452 research outputs found
On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts
Tewodros Amdeberhan and Armin Straub initiated the study of enumerating
subfamilies of the set of (s,t)-core partitions. While the enumeration of
(n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it
equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core
partitions into odd parts remains elusive.
Straub computed the first eleven terms of that sequence, and asked for a
"formula," or at least a fast way, to compute many terms. While we are unable
to find a "fast" algorithm, we did manage to find a "faster" algorithm, which
enabled us to compute 23 terms of this intriguing sequence. We strongly believe
that this sequence has an algebraic generating function, since a "sister
sequence" (see the article), is OEIS sequence A047749 that does have an
algebraic generating function. One of us (DZ) is pledging a donation of 100
dollars to the OEIS, in honor of the first person to generate sufficiently many
terms to conjecture (and prove non-rigorously) an algebraic equation for the
generating function of this sequence, and another 100 dollars for a rigorous
proof of that conjecture.
Finally, we also develop algorithms that find explicit generating functions
for other, more tractable, families of (n+1,n+2)-core partitions.Comment: 12 pages, accompanied by Maple package. This version announces that
our questions were all answered by Paul Johnson, and a donation to the OEIS,
in his honor, has been mad
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area
The Koch snowflake is one of the first fractals that were mathematically
described. It is interesting because it has an infinite perimeter in the limit
but its limit area is finite. In this paper, a recently proposed computational
methodology allowing one to execute numerical computations with infinities
and infinitesimals is applied to study the Koch snowflake at infinity. Numerical
computations with actual infinite and infinitesimal numbers can be
executed on the Infinity Computer being a new supercomputer patented in
USA and EU. It is revealed in the paper that at infinity the snowflake is not
unique, i.e., different snowflakes can be distinguished for different infinite
numbers of steps executed during the process of their generation. It is then
shown that for any given infinite number n of steps it becomes possible to
calculate the exact infinite number, Nn, of sides of the snowflake, the exact
infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn,
of the Koch snowflake as the result of multiplication of the infinite Nn by
the infinitesimal Ln. It is established that for different infinite n and k the
infinite perimeters Pn and Pk are also different and the difference can be infinite.
It is shown that the finite areas An and Ak of the snowflakes can be
also calculated exactly (up to infinitesimals) for different infinite n and k and
the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed
starting from different initial conditions are also studied and their
quantitative characteristics at infinity are computed
Extended Rate, more GFUN
We present a software package that guesses formulae for sequences of, for
example, rational numbers or rational functions, given the first few terms. We
implement an algorithm due to Bernhard Beckermann and George Labahn, together
with some enhancements to render our package efficient. Thus we extend and
complement Christian Krattenthaler's program Rate, the parts concerned with
guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of
Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's
qGeneratingFunctions.m.Comment: 26 page
Resource optimization for fault-tolerant quantum computing
In this thesis we examine a variety of techniques for reducing the resources
required for fault-tolerant quantum computation. First, we show how to simplify
universal encoded computation by using only transversal gates and standard
error correction procedures, circumventing existing no-go theorems. We then
show how to simplify ancilla preparation, reducing the cost of error correction
by more than a factor of four. Using this optimized ancilla preparation, we
develop improved techniques for proving rigorous lower bounds on the noise
threshold.
Additional overhead can be incurred because quantum algorithms must be
translated into sequences of gates that are actually available in the quantum
computer. In particular, arbitrary single-qubit rotations must be decomposed
into a discrete set of fault-tolerant gates. We find that by using a special
class of non-deterministic circuits, the cost of decomposition can be reduced
by as much as a factor of four over state-of-the-art techniques, which
typically use deterministic circuits.
Finally, we examine global optimization of fault-tolerant quantum circuits
under physical connectivity constraints. We adapt techniques from VLSI in order
to minimize time and space usage for computations in the surface code, and we
develop a software prototype to demonstrate the potential savings.Comment: 231 pages, Ph.D. thesis, University of Waterlo
- …