Finch St. R. Mathematical constants (CUP, 2003)(T)(618s)_M

Finch St. R. Mathematical constants (CUP, 2003)(T)(618s)_M_.djvu

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Date Jun 22, 2005

By
the degree (or valency) of a vertex, we mean the number of vertices that are adjacent
to it...
If we
randomly select a binary tree x with In + 1 vertices, then the asymptotics of E(ht(r))
involve 2+/7rn as mentioned in [1.4], and those of E(rg(r)) involve ln(n)/ lnD) plus a
zero mean oscillating function [2.16]...
The probability that exactly two complex components emerge
is 507T/1296 = 0.1212..., but the probability (> 0.9938 ...) that the evolving graph
never has more than two complex components at any time is not precisely known [93]...
= 3
k=\
riwru;rujrB4;=l-5163860591--
Hence the escape probability for a random walk on the three-dimensional cubic lattice
is 1 - ;?3 = 0.6594626704 In these expressions, K denotes the complete elliptic
integral of the first kind [1.4.6] and V denotes the gamma function [1.5.4]...
It is conjectured that sn ~ En2v and tn ~ Fn2v,
where v is the same exponent as for rn, and E/B = 0.14026..., F/B = 0.43961 .....
phase transition in the sense that
0 < lim Pk(x)* < 1 exists for 0 < x < \x~x = 0.3790522777 ...,
k—>3C
lim /K/x-1); = 1,
A'—>-oo
1 < lim Pk(x)~? < oo exists for x > n~x...
Observe that the product enforces the nearest neighbor exclusion: If a configuration
has two particles next to each other, then zero contribution is made to the partition
function...
Given a vector V = (v\, vi,..., v^) ofk distinct integers, define two subvectors Vi
and VR by
VL = (vj : vj <vu 2<j< k), VR = (vj : Vj >vu2<j< k)...
The fact that limiting values for h(V)/\n(n) and
s( V)/ ln(n) are equal means that the trees are almost perfect (with only a small "fringe"
around Iog2(«))...
Yet another objective, however, might be to maximize the hiree's expected quality Qn
itself (the kih applicant has quality s^...
n—»oc
A word is overlap-free if it contains no factor of the form xyxyx, with x nonempty...
More precisely, Tracy & Widom [15-17] derived a
certain probability distribution function F(x) characterizing the largest eigenvalue of
a random Hermitian matrix, generated according to the Gaussian Unitary Ensemble
(GUE) probability law...
The range 0 < T < oo corresponds to unequal weighting, accentu-
accentuating the states with small p and q...
Q and P Moments
Let us return to the random coloring problem, suitably generalized to incorporate
temperature...
Different numerical values of the coefficients apply for T < Tc,
as well as for the antiferromagnetic case [100,101]...
n-+x V 4
The problem of computing /„ and gn is the same as counting nowhere-zero flows modulo
\7
W W
f
Figure 5.24...
In actuality, however, there are several
kinds of three-dimensional ice, depending on temperature and pressure [24,25]...
A connected planar graph G determines a subdivision of the 2-sphere (under stere-
ographic projection) into simply connected regions (faces)...
A heuristic explanation of the Beraha conjecture in [27] is
insightful but is not a rigorous proof [8]...
Other representations of L are
where K denotes the complete elliptic integral of the first kind [1.4.6], T(x) is the
Euler gamma function [1.5.4], y is the Euler-Mascheroni constant [1.5], and fi(x) is
Dirichlet's beta function [1.7]...
Moreover,
the exceptional subset possesses a minimum element that can be characterized exactly
[22]...
While on the subject of artificial constants, let us mention the Champernowne
number [6]
C = 0.12345678910111213141516171819202122232425...,
which is constructed by concatenating the digits of all positive integers, and the
Copeland-Erdos number [7]
0.2357111317192329313741434753596167717379
which is likewise constructed by concatenating the digits of all primes...