Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

On the qualitative behaviors of solutions of third order nonlinear differential equations

We present some new results about oscillation and asymptotic behavior of solutions of third order nonlinear differential equations of the form

(r₂(t)(r₁(t)y^′)^′)^′+p(t)y^′+q(t)f(y(g(t)))=0.     

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G; x, y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x, y) is on the hyperbola H _q given by (x ? 1)(y ? 1) = q for q = 1 or q = 2 or if (x, y) is one of the two special points (x, y) = (?1, ?1) or (x, y) = (1, 1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G; x, y), in the usual sense of “fully polynomial randomized approximation scheme” or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x, y) plane. In particular, there is no FPRAS if x > 1, y < ?1 or if y > 1, x < ?1 or if x < 0, y < 0 and q > 5. Also, there is no FPRAS if x < 1, y < 1 and q = 3. For q > 5, our result is intriguing because it shows that there is no FPRAS at (x, y) =?(1 ? q/(1 + ε), ?ε) for any positive ε but it leaves open the limit point ε =?0, which corresponds to approximately counting q-colorings of a planar graph.     

The complexity of changing colourings with bounded maximum degree

Let G be a graph, x,y∈V(G), and ?:V(G)→[k] a k-colouring of G such that ?(x)=?(y). If then the following question is NP-complete: Does there exist a k-colouring ?^′ of G such that ?^′(x)≠?^′(y)? Conversely, if then the problem is polynomial time.     

Oscillation theorems for second-order nonlinear neutral differential equations

The purpose of this paper is to study the oscillation of the second-order neutral differential equations of the form

(a(t)[z^′(t)]^γ)^′+q(t)x^β(σ(t))=0,     

The existence of positive solutions of singular fractional boundary value problems

We discuss the existence of positive solutions for the singular fractional boundary value problem D^αu+f(t,u,u^′,D^μu)=0, u(0)=0, u^′(0)=u^′(1)=0, where 2<α<3, 0<μ<1. Here D^α is the standard Riemann-Liouville fractional derivative of order α, f is a Carathéodory function and f(t,x,y,z) is singular at the value 0 of its arguments x,y,z.     

Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

Assume that n is a positive integer with n?2. It is proved that between any two different vertices x and y of Qn there exists a path Pl(x,y) of length l for any l with h(x,y)?l?n²−1 and 2|(l−h(x,y)). We expect such path Pl(x,y) can be further extended by including the vertices not in Pl(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that for any two vertices x and z from different partite set of n-dimensional hypercube Qn, for any vertex y∈V(Qn)−{x,z}, and for any integer l with h(x,y)?l?n²−1−h(y,z) and 2|(l−h(x,y)), there exists a hamiltonian path R(x,y,z;l) from x to z such that dR(x,y,z;l)(x,y)=l. Moreover, for any two distinct vertices x and y of Qn and for any integer l with h(x,y)?l?²n−1 and 2|(l−h(x,y)), there exists a hamiltonian cycle S(x,y;l) such that dS(x,y;l)(x,y)=l.     

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

Projection methods,algorithms, and a new system of nonlinear variational inequalities

The convergence of projection methods is based on a new iterative algorithm for the approximation-solvability of the following system of nonlinear variational inequalities (SNVI): determine elements x^*, y^* ϵ K such that 〈ϱT(y^*) + x^* − y^*, x − x^*〉 ≥ 0, for all x ϵ K and for ϱ > 0, and 〈γT(x^*) + y^* − x^*, x − y^*〉 ≥ 0, for all x ∈ K and for γ > 0, where T : K → H is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. This new class of generalized nonlinear variational inequalities reduces to standard class of nonlinear variational inequalities, which are widely studied and applied to various problems arising from mathematical sciences, optimization and control theory, and other related fields.     

Complexity aspects of generalized Helly hypergraphs

In [V.I. Voloshin, On the upper chromatic number of a hypergraph, Australas. J. Combin. 11 (1995) 25-45], Voloshin proposed the following generalization of the Helly property. Let p?1, q?0 and s?0. A hypergraph H is (p,q)-intersecting when every partial hypergraph H^′⊆H formed by p or less hyperedges has intersection of cardinality at least q. A hypergraph H is (p,q,s)-Helly when every partial (p,q)-intersecting hypergraph H^′⊆H has intersection of cardinality at least s. In this work, we study the complexity of determining whether H is (p,q,s)-Helly.     

