On The Determination of Sets by Sets of Sums of Fixed Order

1968 ◽  
Vol 20 ◽  
pp. 596-611 ◽  
Author(s):  
John A. Ewell
Keyword(s):  
Abelian Group ◽  
General Sense ◽  
Fixed Order

The present investigation is based on two papers: “On the determination of numbers by their sums of a fixed order,” by J. L. Self ridge and E. G. Straus (4), and “On the determination of sets by the sets of sums of a certain order,” by B. Gordon, A. S. Fraenkel, and E. G. Straus (2).First of all, we explain the terms implicit in the above titles. Throughout these considerations we use the term “set” to mean “a totality having possible multiplicities,” so that two sets will be counted as equal if, and only if, they have the same elements with identical multiplicities. In the most general sense the term “numbers” of (4) can be replaced by “elements of any given torsioniree Abelian group.”

An Improved Quantitative Image Analyser

1977 ◽  
Vol 35 ◽  
pp. 226-227
Author(s):  
J.P. Fallon ◽  
P.J. Gregory ◽  
C.J. Taylor
Keyword(s):  
Feature Extraction ◽  
Quantitative Analysis ◽  
Frequency Content ◽  
General Sense ◽  
Physical Measurements ◽  
Quantitative Image ◽  
Image Analyser ◽  
Automatic Methods ◽  
Quantitative Results

Quantitative image analysis systems have been used for several years in research and quality control applications in various fields including metallurgy and medicine. The technique has been applied as an extension of subjective microscopy to problems requiring quantitative results and which are amenable to automatic methods of interpretation.Feature extraction. In the most general sense, a feature can be defined as a portion of the image which differs in some consistent way from the background. A feature may be characterized by the density difference between itself and the background, by an edge gradient, or by the spatial frequency content (texture) within its boundaries. The task of feature extraction includes recognition of features and encoding of the associated information for quantitative analysis.Quantitative Analysis. Quantitative analysis is the determination of one or more physical measurements of each feature. These measurements may be straightforward ones such as area, length, or perimeter, or more complex stereological measurements such as convex perimeter or Feret's diameter.

scholarly journals Green’s Classifications and Evolutions of Fixed-Order Networks

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 174
Author(s):  
Allen D. Parks
Keyword(s):  
Computational Complexity ◽  
Equivalence Class ◽  
Network Evolution ◽  
Binary Relations ◽  
Internal Dynamics ◽  
Initial Network ◽  
Fixed Order ◽  
Symmetry Problem ◽  
Structural Invariants

It is shown that the set of all networks of fixed order n form a semigroup that is isomorphic to the semigroup BX of binary relations on a set X of cardinality n. Consequently, BX provides for Green’s L,R,H, and D equivalence classifications of all networks of fixed order n. These classifications reveal that a fixed-order network which evolves within a Green’s equivalence class maintains certain structural invariants during its evolution. The “Green’s symmetry problem” is introduced and is defined as the determination of all symmetries (i.e., transformations) that produce an evolution between an initial and final network within an L or an R class such that each symmetry preserves the required structural invariants. Such symmetries are shown to be solutions to special Boolean equations specific to each class. The satisfiability and computational complexity of the “Green’s symmetry problem” are discussed and it is demonstrated that such symmetries encode information about which node neighborhoods in the initial network can be joined to form node neighborhoods in the final network such that the structural invariants required by the evolution are preserved, i.e., the internal dynamics of the evolution. The notion of “propensity” is also introduced. It is a measure of the tendency of node neighborhoods to join to form new neighborhoods during a network evolution and is used to define “energy”, which quantifies the complexity of the internal dynamics of a network evolution.

scholarly journals New approach to hard corrections in precision QCD for LHC and FCC physics

2016 ◽  
Vol 31 (22) ◽  
pp. 1650126
Author(s):  
B. F. L. Ward
Keyword(s):  
Hadron Collider ◽  
High Energy ◽  
Lepton Pairs ◽  
General Applicability ◽  
New Approach ◽  
Lhc Physics ◽  
Fixed Order ◽  
Obvious Generalization ◽  
Single Formula

We present a new approach to the realization of hard fixed-order corrections in predictions for the processes probed in high energy colliding hadron beam devices, with some emphasis on the large hadron collider (LHC) and the future circular collider (FCC) devices. We show that the usual unphysical divergence of such corrections as one approaches the soft limit is removed in our approach, so that we would render the standard results to be closer to the observed exclusive distributions. We use the single [Formula: see text] production and decay to lepton pairs as our prototypical example, but we stress that the approach has general applicability. In this way, we open another part of the way to rigorous baselines for the determination of the theoretical precision tags for LHC physics, with an obvious generalization to the FCC as well.

On the Determination of Numbers by Their Sums of Fixed Order

1965 ◽  
Vol 72 (8) ◽  
pp. 873
Author(s):  
Nathan Eljoseph
Keyword(s):  
Fixed Order

scholarly journals On the determination of numbers by their sums of a fixed order

1958 ◽  
Vol 8 (4) ◽  
pp. 847-856 ◽  
Author(s):  
John Selfridge ◽  
Ernst Straus
Keyword(s):  
Fixed Order

THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS

2008 ◽  
Vol 04 (06) ◽  
pp. 927-958 ◽  
Author(s):  
ÉRIC BALANDRAUD
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Additive Number Theory ◽  
Alternating Groups ◽  
Additive Number ◽  
New Interpretation ◽  
Set Addition

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.

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