The 2-pebbling property for dense graphs

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f(G) is the smallest number m such that for every distribution of m pebbles and every vertex v,a pebble can be moved to v. A graph G is said to have the 2-pebbling property if for any distribution with more than 2f(G) q pebbles, where q is the number of vertices with at least one pebble, it is possible,using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G(s,t) has the 2-pebbling property, where G(s, t) is a bipartite graph with partite sets of size s and t (s ≥ t). Similarly, the-pebbling number f (G) is the smallest number m such that for every distribution of m pebbles and every vertex v, pebbles can be moved to v. Herscovici et al. conjectured that f(G) ≤ 1.5n + 8-6 for the graph G with diameter 3, where n = |V (G)|. In this paper, we prove that if s ≥ 15 and G(s, t) has minimum degree at least (s+1)/ 2 , then f (G(s, t)) = s + t, G(s, t) has the 2-pebbling property and f (G(s, t)) ≤ s + t + 8(-1). In other words, we extend a result due to Czygrinow and Hurlbert, and show that the above Snevily conjecture and Herscovici et al. conjecture are true for G(s, t) with s ≥ 15 and minimum degree at least (s+1)/ 2 .     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

On the polynomiality and asymptotics of moments of sizes for random(n,dn ± 1)-core partitions with distinct parts

Amdeberhan’s conjectures on the enumeration,the average size,and the largest size of(n,n+1)-core partitions with distinct parts have motivated many research on this topic.Recently,Straub(2016)and Nath and Sellers(2017)obtained formulas for the numbers of(n,dn-1)-and(n,dn+1)-core partitions with distinct parts,respectively.Let X_s,t be the size of a uniform random(s,t)-core partition with distinct parts when s and t are coprime to each other.Some explicit formulas for the k-th moments E[X_n,n+1^k]and E[X_(2 n+1,2 n+3)^k]were given by Zaleski and Zeilberger(2017)when k is small.Zaleski(2017)also studied the expectation and higher moments of X_n,dn-1 and conjectured some polynomiality properties concerning them in ar Xiv:1702.05634.Motivated by the above works,we derive several polynomiality results and asymptotic formulas for the k-th moments of X_n,dn+1 and X_n,dn-1 in this paper,by studying theβ-sets of core partitions.In particular,we show that these k-th moments are asymptotically some polynomials of n with degrees at most 2 k,when d is given and n tends to infinity.Moreover,when d=1,we derive that the k-th moment E[X_n,n+1^k]of X_n,n+1 is asymptotically equal to(n²/10)^kwhen n tends to infinity.The explicit formulas for the expectations E[X_n,dn+1]and E[X_n,dn-1]are also given.The(n,dn-1)-core case in our results proves several conjectures of Zaleski(2017)on the polynomiality of the expectation and higher moments of X_n,dn-1.     

Nearly Comonotone Approximation of Periodic Functions

Suppose that a continuous 27r-periodic function f on the real axis changes its monotonicity at points y_1:-π≤ y_(2s)y_(2s-1)… y_1 π,s ∈ IN.In this PaPer,for each n≥N,a trigonometric polynomial P_n of order cn is found such that:P_n has the same monotonicity as f,everywhere except,perhaps,the small intervals(y_i-π/n,y_i+π/n)and‖f-P_n‖c(s)ω_3(f,π/n),where N is a constant depending only on mini=1,...,2s {y_i-y_(i+1)},c,c(s) are constants depending only on s,ω_3(f_1,·) is the modulus of smoothness of the 3-rd order of the function f,and ||·|| is the max-norm.     

The largest eigenvalue distribution of the laguerre unitary ensemble

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in(0, t), that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, α_n(t) and β_n(t), as well as three auxiliary quantities, denoted by r_n(t), R_n(t), and σ_n(t). We establish the second order differential equations for both β_n(t) and r_n(t). By investigating the soft edge scaling limit when α = O(n) as n →∞ or α is finite, we derive a P_Ⅱ, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials P_n(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n,asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy's equation for ρ(t) = Ξ′(t) with Ξ(t) satisfying the σ-form of P_Ⅳ or P_Ⅴ.     

WEAKLY ALMOST PERIODIC POINT AND ERGODIC MEASURE

Let X be a compact metric space and f: X→X be continuous.This pape introduces the notion of weakly almost periodic point, which is a generalization of the notion of almost periodic point, proves that each of f-invariant ergodic measures can be generated by a weakly almost periodic point of f and gives some equivalent conditions for that f has an invariant ergodic measure whose support is X and ones for that f has no non-atomic invariant ergodic measure, the latter is a generalization of the Blokh's work on self-maps of the interval. Also two formulae for calculating the togological entropy are obtained.     

