scholarly journals Supersimple structures with a dense independent subset

2017 ◽  
Vol 63 (6) ◽  
pp. 552-573
Author(s):  
Alexander Berenstein ◽  
Juan Felipe Carmona ◽  
Evgueni Vassiliev
Keyword(s):  
Independent Subset

scholarly journals Geometric structures with a dense independent subset

2015 ◽  
Vol 22 (1) ◽  
pp. 191-225 ◽  
Author(s):  
Alexander Berenstein ◽  
Evgueni Vassiliev
Keyword(s):  
Independent Subset ◽  
Geometric Structures

scholarly journals A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

10.37236/484 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ondřej Bílka ◽  
Kevin Buchin ◽  
Radoslav Fulek ◽  
Masashi Kiyomi ◽  
Yoshio Okamoto ◽  
...  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minkowski Sum ◽  
Independent Subset ◽  
Point Sets ◽  
Planar Point ◽  
Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

scholarly journals Central elements of the Jennings basis and certain Morita invariants

2019 ◽  
Vol 19 (08) ◽  
pp. 2050160 ◽  
Author(s):  
Taro Sakurai
Keyword(s):  
Group Algebra ◽  
Positive Characteristic ◽  
Finite Dimensional Algebra ◽  
Independent Subset ◽  
Dimensional Algebra ◽  
Ideal Formula ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Morita Invariant ◽  
Central Elements

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the [Formula: see text]th (right) socle [Formula: see text] of a finite-dimensional algebra [Formula: see text] is a Morita invariant; this is a generalization of important Morita invariants — the center [Formula: see text] and the Reynolds ideal [Formula: see text]. As an example, we also studied [Formula: see text] for the group algebra FG of a finite [Formula: see text]-group [Formula: see text] over a field [Formula: see text] of positive characteristic [Formula: see text]. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent subset of [Formula: see text]. In fact, such elements form a basis of [Formula: see text] for every integer [Formula: see text] if [Formula: see text] is powerful. As a corollary we have [Formula: see text] if [Formula: see text] is powerful.

scholarly journals An Independent Subset of TLR Expressing CCR2-Dependent Macrophages Promotes Colonic Inflammation

2010 ◽  
Vol 184 (12) ◽  
pp. 6843-6854 ◽  
Author(s):  
Andrew M. Platt ◽  
Calum C. Bain ◽  
Yvonne Bordon ◽  
David P. Sester ◽  
Allan McI. Mowat
Keyword(s):  
Independent Subset ◽  
Colonic Inflammation

scholarly journals Common Independence in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1411
Author(s):  
Magda Dettlaff ◽  
Magdalena Lemańska ◽  
Jerzy Topp
Keyword(s):  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Subset ◽  
Block Graphs ◽  
Independent Domination ◽  
The Common ◽  
Independent Dominating Set

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

scholarly journals Affine Independence in Vector Spaces

2010 ◽  
Vol 18 (1) ◽  
pp. 87-93 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Linear Space ◽  
Vector Spaces ◽  
Barycentric Coordinates ◽  
Independent Subset ◽  
Linear Combinations ◽  
Affine Hull ◽  
Real Linear Space ◽  
Real Linear ◽  
The Given

Affine Independence in Vector Spaces In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties of the barycentric coordinates.

scholarly journals Searching for the causal effects of BMI in over 300 000 individuals, using Mendelian randomization

2017 ◽  
Author(s):  
Louise A C Millard ◽  
Neil M Davies ◽  
Kate Tilling ◽  
Tom R Gaunt ◽  
George Davey Smith
Keyword(s):  
Body Mass Index ◽  
Mendelian Randomization ◽  
Causal Effect ◽  
Causal Effects ◽  
P Value ◽  
Protective Effects ◽  
Independent Subset ◽  
Uk Biobank ◽  
False Discovery ◽  
Causal Risk Factor

ABSTRACTMendelian randomization (MR) has been used to estimate the causal effect of body mass index (BMI) on particular traits thought to be affected by BMI. However, BMI may also be a modifiable, causal risk factor for outcomes where there is no prior reason to suggest that a causal effect exists. We perform a MR phenome-wide association study (MR-pheWAS) to search for the causal effects of BMI in UK Biobank (n=334 968), using the PHESANT open-source phenome scan tool. Of the 20 461 tests performed, our MR-pheWAS identified 519 associations below a stringent P value threshold corresponding to a 5% estimated false discovery rate, including many previously identified causal effects. We also identified several novel effects, including protective effects of higher BMI on a set of psychosocial traits, identified initially in our preliminary MR-pheWAS and replicated in an independent subset of UK Biobank. Such associations need replicating in an independent sample.

Maximal irredundance and maximal ideal independence in Boolean algebras

2008 ◽  
Vol 73 (1) ◽  
pp. 261-275 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Boolean Algebra ◽  
Maximal Ideal ◽  
Boolean Algebras ◽  
Independent Subset ◽  
Cardinal Invariant ◽  
Associated Functions ◽  
Atomless Boolean Algebra ◽  
Relationship Of ◽  
The Relationship ◽  
The Ideal

Recall that a subset X of an algebra A is irredundant iff x ∉ 〈X∖{x}〉 for all x ϵ X, where 〈X∖{x}) is the subalgebra generated by X∖{x}. By Zorn's lemma there is always a maximal irredundant set in an algebra. This gives rise to a natural cardinal function Irrmm(A) = min{∣X∣: X is a maximal irredundant subset of A}. The first half of this article is devoted to proving that there is an atomless Boolean algebra A of size 2ω for which Irrmm(A) = ω.A subset X of a BA A is ideal independent iff x ∉ (X∖{x}〉id for all x ϵ X, where 〈X∖{x}〉id is the ideal generated by X∖{x}. Again, by Zorn's lemma there is always a maximal ideal independent subset of any Boolean algebra. We then consider two associated functions. A spectrum functionSspect(A) = {∣X∣: X is a maximal ideal independent subset of A}and the least element of this set, smm(A). We show that many sets of infinite cardinals can appear as Sspect(A). The relationship of Smm to similar “continuum cardinals” is investigated. It is shown that it is relatively consistent that Smm/fin) < 2ω.We use the letter s here because of the relationship of ideal independence with the well-known cardinal invariant spread; see Monk [5]. Namely, sup{∣X∣: X is ideal independent in A} is the same as the spread of the Stone space Ult(A); the spread of a topological space X is the supremum of cardinalities of discrete subspaces.

An Application of Ultraproducts to Lattice-Ordered Groups

1972 ◽  
Vol 24 (6) ◽  
pp. 1063-1064
Author(s):  
A. M. W. Glass
Keyword(s):  
Total Order ◽  
Lattice Ordered Group ◽  
Independent Subset ◽  
Ordered Group ◽  
Ordered Groups ◽  
Partial Converse ◽  
Lattice Ordered Groups ◽  
Total Orders

Using ultraproducts, N. R. Reilly proved that if G is a representable lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a total order which induces ≺ on J (see [5]). In [4], H. A. Hollister proved that a group G admits a total order if and only if it admits a representable order and, moreover, every latticeorder on a group is the intersection of right total orders. The purpose of this paper is to provide a partial converse, viz: if G is a lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a right total order which induces ≺ on J.

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