What is an Elegant Proof? Some Coloring Theorems

2018 ◽  
pp. 133-141
Author(s):  
William Gasarch ◽  
Clyde Kruskal
Keyword(s):  
Elegant Proof

On a Theorem of Erdös and Szekeres

1968 ◽  
Vol 11 (4) ◽  
pp. 597-598
Author(s):  
M.V. Subbarao
Keyword(s):  
General Setting ◽  
Real Numbers ◽  
Elegant Proof

Given a sequence of r distinct real numbers such that the number of terms of every decreasing subsequence is at most m, then there exists an increasing subsequence of more than n terms, where n is the largest integer less than r/m.An extremely simple and elegant proof of the theorem was given by A. Seidenberg [2]. This note is intended to point out that a result analogous to the above holds under a more general setting.

scholarly journals A NOTE ON BARKER POLYNOMIALS

2013 ◽  
Vol 09 (03) ◽  
pp. 759-767 ◽  
Author(s):  
PETER BORWEIN ◽  
TAMÁS ERDÉLYI
Keyword(s):  
Newton’S Identities ◽  
Polynomial Formula ◽  
Barker Codes ◽  
Elegant Proof

We call the polynomial [Formula: see text] a Barker polynomial of degree n-1 if each aj ∈{-1, 1} and [Formula: see text] Properties of Barker polynomials were studied by Turyn and Storer thoroughly in the early sixties, and by Saffari in the late eighties. In the last few years P. Borwein and his collaborators revived interest in the study of Barker polynomials (Barker codes, Barker sequences). In this paper we give a new proof of the fact that there is no Barker polynomial of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence of the following new result. Theorem.Ifn ≔ 2m + 1 > 13and[Formula: see text]where eachbj ∈{-1, 0, 1}for even values of j, each bj is an integer divisible by 4 for odd values of j, then there is no polynomial[Formula: see text]such that[Formula: see text], where[Formula: see text]and[Formula: see text]denotes the collection of all polynomials of degree 2m with each of their coefficients in {-1, 1}. A clever usage of Newton's identities plays a central role in our elegant proof.

scholarly journals Minimum Variance Reinsurance

1962 ◽  
Vol 2 (2) ◽  
pp. 257-260 ◽  
Author(s):  
Stefan Vajda
Keyword(s):  
Minimum Variance ◽  
Point Of View ◽  
Stieltjes Integral ◽  
Optimum Amount ◽  
Natural Conditions ◽  
Excess Loss ◽  
Total Claim ◽  
Image Position ◽  
Stop Loss ◽  
Elegant Proof

In a paper entitled “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance” (XVIth Int. Congr. Act. Bruxelles 1960) Karl Borch has shown that, if the reinsurance premium is given, the smallest variance of the cedent's payments is obtained by a stop-loss reinsurance contract. Paul Markham Kahn, in “Some Remarks on a Recent Paper by Borch”, a paper read to the 1961 Astin Colloquium, has given an elegant proof of this theorem which appears to apply also to cases not considered by Borch. In this paper we study the problem from the reinsurer's point of view and it will be seen that, under natural conditions which are also used in the proof of the Borch-Kahn theorem, the minimum variance of the reinsurer's payments is obtained by a quota contract. This focusses attention on a peculiar opposition of interests of the two partners of a reinsurance contract. However, we do not enter any further into the investigation of a possible resolution of this conflict.We study a problem concerning the division of risk between a cedent and his reinsurer. The risk may refer to a whole portfolio (in which case one might consider a Stop-Loss contract), or to a single contract (when an Excess-Loss contract is a possibility). We shall here use the nomenclature of a portfolio reinsurance.Let it be assumed that a function F(x) is known which gives the probability of a total claim not exceeding x. We have then in Stieltjes integral notationThe two partners to a reinsurance arrangement agree that the reinsurer reimburses m(x).x out of a claim of x, where m(x) is a continuous and differentiate function of x and o ≤ m(x) ≤ 1.

Number theory and other reminiscences of Viscount Cherwell

1988 ◽  
Vol 42 (2) ◽  
pp. 197-204 ◽  
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Fundamental Theorem ◽  
Prime Numbers ◽  
Christ Church ◽  
Active Interest ◽  
Fundamental Theorem Of Arithmetic ◽  
Elegant Proof ◽  
Theory Of Numbers

Lord Cherwell (i) was, of course, a very distinguished ex-perimental physicist but he had (like many others) a considerable active interest in the theory of numbers. I met him in 1930 when Christ Church, Oxford, elected me to a Senior (postgraduate) Scholarship and I migrated there from my original college. Cherwell’s first published work (2) in the theory of numbers was a very simple and elegant proof of the fundamental theorem of arithmetic, that any positive integer can be expressed as a product of prime numbers in just one way (apart from a possible rearrangement of the order of the factors). (A prime is a positive integer greater than 1 whose only factors are 1 and itself.) His proof is by the method of descent (used by Fermat, but not for this problem). Assume the fundamental theorem false and call any number that can be expressed as a product of primes in two or more ways abnormal.

