Finch С-. R. Математические константы (2003), Finch St. R. Mathematical constants (CUP, 2003)(T)(618s).djvu:

Finch С-. R. Математические константы (2003)

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

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Date Jun 22, 2005

Cites: If Qn is the number of constitutional isomers of Q,H2,i+i OH without
chiral C atoms, then [18,24]
where %Ql = 0.5947539639.....
If we label the vertices of a graph distinctly with the integers 1, 2,...,«, the cor-
corresponding enumeration problems often simplify; for example, there are exactly n"~2
labeled free trees and n"~l labeled rooted trees...
A cactus is a
connected graph in which no edge lies on more than one (minimal) cycle [52-54]...
A leftist tree of size n is an ordered binary tree with n leaves such that, in any
subtree a, the leaf closest to the root of a is in the right subtree of a...
It is remarkable that this complex com-
component will usually remain unique throughout the entire process, and the probability
that this is true is 57r/18 = 0.8726.....
This endows
the set of all subsets of S with a partial ordering; in fact, it is a lattice with maximum
element S and minimum element 0...
Lengyel [ 19] formulated a different approach to prove that the quotient
rn =
must be bounded between two positive constants as n —>¦ 00...
Shackell, Asymptotics of the Stirling numbers of the second kind, Studies
in Automatic Combinatorics, v...
jt=O
Prellberg [4] observed that the following limit exists:
c = lim \ = 2.2394331040...,
"-*oo Bn exp (\Wf)
where Wn exp(^) = n are special values of the Lambert W function [6.11]...
For simplicity's sake, we have suppressed the dependence of cn on d\ we will do this
for associated constants too whenever possible...
See [5.19] for discussion of lattice animals or polyominoes, which are related
to self-avoiding polygons...
n-00 Yn
A generalization of this, in which the girls must appear in groups of at least k, is given
in [12,13]...
= exp@.3332427219 ...)
is algebraic (in fact, is solvable in radicals [17-22]) with minimal integer polynomial
[23]
2593742460 lx24 + 2013290651222784*22 +2505062311720673 792x20
+ 797726698866658379776x18+7449488310131083100160x16
+ 2958015038376958230528*14 - 72405670285649161617408*12
+107155448150443388043264*10 - 71220809441400405884928*8
- 73347491183630103871488*6 + 97143135277377575190528x4
-32751691810479015985152...
Hard Square Entropy Constant 345
where the sum is over all 2'v possible values of the vector a = {o\, ai, cri, ..., <yN)
and the product is over all edges of the lattice (sites / and j are distinct and adjacent)...
Now, over all integers x, define the recursive function
f(x,V) =
0
if V = 0
@ is the empty vector),
\+f(x, Vi) ifx < v\, otherwise {v\ is the first vector component),
l+f(x,VR) i
Clearly 0 < f(x, V) < k always and the ordering of v\, vi, ..., Vk is crucial
in determining the value of /(x, V), For example, /G, C, 9, 5, 1, 7)) = 4 and
/D, C, 9, 5, 1,7)) = 3...
+ S(n),
and in an unsuccessful search (random x ^ m,- for all i) the expected number is
E(/(x, M, 1)) = -L- ln(«) + I + JL- - a + S(n) + O
lnB) 2 lnB)
Iog2(/i) - 0.273948.....
Products such as Q and K, however, have no known closed-form expression except
when q — exp(—tc%), where ? > 0 is an algebraic number [26,41]...
Note that clearly
lim ?{n {Mn - 1) < v) = lim P (Mn < 1 + -y) = { f %y ^ °'
This asymptotic result is a special case of a far more general theorem due to Fisher &
Tippett [4] and Gnedenko [5]...
Pleasants [ 17] proved that arbitrarily long abelian
square-free words, based on five letters, exist...
An ^-cluster is an isolated grouping of s adjacent Is in M, where adjacency means
horizontal or vertical neighbors (not diagonal)...