458 research outputs found
On the number of pancake stacks requiring four flips to be sorted
Using existing classification results for the 7- and 8-cycles in the pancake
graph, we determine the number of permutations that require 4 pancake flips
(prefix reversals) to be sorted. A similar characterization of the 8-cycles in
the burnt pancake graph, due to the authors, is used to derive a formula for
the number of signed permutations requiring 4 (burnt) pancake flips to be
sorted. We furthermore provide an analogous characterization of the 9-cycles in
the burnt pancake graph. Finally we present numerical evidence that polynomial
formulae exist giving the number of signed permutations that require flips
to be sorted, with .Comment: We have finalized for the paper for publication in DMTCS, updated a
reference to its published version, moved the abstract to its proper
location, and added a thank you to the referees. The paper has 27 pages, 6
figures, and 2 table
Cycles in the burnt pancake graphs
The pancake graph is the Cayley graph of the symmetric group on
elements generated by prefix reversals. has been shown to have
properties that makes it a useful network scheme for parallel processors. For
example, it is -regular, vertex-transitive, and one can embed cycles in
it of length with . The burnt pancake graph ,
which is the Cayley graph of the group of signed permutations using
prefix reversals as generators, has similar properties. Indeed, is
-regular and vertex-transitive. In this paper, we show that has every
cycle of length with . The proof given is a
constructive one that utilizes the recursive structure of . We also
present a complete characterization of all the -cycles in for , which are the smallest cycles embeddable in , by presenting their
canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42
pages, 9 figures, 20 pages of appendice
Fidelity decay in interacting two-level boson systems: Freezing and revivals
We study the fidelity decay in the -body embedded ensembles of random
matrices for bosons distributed in two single-particle states, considering the
reference or unperturbed Hamiltonian as the one-body terms and the diagonal
part of the -body embedded ensemble of random matrices, and the perturbation
as the residual off-diagonal part of the interaction. We calculate the
ensemble-averaged fidelity with respect to an initial random state within
linear response theory to second order on the perturbation strength, and
demonstrate that it displays the freeze of the fidelity. During the freeze, the
average fidelity exhibits periodic revivals at integer values of the Heisenberg
time . By selecting specific -body terms of the residual interaction,
we find that the periodicity of the revivals during the freeze of fidelity is
an integer fraction of , thus relating the period of the revivals with the
range of the interaction of the perturbing terms. Numerical calculations
confirm the analytical results
Effects of Human vs. Automatic Feedback on Students' Understanding of AI Concepts and Programming Style
The use of automatic grading tools has become nearly ubiquitous in large
undergraduate programming courses, and recent work has focused on improving the
quality of automatically generated feedback. However, there is a relative lack
of data directly comparing student outcomes when receiving computer-generated
feedback and human-written feedback. This paper addresses this gap by splitting
one 90-student class into two feedback groups and analyzing differences in the
two cohorts' performance. The class is an intro to AI with programming HW
assignments. One group of students received detailed computer-generated
feedback on their programming assignments describing which parts of the
algorithms' logic was missing; the other group additionally received
human-written feedback describing how their programs' syntax relates to issues
with their logic, and qualitative (style) recommendations for improving their
code. Results on quizzes and exam questions suggest that human feedback helps
students obtain a better conceptual understanding, but analyses found no
difference between the groups' ability to collaborate on the final project. The
course grade distribution revealed that students who received human-written
feedback performed better overall; this effect was the most pronounced in the
middle two quartiles of each group. These results suggest that feedback about
the syntax-logic relation may be a primary mechanism by which human feedback
improves student outcomes.Comment: Published in SIGCSE '20: Proceedings of the 51st ACM Technical
Symposium on Computer Science Educatio
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