Exact upper bound for sorting Rn with LE

2020 ◽  
Vol 12 (03) ◽  
pp. 2050033
Author(s):  
Sai Satwik Kuppili ◽  
Bhadrachalam Chitturi
Keyword(s):  
Symmetric Group ◽  
Upper Bound ◽  
Optimum Number ◽  
Identity Permutation ◽  
Pair Exchange ◽  
The Right

A permutation over alphabet [Formula: see text] is a sequence over [Formula: see text], where every element occurs exactly once. [Formula: see text] denotes symmetric group defined over [Formula: see text]. [Formula: see text] denotes the Identity permutation. [Formula: see text] is the reverse permutation i.e., [Formula: see text]. An operation, that we call as an LE operation, has been defined which consists of exactly two generators: set-rotate that we call Rotate and pair-exchange that we call Exchange (OEIS). At least two generators are the required to generate [Formula: see text]. Rotate rotates all elements to the left (moves leftmost element to the right end) and Exchange is the pair-wise exchange of the two leftmost elements. The optimum number of moves for transforming [Formula: see text] into [Formula: see text] with LE operation are known for [Formula: see text]; as listed in OEIS with identity A048200. However, no general upper bound is known. The contributions of this article are: (a) a novel upper bound for the number of moves required to sort [Formula: see text] with LE has been derived; (b) the optimum number of moves to sort the next larger [Formula: see text] i.e., [Formula: see text] has been computed; (c) an algorithm conjectured to compute the optimum number of moves to sort a given [Formula: see text] has been designed.

scholarly journals Coefficient inequalities for a class of analytic functions associated with the lemniscate of Bernoulli

2018 ◽  
Vol 37 (4) ◽  
pp. 83-95
Author(s):  
Trailokya Panigrahi ◽  
Janusz Sokól
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
Toeplitz Determinant ◽  
Second Hankel Determinant ◽  
Coefficient Inequalities ◽  
The Right

In this paper, a new subclass of analytic functions ML_{\lambda}^{*}  associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}|  for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

scholarly journals Congruences induced by transitive representations of inverse semigroups

1987 ◽  
Vol 29 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Mario Petrich ◽  
Stuart Rankin
Keyword(s):  
General Theory ◽  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transitive Group ◽  
Inverse Semigroups ◽  
Group Representations ◽  
Symmetric Inverse Semigroup ◽  
The Right ◽  
Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

scholarly journals Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

10.37236/6960 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Symmetric Group ◽  
Finite Subset ◽  
Upper Bound ◽  
Cohomology Class ◽  
Symmetric Functions ◽  
A Priori ◽  
Specht Module ◽  
Group A ◽  
Schubert Classes ◽  
Stanley Symmetric Functions

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

scholarly journals A remark on covering of compact Kähler manifolds and applications

2021 ◽  
Vol 73 (1) ◽  
pp. 138-148
Author(s):  
V. V. Hung ◽  
H. N. Quy
Keyword(s):  
Upper Bound ◽  
Kähler Manifold ◽  
Kahler Manifolds ◽  
Hölder Continuous ◽  
Kahler Manifold ◽  
Right Hand ◽  
Regularization Techniques ◽  
The Right ◽  
Compact Kähler Manifold ◽  
Monge Ampere

UDC 517.9 Recently, Kolodziej proved that, on a compact Kähler manifold the solutions to the complex Monge – Ampére equation with the right-hand side in are Hölder continuous with the exponent depending on and (see [Math. Ann., <strong>342</strong>, 379-386 (2008)]).Then, by the regularization techniques in[J. Algebraic Geom., <strong>1</strong>, 361-409 (1992)], the authors in [J. Eur. Math. Soc., <strong>16</strong>, 619-647 (2014)] have found the optimal exponent of the solutions.In this paper, we construct a cover of the compact Kähler manifold which only depends on curvature of Then, as an application, base on the arguments in[Math. Ann., <strong>342</strong>, 379-386 (2008)], we show that the solutions are Hölder continuous with the exponent just depending on the function in the right-hand side and upper bound of curvature of  

A Remark by Philip Hall

1958 ◽  
Vol 1 (1) ◽  
pp. 21-23 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Linear Transformations ◽  
Modern Physics ◽  
The Right ◽  
The Relationship

