scholarly journals On the shadow of squashed families of $k$-sets

10.37236/1210 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Frédéric Maire
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Symmetric Difference ◽  
The Family

The shadow of a collection ${\cal A}$ of $k$-sets is defined as the collection of the $(k-1)$-sets which are contained in at least one $k$-set of ${\cal A}$. Given $|{\cal A}|$, the size of the shadow is minimum when ${\cal A}$ is the family of the first $k$-sets in squashed order (by definition, a $k$-set $A$ is smaller than a $k$-set $B$ in the squashed order if the largest element of the symmetric difference of $A$ and $B$ is in $B$). We give a tight upper bound and an asymptotic formula for the size of the shadow of squashed families of $k$-sets.

scholarly journals Durfee Polynomials

10.37236/1370 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
E. Rodney Canfield ◽  
Sylvie Corteel ◽  
Carla D. Savage
Keyword(s):  
Asymptotic Formula ◽  
Experimental Evidence ◽  
Upper Bound ◽  
Asymptotic Form ◽  
Asymptotically Normal ◽  
The Family ◽  
Average Size ◽  
Durfee Square ◽  
Analytical Expressions ◽  
Addition Formulas

Let ${\bf F}(n)$ be a family of partitions of $n$ and let ${\bf F}(n,d)$ denote the set of partitions in ${\bf F}(n)$ with Durfee square of size $d$. We define the Durfee polynomial of ${\bf F}(n)$ to be the polynomial $P_{{\bf F},n}= \sum |{\bf F}(n,d)|y^d$, where $ 0 \leq d \leq \lfloor \sqrt{n} \rfloor.$ The work in this paper is motivated by empirical evidence which suggests that for several families ${\bf F}$, all roots of the Durfee polynomial are real. Such a result would imply that the corresponding sequence of coefficients $\{|{\bf F}(n,d)|\}$ is log-concave and unimodal and that, over all partitions in ${\bf F}(n)$ for fixed $n$, the average size of the Durfee square, $a_{{\bf F}}(n)$, and the most likely size of the Durfee square, $m_{{\bf F}}(n)$, differ by less than 1. In this paper, we prove results in support of the conjecture that for the family of ordinary partitions, ${\bf P}(n)$, the Durfee polynomial has all roots real. Specifically, we find an asymptotic formula for $|{\bf P}(n,d)|$, deriving in the process a simple upper bound on the number of partitions of $n$ with at most $k$ parts which generalizes the upper bound of Erdös for $|{\bf P}(n)|$. We show that as $n$ tends to infinity, the sequence $\{|{\bf P}(n,d)|\},\ 1 \leq d \leq \sqrt{n},$ is asymptotically normal, unimodal, and log concave; in addition, formulas are found for $a_{{\bf P}}(n)$ and $m_{{\bf P}}(n)$ which differ asymptotically by at most 1. Experimental evidence also suggests that for several families ${\bf F}(n)$ which satisfy a recurrence of a certain form, $m_{{\bf F}}(n)$ grows as $c \sqrt{n}$, for an appropriate constant $c=c_{{\bf F}}$. We prove this under an assumption about the asymptotic form of $|{\bf F}(n,d)|$ and show how to produce, from recurrences for the $|{\bf F}(n,d)|$, analytical expressions for the constants $c_{{\bf F}}$ which agree numerically with the observed values.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

scholarly journals State Complexity of Catenation Combined with a Boolean Operation: A Unified Approach

2016 ◽  
Vol 27 (06) ◽  
pp. 675-703 ◽  
Author(s):  
Pascal Caron ◽  
Jean-Gabriel Luque ◽  
Ludovic Mignot ◽  
Bruno Patrou
Keyword(s):  
Upper Bound ◽  
The State ◽  
Symmetric Difference ◽  
Boolean Operation ◽  
Unified Approach ◽  
State Complexity

We study the state complexity of catenation combined with symmetric difference. First, an upper bound is computed using some combinatoric tools. Then, this bound is shown to be tight by giving a witness for it. Moreover, we relate this work with the study of state complexity for two other combinations: catenation with union and catenation with intersection. We extract a unified approach which allows to obtain the state complexity of any combination involving catenation and a binary boolean operation.

scholarly journals FURTHER RESULTS ON P SYSTEMS WITH PROMOTERS/INHIBITORS

2006 ◽  
Vol 17 (01) ◽  
pp. 205-221 ◽  
Author(s):  
DRAGOŞ SBURLAN
Keyword(s):  
Upper Bound ◽  
P Systems ◽  
Not Given ◽  
The Family

This paper presents several results regarding P systems with non-cooperative rules and promoters/inhibitors at the level of rules. For the class of P systems using inhibitors, generating families of sets of vectors of numbers, a characterization of the family of Parikh sets of ET0L languages is shown. In the case of P systems with non-cooperative promoted rules even if an upper bound was not given, the inclusion of the family PsET0L was proved. Moreover, a characterization of such systems by means of a particular form of random context grammars, therefore a sequential formal device, is proposed.

On the Approximation Rate of Hierarchical Mixtures-of-Experts for Generalized Linear Models

1999 ◽  
Vol 11 (5) ◽  
pp. 1183-1198 ◽  
Author(s):  
Wenxin Jiang ◽  
Martin A. Tanner
Keyword(s):  
Generalized Linear Models ◽  
Upper Bound ◽  
Linear Models ◽  
Sobolev Class ◽  
Link Function ◽  
Approximation Rate ◽  
Smooth Functions ◽  
Mixtures Of Experts ◽  
The Family

We investigate a class of hierarchical mixtures-of-experts (HME) models where generalized linear models with nonlinear mean functions of the form ψ(α + xTβ) are mixed. Here ψ(·) is the inverse link function. It is shown that mixtures of such mean functions can approximate a class of smooth functions of the form ψ(h(x)), where h(·) ε W∞2;k (a Sobolev class over [0, 1]s, as the number of experts m in the network increases. An upper bound of the approximation rate is given as O(m−2/s) in Lp norm. This rate can be achieved within the family of HME structures with no more than s-layers, where s is the dimension of the predictor x.

The Family of Lodato Proximities Compatible With a Given Topological Space

1974 ◽  
Vol 26 (02) ◽  
pp. 388-404 ◽  
Author(s):  
W. J. Thron ◽  
R. H. Warren
Keyword(s):  
Lower Bound ◽  
Topological Space ◽  
Distributive Lattice ◽  
Structural Properties ◽  
Upper Bound ◽  
Set Inclusion ◽  
The Family

Let (X, ) be a topological space. By we denote the family of all Lodato proximities on X which induce . We show that is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of are developed. By means of one of these constructions, all covers of any member of can be obtained. Several examples are given which relate to the lattice of all compatible proximities of Čech and the family of all compatible proximities of Efremovič. The paper concludes with a chart which summarizes many of the structural properties of , and .

On Riesz means of the coefficients of the Rankin–Selberg series

1999 ◽  
Vol 127 (1) ◽  
pp. 117-131 ◽  
Author(s):  
ALEKSANDAR IVIĆ ◽  
KOHJI MATSUMOTO ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Error Term ◽  
Weighted Sum ◽  
Mean Square ◽  
Riesz Means ◽  
The Mean ◽  
Voronoi Formula

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

scholarly journals Induced Matchings and the v-Number of Graded Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2860
Author(s):  
Gonzalo Grisalde ◽  
Enrique Reyes ◽  
Rafael H. Villarreal
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Simplicial Partition ◽  
The Family ◽  
Induced Matchings ◽  
Edge Ring ◽  
Insight Into

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

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