31,064 research outputs found
Addison-type series representation for the Stieltjes constants
The Stieltjes constants appear in the coefficients in the
regular part of the Laurent expansion of the Hurwitz zeta function
about its only pole at . We generalize a technique of Addison for the
Euler constant to show its application to finding series
representations for these constants. Other generalizations of representations
of are given.Comment: 21 pages, no figure
A Corollary for Nonsmooth Systems
In this note, two generalized corollaries to the LaSalle-Yoshizawa Theorem
are presented for nonautonomous systems described by nonlinear differential
equations with discontinuous right-hand sides. Lyapunov-based analysis methods
are developed using differential inclusions to achieve asymptotic convergence
when the candidate Lyapunov derivative is upper bounded by a negative
semi-definite function
Summation of Series Defined by Counting Blocks of Digits
We discuss the summation of certain series defined by counting blocks of
digits in the -ary expansion of an integer. For example, if denotes
the sum of the base-2 digits of , we show that . We recover this previous
result of Sondow in math.NT/0508042 and provide several generalizations.Comment: 12 pages, Introduction expanded, references added, accepted by J.
Number Theor
Upper Bound on the Products of Particle Interactions in Cellular Automata
Particle-like objects are observed to propagate and interact in many
spatially extended dynamical systems. For one of the simplest classes of such
systems, one-dimensional cellular automata, we establish a rigorous upper bound
on the number of distinct products that these interactions can generate. The
upper bound is controlled by the structural complexity of the interacting
particles---a quantity which is defined here and which measures the amount of
spatio-temporal information that a particle stores. Along the way we establish
a number of properties of domains and particles that follow from the
computational mechanics analysis of cellular automata; thereby elucidating why
that approach is of general utility. The upper bound is tested against several
relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables,
http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and
accompanying text modified, to comply with legal demands arising from
on-going intellectual property litigation among third parties. V3: Accepted
for publication in Physica D. References added and other small changes made
per referee suggestion
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