Asymptotic behavior of nonoscillatory solutions of neutral difference equations

The authors consider the m^th-order neutral difference equation D_m(y(n) + p(n)y(n − k) + q(n)f(y(σ(n))) = e(n), where m ≥ 1, {p(n)}, {q(n)}, {e(n)}, and {a₁(n)}, {a₂(n)}, …, {a_m−1(n)} are real sequences, a_i(n) > 0 for i = 1,2,…, m−1, a_m(n) ≡ 1, D₀z(n) = y(n)+p(n)y(n − k), D_iz(n) = a_i(n)ΔD_i−1z(n) for i = 1,2, …, m, k is a positive integer, {σ(n)} → ∞ is a sequence of positive integers, and R → R is continuous with u f(u) > 0 for u ≠ 0. In the case where {q(n)} is allowed to oscillate, they obtain sufficient conditions for all bounded nonoscillatory solutions to converge to zero, and if {q(n)} is a nonnegative sequence, they establish sufficient conditions for all nonoscillatory solutions to converge to zero. Examples illustrating the results are included throughout the paper.     

Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V(G)| inclusive. It has been shown that Q_n is bipancyclic if and only if n?2. In this paper, we improve this result by showing that every edge of Q_n−E′ lies on a cycle of every even length from 4 to |V(G)| inclusive where E′ is a subset of E(Q_n) with |E′|?n−2. The result is proved to be optimal. To get this result, we also prove that there exists a path of length l joining any two different vertices x and y of Q_n when h(x,y)?l?|V(G)|−1 and l−h(x,y) is even where h(x,y) is the Hamming distance between x and y.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Two point hermite approximations for the solution of linear initial value and boundary value problems

Application of an idea originally due to Ch. Hermite allows the derivation of an approximate formula for expressing the integral ∫xixi?1y(x)dx as a linear combination of y(xi?1), y(xi), and their derivatives y(v)(xi?1) up to order v = α and y(v)(xi) up to order v = β. In addition to this integro-differential form a purely differential form of the 2-point Hermite approximation will be derived. Both types will be denoted by Hαβ-approximation. It will be shown that the well-known Obreschkoff-formulas contain no new elements compared to the much older Hαβ-method.The Hαβ-approximation will be applied to the solution of systems of ordinary differential equations of the type y'(x) = M(x)y(x) + q(x), and both initial value and boundary value problems will be treated. Function values at intermediate points x? (xi?1, xi) are obtained by the use of an interpolation formula given in this paper.An advantage of the Hαβ-method is the fact that high orders of approximation (α, β) allow an increase in step size hi. This will be demonstrated by the results of several test calculations.     

CONFLUENT POSSIBILITY DISTRIBUTIONS AND THEIR REPRESENTATIONS

Possibilistic distributions admit both measures of uncertainty and (metric) distances defining their information closeness. For general pairs of distributions these measures and metrics were first introduced in the form of integral expressions. Particularly important are pairs of distributions p and q which have consonant ordering—for any two events x and y in the domain of discourse p(x)&lesseqqgtr; p(y) if and only if q(x) &lesseqqgtr; q(y). We call such distributions confluent and study their information distances.

This paper presents discrete sum form of uncertainty measures of arbitrary distributions, and uses it to obtain similar representations of metrics on the space of confluent distributions. Using these representations, a number of properties like additivity. monotonicity and a form of distributivity are proven. Finally, a branching property is introduced, which will serve (in a separate paper) to characterize axiomatically possibilistic information distances.     

A recursive and a grammatical characterization of the exponential-time languages

We characterize the class of all languages which are acceptable in exponential time by means of recursive and grammatical methods. (i) The class of all languages which are acceptable in exponential time is uniquely characterized by the class of all (0-1)-functions which can be generated, starting with the initial functions of the Grzegorczyk-class $\text{E}$², by means of subtitution and limited recursion of the form f(x, y + 1) = h(x, y), f(x, y), f(x, l(x, y))), l(x, y) ? y. (ii) The class of all languages which are acceptable in exponential time is equal to the class of all languages generated by context-sensitive grammars with context-free control sets.     

An algorithm for the decomposition of differential polynomials in the general case

The theory of decomposition of differential polynomials (DPs) is considered. For a given differential algebraic equation $P(x,y(x), y'(x), \ldots ,y^{(n)} (x)) = 0,$ the possibility to represent the differential polynomial P as the composition P = Q(x, R, R′, ..., R ^(q)), R = R(x, y(x), ..., ^y (r)(x)) of DPs Q and R is studied. It is shown that the decomposition problem is reduced to the factorization of linear ordinary differential operators (LODOs). A generic decomposition algorithm for DPs is described based on the Vandermonde differential theorem.     

Positive solutions of singular Dirichlet and periodic boundary value problems

Let A and T be positive numbers. The singular differential equation (r(x)x′)′ = μq(t)f(t, x) is considered. Here r > 0 on (0, A] may be singular at x = 0, and f(t, x) ≤ 0 may be singular at x = 0 and x = A. Effective sufficient conditions imposed on r, μ, q, and f are given for the existence of a solution x to the above equation satisfying either the Dirichlet conditions x(0) = x(T) = 0 or the periodic conditions x(0) = x(T), x′(0) = x′(T), and, in addition, 0 < x < A on (0, T).