514 research outputs found
An Overview of Planar Flow Casting of Thin Metallic Glasses and its Relation to Slot Coating of Liquid Films
Planar flow casting (PFC) is a method that can be used to make thin, long, and wide metallic alloy foils by extruding molten liquid through a thin and wide nozzle and immediately quenching on a moving roller. The quenching rates are high enough that amorphous metallic glasses may be formed which have many desirable properties for a wide variety of applications. This paper reviews how PFC processes were developed, examines the typical operability range of PFC, and reviews the defects that commonly form. The geometrical similarities between PFC and slot coating process are apparent, and this paper highlights differences between the operability ranges of both processes
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
Two well studied Ramsey-theoretic problems consider subsets of the natural
numbers which either contain no three elements in arithmetic progression, or in
geometric progression. We study generalizations of this problem, by varying the
kinds of progressions to be avoided and the metrics used to evaluate the
density of the resulting subsets. One can view a 3-term arithmetic progression
as a sequence , where , a nonzero
integer. Thus avoiding three-term arithmetic progressions is equivalent to
containing no three elements of the form with , the set of integer translations. One can similarly
construct related progressions using different families of functions. We
investigate several such families, including geometric progressions ( with a natural number) and exponential progressions ().
Progression-free sets are often constructed "greedily," including every
number so long as it is not in progression with any of the previous elements.
Rankin characterized the greedy geometric-progression-free set in terms of the
greedy arithmetic set. We characterize the greedy exponential set and prove
that it has asymptotic density 1, and then discuss how the optimality of the
greedy set depends on the family of functions used to define progressions.
Traditionally, the size of a progression-free set is measured using the (upper)
asymptotic density, however we consider several different notions of density,
including the uniform and exponential densities.Comment: Version 1.0, 13 page
Benford Behavior of Zeckendorf Decompositions
A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as the sum of non-consecutive Fibonacci numbers . A set is said to satisfy Benford's law if
the density of the elements in with leading digit is
; in other words, smaller leading digits are more
likely to occur. We prove that, as , for a randomly selected
integer in the distribution of the leading digits of the
Fibonacci summands in its Zeckendorf decomposition converge to Benford's law
almost surely. Our results hold more generally, and instead of looking at the
distribution of leading digits one obtains similar theorems concerning how
often values in sets with density are attained.Comment: Version 1.0, 12 pages, 1 figur
Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals
Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers , with initial
terms . We consider the distribution of the number of
summands involved in such decompositions. Previous work proved that as the distribution of the number of summands in the Zeckendorf
decompositions of , appropriately normalized, converges
to the standard normal. The proofs crucially used the fact that all integers in
share the same potential summands.
We generalize these results to subintervals of as ; the analysis is significantly more involved here as different integers
have different sets of potential summands. Explicitly, fix an integer sequence
. As , for almost all the distribution of the number of summands in the Zeckendorf
decompositions of integers in the subintervals ,
appropriately normalized, converges to the standard normal. The proof follows
by showing that, with probability tending to , has at least one
appropriately located large gap between indices in its decomposition. We then
use a correspondence between this interval and to obtain
the result, since the summands are known to have Gaussian behavior in the
latter interval. % We also prove the same result for more general linear
recurrences.Comment: Version 1.0, 8 page
On the response of neutrally stable flows to oscillatory forcing with application to liquid sheets
Industrial coating processes create thin liquid films with tight thickness
tolerances, and thus models that predict the response to inevitable
disturbances are essential. The mathematical modeling complexities are reduced
through linearization as even small thickness variations in films can render a
product unsalable. The signaling problem, considered in this paper, is perhaps
the simplest model that incorporates the effects of repetitive (oscillatory)
disturbances that are initiated, for example, by room vibrations and pump
drives. In prior work, Gordillo and P\'erez (Phys. Fluids 14, 2002) examined
the structure of the signaling response for linear operators that admit
exponentially growing or damped solutions, i.e., the medium is classified as
unstable or stable via classical stability analysis. The signaling problem
admits two portions of the solution, the transient behavior due to the start-up
of the disturbance and the long-time behavior that is continually forced; the
superposition reveals how the forced solution evolves through the passage of a
transient. In this paper, we examine signaling for the linear operator examined
by King et al. (King et al. 2016, Phys. Rev. Fluids 1(7)) that governs varicose
waves in a thin liquid sheet and that can admit solutions having algebraic
growth although the underlying medium is classified as being neutrally stable.
Long-time asymptotic methods are used to extract critical velocities that
partition the response into distinct regions having markedly different
location-dependent responses, and the amplitudes of oscillatory responses in
these regions are determined. Together, these characterize the magnitude and
breadth of the solution response. Results indicate that the signaling response
in neutrally stable flows (that admit algebraic growth) is significantly
different from that in exponentially unstable systems
- β¦