An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

Weight distribution of the bases of a binary matroid

Let M be a weighted binary matroid and w₁ < … < w_m be the increasing sequence of all possible distinct weights of bases of M. We give a sufficient condition for the property that w₁, …, w_m is an arithmetical progression of common difference d. We also give conditions which guarantee that w_i+1 − w_i ≤ d, 1 ≤ i ≤ m −1. Dual forms for these results are given also.     

Balanced partitions of two sets of points in the plane

We prove the following theorem. Let m≥2 and q≥1 be integers and let S and T be two disjoint sets of points in the plane such that no three points of ST are on the same line, |S|=2q and |T|=mq. Then ST can be partitioned into q disjoint subsets P₁,P₂,…,P_q satisfying the following two conditions: (i) conv(P_i)∩conv(P_j)=φ for all 1≤i<j≤q, where conv(P_i) denotes the convex hull of P_i; and (ii) |P_i∩S|=2 and |P_i∩T|=m for all 1≤i≤q.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

Cycle decompositions of crowns

For an integer n3, the crown S_n⁰ is defined to be the graph with vertex set {x₀,x₁,…,x_n−1,y₀,y₁,…,y_n−1} and edge set {x_iy_j: 0i,jn−1, i≠j}. In this paper we give some sufficient conditions for the edge decomposition of the crown into isomorphic cycles.     

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes

in the space , where fL²([0,T]ⁿ) is a given function, and {η(t)}_>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t₁,…,t_n)=1_{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t₁,…,t_n)=f₁(t₁)f_n(t_n)1_{t1<t2<<tn}, with f_iL²([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.     

A dense planar point set from iterated line intersections

Given S₁, a starting set of points in the plane, not all on a line, we define a sequence of planar point sets {S_i}_i=1^∞ as follows. With S_i already determined, let L_i be the set of all the lines determined by pairs of points from S_i, and let S_i+1 be the set of all the intersection points of lines in L_i. We show that with the exception of some very particular starting configurations, the limiting point set _i=1^∞S_i is everywhere dense in the plane.     

On the degree of regularity of some equations

In this paper we investigate the behaviour of the solutions of equations Σ_I=1ⁿ a_ix_i = b, where Σ_i=1ⁿ, a_i = 0 and b ≠ 0, with respect to colorings of the set N of positive integers. It turns out that for any b ≠ 0 there exists an 8-coloring of N, admitting no monochromatic solution of x₃ − x₂ = x₂ − x₁ + b. For this equation, for b odd and 2-colorings, only an odd-even coloring prevents a monochromatic solution. For b even and 2-colorings, always monochromatic solutions can be found, and bounds for the corresponding Rado numbers are given. If one imposes the ordering x₁ < x₂ < x₃, then there exists already a 4-coloring of N, which prevents a monochromatic solution of x₃ − x₂ = x₂ − x₁ + b, where b ε N.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

On certain incomplete statistics

The foundations of the incomplete statistics recently proposed by Wang is rediscussed in the context of the canonical statistical ensemble. It is found that the incomplete normalization condition, ∑p^q_i=1 (i=1,…,w), where p_i is the probability of a given microstate, is not compatible with the entropic non-extensive formula proposed by Tsallis. It is proved that the entropic function proposed by Wang must be written as S_q=−k_B∑p_i^2q−1ln_qp_i, whereas the form proposed by Tsallis namely, S_q=−k_B∑p_i^qln_qp_i, is directly associated with the standard normalization condition (∑_ip_i=1).     

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.     

A sufficient and necessary condition for an OWA bag mapping having the self-identity

A mapping ƒ : _n=1^∞Iⁿ → I is called a bag mapping having the self-identity if for every (x₁,…,x_n) ε _i=1^∞Iⁿ we have (1) ƒ(x₁,…,x_n) = ƒ(x_i1,…,x_in) for any arrangement (i₁,…,i_n) of {1,…,n}; monotonic; (3) ƒ(x₁,…,x_n, ƒ(x₁,…,x_n)) = ƒ(x₁,…,x_n). Let {ω_i,n : I = 1,…,n;n = 1,2,…} be a family of non-negative real numbers satisfying Σ_i=1ⁿω_i,n = 1 for every n. Then one calls the mapping ƒ : _i=1^∞Iⁿ → I defined as follows an OWA bag mapping: for every (x₁,…,x_n) ε _i=1^∞Iⁿ, ƒ(x₁,…,x_n) = Σ_i=1ⁿω_i,ny_i, where y_i is the it largest element in the set {x₁,…,x_n}. In this paper, we give a sufficient and necessary condition for an OWA bag mapping having the self-identity.     

The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f₁,f₂,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size f_i. It is proved that there exist constants c=c(n) and k₀=k₀(n) such that f_k+1=f_k+f_k−c for all k>k₀, and lim_n→∞ c(n)/(nlogn)=1.     

About “On certain incomplete statistics” by Lima et al.

Lima et al. recently claim that (Chaos, Solitons & Fractals 19 (2004) 1005) the entropy for the incomplete statistics based on the normalization ∑_ip_i^q=1 should be S=−∑_ip_i^2q−1ln_qp_i instead of S=−∑_ip_i^qln_qp_i initially proposed by Wang. We indicate here that this conclusion is a result of erroneous use of temperature definition for the incomplete statistics.     

Loopless generation of up–down permutations

An up–down permutation P=(p₁,p₂,…,p_n) is a permutation of the integers 1 to n which satisfies constraints specified by a sequence C=(c₁,c₂,…,c_n−1) of U's and D's of length n−1. If c_i is U then p_i<p_i+1 otherwise p_i−1>p_i. A loopless algorithm is developed for generating all the up–down permutations satisfying any sequence C. Ranking and unranking algorithms are discussed.     

Embeddings of the base and bundle isomorphisms

Let π_i :E_i→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections s_i :M→E_i. Suppose that U is an open neighborhood of s₁(M) in E₁ and F :U→E₂ a smooth embedding so that π₂Fs₁ :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E₁ and the induced bundle f^*E₂ are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<kn we give examples of nonisomorphic oriented bundles E₁ and E₂ of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E₁ and f^*E₂ are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.