On the polynomial mixed 0–1 fractional programming problems

This paper proposes a new method of solving polynomial mixed 0–1 fractional programming (P01FP) problems to obtain a global optimum. Given a polynomial 0–1 term x₁x₂,…,x_ny, where x_i is a 0–1 variable and y is a continuous variable; we develop a linearization technique to transfer the x₁x₂,…,x_ny term into a set of mixed 0–1 linear inequalities. Based on this technique, the P01FP can then be solved by a branch-and-bound method to obtain the global solution.     

Rainbow solutions to the Sidon equation

We prove that for every 4-coloring of {1,2,…,n}, with each color class having cardinality more than (n+1)/6, there exists a solution of the equation x+y=z+w with x, y, z and w belonging to different color classes. The lower bound on a color class cardinality is tight.     

Bitableau bases for Garsia-Haiman modules of hollow type

Garsia-Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x₁,x₂,…,xn} and Yn={y₁,y₂,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1?j?n. A bitableau basis for the Garsia-Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.     

Walking in circles

Consider the unit circle S¹ with distance function d measured along the circle. We show that for every selection of 2n points x₁,…,x_n,y₁,…,y_n∈S¹ there exists i∈{1,…,n} such that . We also discuss a game theoretic interpretation of this result.     

Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

We modify the classical Paley-Wiener spaces PWx of entire functions of finite exponential type at most x>0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z₁,…,zn. We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z₁,…,zn are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions.     

An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order

We prove that each OBDD (ordered binary decision diagram) for the middle bit of n-bit integer multiplication for one of the variable orders which so far achieve the smallest OBDD sizes with respect to asymptotic order of growth, namely the pairwise ascending order x₀,y₀,…,x_n−1,y_n−1, requires a size of Ω(2^(6/5)n). This is asymptotically optimal due to a bound of the same order by Amano and Maruoka (2007) [1].     

The rank-width of the square grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if, and only if, it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width.The n×nsquare grid, denoted by G_n,n, is a graph on the vertex set {1,2,…,n}×{1,2,…,n}, where a vertex (x,y) is connected by an edge to a vertex (x^′,y^′) if and only if |x−x^′|+|y−y^′|=1.We prove that the rank-width of G_n,n is equal to n−1, thus solving an open problem of Oum.     

The existence and uniqueness of strong kings in tournaments

A king x in a tournament T is a player who beats any other player y directly (i.e., x→y) or indirectly through a third player z (i.e., x→z and z→y). For x,y∈V(T), let b(x,y) denote the number of third players through which x beats y indirectly. Then, a king x is strong if the following condition is fulfilled: b(x,y)>b(y,x) whenever y→x. In this paper, a result shows that for a tournament on n players there exist exactly k strong kings, 1?k?n, with the following exceptions: k=n-1 when n is odd and k=n when n is even. Moreover, we completely determine the uniqueness of tournaments.     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Oberwolfach rectangular table negotiation problem

We completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x₁,x₂,…,x_k},Y={y₁,y₂,…,y_k} and edges x₁y₁,x₁y₂,x_ky_k−1,x_ky_k, and x_iy_i−1,x_iy_i,x_iy_i+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs K_n,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3).     

Oscillation of partial difference equations with continuous variables

In this paper, we consider the partial difference equation with continuous variables of the form P₁z(x + a, y + b) + p₂z (x + a, y) + p₃z (x, y + b) − p₄z (x, y) + P (x, y) z (x − τ, y − σ) = 0, where P ϵ C(R⁺ × R⁺, R⁺ − {0}), a, b, τ, σ are real numbers and p_i (i = 1, 2, 3, 4) are nonnegative constants. Some sufficient conditions for all solutions of this equation to be oscillatory are obtained.     

Bivariate Hermite Interpolation and Numerical Curves

In this paper, Hermite interpolation by bivariate algebraic polynomials of total degree ?nis considered. The interpolation parameters are the values of a function and its partial derivatives up to some ordern_ν−1 at the nodesz_ν=(x_ν, y_ν),ν=1, …, s, wheren_νis the multiplicity ofz_ν. The sequence ={n₁, …, n_s; n} of multiplicities associated with the degree of interpolating polynomials is investigated. Some results of the paper were announced in [GHS93].     

Nevanlinna-Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z₁,…,z_k in the open unit disk D and a set of complex numbers {f_i,0,f_i,1,…,f_i,ni−1} assigned to each z_i, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at z_i has f_i,0,f_i,1,…,f_i,ni−1 as its first n_i coefficients.     

Representation of group elements as subsequence sums

Let G be a finite (additive written) abelian group of order n. Let w₁,…,w_n be integers coprime to n such that w₁+w₂+?+w_n≡0 (mod n). Let I be a set of cardinality 2n-1 and let ξ={x_i:i∈I} be a sequence of elements of G. Suppose that for every subgroup H of G and every a∈G, ξ contains at most terms in a+H.Then, for every y∈G, there is a subsequence {y₁,…,y_n} of ξ such that y=w₁y₁+?+w_ny_n.Our result implies some known generalizations of the Erd?s-Ginzburg-Ziv Theorem.     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

Bitableaux Bases for the Diagonally Invariant Polynomial Quotient Rings

LetR=Q[x₁, x₂, …, x_n,y₁, y₂, …, y_n,z₁, …, z_n,w₁, …, w_n], letR^S_n={P∈R:σP=P∀σ∈S_n} and letμandνbe hook shape partitions ofn. WithΔ_μ(X, Y) andΔ_ν(Z, W) being appropriately defined determinants, ∂_xibeing the partial derivative operator with respect tox_iandP(∂)=P(∂_x1, …, ∂_xn, ∂_y1, …, ∂_wn), define _{μ, ν}={P∈R^S_n:P(∂)Δ_μ(X, Y)Δ_ν(Z, W)=0}. A basis is constructed for the polynomial quotient ringR^S_n/_{μ, ν}that is indexed by pairs of standard tableaux. The Hilbert series ofR^S_n/_{μ, ν}is related to the Macdonaldq, t-Kostka coefficients.