Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

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).     

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq"(t)+ Bq(t)=f(q(t)), where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q)=-▽_qV(q) for a real-valued function V(q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q', q)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.     

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.     

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.     

Numerical method for solving 3D inverse scattering problems

Let q(x) L²(D), D R³ is a bounded domain, q = 0 outside D, q is real-valued. Assume that A(\Gj;\t';,\Gj;,k) A(\Gj;\t';,\Gj), the scattering amplitude, is known for all \Gj;|t',\Gj; S², S² is the unit sphere, an d a fixed k \r>0. These data determine q(x) uniquely and a numerical method is given for computing q(x).     

On an optimal stopping problem of Gusein-Zade

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n→∞) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1−t^*)^r, where t^*≈0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule τ_r,n lets approximately t^*n arrivals go by and then stops ‘almost immediately’, in the sense that τ_r,n/n→t^* in probability as n→∞, r→∞     

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 embedding an outer-planar graph in a point set

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog³n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n²) time. Our algorithm is near-optimal as there is an Ω(nlogn) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where lognd2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(nlogn) and O(n²) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(nlogn) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.     

Maximal partial spreads and the modular n-queen problem

We prove that for any integer n in the interval there is a maximal partial spread of size n in PG (3, q) where q is odd and q7. We also prove that there are maximal partial spreads of size (q²+3)/2 when gcd(q+1,24)=2 or 4 and of size (q²+5)/2 when gcd(q+1,24)=4.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Partitions of finite affine geometries into caps

In a finite geometry of order q² we define a (q^mq^r)-affine cap to be a set of cardinality q^m which is a disjoint union ot q^m affine subgeometrics AG(r,q). such that no three points are coliinear unless contained in the same AG(r,q).

Given a PG(n,q²), where n = 2t + 1 or 2t + 2, and an n + 1 by n + 1 Hermitian matrix H over Gh(q²) with minimal polynomial (x - λ)^{n + 1}. we show that H induces a partition of the AG(n, q²) obtained by deleting a distinguished hyperplane from the PG, into (qⁿ,q^{l + 1})-affine caps; these caps can be viewed as the "large points" of an AG (n,q) with a natural incidence relation. It is also shown that H induces another partition of AG(n,q²), into q^{n - l 1}-caps, constituting the "large points" of an affine geometry AG(n + t + 1,q).

Also, the collineation C of PG(n, q²) given by x^c = H^Tx induces collineations on the AG(n,q) and AG(n + t + 1,q).     

On Duality in Filtered Cohomologies

For a symplectic manifold(M~(2n), ω) without boundary(not necessarily compact), we prove Poincaré type duality in filtered cohomology rings of differential forms on M, and we use this result to obtain duality between(d + d~Λ)-and dd~Λ-cohomologies.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

Impulsive effects on global existence of solutions for degenerate semilinear parabolic equations

Let q be a nonnegative real number, and λ and σ be positive constants. This article studies the following impulsive problem: for n = 1, 2, 3,…,

. The number λ^* is called the critical value if the problem has a unique global solution u for λ < λ^*, and the solution blows up in a finite time for λ > λ^*. For σ < 1, existence of a unique λ^* is established, and a criterion for the solution to decay to zero is studied. For σ > 1, existence of a unique λ^* and three criteria for the blow-up of the solution in a finite time are given respectively. It is also shown that there exists a unique T^* such that u exists globally for T> T^*, and u blows up in a finite time for T < T^*.     

ContributionsSign-balanced covering matrices

A q × n array with entries from 0, 1,…,q − 1 is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of [0, 1,… q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1]^t, a λq × n array will be said to form a (t, q, λ, Δ) sign-balanced matrix if for each choice C₁, C₂,…, C_t of t columns and for each choice = (₁,…,_t) Δ of signs, the linear combination ∑_j=1^t _jC_j contains (mod q) each entry of [0, 1,…, q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k(n, t, q, λ, δ) of rows have to be so that for each choice of t columns and for each choice (₁, …, _t) of signs in δ, the linear combination ∑_j=1^t _jC_j contains each entry of [0, 1,…, q t- 1] at least λ times? We use probabilistic methods, in particular the Lovász local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k(n, t, q, λ, δ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described.     

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.     

Nonexistence of a code for

In this paper, we shall prove that there is no [3q⁴-q³-q²-3q-1,5,3q⁴-4q³-2q+1]_q code over the finite field for q11. Thus, we conclude the nonexistence of a [g_q(5,d),5,d]_q code for 3q⁴-4q³-2q+1d3q⁴-4q³-q.     

On oscillation of second order neutral type delay differential equations

Oscillation criteria are obtained by using the so called H-method for the second order neutral type delay differential equations of the form

(r(t)ψ(x(t))z^′(t))^′+q(t)f(x(σ(t)))=0, tt₀,

where z(t)=x(t)+p(t)x(τ(t)), r, p, q, τ, σ, C([t₀,∞),R) and f,ψC(R,R).

The results of the paper contains several results obtained previously as special cases. Furthermore, we are also able to fix an error in a recent paper related to the oscillation of second order nonneutral delay differential equations.     

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.