A New Approach to Synchronization Analysis of Linearly Coupled Map Lattices

In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector x(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t 1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each xi(t) for i = 1, ... ,m, but not a solution s(t) satisfying s(t 1) = f(s(t)).     

Pansystems Whole-Part Relation Analysis and Triple Covering Systems

Let A_n={1,2,…,n} and let ％={B_1,B_2,…,B_r} where B_1、B_2,…, B_r are subsets of A_n each of size m. is said to cover all the triples (i, j, k), 1≤i2/3, then V(m, n) is a function of the fraction m/n only and the values of V(m, n) are determined for all m, n with m/n≥2/3. The value of V(m, n) for m/n<2/3 is also discussed.     

DOUBLE φ-INEQUALITIES FOR BANACH-SPACE-VALUED MARTINGALES

Let B be a Banach space, Φ1 , Φ2 be two generalized convex Φ-functions and Ψ 1 , Ψ 2 the Young complementary functions of Φ1 , Φ2 respectively with ∫t t 0 ψ2 (s) s ds ≤ c 0 ψ1 (c 0 t) (t > t 0 ) for some constants c 0 > 0 and t 0 > 0, where ψ1 and ψ2 are the left-continuous derivative functions of Ψ 1 and Ψ 2 , respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (f n ) n ≥ 0 , ‖f*‖Φ1 ≤ c‖S (p) (f ) ‖Φ2 (or ‖S (q) (f )‖Φ1 ≤ c‖f*‖Φ2 , respectively), where f and S (p) (f ) are the maximal function and the p-variation function of f respec- tively; (ii) If B is a UMD space, T v f is the martingale transform of f with respect to v = (v n ) n ≥ 0 (v*≤ 1), then ‖(T v f )*‖Φ1 ≤ c ‖f *‖Φ2 .     

Effective computation of singularities of parametric affine curves

Let k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x₁=f₁(t),…,x_n=f_n(t) over an arbitrary ground field k, where f₁,…,f_n∈k[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.     

On oscillation of solutions of higher order forced functional differential inequalities

Oscillation criteria for the class of forced functional differential inequalities x(t){L_nx(t) + f(t, x(t), x[g₁(t)],…, x[g_m(t)]) ? h(t)} ? 0, for n even, and x(t){L_nx(t) ? f(t, x(t), x[g₁(t)],…, x[g_m(t)]) ? h(t)} ? 0, for n odd, are established.     

Oscillation and asymptotic behavior of bounded solutions of functional differential equations with delay

Conditions on a(t), g(t), and f(t) have been found under which the bounded nonoscillatory solutions of the equation y⁽ⁿ⁾(t) ? a(t) y(g(t)) = f(t) approach zero. For the even order equation y⁽²ⁿ⁾(t) ? a(t) y(g(t)) = f(t) the delay is shown to be causing the oscillatory behavior.     

On oscillation and nonoscillation of general functional differential equations

Oscillation and nonoscillation criteria for the nth order nonlinear functional differential equation L_nx(t) + f(t, x(t), x[g(t)]) = h(t) are established. Some illustrative examples are also included.     

Approximation of curves by polygons

Let A be a smooth curve in a Euclidean space E given by an arc length parametrization f: [0, 1] → E. Let π_n = {0 = t₀ ≤ t₁ ≤ … ≤ t_n = 1} be a partition of [0, 1] and let P_n be the polygon with vertices f(t₀), f(t₁),…, f(t_n). Let L(A) and L(P_n) denote the lengths of A and P_n, respectively. The paper investigates the behavior of n² |L(A) ? L(P_n)| when the partition π_n is induced by the sequence nθ(mod 1) for some irrational number θ. It turns out that this behavior depends on the partial quotients of the continued fraction expansion of θ.     

On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT _t (n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with α _t (G) ≤ f(n), where α _t (G) is the maximum number of vertices in a K _t -free induced subgraph of G. Erd?s, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RT _t (n,K _s , o(n)), among them finding the minimum ? such that RT _t (n,K _t+? , o(n)) = Ω(n ²), where it is easy to see that RT _t (n,K _t+1, o(n)) = o(n ²). In this paper, we answer this question by proving that RT _t (n,K _t+2, o(n)) = Ω(n ²); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.     

Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

On the measurability of functions with quasi-continuous and upper semi-continuous vertical sections

Let f: ?² → ? be a function with upper semicontinuous and quasi-continuous vertical sections f _x (t) = f(x, t), t, x ∈ ?. It is proved that if the horizontal sections f ^y (t) = f(t, y), y, t ∈ ?, are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.