Cyclicity of several planar graphics and ensembles through three singular points without generic conditions

This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node P₀ and two hyperbolic saddles P₁ and P₂ for the non-generic cases of r₁(0)=1, r₂(0)≠1 and r₁(0)≠1, r₂(0)=1, where r₁(0) and r₂(0) are the hyperbolicity ratio of the saddles P₁ and P₂, respectively. For the case of r₁(0)=1, r₂(0)≠1, we suppose that the connection from P₀ to P₂ and the connection from P₀ to P₁ keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle P₁ if P₂ is contractive and r₂(0)∈Q. We also show that the nearer r₂(0) is close to 1, the more the limit cycles are bifurcated. For the case of r₁(0)≠1, r₂(0)=1, we obtain that these graphics and ensembles are of finite cyclicity respectively if P₁ is of finite order and the hp-connection from P₀ to P₂ keeps unbroken.     

Optimal tristance anticodes in certain graphs

For z₁,z₂,z₃∈Zⁿ, the tristance d₃(z₁,z₂,z₃) is a generalization of the L₁-distance on Zⁿ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticodeA_d of diameter d is a subset of Zⁿ with the property that d₃(z₁,z₂,z₃)?d for all z₁,z₂,z₃∈A_d. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z² for all diameters d?1. We then generalize this result to two related distance models: a different distance structure on Z² where d(z₁,z₂)=1 if z₁,z₂ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z² is replaced by the hexagonal lattice A₂. We also investigate optimal tristance anticodes in Z³ and optimal quadristance anticodes in Z², and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.     

Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.     

Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family $\text{J}$. Consider a bivariate Gumbel Type A survival distribution, S₁₂(x₁, x₂), defined in (2.1), for which both marginal distributions, S₁(x₁), S₂(x₂), belong to the same family, $\text{J}$ of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h₁(t) and h₂(t), are proportional if and only if the marginal hazard rates, λ₁(t) and λ₂(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X₁, X₂), and of the identified minimum, W_i = X_i, for X_i < X_j, j ≠ i, belong to the same family $\text{J}$ as do S₁(x₁), S₂(x₂) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.     

Nonzero decomposable symmetrized tensors

Suppose k₁ ? ? ? k_t ? 1, m₁ ? ?? m_r ? 1, k₁+ ? +k_t = m₁+ ? +m_r = m. Let λ=(k₁,…,k_t) be a character of the symmetric group S_m. The restriction of λ to S_m1X…XS_mr contains the principal character as a component if and only if λ majorizes (m₁,…,m_r). This result is used to characterize the index set of the nonzero decomposable symmetrized tensors, corresponding to S_m and λ, which are induced from a basis of the underlying vector space.     

An explicit expression for the cost of a class of Huffman trees

Let T_n denote a binary tree with n terminal nodes V={υ₁,…,υ_n} and let l_i denote the path length from the root to υ_i. Consider a set of nonnegative numbers W={w₁,…,w_n} and for a permutation π of {1,…,n} to {1,…,n}, associate the weight w_i to the node υ_π(i). The cost of T_n is defined as C(T_n∣W)=Min_π∑ⁿ_i=1w_il_π(i).A Huffman tree H_n is a binary tree which minimizes C(T_n∣W) over all possible T_n. In this note, we give an explicit expression for C(H_n∣W) when W assumes the form: w_i=k for i=1,…,n?m; w_i=x for i=n?m+1,…,n. This simplifies and generalizes earlier results in the literature.     

On Rado's Boundedness Conjecture

We prove that Rado's Boundedness Conjecture from Richard Rado's 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado's Boundedness Conjecture: if a₁,a₂, and a₃ are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a₁x₁+a₂x₂+a₃x₃=0, then for every finite coloring of the positive integers there is a monochromatic solution to a₁x₁+a₂x₂+a₃x₃=0.     

Bifurcation formulae derived from center manifold theory

Applicable formulae for the parameters μ₂, β₂, τ₂, μ₄, β₄ and τ₄ of N-dimensional Hopf bifurcation theory are presented. The center manifold theorem is used to reduce the system from N dimensions to 2 dimensions. Approximate solution of the system in Poincaré normal form provides the formulae. The formulae are explicit so that the parameters may be computed directly from partial derivatives of the system in real canonical form. The formula for μ₂ is shown to be identical to that of I. D. Hsu and N. D. Kazarinoff. Formulae for the parameters μ₁, μ₂, μ₃, τ₁, τ₂, τ₃, β₂, β₃ and β₄ in the “tangency” case Re λ₁′(v_c) = 0, Re λ₁′'(v_c) ≠ 0 are also presented.     

