scholarly journals Optimal and Nonoptimal Gronwall Lemmas

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1728
Author(s):  
Daniela Marian ◽  
Sorina Anamaria Ciplea ◽  
Nicolaie Lungu
Keyword(s):  
Upper Bound ◽  
Partial Answer ◽  
Optimal Inequalities

In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. A. Rus. This paper is also motivated by the fact that, in many inequalities, the upper bound is not an optimal one.

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 B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

2006 ◽  
Vol 74 (1) ◽  
pp. 121-132 ◽  
Author(s):  
A. Abdollahi ◽  
A. Azad ◽  
A. Mohammadi Hassanabadi ◽  
M. Zarrin
Keyword(s):  
Finite Group ◽  
Exponential Function ◽  
Finite Groups ◽  
Upper Bound ◽  
Partial Answer ◽  
A Finite Group

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

scholarly journals Set-Systems with Restricted Multiple Intersections

10.37236/1625 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Upper Bound ◽  
Explicit Construction ◽  
Partial Answer ◽  
Residue Classes ◽  
Set Systems ◽  
Element Set ◽  
Multiple Intersections

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if ${\cal H}$ is a set-system, which satisfies that for some $k$, the $k$-wise intersections occupy only $\ell$ residue-classes modulo a $p$ prime, while the sizes of the members of ${\cal H}$ are not in these residue classes, then the size of ${\cal H}$ is at most $$(k-1)\sum_{i=0}^{\ell}{n\choose i}$$ This result considerably strengthens an upper bound of Füredi (1983), and gives partial answer to a question of T. Sós (1976). As an application, we give a direct, explicit construction for coloring the $k$-subsets of an $n$ element set with $t$ colors, such that no monochromatic complete hypergraph on $$\exp{(c(\log m)^{1/t}(\log \log m)^{1/(t-1)})}$$ vertices exists.

On the maximum number of the pairwise noncommuting elements in a finite group

2016 ◽  
Vol 15 (10) ◽  
pp. 1650197 ◽  
Author(s):  
Seyyed Majid Jafarian Amiri ◽  
Halimeh Madadi
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Structural Properties ◽  
Finite Groups ◽  
Upper Bound ◽  
Frobenius Group ◽  
Maximum Size ◽  
Partial Answer ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] be the maximum size of a set of pairwise noncommuting elements in [Formula: see text]. In this paper, we give an upper bound of [Formula: see text] for an arbitrary nilpotent group [Formula: see text]. As an application of this result, we give a partial answer to Question 2.8 of [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000) 139–146]. Also we compute [Formula: see text] when [Formula: see text] is a Frobenius group. Finally we describe structural properties of all groups [Formula: see text] with [Formula: see text].

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