scholarly journals An Approximate Vertex-Isoperimetric Inequality for $r$-sets

10.37236/2458 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Demetres Christofides ◽  
David Ellis ◽  
Peter Keevash
Keyword(s):  
Isoperimetric Inequality ◽  
Absolute Constant ◽  
Constant Factor ◽  
Positive Absolute Constant

We prove a vertex-isoperimetric inequality for \([n]^{(r)}\), the set of all \(r\)-element subsets of \(\{1,2,\ldots,n\}\), where \(x,y \in [n]^{(r)}\) are adjacent if \(|x \Delta y|=2\). Namely, if \(\mathcal{A} \subset [n]^{(r)}\) with \(|\mathcal{A}|=\alpha {n \choose r}\), then its vertex-boundary \(b(\mathcal{A})\) satisfies\[|b(\mathcal{A})| \geq c\sqrt{\frac{n}{r(n-r)}} \alpha(1-\alpha) {n \choose r},\]where \(c\) is a positive absolute constant. For \(\alpha\) bounded away from 0 and 1, this is sharp up to a constant factor (independent of \(n\) and \(r\)).

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

1969 ◽  
Vol 12 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Asymptotic Formula ◽  
Absolute Constant ◽  
Log P ◽  
Positive Absolute Constant ◽  
Image Position

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

Probabilities of ruin when the safety loading tends to zero

2000 ◽  
Vol 32 (3) ◽  
pp. 885-923 ◽  
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Finite Time ◽  
Absolute Constant ◽  
Classical Result ◽  
Risk Theory ◽  
Premium Rate ◽  
Collective Risk ◽  
Time Period ◽  
Positive Absolute Constant ◽  
Uniform Approximations ◽  
Insurance Business

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

Probabilities of ruin when the safety loading tends to zero

2000 ◽  
Vol 32 (03) ◽  
pp. 885-923 ◽  
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Finite Time ◽  
Absolute Constant ◽  
Classical Result ◽  
Risk Theory ◽  
Premium Rate ◽  
Collective Risk ◽  
Time Period ◽  
Positive Absolute Constant ◽  
Uniform Approximations ◽  
Insurance Business

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

scholarly journals Directed Graphs Without Short Cycles

2009 ◽  
Vol 19 (2) ◽  
pp. 285-301 ◽  
Author(s):  
JACOB FOX ◽  
PETER KEEVASH ◽  
BENNY SUDAKOV
Keyword(s):  
Directed Graph ◽  
Absolute Constant ◽  
Simple Structure ◽  
Directed Graphs ◽  
Constant Factor ◽  
Directed Cycle ◽  
Strong Component ◽  
Feedback Arc Set ◽  
Directed Cycles

For a directed graph G without loops or parallel edges, let β(G) denote the size of the smallest feedback arc set, i.e., the smallest subset X ⊂ E(G) such that G ∖ X has no directed cycles. Let γ(G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r2, where c is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour and Sullivan.This result can also be used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn2, then for any 0 ≤ m ≤ θn − o(n) it contains a directed cycle whose length is between m and m + 6θ−1/2. Moreover, there is a constant C such that either G contains directed cycles of every length between C and θn − o(n) or it is close to a digraph G′ with a simple structure: every strong component of G′ is periodic. These results are also tight up to the constant factors.

Decompositions into Subgraphs of Small Diameter

2010 ◽  
Vol 19 (5-6) ◽  
pp. 753-774 ◽  
Author(s):  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Absolute Constant ◽  
Minimum Degree ◽  
Small Diameter ◽  
Constant Factor ◽  
Regularity Lemma ◽  
Small Worlds ◽  
Dense Graph ◽  
Edge Partition ◽  
Szemeredi's Regularity Lemma

