scholarly journals On Orthogonal Symmetric Chain Decompositions

10.37236/8531 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Karl Däubel ◽  
Sven Jäger ◽  
Torsten Mütze ◽  
Manfred Scheucher
Keyword(s):  
Recent Result ◽  
Single Element ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Orthogonal Decompositions ◽  
Symmetric Chain

The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has three pairwise orthogonal chain decompositions for $n\geq 24$. In this paper, we construct four pairwise orthogonal chain decompositions of the $n$-cube for $n\geq 60$. We also construct five pairwise edge-disjoint symmetric chain decompositions of the $n$-cube for $n\geq 90$, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, Jäger, Mütze, Sawada, and Wille.  

Rapid Quantitative Analysis by X-Ray Spectrometry

1971 ◽  
Vol 15 ◽  
pp. 164-175 ◽  
Author(s):  
Robert D. Giauque ◽  
Joseph M. Jaklevic
Keyword(s):  
Cross Sections ◽  
Fluorescence Analysis ◽  
Selective Excitation ◽  
Single Element ◽  
X Rays ◽  
Analysis Method ◽  
Yield Data ◽  
X Ray ◽  
Minimum Number ◽  
Sample Dilution

An x-ray fluorescence analysis method applicable to the case of fluorescent spectra excited with monoenergetic x-rays has been developed. The technique employs a minimum number of calibration steps using single element thin film standards and depends upon theoretical cross sections and fluorescent yield data to interpolate from element to element. The samples are treated as thin films and corrections for absorption effects are easily determined- Enhancement effects, if not negligible, are minimized by sample dilution techniques or by selective excitation.

scholarly journals Powers of a product of commutators as products of squares

2004 ◽  
Vol 2004 (7) ◽  
pp. 373-375 ◽  
Author(s):  
Alireza Abdollahi
Keyword(s):  
Recent Result ◽  
Free Group ◽  
Infinite Rank ◽  
Minimum Number

We prove that for any odd integerNand any integern>0, theNth power of a product ofncommutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of2n+1elements and, for all such oddNand integersn, there are commutators for which the number2n+1of squares is the minimum number such that theNth power of its product can be written as a product of squares. This generalizes a recent result of Akhavan-Malayeri.

scholarly journals REPARTITORS, SELECTORS AND SUPERSELECTORS

2006 ◽  
Vol 07 (03) ◽  
pp. 391-415 ◽  
Author(s):  
FRÉDÉRIC HAVET
Keyword(s):  
Lower Bounds ◽  
Disjoint Paths ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Vertex Set ◽  
Optimal Networks

An (n, p, f)-network G is a graph (V, E) where the vertex set V is partitioned into four subsets [Formula: see text] and [Formula: see text] called respectively the priorities, the ordinary inputs, the outputs and the switches, satisfying the following constraints: there are p priorities, n - p ordinary inputs and n + f outputs; each priority, each ordinary input and each output is connected to exactly one switch; switches have degree at most 4. An (n, p, f)-network is an (n, p, f)-repartitor if for any disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist in G, n edge-disjoint paths, p of them from [Formula: see text] to [Formula: see text] and the n - p others joining [Formula: see text] to [Formula: see text]. The problem is to determine the minimum number R(n, p, f) of switches of an (n, p, f)-repartitor and to construct a repartitor with the smallest number of switches. In this paper, we show how to build general repartitors from (n, 0, f)-repartitors also called (n, n + f)-selectors. We then consrtuct selectors using more powerful networks called superselectors. An (n, 0, 0)-network is an n-superselector if for any subsets [Formula: see text] and [Formula: see text] with [Formula: see text], there exist in G, [Formula: see text] edge-disjoint paths joining [Formula: see text] to [Formula: see text]. We show that the minimum number of switches of an n-superselector S+ (n) is at most 17n + O(log(n)). We then deduce that [Formula: see text] if [Formula: see text], R(n, p, f) ≤ 18n + 34f + O( log (n + f)), if [Formula: see text] and [Formula: see text] if [Formula: see text]. Finally, we give lower bounds for R(n, 0, f) and S+ (n) and show optimal networks for small value of n.

scholarly journals A study on the pendant number of graph products

2019 ◽  
Vol 11 (1) ◽  
pp. 24-40
Author(s):  
Jomon K. Sebastian ◽  
Joseph Varghese Kureethara ◽  
Sudev Naduvath ◽  
Charles Dominic
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Disjoint Paths ◽  
Graph Products ◽  
Path Decomposition ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Number Π

Abstract A path decomposition of a graph is a collection of its edge disjoint paths whose union is G. The pendant number Πp is the minimum number of end vertices of paths in a path decomposition of G. In this paper, we determine the pendant number of corona products and rooted products of paths and cycles and obtain some bounds for the pendant number for some specific derived graphs. Further, for any natural number n, the existence of a connected graph with pendant number n has also been established.

