Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions

We describe high-precision computations of the Pearcey integral Pe(x,y) for real x and y by means of Hadamard expansions. Numerical results for (x,y) situated in different regions of the x,y-plane are given to illustrate the levels of precision that can be achieved. Particular emphasis is given to computation in the neighbourhood of the two cusped curves associated with Pe(x,y) across which there is either a coalescence of saddles or a Stokes phenomenon.     

Ore-type degree condition for heavy paths in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that d^w(u)+d^w(v)2d for every pair of non-adjacent vertices u and vV(G) x,y . Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.     

Single step formulas and multi-step formulas of the integration method for solving the initial value problem of ordinary differential equation

Adam–Bashforth method and Adam–Moulton method are two known multi-step methods for finding the numerical solution of the initial value problem of ordinary differential equation. These two methods used the Newton backward difference method to approximate the value of f(x,y) in the integral equation which is equivalent to the given differential equation.     

An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality

We consider the asymmetric traveling salesperson problem with γ-parameterized triangle inequality for γ[1/2,1). That means, the edge weights fulfill w(u,v)γ(w(u,x)+w(x,v)) for all nodes u,v,x. Chandran and Ram [L.S. Chandran, L.S. Ram, Approximations for ATSP with parametrized triangle inequality, in: Proc. 19th Int. Symp. on Theoret. Aspects of Comput. Sci. (STACS), in: Lecture Notes in Comput. Sci., vol. 2285, Springer, Berlin, 2002, pp. 227–237] gave the first constant factor approximation algorithm with polynomial running time for this problem. They achieve performance ratio γ/(1−γ). We devise an approximation algorithm with performance ratio (1+γ)/(2−γ−γ³), which is better for γ[0.5437,1), that is, for the particularly interesting large values of γ.     

Uniqueness implies existence for three-point boundary value problems for dynamic equations

Shooting methods are used to obtain solutions of the three-point boundary value problem for the second-order dynamic equation, y^ΔΔ = f (x, y, y^Δ), y(x₁) = y₁, y(x₃) − y(x₂) = y₂, where f : (a, b)_T × ² → is continuous, x₁ < x₂ < x₃ in (a, b)_T, y₁, y₂ ε , and T is a time scale. It is assumed such solutions are unique when they exist.     

Starlikeness of Hadamard product of certain analytic functions

An interesting criterion was given by Lashin (A.Y. Lashin, Some convolution properties of analytic functions, Appl. Math. Lett. 18 (2005) 135–138) to be starlike for convolution of analytic functions f, g such that Re[f^′(z)],Re[g^′(z)]β,,β<1 in the unit disc U. In this paper we shall improve this criterion.     

An algorithm for calculating the independence and vertex-cover polynomials of a graph

The independence polynomial, ω(G,x)=∑w_kx^k, of a graph, G, has coefficients, w_k, that enumerate the ways of selecting k vertices from G so that no two selected vertices share an edge. The independence number of G is the largest value of k for which w_k≠0. Little is known of less straightforward relationships between graph structure and the properties of ω(G,x), in part because of the difficulty of calculating values of w_k for specific graphs. This study presents a new algorithm for these calculations which is both faster than existing ones and easily adaptable to high-level computer languages.     

Analytic solutions of an iterative functional differential equation

This paper is concerned with a functional differential equation x^′(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b²y^′(z)=(y(μ²z)−ay(μz))(μy^′(μz)−ay^′(z)), analytic solutions of the form for the original differential equation are obtained.     

A chromatic partition polynomial

A polynomial in two variables is defined by C_n(x,t)=Σ_πΠnx(Gπ,x)t^|π|, where Π_n is the lattice of partitions of the set {1, 2, …, n}, G_π is a certain interval graph defined in terms of the partition gp, χ(G_π, x) is the chromatic polynomial of G_π and |π| is the number of blocks in π. It is shown that , where S(n, i) is the Stirling number of the second kind and (x)_i = x(x − 1) ··· (x − i + 1). As a special case, C_n(−1, −t) = A_n(t), where A_n(t) is the nth Eulerian polynomial. Moreover, A_n(t)=Σ_πΠna_πt^|π| where a_π is the number of acyclic orientations of G_π.     

