199 research outputs found
Contractions of Low-Dimensional Lie Algebras
Theoretical background of continuous contractions of finite-dimensional Lie
algebras is rigorously formulated and developed. In particular, known necessary
criteria of contractions are collected and new criteria are proposed. A number
of requisite invariant and semi-invariant quantities are calculated for wide
classes of Lie algebras including all low-dimensional Lie algebras.
An algorithm that allows one to handle one-parametric contractions is
presented and applied to low-dimensional Lie algebras. As a result, all
one-parametric continuous contractions for the both complex and real Lie
algebras of dimensions not greater than four are constructed with intensive
usage of necessary criteria of contractions and with studying correspondence
between real and complex cases.
Levels and co-levels of low-dimensional Lie algebras are discussed in detail.
Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio
Characterization of Knots and Links Arising From Site-specific Recombination on Twist Knots
We develop a model characterizing all possible knots and links arising from
recombination starting with a twist knot substrate, extending previous work of
Buck and Flapan. We show that all knot or link products fall into three
well-understood families of knots and links, and prove that given a positive
integer , the number of product knots and links with minimal crossing number
equal to grows proportionally to . In the (common) case of twist knot
substrates whose products have minimal crossing number one more than the
substrate, we prove that the types of products are tightly prescribed. Finally,
we give two simple examples to illustrate how this model can help determine
previously uncharacterized experimental data.Comment: 32 pages, 7 tables, 27 figures, revised: figures re-arranged, and
minor corrections. To appear in Journal of Physics
Special symplectic Lie groups and hypersymplectic Lie groups
A special symplectic Lie group is a triple such that
is a finite-dimensional real Lie group and is a left invariant
symplectic form on which is parallel with respect to a left invariant
affine structure . In this paper starting from a special symplectic Lie
group we show how to ``deform" the standard Lie group structure on the
(co)tangent bundle through the left invariant affine structure such
that the resulting Lie group admits families of left invariant hypersymplectic
structures and thus becomes a hypersymplectic Lie group. We consider the affine
cotangent extension problem and then introduce notions of post-affine structure
and post-left-symmetric algebra which is the underlying algebraic structure of
a special symplectic Lie algebra. Furthermore, we give a kind of double
extensions of special symplectic Lie groups in terms of post-left-symmetric
algebras.Comment: 32 page
The Compressibility of Minimal Lattice Knots
The (isothermic) compressibility of lattice knots can be examined as a model
of the effects of topology and geometry on the compressibility of ring
polymers. In this paper, the compressibility of minimal length lattice knots in
the simple cubic, face centered cubic and body centered cubic lattices are
determined. Our results show that the compressibility is generally not
monotonic, but in some cases increases with pressure. Differences of the
compressibility for different knot types show that topology is a factor
determining the compressibility of a lattice knot, and differences between the
three lattices show that compressibility is also a function of geometry.Comment: Submitted to J. Stat. Mec
On diagrammatic bounds of knot volumes and spectral invariants
In recent years, several families of hyperbolic knots have been shown to have
both volume and (first eigenvalue of the Laplacian) bounded in
terms of the twist number of a diagram, while other families of knots have
volume bounded by a generalized twist number. We show that for general knots,
neither the twist number nor the generalized twist number of a diagram can
provide two-sided bounds on either the volume or . We do so by
studying the geometry of a family of hyperbolic knots that we call double coil
knots, and finding two-sided bounds in terms of the knot diagrams on both the
volume and on . We also extend a result of Lackenby to show that a
collection of double coil knot complements forms an expanding family iff their
volume is bounded.Comment: 16 pages, 7 figure
Minimal knotted polygons in cubic lattices
An implementation of BFACF-style algorithms on knotted polygons in the simple
cubic, face centered cubic and body centered cubic lattice is used to estimate
the statistics and writhe of minimal length knotted polygons in each of the
lattices. Data are collected and analysed on minimal length knotted polygons,
their entropy, and their lattice curvature and writhe
Predicting mental imagery based BCI performance from personality, cognitive profile and neurophysiological patterns
Mental-Imagery based Brain-Computer Interfaces (MI-BCIs) allow their users to send commands
to a computer using their brain-activity alone (typically measured by ElectroEncephaloGraphy—
EEG), which is processed while they perform specific mental tasks. While very
promising, MI-BCIs remain barely used outside laboratories because of the difficulty
encountered by users to control them. Indeed, although some users obtain good control
performances after training, a substantial proportion remains unable to reliably control an
MI-BCI. This huge variability in user-performance led the community to look for predictors of
MI-BCI control ability. However, these predictors were only explored for motor-imagery
based BCIs, and mostly for a single training session per subject. In this study, 18 participants
were instructed to learn to control an EEG-based MI-BCI by performing 3 MI-tasks, 2
of which were non-motor tasks, across 6 training sessions, on 6 different days. Relationships
between the participants’ BCI control performances and their personality, cognitive
profile and neurophysiological markers were explored. While no relevant relationships with
neurophysiological markers were found, strong correlations between MI-BCI performances
and mental-rotation scores (reflecting spatial abilities) were revealed. Also, a predictive
model of MI-BCI performance based on psychometric questionnaire scores was proposed.
A leave-one-subject-out cross validation process revealed the stability and reliability of this
model: it enabled to predict participants’ performance with a mean error of less than 3
points. This study determined how users’ profiles impact their MI-BCI control ability and
thus clears the way for designing novel MI-BCI training protocols, adapted to the profile of
each user
Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory
We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research
Discovery of Tantalum, Rhenium, Osmium, and Iridium Isotopes
Currently, thirty-eight tantalum, thirty-eight rhenium, thirty-nine osmium,
and thirty-eight iridium, isotopes have been observed and the discovery of
these isotopes is discussed here. For each isotope a brief synopsis of the
first refereed publication, including the production and identification method,
is presented.Comment: To be published in At. Data Nucl. Data Table
Loop Quantum Gravity a la Aharonov-Bohm
The state space of Loop Quantum Gravity admits a decomposition into
orthogonal subspaces associated to diffeomorphism equivalence classes of
spin-network graphs. In this paper I investigate the possibility of obtaining
this state space from the quantization of a topological field theory with many
degrees of freedom. The starting point is a 3-manifold with a network of
defect-lines. A locally-flat connection on this manifold can have non-trivial
holonomy around non-contractible loops. This is in fact the mathematical origin
of the Aharonov-Bohm effect. I quantize this theory using standard field
theoretical methods. The functional integral defining the scalar product is
shown to reduce to a finite dimensional integral over moduli space. A
non-trivial measure given by the Faddeev-Popov determinant is derived. I argue
that the scalar product obtained coincides with the one used in Loop Quantum
Gravity. I provide an explicit derivation in the case of a single defect-line,
corresponding to a single loop in Loop Quantum Gravity. Moreover, I discuss the
relation with spin-networks as used in the context of spin foam models.Comment: 19 pages, 1 figure; v2: corrected typos, section 4 expanded
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