Homology of certain algebras defined by graphs

Let K(G) for a finite graph G with vertices v₁,...,v_n denote the K-algebra with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G. We describe centralizers of monomials, show that the centralizer of a monomial is again a graph algebra, prove a unique factorization theorem for factorizations of monomials into commuting factors, compute the homology of K(G), and show that K(G) is the homology ring of a certain loop space. We also construct a K(π, 1) explicitly where π is the group with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f_t(G) as the smallest number such that whenever f_t(G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f₁(G×H)≤f₁(G)f₁(H) for any connected G and H. We define the α-pebbling number α(G) and prove that α(C_pj×?×C_p2×C_p1×G)≤α(C_pj)?α(C_p2)α(C_p1)α(G) when none of the cycles is C₅, and G satisfies one more criterion. We also apply this result with G=C₅×C₅ by showing that C₅×C₅ satisfies Chung’s two-pebbling property, and establishing bounds for f_t(C₅×C₅).     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Properties of Regular Uniform k—Stratified Graphs

A regular graph G = (V, E) is a k-stratified graph if V is partitioned into V₁, V₂, …, V_k subsets called strata. The stratification splits the degree d_v ∀v ϵ V into k-integers d_v1, d_v2, …, d_vk each one corresponding to a stratum. If d_v1 = d_v2 = … = d_vk ∀v ϵ V then G is called regular uniform k-stratified, RU^k_s(n, d) where n is the cardinality of the vertex set in each stratum and d is the degree of every vertex in each stratum. For every k, the class RU^k_s(n, d) has a unique graph generator class RU^l_s(n, d) derived by decomposition of graphs in RU^k_s(n, d). We investigate the minimization of the cardinality of V, the colorability, vertex coloring and the diameter of the graphs in the class. We also deal with complexity questions concerning RU^k_s(n, d). Some well-known computer network models such as barrel shifters and hypercubes are shown to belong in RU^k_s(n, d).     

The diminishing segment process

Let Ξ₀=[−1,1], and define the segments Ξ_n recursively in the following manner: for every n=0,1,…, let Ξ_n+1=Ξ_n∩[a_n+1−1,a_n+1+1], where the point a_n+1 is chosen randomly on the segment Ξ_n with uniform distribution. For the radius ρ_n of Ξ_n, we prove that n(ρ_n−1/2) converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.     

Permanents of d-dimensional matrices

Let A = (A_i1i2…_id) be an n₁ × n₂ × · × n_d matrix over a commutative ring. The permanent of A is defined by per (A) = ∑πⁿ_1i = ₁A_iσ2(_i)σ₃(_i)…σ_d(_i), where the summation ranges over all one-to-one functions σ_k from {1,2,…, n₁} to {1,2,…, n_k}, k = 2,3,…, d. In this paper it is shown that a number of properties of permanents of 2-dimensional matrices extend to higher-dimensional matrices. In particular, permanents of nonnegative d-dimensional matrices with constant hyperplane sums are investigated. The paper concludes by introducing s-permanents, which generalize the definition above that the permanent becomes the 1-permanent, and showing that an s-permanent can always be converted into a 1-permanent.     

On a nonlinear p-adic dynamical system

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ? _p , when there exists a unique fixed point x ₀. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x ₀). We show that there exists a radius r depending on parameters of the rational function such that: when x ₀ is an attracting point then the trajectory of an inner point from the ball U _r (x ₀) goes to x ₀ and each sphere with a radius > r (with the center at x ₀) is invariant; When x ₀ is a repeller point then the trajectory of an inner point from a ball U _r (x ₀) goes forward to the sphere S _r (x ₀). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U _r (x ₀) or stays in S _r (x ₀) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U _r(x ₀) it will stay (for all the rest of time) in the sphere (outside of U _r(x ₀)) that it reached first.     

Viscosity analysis on the Boltzmann equation

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation tfε + v ·▽xfε = Q (fε,fε ) + εΔvfε as ε→ 0+ . We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L1 ((0 , T ) × RN × RN ). The proof is based on compactness analysis and velocity averaging theory.     

Sharpness of the Calderón-Lozanovskii interpolation construction

We present a short proof of the sharpness of the Calderón-Lozanovskii interpolation construction in couples of weighted L _p spaces in the “lower triangle,” i.e., for operators from a couple { L _p0 (V ₀), L _p1 (V ₁)} to a couple {L _q0 (U ₀), L _q1 (U ₁)} with p ₀ ? q ₀ and p ₁ ? q ₁. This generalizes the well-known result due to Dmitriev and Semenov on the sharpness of the Riesz-Thorin interpolation theorem in the “lower triangle” for L _p spaces on intervals.