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively.

In this paper we study 1-spanners under the Manhattan (or L₁-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(nlogn) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P.

We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.     

A new formula for an evaluation of the Tutte polynomial of a matroid

Given a matroid M and its Tutte polynomial T_M(x,y), T_M(0,1) is an invariant of M with various interesting combinatorial and topological interpretations. Being a Tutte–Grothendieck invariant, T_M(0,1) may be computed via deletion–contraction recursions. In this note we derive a new recursion formula for this invariant that involves contractions of M through the circuits containing a fixed element of M.     

Integral methods for computing solutions of a class of singular two-point boundary value problems

In this paper, we establish three iteration methods to compute solutions for a class of (weakly) singular two-point boundary value problems (xy^′)^′=f(x,y), where x(0,1) and <2. We obtain the sufficient conditions for existence of a unique solution on . Finally, we given some numerical examples.     

Geometric mesh FDM for self-adjoint singular perturbation boundary value problems

A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors max_i|y(x_i)-y_i|, for different values of parameter , number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.     

3-Homogeneous latin trades

Let T be a partial latin square and L be a latin square with TL. We say that T is a latin trade if there exists a partial latin square T^′ with T^′∩T= such that (LT)T^′ is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 3-homogeneous latin trades from hexagonal packings of the plane with circles. We show that 3-homogeneous latin trades of size 3 m exist for each m3. This paper discusses existence results for latin trades and provides a glueing construction which is subsequently used to construct all latin trades of finite order greater than three.     

The minimum weight triangulation problem with few inner points

We look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n³) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(6^kn⁵logn) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k=O(logn). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k inner points.     

A note on exponential dispersion models which are invariant under length-biased sampling

Length-biased sampling (LBS) situations may occur in clinical trials, reliability, queueing models, survival analysis and population studies where a proper sampling frame is absent. In such situations items are sampled at rate proportional to their “length” so that larger values of the quantity being measured are sampled with higher probabilities. More specifically, if f(x) is a p.d.f. presenting a parent population composed of non-negative valued items then the sample is practically drawn from a distribution with p.d.f. g(x)=xf(x)/E(X) describing the length-biased population. In this case the distribution associated with g is termed a length-biased distribution. In this note, we present a unified approach for characterizing exponential dispersion models which are invariant, up to translations, under various types of LBS. The approach is rather simple as it reduces such invariance problems into differential equations in terms of the derivatives of the associated variance functions.     

Visibility maps of segments and triangles in 3D

Let T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T. We consider the subdivision of T based on (in)visibility from s; this is the visibility map of the segment s with respect to T. The visibility map of the triangle t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω(n²) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O(n²(n)) upper bound for both structures, where (n) is the extremely slowly increasing inverse Ackermann function. Furthermore, we prove that the weak visibility map of s has complexity Θ(n⁵), and the weak visibility map of t has complexity Θ(n⁷). If T is a polyhedral terrain, the complexity of the weak visibility map is Ω(n⁴) and O(n⁵), both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures.     

Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem

when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by ε^N+1 multiplied by a constant depending on T but not on x and t.     

Exact algorithms and applications for Tree-like Weighted Set Cover

We introduce an NP-complete special case of the Weighted Set Cover problem and show its fixed-parameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a base set S should be “tree-like”. That is, the subsets in C can be organized in a tree T such that every subset one-to-one corresponds to a tree node and, for each element s of S, the nodes corresponding to the subsets containing s induce a subtree of T. This is equivalent to the problem of finding a minimum edge cover in an edge-weighted acyclic hypergraph. Our main result is an algorithm running in O(3^kmn) time where k denotes the maximum subset size, n:=|S|, and m:=|C|. The algorithm also implies a fixed-parameter tractability result for the NP-complete Multicut in Trees problem, complementing previous approximation results. Our results find applications in computational biology in phylogenomics and for saving memory in tree decomposition based graph algorithms.