomes suggests that the striking similarities between noncoding DNA and certain linguistics units could be anecdotal for understanding the recurrence of that statistical law. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Benford's law and distractors in multiple choice exams

Suppose that in a multiple choice examination the leading significant digit of the correct options follows Benford's Law, while the leading digit of the distractors is uniform. Consider a strategy for guessing at answers that selects the option with the lowest leading digit with ties broken at random. We provide an expression for both the probability that this strategy selects the correct option and also the generalization to the probability of selecting the option with the lowest r significant digit string.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

On the strong circular 5‐flow conjecture

The Strong Circular 5‐flow Conjecture of Mohar claims that each snark—with the sole exception of the Petersen graph—has circular flow number smaller than 5. We disprove this conjecture by constructing an infinite family of cyclically 4‐edge connected snarks whose circular flow number equals 5. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Newton's second law in field theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.     

Compression and the origins of Zipf's law for word frequencies

Here we sketch a new derivation of Zipf's law for word frequencies based on optimal coding. The structure of the derivation is reminiscent of Mandelbrot's random typing model but it has multiple advantages over random typing: (1) it starts from realistic cognitive pressures, (2) it does not require fine tuning of parameters, and (3) it sheds light on the origins of other statistical laws of language and thus can lead to a compact theory of linguistic laws. Our findings suggest that the recurrence of Zipf's law in human languages could originate from pressure for easy and fast communication. © 2016 Wiley Periodicals, Inc. Complexity 21: 409–411, 2016     

Darcy's laws for non‐stationary viscous fluid flow in a thin porous medium

We consider a non‐stationary Stokes system in a thin porous medium Ω_? of thickness ? which is perforated by periodically solid cylinders of size a _? . We are interested here to give the limit behavior when ? goes to zero. To do so, we apply an adaptation of the unfolding method. Time‐dependent Darcy's laws are rigorously derived from this model depending on the comparison between a _? and ? . Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence versus blow‐up results for one dimensional compressible Navier–Stokes equations with Maxwell's law

We consider one dimensional isentropic compressible Navier–Stokes equations with constitutive relation of Maxwell's law instead of Newtonion law. For this new model, we show that for small initial data, a unique smooth solution exists globally and converges to the equilibrium state as time goes to infinity. For some large data, in contrast to the situation for classical compressible Navier–Stokes equations, which admits global solutions, we show finite time blow up of solutions for the relaxed system. Moreover, we prove the compatibility of the two systems in the sense that, for vanishing relaxation parameters, the solutions to the relaxed system are shown to converge to the solutions of classical system.     

List multicoloring problems involving the k‐fold Hall numbers

We show that the four‐cycle has a k‐fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ?5k/3?; furthermore, the same is not true with shorter list lengths. In terms of h^(k)(G), the k ‐fold Hall number of a graph G, this result is stated as h^(k)(C₄)=2k??k/3?. For longer cycles it is known that h^(k)(C_n)=2k, for n odd, and 2k??k/(n?1)?≤h^(k)(C_n)≤2k, for n even. Here we show the lower bound for n even, and conjecture that this is the right value (just as for C₄). We prove that if G is the diamond (a four‐cycle with a diagonal), then h^(k)(G)=2k. Combining these results with those published earlier we obtain a characterization of graphs G with h^(k)(G)=k. As a tool in the proofs we obtain and apply an elementary generalization of the classical Hall–Rado–Halmos–Vaughan theorem on pairwise disjoint subset representatives with prescribed cardinalities. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 16–34, 2010.     

Exact traveling wave solutions to the fourth‐order dispersive nonlinear Schrödinger equation with dual‐power law nonlinearity

In this paper, we investigate exact traveling wave solutions of the fourth‐order nonlinear Schrödinger equation with dual‐power law nonlinearity through Kudryashov method and (G'/G)‐expansion method. We obtain miscellaneous traveling waves including kink, antikink, and breather solutions. These solutions may be useful in the explanation and understanding of physical behavior of the wave propagation in a highly dispersive optical medium. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Green function in visco‐elastic media obeying a frequency power‐law

In this work, we present an explicit expression for the Green function in a visco‐elastic medium. We choose Szabo and Wu's frequency power law model to describe the visco‐elastic properties and derive a generalized visco‐elastic wave equation. We express the ideal Green function (without any viscous effect) in terms of the viscous Green function using an attenuation operator. By means of an approximation of the ideal Green function, we address the problem of reconstructing a small anomaly in a visco‐elastic medium from wavefield measurements. Copyright © 2011 John Wiley & Sons, Ltd.     

Hadwiger's Conjecture for ℓ‐Link Graphs

In this article, we define and study a new family of graphs that generalizes the notions of line graphs and path graphs. Let G be a graph with no loops but possibly with parallel edges. An ?‐link of G is a walk of G of length in which consecutive edges are different. The ?‐link graph of G is the graph with vertices the ?‐links of G , such that two vertices are joined by edges in if they correspond to two subsequences of each of μ ‐links of G . By revealing a recursive structure, we bound from above the chromatic number of ?‐link graphs. As a corollary, for a given graph G and large enough ?, is 3‐colorable. By investigating the shunting of ?‐links in G , we show that the Hadwiger number of a nonempty is greater or equal to that of G . Hadwiger's conjecture states that the Hadwiger number of a graph is at least the chromatic number of that graph. The conjecture has been proved by Reed and Seymour (Eur J Combin 25(6) (2004), 873–876) for line graphs, and hence 1‐link graphs. We prove the conjecture for a wide class of ?‐link graphs.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

Claw‐Free Graphs,Skeletal Graphs,and a Stronger Conjecture on ω, Δ, and χ

The second author's (B.A.R.) ω, Δ, χ conjecture proposes that every graph satisfies . In this article, we prove that the conjecture holds for all claw‐free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way, we discuss a stronger local conjecture, and prove that it holds for claw‐free graphs with a three‐colorable complement. To prove our results, we introduce a very useful χ‐preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so‐called skeletal graphs.     

On the probability distribution of data at points in real complete intersection varieties

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multi-variate polynomial equations and convergence radius of Newton's operator for under-determined system of multi-variate polynomial equations.     

A robust approach for assessing misclassification rates under the two‐component measurement error model

The majority of actions designed to improve processes and quality include the assessment of the capability of a measurement system. The statistical model relating the measured value to the true, but not observable, value of a product characteristic is usually Gaussian and additive. In this paper we propose to extend the said model to a more general formulation by introducing the structure of the two‐component error model. An approximated method for evaluating the misclassification rates under the two‐component error model is proposed and assessed. Copyright © 2009 John Wiley & Sons, Ltd.     

Chromatic numbers of products of graphs: The directed and undirected versions of the Poljak‐Rödl function

Let f(n) = min{χ(G × H) : G and H are n‐chromatic digraphs} and g(n) = min{χ(G × H) : G and H are n‐chromatic graphs}. We prove that f is bounded if and only if g is bounded. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Analysis of the energy‐conserved S‐FDTD scheme for variable coefficient Maxwell's equations in disk domains

In this paper, we analyze the energy‐conserved splitting finite‐difference time‐domain (FDTD) scheme for variable coefficient Maxwell's equations in two‐dimensional disk domains. The approach is energy‐conserved, unconditionally stable, and effective. We strictly prove that the EC‐S‐FDTD scheme for the variable coefficient Maxwell's equations in disk domains is of second order accuracy both in time and space. It is also strictly proved that the scheme is energy‐conserved, and the discrete divergence‐free is of second order convergence. Numerical experiments confirm the theoretical results, and practical test is simulated as well to demonstrate the efficiency of the proposed EC‐S‐FDTD scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011