191 research outputs found
Generating strings for bipartite Steinhaus graphs
AbstractLet b(n) be the number of bipartite Steinhaus graphs with n vertices. We show that b(n) satisfies the recurrence, b(2) = 2, b(3) = 4, and for k ⩾ 2, b(2k + 1) = 2b(k + 1) + 1, b(2k) = b(k) + b(k + 1). Thus b(n) ⩽ 52n − 72 with equality when n is one more than a power of two. To prove this recurrence, we describe the possible generating strings for these bipartite graphs
On skew loops, skew branes and quadratic hypersurfaces
A skew brane is an immersed codimension 2 submanifold in affine space, free
from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew
brane cannot lie on a quadratic hypersurface. We also prove that there are no
skew loops on embedded ruled developable discs in 3-space. The paper extends
recent work by M. Ghomi and B. Solomon.Comment: 13 pages, 2 figure
Recommended from our members
Cycles and Cliques in Steinhaus Graphs
In this dissertation several results in Steinhaus graphs are investigated. First under some further conditions imposed on the induced cycles in steinhaus graphs, the order of induced cycles in Steinhaus graphs is at most [(n+3)/2]. Next the results of maximum clique size in Steinhaus graphs are used to enumerate the Steinhaus graphs having maximal cliques. Finally the concept of jumbled graphs and Posa's Lemma are used to show that almost all Steinhaus graphs are Hamiltonian
On the Hardness of Gray Code Problems for Combinatorial Objects
Can a list of binary strings be ordered so that consecutive strings differ in
a single bit? Can a list of permutations be ordered so that consecutive
permutations differ by a swap? Can a list of non-crossing set partitions be
ordered so that consecutive partitions differ by refinement? These are examples
of Gray coding problems: Can a list of combinatorial objects (of a particular
type and size) be ordered so that consecutive objects differ by a flip (of a
particular type)? For example, 000, 001, 010, 100 is a no instance of the first
question, while 1234, 1324, 1243 is a yes instance of the second question due
to the order 1243, 1234, 1324. We prove that a variety of Gray coding problems
are NP-complete using a new tool we call a Gray code reduction.Comment: 15 pages, 5 figures, WALCOM 202
Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions
Kernel mean embeddings have recently attracted the attention of the machine
learning community. They map measures from some set to functions in a
reproducing kernel Hilbert space (RKHS) with kernel . The RKHS distance of
two mapped measures is a semi-metric over . We study three questions.
(I) For a given kernel, what sets can be embedded? (II) When is the
embedding injective over (in which case is a metric)? (III) How does
the -induced topology compare to other topologies on ? The existing
machine learning literature has addressed these questions in cases where is
(a subset of) the finite regular Borel measures. We unify, improve and
generalise those results. Our approach naturally leads to continuous and
possibly even injective embeddings of (Schwartz-) distributions, i.e.,
generalised measures, but the reader is free to focus on measures only. In
particular, we systemise and extend various (partly known) equivalences between
different notions of universal, characteristic and strictly positive definite
kernels, and show that on an underlying locally compact Hausdorff space,
metrises the weak convergence of probability measures if and only if is
continuous and characteristic.Comment: Old and longer version of the JMLR paper with same title (published
2018). Please start with the JMLR version. 55 pages (33 pages main text, 22
pages appendix), 2 tables, 1 figure (in appendix
- …