scholarly journals Minimum degree and minimum number of edge-disjoint trees

2004 ◽  
Vol 275 (1-3) ◽  
pp. 195-205
Author(s):  
A. Lladó ◽  
S.C. Lopez
Keyword(s):  
Minimum Degree ◽  
Edge Disjoint ◽  
Minimum Number

Multielement Analysis of Rubber Samples by X-Ray Fluorescence

1996 ◽  
Vol 50 (11) ◽  
pp. 1373-1377 ◽  
Author(s):  
Peter Kump ◽  
Marijan Nečemer ◽  
Borut Smodiš ◽  
Radojko Jač'Imović
Keyword(s):  
Reference Materials ◽  
Fluorescence Analysis ◽  
Standard Reference Materials ◽  
Single Element ◽  
Rubber Industry ◽  
X Ray ◽  
Minimum Number ◽  
Cheap Alternative ◽  
Rapid Tool ◽  
Absorption Measurements

Destructive elemental analysis of rubber samples for major and trace element constituents is a rather demanding task, mainly because of the need for sophisticated and time-consuming sample preparation procedures. X-ray analysis has so far been used in the rubber industry primarily as a rapid tool for qualitative analysis, but a more realistic estimate of the accuracy of this, in many aspects, advantageous technique indicates that it could qualify as a quantitative method. In this work rubber samples were analyzed by the X-ray fluorescence analysis (XRFA) technique, utilizing Cd-109 and Fe-55 for excitation radioactive sources. The quantification procedure used employed a minimum number of calibration steps, utilizing only single-element-thick standards and stable compounds, or standard reference materials. Matrix correction calculations employed known fundamental constants and absorption measurements on a particular sample. In order to validate the results, and to establish the accuracy of analysis, some samples were also analyzed by neutron activation analysis (NAA). For the same purpose, the analysis of some standard reference materials of biological and inorganic matrices were performed. The accuracy of 5 to 6% achieved by XRFA means that it should be considered in the rubber industry as a rapid, simple, and cheap alternative to the analytical methods usually adopted.

scholarly journals Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

10.37236/1755 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jerrold Griggs ◽  
Charles E. Killian ◽  
Carla D. Savage
Keyword(s):  
Boolean Lattice ◽  
Constructive Proof ◽  
Existence Proof ◽  
Chain Decomposition ◽  
Venn Diagrams ◽  
Minimum Number ◽  
Symmetric Chain Decomposition ◽  
Open Question ◽  
Symmetric Chain

We show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice ${\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.

scholarly journals Elemental analysis of Scottish populations of the ectoparasitic copepod Lepeophtheirus salmonis

2000 ◽  
Vol 69 (1-2) ◽  
pp. 79-87 ◽  
Author(s):  
A.P. Shinn ◽  
J.E. Bron ◽  
D.J. Gray ◽  
C. Sommervill
Keyword(s):  
Elemental Analysis ◽  
Inductively Coupled Plasma ◽  
Sea Lice ◽  
Single Element ◽  
Lepeophtheirus Salmonis ◽  
Correct Classification ◽  
Fish Farms ◽  
Multivariate Statistical ◽  
Inductively Coupled ◽  
Minimum Number

Conventional nebulisation ICPMS (Inductively Coupled Plasma Mass Spectrometry), was used to determine the concentration of a broad range of elements in the salmon louse Lepeophtheirus salmonis. Lice samples were collected from Atlantic salmon in seven localities (4 fish farms and 3 wild salmon fisheries) on two separate sampling occasions and prepared for analysis. Sixty six elements were measured, 35 of these were found to be variable and were subjected to univariate and multivariate statistical analysis. The results of the single element comparisons showed that not all individual sites could be discriminated from each other. Sea lice collected from cultured salmonids could be discriminated from those on wild salmonids at the same site using the elements magnesium (<0.05%), vanadium (<0.01%) and uranium (<0.05%). Using discriminant analysis based on 28 elements, the separation of all sampled sea lice localities from each other was clear (100% correct classification) giving each an individual signature. Further analysis examined the effects of sequentially removing elements from the discrimination model in order to determine the minimum number of elements required to obtain satisfactory discrimination of populations. It was found that 16 elements could still provide 100% correct classification, whilst 12 elements still provided 97.30% correct classification. This pilot study has shown elemental analysis to be a potentially successful method for the discrimination of populations of L. salmonis, although the biological basis of the elemental signatures derived remains to be established.

scholarly journals The packing and covering of the complete graph I: the forests of order five

1986 ◽  
Vol 9 (2) ◽  
pp. 277-282 ◽  
Author(s):  
Y. Roditty
Keyword(s):  
Complete Graph ◽  
Packing And Covering ◽  
Edge Disjoint ◽  
Minimum Number

The maximum number of pairwise edge disjoint forests of order five in the complete graphKn, and the minimum number of forests of order five whose union isKn, are determined.

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