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

Finch С-. R. Математические константы (2003) М.

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

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Date Jun 22, 2005

Cites: — ?' — =0.1924225474...,
hti \%kH
lim^ ~ = 2jt?h{$h + \Wh = 0.6844472720
where ?^! = 0.4567332095 .....
The (transformed) exponential generating function of
310 5 Constants Associated with Enumerating Discrete Structures
labeled acyclic digraphs is [65,71-74]
*n n_ 3 2 25 3 543 4 29281
where ^ = 1.4880785456.....
An efficient way of doing this (maintaining equivalence classes that change over
time) is found in QF and QFW, two of a class of union-find algorithms in computer
science...
Knuth [1,3] studied the Takeuchi numbers Tn = T(n, 0, n + 1):
To = 0, Ti = 1, T2 = 4, T3 = 14, T4 = 53, T5 = 223,.....
What is the likelihood that
con = 0 for some n > 0? That is, what is the return probability p<p-
P61ya[ 1-4] proved the remarkable fact that p\ = p2 = \ butpj < 1 fore/ > 2...
This can be rephrased as the asymptotic number of times a drunkard drops
by his favorite watering hole [41,42]...
Regular holonomic
constants are values of / at algebraic points zq where each r* is analytic; / can
be proved to be analytic at zq as well...
Let wnj denote the probability that no
cloudy intervals of length k > 1 occur, and assume that initially ?(X0 = 1) = 6\...
In both cases, the number of boundary sites, relative to N, is vanishingly small
as n -> oo, so this convention does not lead to any error...
Define hn to be the smallest positive integer not
representable as a sum of n integers of the form 2'3-/, / >0,j> 0, that is, ho =
I, h\ = 5, h2 = 23, h3 = 431,.....
Assume s\, S2,..., sn are independent, uniformly
distributed variables on the interval [0, N]...
We
ask the following question: How many square-free words of length n are there?
Over a two-letter alphabet, the only square-free words are a, b, ab, ba,aba, and bab;
thus binary square-free words are not interesting...
Observe that a square-free word must be overlap-free, and that an
overlap-free word must be cube-free...
For
example, if 9(p) denotes the percolation probability, that is, the probability that an co-
cluster contains a prescribed site or bond, then pc is the unique point for which p < pc
implies 0(p) = 0, and p > pc implies 9(p) > 0...
Let Tn denote the shortest
length of all lattice path lengths starting at the origin @, 0) and ending at («, 0); then it
can be proved that the limit
E(Tn) ...
It turns out that the growth constant a for convex polyominoes is the same as the
growth constant a' for parallelogram polyominoes, that is, polyominoes whose left
and right boundaries both climb in a northeasterly direction:
A'{\) = 1, A\2) = 2, AfC) = 4, A'(A) = 9, A'E) = 20,
A'F) = 46, A'{1)= 105,......
An increasing sub-
subsequence of n is a sequence (tc(J\), 7t(J2), .-., x{jk)) satisfying both 1 < j\ < jj <
• • • < jk 5 N and tt(j\ ) < n(j2) < .....
When d = 3
[11,25,29,33-36],
exp()8(z)) = 1 + 3z4 + 22z6 + 192z8 + 2046z10 + 24853z12 + 329334z
but again our knowledge of the series coefficients is limited...
Unlike
other lattice models (see [5.12], [5.18], and [5.22]), monomer-dimer systems do not
have a phase transition [7]...
th Southeastern Conf on Combinatorics, Graph Theory and Computing, Boca Raton,
1997, Congr...
Tile 1
Tile 2
Tile 1: Alternating black and white segments join the centers of the consecutive edges around
the square; west-to-north segment is black, north-to-east is white, east-to-south is black, and
south-to-west is white...
These complicated three-
dimensional lattices are not the same as the simple models mathematicians tend to
focus on...