Rational approximation with varying weights I

We investigate two problems concerning uniform approximation by weighted rationals {w ⁿr_n～ _∞ ⁿ⁼¹ }, wherer _n=p_n Namely, forw(x):=e ^x we prove that uniform convergence to 1 ofw ⁿr_n is not possible on any interval [0,a] witha>2π. Forw(x):=x ^?, ?>1, we show that uniform convergence to 1 ofw ⁿr_n is not possible on any interval [b, 1] withb<tan ⁴(π(??1)/4?). (The latter result can be interpreted as a rational analogue of results concerning “incomplete polynomials.”) More generally, for α≥0, β≥0, α+β>0, we investigate forw(x)=e ^x andw(x)=x ^?, the possibility of approximation byw ⁿ p_n/q_n～ _∞ ⁿ⁼¹ , where depp _n≤αn, degq _n≤βn. The analysis utilizes potential theoretic methods. These are essentially sharp results though this will not be established in this paper.     

Order of magnitude of multiple Fourier coefficients of functions of bounded <Emphasis Type="Italic">p</Emphasis>-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \(\mathbb{T}^n \):= [0, 2π) ⁿ , let \(\hat f\)(k) denote the Fourier coefficient of f, where k = (k ₁, … k _n ) ∈ ? ⁿ . In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π] ⁿ to ? in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Móricz [5] and Fülöp and Móricz [3].     

Some results on Hu's conjecture concerning binary trees

Given n weights, w₁, w₂,…, w_n, such that 0?w₁?w₂???w₁, we examine a property of permutation π¹, where π¹=(w₁, w_n, w₂, w_n?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by V_u(π).This paper asserts that V_u(π¹)?V_u(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π¹)?V(π) for all π.     

Partitions of a set satisfying certain set of conditions

In this paper we present a new combinatorial class enumerated by Catalan numbers. More precisely, we establish a bijection between the set of partitions π₁π₂?π_n of [n] such that π_i+1−π_i≤1 for all i=,1,2…,n−1, and the set of Dyck paths of semilength n. Moreover, we find an explicit formula for the generating function for the number of partitions π₁π₂?π_n of [n] such that either π_i+?−π_i≤1 for all i=1,2,…,n−?, or π_i+1−π_i≤m for all i=1,2,…,n−1.     

Quasi-symmetric functions and up–down compositions

Carlitz (1973) [5] and Rawlings (2000) [13] studied two different analogues of up–down permutations for compositions with parts in {1,…,n}. Cristea and Prodinger (2008/2009) [7] studied additional analogues for compositions with unbounded parts. We show that the results of Carlitz, Rawlings, and Cristea and Prodinger on up–down compositions are special cases of four different analogues of generalized Euler numbers for compositions. That is, for any s≥2, we consider classes of compositions that can be divided into an initial set of blocks of size s followed by a block of size j where 0≤j≤s−1. We then consider the classes of such compositions where all the blocks are strictly increasing (weakly increasing) and there are strict (weak) decreases between blocks. We show that the weight generating functions of such compositions w=w₁?w_m, where the weight of w is , are always the quotients of sums of quasi-symmetric functions. Moreover, we give a direct combinatorial proof of our results via simple involutions.     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

On the minimum number of completely 3-scrambling permutations

A family P={π₁,…,π_q} of permutations of [n]={1,…,n} is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x₁,…,x_k∈[n], permutations π_i's in P produce all k! possible orders on π_i(x₁),…,π_i(x_k). Let N^*(n,k) be the minimum size of such a family. This paper focuses on the case k=3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison.

     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.     

Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V₀={p₁,p₂,p₃} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw₁uw₂?uwn for any sequence w=(w₁,w₂,…,wn)∈n{1,2,3}. The union of the images of V₀ under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that

f(uw(x))=αwf(x)+hw(x)     

A note on the continuous mixing set

The continuous mixing set is , where w₁,…,w_n>0 and f₁,…,f_n∈ℜ. Let m=|{w₁,…,w_n}|. We show that when w₁|?|w_n, optimization over S can be performed in time O(n^m+1), and in time O(nlogn) when w₁=?=w_n=1.     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

On coherent families of uniformizing elements in some towers of Abelian extensions of local number fields

For a local number field K with the ring of integers \( {\mathcal{O}_K} \), the residue field \( {\mathbb{F}_q} \), and uniformizing π, we consider the Lubin–Tate tower \( {K_\pi } = \bigcap\limits_{n \geqslant 0} {{K_n}} \), where K _n = K(π _n ), f(π0) = 0, and f(π _n +1) = π _n . Here f(X) defines the endomorphism [π] of the Lubin–Tate group. If q ≠ 2, then for any formal power series \( g(X) \in {\mathcal{O}_K}\left[ {\left[ X \right]} \right] \) the following equality holds: \( \sum\limits_{n = 0}^\infty {{\text{SP}}{{{K_n}} \mathord{\left/{\vphantom {{{K_n}} K}} \right.} K}} g\left( {{\pi_n}} \right) = - g(0) \). One has a similar equality in the case q = 2.     

The role of the endpoint in weighted polynomial approximation with varying weights

For a weight functionw: [a, b]→(0, ∞), we consider weighted polynomials of the formw ⁿ P_n where the degree ofP _n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t?a) ^β ast→a with β>?1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials.     

Small alliances in a weighted graph

We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑_{x∈NG(v)∩S}w(x)+w(v)≥∑_{x∈NG(v)−S}w(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3].     

On approximation by polynomials increasing to the right of the interval

We obtain sufficient conditions on a real valued function ?, continuous on [0, + ∞), to insure that, for some nonnegative integer n, there is a nonnegative number r(n) so that for any r ? r(n), the polynomial of best approximation to ? on [0, r] from π_n is increasing and nonnegative on [r, + ∞). Here, π_n denotes the set of all real polynomials of degree n or less. The proofs of Theorems 1 and 2 use only properties of Lagrange interpolation while that of Theorem 3 employs results on the location of interpolation points in Chebyshev approximation.     

Embeddings between weak Orlicz martingale spaces

Let Φ be an increasing and convex function on [0,∞) with Φ(0)=0 satisfying that for any α>0, there exists a positive constant Cα such that Φ(αt)?CαΦ(t), t>0. Let wLΦ denote the corresponding weak Orlicz space. We obtain some embeddings between vector-valued weak Orlicz martingale spaces by establishing the wLΦ-inequalities for martingale transform operators with operator-valued multiplying sequences. These embeddings are closely related to the geometric properties of the underlying Banach space. In particular, for any scalar valued martingale f=(fn)_n?1, we claim that

     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).     

A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.