Tellegen's Proof of Thevenim's Theorem

1978 ◽  
Vol 15 (1) ◽  
pp. 83-85
Author(s):  
W. Buijze
Keyword(s):  
Elegant Proof

A very simple and elegant proof of Thévenin and Norton's theorem is given.

scholarly journals Colourability and word-representability of near-triangulations

2019 ◽  
Vol 28 (1) ◽  
pp. 70-76
Author(s):  
Marc Elliot Glen
Keyword(s):  
Chromatic Number ◽  
Open Problems ◽  
Elegant Proof

Abstract A graph G = (V;E) is word-representable if there is a word w over the alphabet V such that x and y alternate in w if and only if the edge (x; y) is in G. It is known [6] that all 3-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not. There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al. [1] showed that a triangulation of a convex polyomino is word-representable if and only if it is 3-colourable; and Glen and Kitaev [5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed. It was shown in [4] that a near-triangulation is 3-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5].

scholarly journals Tiling the Line with Triples

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Aaron Meyerowitz
Keyword(s):  
Greedy Algorithm ◽  
Direct Proof ◽  
Open Problems ◽  
One Dimensional ◽  
Proof By Contradiction ◽  
International Audience ◽  
The Subject ◽  
The One ◽  
Finite Digraph ◽  
Elegant Proof

International audience It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.

scholarly journals Combinatorial Laplacian of the Matching Complex

10.37236/1634 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Xun Dong ◽  
Michelle L. Wachs
Keyword(s):  
Symmetric Group ◽  
Combinatorial Laplacian ◽  
Striking Result ◽  
Matching Complex ◽  
Spectrum Of The Laplacian ◽  
Elegant Proof

A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral.

scholarly journals Application of Reliability Theory to Insurance

1971 ◽  
Vol 6 (2) ◽  
pp. 97-107 ◽  
Author(s):  
E. Straub
Keyword(s):  
Stability Criterion ◽  
General Rule ◽  
Reliability Theory ◽  
Exact Calculation ◽  
Stanford University ◽  
Risk Models ◽  
Loss Probabilities ◽  
Alternative Approach ◽  
The Future ◽  
Elegant Proof

1. There is a general rule applicable to all insurance and reinsurance fields according to which the level of the so-called technical minimum premium should be fixed such that a certain stability criterion is satisfied for the portfolio under consideration. The two bestknown such criteria are(i) the probability that there is a technical loss in any of the future years should be less than a given percentage(ii) the probability that the company gets “ruined” i.e. initial reserves plus accumulated premiums minus accumulated claims becomes negative at any time of a given period in the future should be less than a tolerated percentage.Confining ourselves to criterion (i) in the present paper we may then say that the problem of calculating technical minimum premiums is broadly spoken equivalent with the problem of estimating loss probabilities. Since an exact calculation of such probabilities is only possible for a few very simple and therefore mostly unrealistic risk models and since e.g. Esscher's method is not always very easy to apply in practice it might be worthwhile to describe in the following an alternative approach using results and techniques from Reliability Theory in order to establish bounds for unknown loss probabilities.It would have been impossible for me to write this paper without having had the opportunity of numerous discussions with the Reliability experts R. Barlow and F. Proschan while I was at Stanford University. In particular I was told the elegant proof of theorem 3 given below by R. Barlow recently.

scholarly journals Equilateral Sets and a Schütte Theorem for the 4-norm

2014 ◽  
Vol 57 (3) ◽  
pp. 640-647
Author(s):  
Konrad J. Swanepoel
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Minimum Distances ◽  
Equilateral Sets ◽  
Sharp Lower Bound ◽  
Elegant Proof

AbstractA well-known theorem of Schütte (1963) gives a sharp lower bound for the ratio of the maximum and minimum distances between n + 2 points in n-dimensional Euclidean space. In this note we adapt Bárány’s elegant proof (1994) of this theorem to the space . This gives a new proof that the largest cardinality of an equilateral set in is n + 1 and gives a constructive bound for an interval (4–εn, 4 + εn) of values of p close to 4 for which it is known that the largest cardinality of an equilateral set in is n + 1.

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