Computing rectangle enclosures

The rectangle enclosure problem is the problem of determining the subset of n iso-oriented planar rectangles that enclose a query rectangle Q. In this paper, we use a three layered data structure which is a combination of Range and Priority search trees and answers both the static and dynamic cases of the problem. Both the cases use O(n> log² n) space. For the static case, the query time is O(log² n log log n + K). The dynamic case is supported in O(log³ n + K) query time using O(log³ n) amortized time per update. K denotes the size of the answer. For the d-dimensional space the results are analogous. The query time is O(log^2d-2 n log log n + K) for the static case and O(log^2d-1 n + K) for the dynamic case. The space used is O(n> log^2d-2 n) and the amortized time for an update is O(log^2d-1 n). The existing bounds given for a class of problems which includes the present one, are O(log^2d n + K) query time, O(log^2d n) time for an insertion and O(log^2d-1 n) time for a deletion.     

The Second Neighbourhood for Quasi-transitive Oriented Graphs

In 2006, Sullivan stated the conjectures:(1) every oriented graph has a vertex x such that d~(++)(x) ≥ d~-(x);(2) every oriented graph has a vertex x such that d~(++)(x) + d~+(x) ≥ 2 d~-(x);(3) every oriented graph has a vertex x such that d~(++)(x) + d~+(x) ≥ 2 · min{d~+(x), d~-(x)}. A vertex x in D satisfying Conjecture(i) is called a Sullivan-i vertex, i = 1, 2, 3. A digraph D is called quasi-transitive if for every pair xy, yz of arcs between distinct vertices x, y, z, xz or zx("or" is inclusive here) is in D. In this paper, we prove that the conjectures hold for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. Furthermore, we show that a quasi-transitive oriented graph with no vertex of in-degree zero has at least three Sullivan-1 vertices and a quasi-transitive oriented graph has at least three Sullivan-3 vertices unless it belongs to an exceptional class of quasitransitive oriented graphs. For Sullivan-2 vertices, we show that an extended tournament, a subclass of quasi-transitive oriented graphs and a superclass of tournaments, has at least two Sullivan-2 vertices unless it belongs to an exceptional class of extended tournaments.     

On induced matchings

Let q^*(G) denote the minimum integer t for which E(G) can be partitioned into t induced matchings of G. Faudree et al. conjectured that q^*(G)d², if G is a bipartite graph and d is the maximum degree of G. In this note, we give an affirmative answer for d=3, the first nontrivial case of this conjecture.     

Wandering subspaces and quasi-wandering subspaces in the Hardy–Sobolev spaces

In this paper, we prove that for-1/2 ≤β≤0.suppose M is an invariant subspaces of the Hardy Sobolev spaces H_β~2(D) for T_z~β, then M() zM is a generating wandering subspace of M, that is,M=[MzM]_T_z~β Moreover, any non-trivial invariant subspace M of H_β~2(D) is also generated by the quasi-wandering subspace P_MT_z~βM~⊥ that is,M=[P_MT_z~βM~⊥]_(T_z~β).     

Enumeration of irreducible binary words

Overlap free words over two letters are called irreducible binary words. Let d(n) denote the number of irreducible binary words of length n. In this paper we show that there are positive constants C₁ and C₂ such that C₁n^1.155<d(n)<C₂n^1.587 holds for all n>0.     

Self-maps of <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> homotopic to a smooth map with a single <Emphasis Type="Italic">n</Emphasis>-periodic point

We show for which (d,n) ∈ Z×N there exists a smooth self-map f:S² → S² so that deg(f)=d and Fix(fⁿ) is a point.     

Convolution integral equations with Gegenbauer function kernel

In this paper we develop a concise and transparent approach for solving Mellin convolution equations where the convolutor is the product of an algebraic function and a Gegenbauer function. Our method is primarily based on

1. the use of fractional integral/differential operators;

2. a formula for Gegenbauer functions which is a fractional extension of the Rodrigues formula for Gegenbauer polynomials (see Theorem 3);

3. an intertwining relation concerning fractional integral/differential operators (see Theorem 1), which in the integer case reads (d/dx)²ⁿ⁺¹ = (x⁻¹ d/dx)ⁿx²ⁿ⁺¹(x⁻¹ d/dx)ⁿ⁺¹.

Thus we cover most of the known results on this type of integral equations and obtain considerable extensions. As a special illustration we present the Gegenbauer transform pair associated to the Radon transformation.     

Weak stability for matrices

An n×n complex matrix A is called weak stable if there exists a matrix W such that W+W^* is positive definite and such that AW+W^*A^* is positive definite. In this note several characterizations for weak stability of a matrix are given, and conditions (on A) allowing W to be a diagonal matrix are also considered. A consequence of our results here is a characterization for nonsingular M-matrices.     