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

scholarly journals Better Distance Preservers and Additive Spanners

2021 ◽  
Vol 17 (4) ◽  
pp. 1-24
Author(s):  
Greg Bodwin ◽  
Virginia Vassilevska Williams
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Error Function ◽  
Shortest Paths ◽  
Prior Work ◽  
Input Graph ◽  
Original Graph ◽  
New Construction ◽  
The Right ◽  
Unweighted Graph

We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers , which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work on distance preservers has exploited only a simple structural property of shortest paths, called consistency , stating that one can break shortest path ties such that no two paths intersect, split apart, and then intersect again later. We prove that consistency alone is not enough to understand distance preservers, by showing both a lower bound on the power of consistency and a new general upper bound that polynomially surpasses it. Specifically, our new upper bound is that any p demand pairs in an n -node undirected unweighted graph have a distance preserver on O( n 2/3 p 2/3 + np 1/3 edges. We leave a conjecture that the right bound is O ( n 2/3 p 2/3 + n ) or better. The second part of this paper leverages these distance preservers in a new construction of additive spanners , which are subgraphs that preserve all pairwise distances up to an additive error function. We give improved error bounds for spanners with relatively few edges; for example, we prove that all graphs have spanners on O(n) edges with + O ( n 3/7 + ε ) error. Our construction can be viewed as an extension of the popular path-buying framework to clusters of larger radii.

scholarly journals Third Hankel Determinant for a Subclass of Univalent Functions Associated with Lemniscate of Bernoulli

2022 ◽  
Vol 6 (1) ◽  
pp. 48
Author(s):  
Najeeb Ullah ◽  
Irfan Ali ◽  
Sardar Muhammad Hussain ◽  
Jong-Suk Ro ◽  
Nazar Khan ◽  
...  
Keyword(s):  
Univalent Function ◽  
Upper Bound ◽  
Univalent Functions ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
Efficient Estimate ◽  
The Right

This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We have find the upper bound of the Hankel determinant H3(1) for this subclass by applying the Carlson–Shaffer operator to it. The present work also deals with certain properties of this newly defined subclass, such as the upper bound of the Hankel determinant of order 3, the co-efficient estimate, etc.

scholarly journals COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

2013 ◽  
Vol 24 (07) ◽  
pp. 1009-1027 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
HELLIS TAMM
Keyword(s):  
Upper Bound ◽  
Regular Language ◽  
Regular Languages ◽  
State Complexity ◽  
Empty Intersection ◽  
Language L ◽  
The Right

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n; for 1 ≤ r ≤ n − 1 the bound is[Formula: see text] For each n ≥ 2, we exhibit a language with 2n atoms which meet these bounds.

scholarly journals Obtaining Reliable Feedback for Sanctioning Reputation Mechanisms

2007 ◽  
Vol 29 ◽  
pp. 391-419 ◽  
Author(s):  
R. Jurca ◽  
B. Faltings
Keyword(s):  
Upper Bound ◽  
Pareto Optimal ◽  
Rational Behavior ◽  
Side Payment ◽  
Business Decisions ◽  
Building Trust ◽  
Cheating Behavior ◽  
Payment Schemes ◽  
Reputation Mechanisms ◽  
The Right

Reputation mechanisms offer an effective alternative to verification authorities for building trust in electronic markets with moral hazard. Future clients guide their business decisions by considering the feedback from past transactions; if truthfully exposed, cheating behavior is sanctioned and thus becomes irrational. It therefore becomes important to ensure that rational clients have the right incentives to report honestly. As an alternative to side-payment schemes that explicitly reward truthful reports, we show that honesty can emerge as a rational behavior when clients have a repeated presence in the market. To this end we describe a mechanism that supports an equilibrium where truthful feedback is obtained. Then we characterize the set of pareto-optimal equilibria of the mechanism, and derive an upper bound on the percentage of false reports that can be recorded by the mechanism. An important role in the existence of this bound is played by the fact that rational clients can establish a reputation for reporting honestly.

scholarly journals Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 848
Author(s):  
Hari M. Srivastava ◽  
Qazi Zahoor Ahmad ◽  
Maslina Darus ◽  
Nazar Khan ◽  
Bilal Khan ◽  
...  
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Convex Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
The Third ◽  
The Right

In this paper, our aim is to define a new subclass of close-to-convex functions in the open unit disk U that are related with the right half of the lemniscate of Bernoulli. For this function class, we obtain the upper bound of the third Hankel determinant. Various other related results are also considered.

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