scholarly journals Drinfeld–Manin solutions of the Yang–Baxter equation coming from cube complexes

2021 ◽  
pp. 1-14
Author(s):  
Alina Vdovina
Keyword(s):  
Geometric Interpretation ◽  
Reidemeister Move ◽  
Baxter Equation ◽  
The Third ◽  
Reidemeister Moves ◽  
Explicit Constructions ◽  
Higher Dimensional ◽  
Dimensional Cube ◽  
Cube Complexes ◽  
By Products

The most common geometric interpretation of the Yang–Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the [Formula: see text]-state Yang–Baxter equation for arbitrarily large [Formula: see text]. More precisely, we introduce explicit constructions of cube complexes covered by products of [Formula: see text] trees and show that these cube complexes lead to new solutions of the Yang–Baxter equations.

scholarly journals (1, 2) AND WEAK (1, 3) HOMOTOPIES ON KNOT PROJECTIONS

2013 ◽  
Vol 22 (14) ◽  
pp. 1350085 ◽  
Author(s):  
NOBORU ITO ◽  
YUSUKE TAKIMURA
Keyword(s):  
Equivalence Relation ◽  
Finite Sequence ◽  
Reidemeister Move ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Homotopy Classes ◽  
The Third ◽  
Reidemeister Moves ◽  
Necessary And Sufficient

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

scholarly journals The third way to 3D gravity

2015 ◽  
Vol 24 (12) ◽  
pp. 1544015 ◽  
Author(s):  
Eric Bergshoeff ◽  
Wout Merbis ◽  
Alasdair J. Routh ◽  
Paul K. Townsend
Keyword(s):  
Equations Of Motion ◽  
Massive Gravity ◽  
De Sitter ◽  
Gravitational Field Equation ◽  
Third Way ◽  
Metric Tensor ◽  
The Third ◽  
Higher Dimensional ◽  
Conservation Condition ◽  
3D Gravity

Consistency of Einstein’s gravitational field equation [Formula: see text] imposes a “conservation condition” on the [Formula: see text]-tensor that is satisfied by (i) matter stress tensors, as a consequence of the matter equations of motion and (ii) identically by certain other tensors, such as the metric tensor. However, there is a third way, overlooked until now because it implies a “nongeometrical” action: one not constructed from the metric and its derivatives alone. The new possibility is exemplified by the 3D “minimal massive gravity” model, which resolves the “bulk versus boundary” unitarity problem of topologically massive gravity with Anti-de Sitter asymptotics. Although all known examples of the third way are in three spacetime dimensions, the idea is general and could, in principle, apply to higher dimensional theories.

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

scholarly journals A multivariate approach to investigate the combined biological effects of multiple exposures

2018 ◽  
Vol 72 (7) ◽  
pp. 564-571 ◽  
Author(s):  
Pooja Jain ◽  
Paolo Vineis ◽  
Benoît Liquet ◽  
Jelle Vlaanderen ◽  
Barbara Bodinier ◽  
...  
Keyword(s):  
Biological Effects ◽  
Epidemiological Studies ◽  
Multivariate Approach ◽  
Immune Markers ◽  
Protein Levels ◽  
Multiple Exposures ◽  
Multiple Observations ◽  
Higher Dimensional ◽  
Before And After ◽  
By Products

Epidemiological studies provide evidence that environmental exposures may affect health through complex mixtures. Formal investigation of the effect of exposure mixtures is usually achieved by modelling interactions, which relies on strong assumptions relating to the identity and the number of the exposures involved in such interactions, and on the order and parametric form of these interactions. These hypotheses become difficult to formulate and justify in an exposome context, where influential exposures are numerous and heterogeneous. To capture both the complexity of the exposome and its possibly pleiotropic effects, models handling multivariate predictors and responses, such as partial least squares (PLS) algorithms, can prove useful. As an illustrative example, we applied PLS models to data from a study investigating the inflammatory response (blood concentration of 13 immune markers) to the exposure to four disinfection by-products (one brominated and three chlorinated compounds), while swimming in a pool. To accommodate the multiple observations per participant (n=60; before and after the swim), we adopted a multilevel extension of PLS algorithms, including sparse PLS models shrinking loadings coefficients of unimportant predictors (exposures) and/or responses (protein levels). Despite the strong correlation among co-occurring exposures, our approach identified a subset of exposures (n=3/4) affecting the exhaled levels of 8 (out of 13) immune markers. PLS algorithms can easily scale to high-dimensional exposures and responses, and prove useful for exposome research to identify sparse sets of exposures jointly affecting a set of (selected) biological markers. Our descriptive work may guide these extensions for higher dimensional data.

