scholarly journals Enumerating Matroids of Fixed Rank

10.37236/5894 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Rudi Pendavingh ◽  
Jorn Van der Pol
Keyword(s):  
Upper Bound ◽  
Fixed Rank ◽  
Almost All ◽  
Concise Description

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that~$s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In an earlier paper, we showed that $\log s(n) \sim \log m(n)$. The bounds that we used for that result were dominated by matroids of rank $r\approx n/2$. In this paper we consider the relation between the number of sparse paving matroids $s(n,r)$ and the number of matroids $m(n,r)$ on a fixed groundset of size $n$ of fixed rank $r$. In particular, we show that $\log s(n,r) \sim \log m(n,r)$ whenever $r\ge 3$, by giving asymptotically matching upper and lower bounds.Our upper bound on $m(n,r)$ relies heavily on the theory of matroid erections as developed by Crapo and Knuth, which we use to encode any matroid as a stack of paving matroids. Our best result is obtained by relating to this stack of paving matroids an antichain that completely determines the matroid. We also obtain that the collection of essential flats and their ranks gives a concise description of matroids.

scholarly journals On the density and the structure of the Peirce-like formulae

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Finite Number ◽  
Upper Bound ◽  
Large Family ◽  
Partial Answer ◽  
International Audience ◽  
Almost All

International audience Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.

scholarly journals TLRs and innate immunity

Blood ◽  
2009 ◽  
Vol 113 (7) ◽  
pp. 1399-1407 ◽  
Author(s):  
Bruce A. Beutler
Keyword(s):  
Immune Response ◽  
Innate Immunity ◽  
Positional Cloning ◽  
Forward Genetics ◽  
Host Responses ◽  
Toll Like Receptors ◽  
Genetic Studies ◽  
The Past ◽  
Almost All ◽  
Concise Description

AbstractOne of the most fundamental questions in immunology pertains to the recognition of non-self, which for the most part means microbes. How do we initially realize that we have been inoculated with microbes, and how is the immune response ignited? Genetic studies have made important inroads into this question during the past decade, and we now know that in mammals, a relatively small number of receptors operate to detect signature molecules that herald infection. One or more of these signature molecules are displayed by almost all microbes. These receptors and the signals they initiate have been studied in depth by random germline mutagenesis and positional cloning (forward genetics). Herein is a concise description of what has been learned about the Toll-like receptors, which play an essential part in the perception of microbes and shape the complex host responses that occur during infection.

scholarly journals Iterative estimation of total π-electron energy

2005 ◽  
Vol 70 (10) ◽  
pp. 1193-1197 ◽  
Author(s):  
Lemi Türker ◽  
Ivan Gutman
Keyword(s):  
Lower Bound ◽  
Electron Energy ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Total Electron ◽  
Iterative Estimation ◽  
Total Electron Energy ◽  
Almost All

In this work, the lower and upper bounds for total ?-electron energy (E) was studied. A method is presented, by means of which, starting with a lower bound EL and an upper bound EU for E, a sequence of auxiliary quantities E0 E1, E2,? is computed, such that E0 = EL, E0 < E1 < E2 < ?, and E = EU. Therefore, an integer k exists, such that Ek E < Ek+1. If the estimates EL and EU are of the McClelland type, then k is called the McClelland number. For almost all benzenoid hydrocarbons, k = 3.

scholarly journals Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

2016 ◽  
Vol 160 (3) ◽  
pp. 477-494 ◽  
Author(s):  
J. CILLERUELO ◽  
M. Z. GARAEV
Keyword(s):  
Upper Bound ◽  
Absolute Constant ◽  
Primitive Root ◽  
Multiple Roots ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Primitive Root Modulo ◽  
Image Position ◽  
Almost All ◽  
Positive Integers

AbstractIn this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n, \end{equation}$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ $\mathbb{Z}$[x] is a fixed polynomial without multiple roots in $\mathbb{C}$, then the congruence xf(x) ≡ 1 (mod p), x ∈ $\mathbb{N}$, x ⩽ p, has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ϵ and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ϵ.

Linear Equations with Small Prime and Almost Prime Solutions

2008 ◽  
Vol 51 (3) ◽  
pp. 399-405
Author(s):  
Xianmeng Meng
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Equations ◽  
Prime Factors ◽  
Almost All ◽  
Sharp Lower Bound ◽  
Almost Prime

AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.

scholarly journals Total Graph Interpretation of the Numbers of the Fibonacci Type

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Urszula Bednarz ◽  
Iwona Włoch ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Total Graph ◽  
Graph Interpretation ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all(A1,2A1)-edge colourings in trees.

ON THE MINIMAL NUMBER OF SMALL ELEMENTS GENERATING FINITE PRIME FIELDS

2017 ◽  
Vol 96 (2) ◽  
pp. 177-184
Author(s):  
MARC MUNSCH
Keyword(s):  
Finite Field ◽  
General Result ◽  
Upper Bound ◽  
Prime Fields ◽  
Almost All ◽  
Small Elements

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

SOME DECISION QUESTIONS CONCERNING THE TIME COMPLEXITY OF LANGUAGE ACCEPTORS

2014 ◽  
Vol 25 (08) ◽  
pp. 1127-1140 ◽  
Author(s):  
OSCAR H. IBARRA ◽  
BALA RAVIKUMAR
Keyword(s):  
Upper Bound ◽  
Time Complexity ◽  
Complexity Class ◽  
Exact Boundary ◽  
Resource Requirements ◽  
Almost All

Almost all the decision questions concerning the resource requirements of a computational device are undecidable. Here we want to understand the exact boundary that separates the undecidable from the decidable cases of such problems by considering the time complexity of very simple devices that include NFAs (1-way and 2-way), PDAs and PDAs augmented with counters - and their unambiguous restrictions. We consider several variations - based on whether the bound holds exactly or as an upper-bound and show decidability as well as undecidability results. In the case of decidable problems, we also attempt to determine more precisely the complexity class to which the problem belongs.

The Matroid Ramsey Number n(6,6)

1999 ◽  
Vol 8 (3) ◽  
pp. 229-235 ◽  
Author(s):  
JOSEPH E. BONIN ◽  
JENNIFER McNULTY ◽  
TALMAGE JAMES REID
Keyword(s):  
Upper Bound ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Connected Matroid ◽  
Six Elements ◽  
Fixed Rank

A tight upper bound on the number of elements in a connected matroid with fixed rank and largest cocircuit size is given. This upper bound is used to show that a connected matroid with at least thirteen elements contains either a circuit or a cocircuit with at least six elements. In the language of matroid Ramsey numbers, n(6, 6) = 13: this is the largest currently known matroid Ramsey number.

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