Commuting pairs and triples of matrices and related varieties

In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M_n(F), and show that it is irreducible of dimension n²−n for n4, but reducible, of dimension greater than n²−n for n7.     

Complexity of projected images of convex subdivisions

Let S be a subdivision of ^d into n convex regions. We consider the combinatorial complexity of the image of the (k - 1)-skeleton of S orthogonally projected into a k-dimensional subspace. We give an upper bound of the complexity of the projected image by reducing it to the complexity of an arrangement of polytopes. If k = d − 1, we construct a subdivision whose projected image has Ω(n^(3d−2)/2) complexity, which is tight when d 4. We also investigate the number of topological changes of the projected image when a three-dimensional subdivision is rotated about a line parallel to the projection plane.     

Cohomology of the Universal Enveloping Algebras of Certain Bigraded Lie Algebras

Let p be an odd prime and q = 2(p-1).Up to total degree t-s max{(5p~3+ 6p~2+ 6 p +4)q-10,p~4q},the generators of H~(s,t)(U(L)),the cohomology of the universal enveloping algebra of a bigraded Lie algebra L,are determined and their convergence is also verified.Furthermore our results reveal that this cohomology satisfies an analogous Poinare duality property.This largely generalizes an earlier classical results due to J.P.May.     

On the number of flats spanned by a set of points in PG(d, q)

It is shown that for fixed 1 r s < d and > 0, if X PG(d, q) contains (1 + )q^s points, then the number of r-flats spanned by X is at least c()q^{(r+1)(s+1−r)}, i.e. a positive fraction of the number of r-flats in PG(s + 1,q).     

Phase Retrieval of Real-valued Functions in Sobolev Space

The Sobolev space H^s(R^d) with s > d/2 contains many important functions such as the bandlimited or rational ones. In this paper we propose a sequence of measurement functions {ϕ_j,k^γ} ⊆ H^-s(R^d) to the phase retrieval problem for the real-valued functions in H^s(R^d). We prove that any real-valued function f ∈ H^s(R^d) can be determined, up to a global sign, by the phaseless measurements {|<f, ϕ_j,k^γ>|}. It is known that phase retrieval is unstable in infinite dimensional spaces with respect to perturbations of the measurement functions. We examine a special type of perturbations that ensures the stability for the phase-retrieval problem for all the real-valued functions in H^s(R^d) ∩ C¹(R^d), and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ H^s(R^d) ∩ C¹(R^d) whose Fourier transform f is L¹-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.     

Linear time algorithms for the weighted tailored 2-partition problem and the weighted 2-center problem under l∞-distance

In this paper, the weighted tailored 2-partition problem and the weighted 2-center problem under l_∞-distance are considered. An O(2^d−1·d·n) algorithm to solve the weighted tailored 2-partition problem and an O(d²·n + d²·log*d) time algorithm to solve the weighted 2-center problem in the d-dimensional case are presented.     

On determining the congruence of point sets in d dimensions

This paper considers the following problem: given two point sets A and B (|A| = |B| = n) in d dimensional Euclidean space, determine whether or not A is congruent to B. This paper presents an O(n^(d−1)/2 log n) time randomized algorithm. The birthday paradox, which is well-known in combinatorics, is used effectively in this algorithm. Although this algorithm is Monte-Carlo type (i.e., it may give a wrong result), this improves a previous O(n^d−2 log n) time deterministic algorithm considerably. This paper also shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too.     

On weights in duadic abelian codes

In this note, we prove that if C is a duadic binary abelian code with splitting μ=μ₋₁ and the minimum odd weight of C satisfies d²−d+1≠n, then d(d−1)n+11. We show by an example that this bound is sharp. A series of open problems on this subject are proposed.     

Characterization of Laborde-Mulder graphs (extended odd graphs)

The identification of diametrical vertices in the d-dimensional hypercube (d 3) leads to a (0, 2)-graph of degree d on 2^d−1 vertices and of diameter d/2 namely the extended odd graph (or Laborde-Mulder graph) for odd values of d, and the half-cube for even values of d. In this paper we prove that the diameter of a (0, 2)-graph of degree d on 2^d−1 vertices is at least d/2 , and when d is odd the equality holds if and only if the graph is a Laborde-Mulder graph.     

Unconditional uniqueness of solution for $$\dot H^{s_c }$$ critical 4th order NLS in high dimensions

In this paper, we study the unconditional uniqueness of solution for the Cauchy problem of Ḣ^s_c(0 ≤ s_c < 2) critical nonlinear fourth-order Schrödinger equations i∂_tu + Δ²u-εu=λ|u|^αu. By employing paraproduct estimates and Strichartz estimates, we prove that unconditional uniqueness of solution holds in C_t(I; Ḣ^s_c(R^d)) for d ≥ 11 and min{1^-, (8)/(d-4)} ≥ α >(-(d-4)+√4(d-4)²+64)/4.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.