scholarly journals On Weighted Pál Type (0,2)-interpolation on the Unit Circle

2021 ◽  
Author(s):  
Swarnima Bahadur ◽  
Sariya Bano
Keyword(s):  
Unit Circle ◽  
Legendre Polynomial ◽  
Pairwise Disjoint ◽  
Explicit Representation ◽  
Disjoint Sets ◽  
Ams Classification

Abstract In this paper, we study the explicit representation of weighted Pál-type (0,2)-interpolation on two pairwise disjoint sets of nodes on the unit circle, which are obtained by projecting vertically the zeros of (1−x2)Pn(x) and Pn′′(x) respectively, where Pn(x) stands for nth Legendre polynomial.AMS Classification (2000): 41A05, 30E10.

An Analogue of a Problem of J. Balázs and P. Turán

1969 ◽  
Vol 21 ◽  
pp. 54-63 ◽  
Author(s):  
J. Prasad ◽  
A. K. Varma
Keyword(s):  
Differential Equation ◽  
Modulus Of Continuity ◽  
Legendre Polynomial ◽  
Existence And Uniqueness ◽  
Explicit Representation ◽  
Lacunary Interpolation ◽  
First Derivative ◽  
Second Derivatives ◽  
Significant Application

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

scholarly journals Families with no s pairwise disjoint sets

2017 ◽  
Vol 95 (3) ◽  
pp. 875-894 ◽  
Author(s):  
Peter Frankl ◽  
Andrey Kupavskii
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Sets

Strongs-Sequences and Variations on Martin's Axiom

1985 ◽  
Vol 37 (4) ◽  
pp. 730-746 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Family ◽  
Uncountable Family ◽  
Martin’S Axiom ◽  
Finite Set ◽  
Almost Disjoint ◽  
Disjoint Sets ◽  
Almost Disjoint Family ◽  
Require Equality ◽  
Single Set

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

scholarly journals On the One Extremal Problem with the Free Poles on the Unit Circle

2016 ◽  
Vol 6 ◽  
pp. 26-31 ◽  
Author(s):  
Andrey L. Targonskii
Keyword(s):  
Finite Number ◽  
Unit Circle ◽  
Extremal Problem ◽  
Pairwise Disjoint ◽  
Sharp Estimates ◽  
Inner Radius ◽  
The One ◽  
Disjoint Domains

The sharp estimates of the product of the inner radius for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of an arbitrary finite number of the free poles on the unit circle for the following functional (see formula in paper)

scholarly journals Fitting classes and lattice formations I

2004 ◽  
Vol 76 (1) ◽  
pp. 93-108 ◽  
Author(s):  
M. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Direct Product ◽  
Pairwise Disjoint ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Closure Properties ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
Disjoint Sets ◽  
Lattice Formation ◽  
Fitting Classes

AbstractA lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.

Two Isomorphic 9 Order Steiner Triple Systems Large Sets

2013 ◽  
Vol 427-429 ◽  
pp. 1237-1240
Author(s):  
Zhao Di Xu ◽  
Xiao Yi Li ◽  
Wan Xi Chou
Keyword(s):  
Pairwise Disjoint ◽  
Permutation Matrix ◽  
Large Set ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Large Sets ◽  
Initial Block ◽  
Disjoint Sets ◽  
Basic Ideas ◽  
Block Permutation

This paper Clarifies the basic ideas of constructing the v order Steiner triple systems. This paper proposed the construction method of pairwise disjoint sets s(i)(v) for Steiner triple systems based on the initial block permutation matrix. And a method of initial block permutation matrix is given. This paper also introduced the entire construction process of two isomorphic 9 order Steiner triple systems large set. At last, this paper proved the number of pairwise disjoint forsi(9)is d(9)=7 .

Some Interpolators Properties of Laguerre Polynomials*

1967 ◽  
Vol 10 (4) ◽  
pp. 559-571 ◽  
Author(s):  
J. Prasad ◽  
R. B. Saxena
Keyword(s):  
Legendre Polynomial ◽  
Laguerre Polynomials ◽  
Explicit Representation ◽  
Second Derivatives

In 1955, J. Suranyi and P. Turán [8] introduced the nomenclature (0, 2)-interpolation for the problem of finding polynomials of degree ≤ 2n-1 whose values and second derivatives are prescribed in certain given nodes. In a series of papers ([2], [3], [8]) Professor Turán and his associates discussed the problems of existence, uniqueness, explicit representation and convergence of such interpolatory 2 polynomials when the nodes are the zeros of (1-x2) P′n- 1(x), Pn- 1(x) being the Legendre polynomial of degree n - 1.

scholarly journals Problem on extremal decomposition of the complex plane

2019 ◽  
Vol 27 (1) ◽  
pp. 61-77
Author(s):  
Iryna Denega ◽  
Yaroslav Zabolotnii
Keyword(s):  
Unit Circle ◽  
General Problem ◽  
Complex Variable ◽  
Pairwise Disjoint ◽  
Function Theory ◽  
Geometric Function Theory ◽  
Extremal Decomposition ◽  
Geometric Function ◽  
Inner Radius ◽  
Disjoint Domains

Abstract In geometric function theory of a complex variable problems on extremal decomposition with free poles on the unit circle are well known. One of such problem is the problem on maximum of the functional $${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$ where B0, B1, B2,..., Bn, n ≥ 2, are pairwise disjoint domains in ¯𝔺, a0 = 0, |ak| = 1, $k = \overline {1,n}$ and γ ∈ 2 (0; n], r(B, a) is the inner radius of the domain, B ⊂ ¯𝔺, with respect to a point a ∈ B. In the paper we consider a more general problem in which restrictions on the geometry of the location of points ak, $k = \overline {1,n}$ are weakened.

scholarly journals Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

2018 ◽  
Vol 2020 (13) ◽  
pp. 4037-4061 ◽  
Author(s):  
Florian Frick
Keyword(s):  
Euclidean Space ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Pairwise Disjoint ◽  
Convex Hulls ◽  
Chromatic Numbers ◽  
Set Systems ◽  
Equivariant Topology ◽  
Disjoint Sets ◽  
Combination Of Methods

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser’s conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.

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