scholarly journals The $11$-element case of Frankl's conjecture

10.37236/812 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ivica Bošnjak ◽  
Petar Marković
Keyword(s):  
Finite Union ◽  
Finite Sets ◽  
Frankl’S Conjecture

In 1979, P. Frankl conjectured that in a finite union-closed family ${\cal F}$ of finite sets, ${\cal F}\neq\{\emptyset\}$, there has to be an element that belongs to at least half of the sets in ${\cal F}$. We prove this when $|\bigcup{\cal F}|\leq 11$.

scholarly journals An attempt at frankl’s conjecture

2007 ◽  
pp. 29-43 ◽  
Author(s):  
Petar Markovic
Keyword(s):  
Finite Union ◽  
Finite Sets ◽  
Frankl’S Conjecture

In 1979 Frankl conjectured that in a finite union-closed family F of finite sets, F _= {?} there has to be an element that belongs to at least half of the sets in F. We prove this when |U F| _ <10.

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

Representation and manipulation of finite sets

1980 ◽  
Vol 10 (4) ◽  
pp. 8-12 ◽  
Author(s):  
B. L. McAllister
Keyword(s):  
Finite Sets

scholarly journals New refinements of the discrete Jensen’s inequality generated by finite or infinite permutations

2019 ◽  
Vol 94 (6) ◽  
pp. 1109-1121
Author(s):  
László Horváth
Keyword(s):  
Banach Spaces ◽  
Jensen’S Inequality ◽  
Vector Spaces ◽  
Real Vector ◽  
Finite Sets ◽  
Jensen's Inequality

AbstractIn this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.

Well-quasi-ordering hereditarily finite sets

2013 ◽  
Vol 90 (6) ◽  
pp. 1278-1291 ◽  
Author(s):  
Alberto Policriti ◽  
Alexandru I. Tomescu
Keyword(s):  
Finite Sets ◽  
Quasi Ordering

Random Mappings and Decompositions of Finite Sets

1972 ◽  
Vol 17 (1) ◽  
pp. 132-145 ◽  
Author(s):  
B. A. Sevast’yanov
Keyword(s):  
Finite Sets ◽  
Random Mappings

PROPERTIES OF QUASI-RELABELING TREE BIMORPHISMS

2010 ◽  
Vol 21 (03) ◽  
pp. 257-276 ◽  
Author(s):  
ANDREAS MALETTI ◽  
CĂTĂLIN IONUŢ TÎRNĂUCĂ
Keyword(s):  
Cartesian Product ◽  
Canonical Representation ◽  
Finite Union ◽  
Top Down ◽  
Tree Language ◽  
Regular Tree ◽  
Fundamental Properties ◽  
Tree Transducers ◽  
Tree Languages ◽  
Context Free

The fundamental properties of the class QUASI of quasi-relabeling relations are investigated. A quasi-relabeling relation is a tree relation that is defined by a tree bimorphism (φ, L, ψ), where φ and ψ are quasi-relabeling tree homomorphisms and L is a regular tree language. Such relations admit a canonical representation, which immediately also yields that QUASI is closed under finite union. However, QUASI is not closed under intersection and complement. In addition, many standard relations on trees (e.g., branches, subtrees, v-product, v-quotient, and f-top-catenation) are not quasi-relabeling relations. If quasi-relabeling relations are considered as string relations (by taking the yields of the trees), then every Cartesian product of two context-free string languages is a quasi-relabeling relation. Finally, the connections between quasi-relabeling relations, alphabetic relations, and classes of tree relations defined by several types of top-down tree transducers are presented. These connections yield that quasi-relabeling relations preserve the regular and algebraic tree languages.

Families of finite sets with three intersections

COMBINATORICA ◽  
1984 ◽  
Vol 4 (2-3) ◽  
pp. 141-148 ◽  
Author(s):  
Peter Frankl
Keyword(s):  
Finite Sets
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