The inverse problem for certain tree parameters

Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We will provide a very general setting for this type of problem over the set of all trees, describe some simple examples and finally consider the interesting parameter “number of subtrees”, where the problem can be reduced to some number-theoretic considerations. Specifically, we will prove that every positive integer, with only 34 exceptions, is the number of subtrees of some tree.     

On a reducibility of the Kampé de Fériet function

The main objective of this research note is to provide an interesting result for the reducibility of the Kampé de Fériet function. The result is derived with the help of two results for the terminating ₃F₂ series very recently obtained by Rakha et al. A few interesting special cases have also been given. Copyright © 2014 John Wiley & Sons, Ltd.     

One interesting family of diophantine triplets

In this note properties of two sequences generated by the recurrence relation G _n+2 = 4G _n+1 ? G _n , are studied. It is shown that one of the sequences leads to a family of diophantine triplets. Some interesting properties of these sequences are also established.     

A Graph b-coloring Framework for Data Clustering

The graph b-coloring is an interesting technique that can be applied to various domains. The proper b-coloring problem is the assignment of colors (classes) to the vertices of one graph so that no two adjacent vertices have the same color, and for each color class there exists at least one dominating vertex which is adjacent (dissimilar) to all other color classes. This paper presents a new graph b-coloring framework for clustering heterogeneous objects into groups. A number of cluster validity indices are also reviewed. Such indices can be used for automatically determining the optimal partition. The proposed approach has interesting properties and gives good results on benchmark data set as well as on real medical data set.     

A Liapunov type inequality for Sugeno integrals

The classical Liapunov inequality shows an interesting upper bound for the Lebesgue integral of the product of two functions. This paper proposes a Liapunov type inequality for Sugeno integrals. That is, we show that holds for some constant H_s,t,r where 0<t<s<r,f:[0,1]→[0,∞) is a non-increasing concave function, and μ is the Lebesgue measure on R. We also present two interesting classes of functions for which the classical Liapunov type inequality for Sugeno integrals with H_s,t,r=1 holds. Some examples are provided to illustrate the validity of the proposed inequality.     

Equivalence classes in the Weyl groups of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We consider two families of equivalence classes in the Weyl groups of type B _n which are suggested by the study of left cells in unequal parameter Iwahori-Hecke algebras. Both families are indexed by a non-negative integer r. It has been shown that the first family coincides with left cells corresponding to the equal parameter Iwahori-Hecke algebra when r=0; the equivalence classes in the second family agree with left cells corresponding to a special class of choices of unequal parameters when r is sufficiently large. Our main result shows that the two families of equivalence classes coincide, suggesting the structure of left cells for remaining choices of the Iwahori-Hecke algebra parameters.      

The Falkner-Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

The Falkner-Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336-394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur.     

Hankel transforms of generalized Motzkin numbers

We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.     

On the p-rank of incidence matrices and a bound of Bruen and Ott

Bruen and Ott have derived an interesting lower bound on the p-rank of the incidence matrix of a partial linear space. We derive two extensions of their result.     

Induced generalized <Emphasis Type="Italic">S</Emphasis>-space-form structures on submanifolds

We study non-anti-invariant slant submanifolds of generalized S-space-forms with two structure vector felds in order to know if they inherit the ambient structure. In this context, we focus on totally geodesic, totally umbilical, totally ƒ-geodesic and totally ƒ-umbilical non-anti-invariant slant submanifolds and obtain some obstructions. Moreover, we present some new interesting examples of generalized S-space-forms.     

Quantum Yang-Baxter H-Module Algebras and Their Braided Products

Abstract

We give a necessary and sufficient condition of the braided product to be a bialgebra or a Hopf algebra and two interesting examples to show that the conditions in Theorem 2.4: “(H1) and (H2)” weaken the commutativity and cocommutativity of H in Caenepeel et al. (Caenepeel, S., Oystaeyen, F. Van, Zhang, Yin-huo (1994). Quantum Yang-Baxter module algebra. K-Theorem 8:231–255.). Dually, we introduce the concept of a braided coproduct and give the distinguished conditions.     

A parachute for the degree of a polynomial in algebraically independent ones

We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or “wildness”, of Nagata’s automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism of K[x ₁, . . . , x _n ] and reprove the tameness of automorphisms of K[x ₁, x ₂].     

Analyzing retrial queues by censoring

In this paper we analyze the M/M/c retrial queue using the censoring technique. This technique allows us to carry out an asymptotic analysis, which leads to interesting and useful asymptotic results. Based on the asymptotic analysis, we develop two methods for obtaining approximations to the stationary probabilities, from which other performance metrics can be obtained. We demonstrate that the two proposed approximations are good alternatives to existing approximation methods. We expect that the technique used here can be applied to other retrial queueing models.     

Some Laws of the Iterated Logarithm for Two Parameter Martingales

We prove some laws of the iterated logarithm for two parameter martingale differences of the following type: E(X _t|(X _s, s ₁<t ₁ or s ₂<t ₂))=0 a.s.     

Involutions and linear systems on holomorphic symplectic manifolds

A K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that similar statement holds for the generic couple (X, H) with X a deformation of (K3)^[n] and H an ample divisor of square 2 for Beauville’s quadratic form. If n = 2 then according to the conjecture X is a double cover of a singular) sextic 4-fold in It follows from the conjecture that a deformation of (K3)^[n] carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two antisymplectic involutions generating an interesting dynamical system, a case Strange duality and what is probably an involution on the moduli space degree-2 quasi-polarized (X, H) where X is a deformation of (K3)^[2]. Received: June 2004 Revision: December 2004 Accepted: January 2005     

On generalized independent subsets of trees

A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ?-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.     

Direct Evaluation of Nonlinear Gradient Elasticity in 2D with C1 Continuous Discretization Methods

In this contribution, we present a study on the performance of four different subparametric triangular finite element formulations for linear and nonlinear strain gradient elastic continua. Fully C¹ and nonconforming C^* finite elements are considered for a numerical analysis. The results are demonstrated by means of two computational examples together with the L², H¹ and H² error norms. An interesting aspect concerning the linear geometry approximation is revealed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Proper S(t,K,v)'s for t ≥ 3,v ≤ 16, |K| > 1 and Their Extensions

We characterize the proper t-wise balanced designs t-(v,K,1) for t ≥ 3, λ = 1 and v ≤ 16 with at least two block sizes. While we do not examine extensions of S(3,4,16)'s, we do determine all other possible extensions of S(3,K,v)'s for v ≤ 16. One very interesting extension is an S(4, {5,6}, 17) design.©1995 John Wiley & Sons, Inc.     

The solution of a recursive sequence arising from a combinatorial problem in botanical epidemiology

One of the central problems in botanical epidemiology is whether disease spreads within crops in a regular pattern or follows a random process. In this study, we consider a row of n plants in which m are infected. We then develop a rigorous mathematical approach to investigate the total number of ways to obtain k isolated individuals among m infected plants. We give a recurrence relation in three parameters that describes the problem, then we find a closed-form solution, and give two different approaches to tackle the proof. Finally, we find interesting formulae for the expectation and variance of the random variable that represents the number of infected and isolated plants.