Diophantus found three rationals ~, -y, ~ with the property that the product of any two of them Increased by the sum of those two gives a perfect square (see [5], pp. 85-86, 215-217), and Euler found four rationals ^ , ^ 4 , -̂ -, j with the same property (see [4], pp. 518-519). We will call a set {x1? x2,..., xm} ofm rationals such that xtXj +xt +Xj is a perfect square for all 1 < i < j < m a E...

We consider the question whether an infinite eulerian graph has a decomposition into circuits and rays if the graph has only finitely many, say n, vertices of infinite degree, and only finitely many finite components after the removal of the vertices of infinite degree. It is known that the answer is affirmative for n < 2 and negative for n > 4. We settle the remaining case n = 3, showing that ...

1. INTRODUCTION. So, you've read your fifty page manuscript a half dozen times and you continue to stumble across annoying typos. Is there any end in sight? Just how many more times do you need to read it before you're satisfied? For purposes of discussion, let's say that your manuscript originally contained 10 errors and that, because of fatigue and a certain level of impatience, the probabili...

We define a noncommutative algebra of flag-enumeration functionals on graded posets and show it to be isomorphic to the free associative algebra on countably many generators. Restricted to Eulerian posets, this ring has a particularly appealing presentation with kernel generated by Euler relations. A consequence is that even on Eulerian posets, the algebra is free, with generators corresponding...

Let f and F be two polynomials satisfying F (x) = u(x)f(x) + v(x)f (x). We characterize the relation between the location and multiplicity of the real zeros of f and F , which generalizes and unifies many known results, including the results of Brenti and Brändén about the q-Eulerian polynomials.

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties....

The Hamiltonian index ofa graph G is defined as h(G) = minim: LIIl(G) is Hamiltonian). In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29-44], we constructed a graph i/(m)(G) from G and prove that ifh(G)?:. 2, then h(G) = minim : i/(m)(G) has a spanning Eulerian subgraphl.

We give counterexamples to two conjectures in \Bill Jackson, Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs, Journal of Graph Theory, 12(3):429{436, 1988" concerning orientations of mixed graphs and splitting oo in digraphs, and prove the rst conjecture in the (di-) Eulerian case(s). Beside that we solve the degree constrained directed augmentation pr...

A Large Eddy Simulation approach for Eulerian-Eulerian dispersed two-phase flow is presented. It is shown that not only a Sub-Grid Scales term modeling the unresolved field in the mesoscopic momentum transport equation of the dispersed phase, but also a stress term modeling the Random-Uncorrelated Motion is needed. Simulations of a non-homogeneous particle laden turbulent gas flow allow to comp...

We exploit Krattenthaler’s bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set Sn(123) of 123-avoiding permutations in Sn. In particular, we show that the descents of a permutation correspond to valleys and triple ascents of the associated Dyck path. We get the Eulerian numbers of Sn(123) by studying the joint distribution of these two ...