Mixing sets and relative entropies for higher-dimensional Markov shifts

1993 ◽  
Vol 13 (4) ◽  
pp. 705-735 ◽  
Author(s):  
Bruce Kitchens ◽  
Klaus Schmidt
Keyword(s):  
Higher Order ◽  
The Third ◽  
Markov Shifts ◽  
Higher Dimensional ◽  
Probability Spaces ◽  
Mixing Behaviour

AbstractWe consider certain measurable isomorphism invariants for measure-preserving ℤd-actions on probability spaces, compute them for a class of d-dimensional Markov shifts, and use them to prove that some of these examples are non-isomorphic. The invariants under discussion are of three kinds: the first is associated with the higher-order mixing behaviour of the ℤd-action, and is related—in this class of examples—to an an arithmetical result by David Masser, the second arises from certain relative entropies associated with the ℤd-action, and the third is a collection of canonical invariant σ-algebras. The results of this paper are generalizations of earlier results by Kitchens and Schmidt, and we include a proof of David Masser's unpublished theorem.

Reidemeister moves and parity polynomials of virtual knot diagrams

2017 ◽  
Vol 26 (10) ◽  
pp. 1750051
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bounds ◽  
Virtual Knot ◽  
The Third ◽  
Reidemeister Moves ◽  
Polynomial Formula ◽  
Knot Diagrams

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial [Formula: see text] of a virtual knot diagram [Formula: see text] and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using [Formula: see text]. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from [Formula: see text].

Nutritional Myths

2005 ◽  
Vol 41 (4) ◽  
pp. 273-276 ◽  
Author(s):  
Ann Wortinger
Keyword(s):  
Popular Press ◽  
Pet Food ◽  
Food Preservatives ◽  
The Third ◽  
Feeding Trials ◽  
Nutritional Facts ◽  
By Products

Owners are sometimes confused or misinformed about nutritional facts pertaining to pet foods, and three common nutritional myths have been propagated in the popular press. The first myth is that meat by-products are of inferior quality compared to whole meat. The second myth is that feeding trials are unnecessary, and the third myth is that pet food preservatives are bad. This paper examines the known facts related to these three myths and discusses the importance of food trials and the different classes and forms of antioxidants used in pet foods.

scholarly journals Lyapunov exponents for higher dimensional random maps

1997 ◽  
Vol 10 (3) ◽  
pp. 209-218
Author(s):  
P. Góra ◽  
A. Boyarsky ◽  
Y. S. Lou
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponents ◽  
Discrete Time ◽  
Quantum Physics ◽  
Interference Effects ◽  
Random Maps ◽  
Discrete Time Dynamical System ◽  
Higher Dimensional ◽  
Dimensional Cube ◽  
The Individual

A random map is a discrete time dynamical system in which one of a number of transformations is selected randomly and implemented. Random maps have been used recently to model interference effects in quantum physics. The main results of this paper deal with the Lyapunov exponents for higher dimensional random maps, where the individual maps are Jabloński maps on the n-dimensional cube.

A Plessner Decomposition Along Transverse Curves

1988 ◽  
Vol 40 (5) ◽  
pp. 1243-1255
Author(s):  
Frank Beatrous ◽  
Songying Li
Keyword(s):  
Harmonic Functions ◽  
Entire Plane ◽  
Cartesian Products ◽  
One Dimensional ◽  
Cluster Set ◽  
Higher Dimensional ◽  
Distinguished Boundary ◽  
The One ◽  
By Products ◽  
Null Set

A classical theorem of Plessner [6] asserts that any holomorphic function f on the unit disk partitions the unit circle, modulo a null set, into two disjoint pieces such that at each point of the first piece, f has a non-tangential limit, and at each point of the second piece, the cluster set of f in any Stolz angle is the entire plane. Higher dimensional versions of this result were first obtained by Calderon [2], who considered holomorphic functions on Cartesian products of half-planes. In this setting, an exact analogue of the one-dimensional result is obtained, in which the circle is replaced by the distinguished boundary, and the Stolz angles are replaced by products of cones in the coordinate half-planes. The ideas of Calderon were further developed by Rudin [8, pp. 79-83], who considered holomorphic and invariant harmonic functions in the ball of Cn. In this case, the circle is replaced by the unit sphere, and the Stolz angles are replaced by the approach regions of Korányi [4].

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