On balanced bases

It is proved that either a given balanced basis of the algebra (n + 1)M₁ M_n or the corresponding complementary basis is of rank n + 1. This result enables us to claim that the algebra (n + 1)M₁ M_n is balanced if and only if the matrix algebra M_n admits a WP-decomposition, i.e., a family of n + 1 subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form tr XY) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra (n + 1)M₁ M_n is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type A_n–1 into the sum of Cartan subalgebras.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 213–218.Original Russian Text Copyright © 2005 by D. N. Ivanov.This revised version was published online in April 2005 with a corrected issue number.     

On balanced bases

It is proved that either a given balanced basis of the algebra (n + 1)M₁ M_n or the corresponding complementary basis is of rank n + 1. This result enables us to claim that the algebra (n + 1)M₁ M_n is balanced if and only if the matrix algebra M_n admits a WP-decomposition, i.e., a family of n + 1 subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form tr XY) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra (n + 1)M₁ M_n is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type A_n–1 into the sum of Cartan subalgebras.     

On homothetic balanced metrics

In this article, we study the set of balanced metrics given in Donaldson’s terminology (J. Diff. Geometry 59:479–522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler–Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179–228, 2006; Ann. Math. 170(2):685–738, 2009).     

On arc reversal in balanced digraphs

In this note we consider closed walks, which are cycles that are not necessarily elementary. We prove that any arc reversal in a balanced multidigraph without loops decreases the number of closed walks. This also proves that arc reversal in a simple balanced digraph decreases the number of closed walks.     

Chaotic dynamics of an unbalanced gyrostat

The free three-dimensional motion of an unbalanced gyrostat about the centre of mass is considered. The perturbed Hamiltonian for the case of small dynamical asymmetry of the rotor is written in Andoyer–Deprit canonical variables. The structure of the phase space of the unperturbed system is analysed, six forms of possible phase portraits are identified, and the equations of the phase trajectories are found analytically. Explicit analytical time dependences of the Andoyer–Deprit variables corresponding to heteroclinic orbits are obtained for all the phase portrait forms. The Melnikov function of the perturbed system is written for heteroclinic separatrix orbits using the analytical solutions obtained, and the presence of simple zeros is shown numerically. This provides evidence of intersections of the stable and unstable manifolds of the hyperbolic points and chaotization of the motion. Illustrations of chaotic modes of motion of the unbalanced gyrostat are presented using Poincaré sections.     

On some balanced, totally balanced and submodular delivery games

This paper studies a class of delivery problems associated with the Chinese postman problem and a corresponding class of delivery games. A delivery problem in this class is determined by a connected graph, a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office. It is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server, who starts at the post office and is expected to traverse all edges in the graph before returning to the post office. A graph G is called Chinese postman-submodular, or, for short, CP-submodular (CP-totally balanced, CP-balanced, respectively) if for each delivery problem in which G is the underlying graph the associated delivery game is submodular (totally balanced, balanced, respectively). For undirected graphs we prove that CP-submodular graphs and CP-totally balanced graphs are weakly cyclic graphs and conversely. An undirected graph is shown to be CP-balanced if and only if it is a weakly Euler graph. For directed graphs, CP-submodular graphs can be characterized by directed weakly cyclic graphs. Further, it is proven that any strongly connected directed graph is CP-balanced. For mixed graphs it is shown that a graph is CP-submodular if and only if it is a mixed weakly cyclic graph. Finally, we note that undirected, directed and mixed weakly cyclic graphs can be recognized in linear time. Received May 20, 1997 / Revised version received August 18, 1998?Published online June 11, 1999     

On generalized balanced optimization problems

In the generalized balanced optimization problem (GBaOP) the objective value \({\max_{e \in S}{|c(e)-k\max(S)|}}\) is minimized over all feasible subsets S of E = {1, . . . , m}. We show that the algorithm proposed in Punnen and Aneja (Oper Res Lett 32:27–30, 2004) can be modified to ensure that the resulting solution is indeed optimal. This modification is attained at the expense of increased worst-case complexity, but still maintains polynomial solvability of various special cases that are of general interest. In particular, we show that GBaOP can be solved in polynomial time if an associated bottleneck problem can be solved in polynomial time. For the solution of this bottleneck problem, we propose two alternative approaches.     

On balanced regular Hjelmslev planes

We consider regular Hjelmslev planes satisfying additional assumptions; in particular, we always assume the planes to be balanced. The main result of the paper is the determination of the spectra of invariants of regular n-uniform H-planes and of balanced minimally uniform regular H-planes. These are determined exactly (not only up to orders of projective planes), the only other spectrum known exactly up to now being the spectrum of desarguesian H-planes. We also prove that our construction method for balanced minimally uniform regular H-planes is canonical.The author acknowledges the hospitality of the University of Florida while doing this research.     

On locally balanced gray codes

We consider locally balanced Gray codes.We say that a Gray code is locally balanced if every “short” subword in its transition sequence contains all letters of the alphabet |1, 2,..., n~. The minimal length of these subwords is the window width of the code. We show that for each n ≥ 3 there exists a Gray code with window width at most n + 3?log n?.     

On balanced colorings of hypergraphs

This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m _k (n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains k vertices of each color. In this paper, we obtain the exact values of m ₂(5) and m ₂(4), and the upper bounds for m ₃(7) and m ₄(9).     

On balanced sets mod p

Let p be an odd prime, and let a₁, a₂,…, a_2n+1 be elements of $\text{Z}\text{pZ}$ such that, if any one of them is removed, the remaining ones can be divided into two sets of n elements with equal sums. This paper gives an estimate of the smallest possible size of n, assuming that not all the a_i agree.     

Forced oscillations of an unbalanced rotor in nonisotropic bearings

The influence of anisotropy of elastic bearings on forced oscillations of a rotor with the static and moment unbalance is studied for the cases of its fastening on a rigid shaft and on a flexible one. The rotor with four degrees of freedom is considered. It is suggested that the shaft is fixed in linear elastic nonisotropic bearings. The differential equations of rotation of the rotor are written in complex variables, and an exact solution to the equation system is found that corresponds to the elliptical synchronous precession. The exact solution is a sum of two vectors, one of which parameterizes a forward precession, while another parameterizes a reverse precession. Amplitude-frequency characteristics of forward and reverse precessions and elliptical trajectories of the rotor axis ends are constructed. It is shown that, in case of nonisotropic bearings, both the forward and reverse precession, as well as the axis motion of nonsimple type (when its one end is moving forward, while another is moving in the reverse direction), can take place. The influence of anisotropy of elastic bearings also manifests itself by change in critical frequencies towards their reduction and by arising of additional critical frequencies in the bottom part of the spectrum, which significantly complicates dynamics of the high-speed rotor at the moment when it reaches the working angular speed.     

On the integrality of an extreme solution to pluperfect graph and balanced systems

Let A be a nonnegative integer matrix, and let e denote the vector all of whose components are equal to 1. The pluperfect graph theorem states that if for all integer vectors b the optimal objective value of the linear program minse′xvbAx ? b, x ? 0 s is integer, then those linear programs possess optimal integer solutions. We strengthen this theorem and show that any lexicomaximal optimal solution to the above linear program (under any arbitrary ordering of the variables) is integral and an extreme point of sxvbAx ? b, x ? 0 s. We note that this extremality property of integer solutions is also shared by covering as well as packing problems defined by a balanced matrix A.     

On the balanced domination of graphs

Czechoslovak Mathematical Journal - Let G = (VG,EG) be a graph and let NG[υ] denote the closed neighbourhood of a vertex υ in G. A function f: VG ? {?1,0,1} is said to be a...     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.