Positive definite rational functions of two variables which are not the sum of three squares

Cassels, Ellison, and Pfister have shown that there is a positive semidefinite function of R(x, y) that is not the sum of three squares. In this paper positive definite functions of R(x, y) are found having the same property. The proof involves showing the nonexistence of points on some elliptic curves defined over C(x), and extends the methods of [1].     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.     

An Improved Interval Linearization for Solving Nonlinear Problems

Let f(x) denote a system of n nonlinear functions in m variables, m≥n. Recently, a linearization of f(x) in a box x has been suggested in the form L(x)=Ax+b where A is a real n×m matrix and b is an interval n-dimensional vector. Here, an improved linearization L(x,y)=Ax+By+b, x∈x, y∈y is proposed where y is a p-dimensional vector belonging to the interval vector y while A and B are real matrices of appropriate dimensions and b is a real vector. The new linearization can be employed in solving various nonlinear problems: global solution of nonlinear systems, bounding the solution set of underdetermined systems of equations or systems of equalities and inequalities, global optimization. Numerical examples illustrating the superiority of L(x,y)=Ax+By+b over L(x)=Ax+b have been solved for the case where the problem is the global solution of a system of nonlinear equations (n=m).     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

More bounds for eigenvalues

A method introduced by Leighton [J. Math. Anal. Appl.35, 381–388 (1971)] for bounding eigenvalues has been extended to include problems of the form ?y″ + p(x) y = λy, when p(x) ? 0 on [0, 1]. The boundary conditions are the general homogeneous conditions y(0) ? ay′(0) = 0 = y(1) + by′(1), where 0 ? a, b ? ∞. Upper and lower bounds for the eigenvalues of these problems are obtained, and these bounds may be made as close together as desired, thereby allowing λ to be estimated precisely.     

Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation: An Improvement

In this paper, we consider a discrete delay problem with negative feedback x(t)=f(x(t), x(t−1)) along with a certain family of time discretizations with stepsize 1/n. In the original problem, the attractor admits a nice Morse decomposition. We proved in (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36-78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995, J. Dynam. Differential Equations7, 141-190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36-78) we proved certain continuity results about the individual Morse sets, including that if f(x, y)=f(y), then the individual Morse sets are upper semicontinuous at n=∞. In this paper we extend this result to the general case; that is, we prove for general f(x, y) with negative feedback that the Morse sets are upper semicontinuous.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).     

Constructive existence and uniqueness for two-point boundary-value problems with a linear gradient term

Constructive existence and uniqueness theorems are established for nonlinear two-point boundary-value problems governed by the equation y″=f(x, y)+p(x)y′, 0?x?1. The existence and uniqueness is established for functions y(x) that satisfy a certain constraint, i.e., that sfncy(x)sfnc is bounded by a known function or that 0?y(x)?M for some M. Applications to scientific problems in the literature are given.     

A lower bound for the first eigenvalue of an elliptic operator

Oscillatory behavior of the solutions of the nth-order delay differential equation L_nx(t) + q(t)f(x[g(t)]) = 0 is discussed. The results obtained are extensions of some of the results by Kim (Proc. Amer. Math. Soc.62 (1977), 77–82) for y⁽ⁿ⁾ + py = 0.     

Cone superadditivity of discrete convex functions

A function f is said to be cone superadditive if there exists a partition of R ⁿ into a family of polyhedral convex cones such that f(z?+?x) + f(z?+?y) ≤ f(z) + f(z?+?x?+?y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by local optimality criterion involving Hilbert bases. This paper shows cone superadditivity of L-convex and M-convex functions with respect to conic partitions that are independent of particular functions. L-convex and M-convex functions in discrete variables (integer vectors) as well as in continuous variables (real vectors) are considered.     

Hamming distance for conjugates

Let x,y be strings of equal length. The Hamming distanceh(x,y) between x and y is the number of positions in which x and y differ. If x is a cyclic shift of y, we say x and y are conjugates. We consider f(x,y), the Hamming distance between the conjugates xy and yx. Over a binary alphabet f(x,y) is always even, and must satisfy a further technical condition. By contrast, over an alphabet of size 3 or greater, f(x,y) can take any value between 0 and |x|+|y|, except 1; furthermore, we can always assume that the smaller string has only one type of letter.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

On the functions counting walks with small steps in the quarter plane

Models of spatially homogeneous walks in the quarter plane $\mathbf{ Z}_{+}^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z)?Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at $(i,j) \in\mathbf{ Z}_{+}^{2}$ after n steps is studied. For all non-singular models of walks, the functions x?Q(x,0;z) and y?Q(0,y;z) are continued as multi-valued functions on C having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of C ², the interval $]0,1/|\mathcal{S}|[$ of variation of z splits into two dense subsets such that the functions x?Q(x,0;z) and y?Q(0,y;z) are shown to be holonomic for any z from the one of them and non-holonomic for any z from the other. This entails the non-holonomy of (x,y,z)?Q(x,y;z), and therefore proves a conjecture of Bousquet-Mélou and Mishna in Contemp. Math. 520:1?C40 (2010).     

Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

A deficient spline function approximation to systems of first-order differential equations: part 2

The problem of stability was discussed in part 1 of this paper (Appl. Math. Modelling 1983, 7, 380). This part looks at the convergence of the spline approximation of deficiency 3 to systems of first-order differential equations. Convergence is shown for m = 4 and 5. In addition, global error bounds of the form: ∥S⁽ⁱ⁾(x) ? y⁽ⁱ⁾(x)∥∞ = 0(h^m+1?i), i = 0(1)m are presented, together with a computational example which illustrates the convergence of the proposed method.     

High-order one-step P-stable methods for the numerical integration of periodic initial value problems

Using Lobatto nodes, one-step methods of order six and eight have been obtained for the second-order differential equation y″ = f(x, y), y(x₀) = y₀, y′(x₀) = y′₀. The methods are shown to be P-stable. If

, then at each integration step a system of dimension 3s, 4s, respectively, has to be solved. The numerical results, for two problems, obtained by using these methods are given in the end.