We investigate decompositions of a graph into a small number of low-diameter subgraphs. Let P(n, ε, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ ⋅⋅⋅ ∪ Ek such that |E0| ≤ εn2, and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi's regularity lemma, Polcyn and Ruciński showed that P(n, ε, 4) is bounded above by a constant depending only on ε. This shows that every dense graph can be partitioned into a small number of ‘small worlds’ provided that a few edges can be ignored. Improving on their result, we determine P(n, ε, d) within an absolute constant factor, showing that P(n, ε, 2) = Θ(n) is unbounded for ε < 1/4, P(n, ε, 3) = Θ(1/ε2) for ε > n−1/2 and P(n, ε, 4) = Θ(1/ε) for ε > n−1. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low-diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, Rödl, Ruciński and Szemerédi.

scholarly journals On Regular Hypergraphs of High Girth

10.37236/3851 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
David Ellis ◽  
Nathan Linial
Keyword(s):  
Symmetric Group ◽  
Lower Bounds ◽  
Upper Bound ◽  
Absolute Constant ◽  
High Probability ◽  
Constant Factor ◽  
Positive Probability ◽  
Uniform Hypergraphs ◽  
Definition Of

We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform 'Cayley' hypergraph on the symmetric group $S_n$ which has girth $\Omega (\sqrt{\log |S_n|})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.

scholarly journals The Quest for Strong Inapproximability Results with Perfect Completeness

2021 ◽  
Vol 17 (3) ◽  
pp. 1-35
Author(s):  
Joshua Brakensiek ◽  
Venkatesan Guruswami
Keyword(s):  
Constraint Satisfaction ◽  
Absolute Constant ◽  
Independent Set ◽  
Constant Factor ◽  
Constraint Satisfaction Problems ◽  
Uniform Hypergraph ◽  
Semidefinite Programming Relaxation ◽  
Worst Case ◽  
Unique Games Conjecture ◽  
Unique Games

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

The growth of univalent functions with an initial gap II

2007 ◽  
Vol 142 (2) ◽  
pp. 305-318
Author(s):  
DOV AHARONOV ◽  
WALTER K. HAYMAN
Keyword(s):  
Unit Disk ◽  
Absolute Constant ◽  
Univalent Functions ◽  
Positive Absolute Constant ◽  
Image Position

AbstractWe consider the class Sp of functions univalent in the unit disk Δ.In [1] it was shown that if f∈Sp and p is large, (0.1) Here we show that there exists f in Sp for p=1,2,. . . such that where C0 is a positive absolute constant.

scholarly journals THE SECOND DERIVATIVE OF A MEROMORPHIC FUNCTION

2001 ◽  
Vol 44 (3) ◽  
pp. 455-478 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Real Axis ◽  
Absolute Constant ◽  
Subject Classification ◽  
List Type ◽  
Second Derivative ◽  
Type Number ◽  
Previous Estimate ◽  
Positive Absolute Constant

AbstractLet $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or$f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.AMS 2000 Mathematics subject classification: Primary 30D35

Role and rationale of Kaala in the field of Ayurvedic Pharmaceutics - A Conceptual Approach

2017 ◽  
Vol 2 (4) ◽  
Author(s):  
Vrinda Bhat ◽  
Surekha S. Medikeri ◽  
Shobha G. Hiremath
Keyword(s):  
Dosage Form ◽  
Pharmaceutical Preparation ◽  
Vital Role ◽  
Constant Factor ◽  
Conceptual Approach ◽  
Safety And Efficacy ◽  
Significant Aspect ◽  
Definition Of

Samskara is defined as a process of bringing about a desired modification or establishing a change of property in a drug or group of drugs. In the process of Aushadhi Nirmana, varied number of procedures (Samskaras) are adopted to inculcate the desired dosage form and efficacy to the medicine. Among all Samskaras, Kaala plays a vital role in Ayurvedic pharmaceutics. Kaala is a constant factor which follows incoherently in every step of Aushadhi Nirmana. Active principles of plants vary in every season and at different quarters of the day. After the collection of drugs for a pharmaceutical preparation, Kaala plays its role during Paka of various formulations. The definition of pharmaceutics does not end with mere production of a dosage form but also includes its safety and efficacy. Kaala has the potential to influence both these factors. Thus, our Acharyas have provided meticulous information on Ayurvedic pharmaceutics giving prime importance to a minute, yet very significant aspect called “Kaala”.

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