Darboux transformations, finite reduction groups and related Yang-Baxter maps

In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present 6-dimensional YB maps corresponding to Darboux transformations for the Nonlinear Schr\"odinger (NLS) equation and the derivative Nonlinear Schr\"odinger (DNLS) equation. These YB maps can be restricted to $4-$dimensional YB maps on invariant leaves. The former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries \cite{Allan-Pavlos}. We give a $6-$ dimensional YB-map corresponding to the Darboux transformation for a deformation of the DNLS equation. We also consider vector generalisations of the YB maps corresponding to the NLS and DNLS equation.Comment: 18 pages, revised version. The format of the paper has changed, we added one sectio

On the Theory for a Spin 2 Particle with Anomalous Magnetic Moment in the Basis of Tensors of 2-nd and 3-rd Ranks

The known Pauli – Fierz theory for spin 2 field is based on the use of second order equations. Equivalent approach based on the first order equations was developed by Fedorov and Regge, this theory assumes the use of 30-component field function. Later it was proposed more complicated 50-component theory which describes the massive spin 2 particle possessing the anomalous magnetic moment. In both theories, there is known transition to massless case. Initially in 50-component theory it was used the following set of tensors: scalar, two vectors, symmetric second rank tensor, symmetric tensor of the third rank, and 3-rank tensor antisymmetric in two indices. In the present paper we transform the initial system of equations to new variables: symmetric 2-rank tensor and 3-rank tensor which is symmetric in two indices. The symmetry properties of the new variables are similar to symmetry properties of the metric tensor and Kristoffel symbols in General Relativity. The matrix 50-component theory in Minkowski space-time is elaborated. It is demonstrated that in presence of external electromagnetic fields, after eliminating the 3rd rank tensor we arrive at the system of equations for symmetric tensor which contains the non-minimal interaction term trough electromagnetic tensor due to anomalous magnetic moment of the spin 2 particle. In accordance with the tetrad method by Tetrode – Weyl – Fock – Ivanenko, extension to any curved space-time is performed

Spin 2 Particle with Anomalous Magnetic Moment in Presence of Uniform Magnetic Field, Exact Solutions and Energy Spectra

The 50-component matrix equation for spin 2 particle with anomalous magnetic moment is studied in presence of external magnetic field. The matrix tetrad based form of equation in the cylindrical coordinates is used. By diagonalizing the operators of energy, of the third projection of the total angular momentum and the third projection of the linear momentum wa derive the system of 50 differential equations of the first order in polar coordinate. In accordance with the method by Fedorov – Gronskiy based on the use of projective operators, we express all the 50 variables trough 7 different functions? equations for them reduce to the confluent hypergeometric functions. In the result, we obtain a 50-component system algebraic equations which should determine the structure of the total wave function. After eliminating the variables related to 40 components of the third rank tensor we derive the homogeneous algebraic system of 10 equations. It is solved, giving 5 independent solutions. There arise 5 different energy spectra as solutions of the 2-nd and the third order equations. They are found in analytical form and studied numerically

50-component Theory for Spin 2 Particle, Plane Wave Solutions, Massive and Massless Cases

It is known a 50-component theory for a spin 2 particle with the anomalous magnetic moment, invariant under the Lorentz group and based on the use of symmetric 2-nd rank tensor and a 3-rd rank tensor symmetric in two indices. Their symmetries are similar to these of the metric tensor and Christoffel symbols of General relativity. By eliminating the 3-rd rank tensor we derive a system of 2-nd order 10 equations for symmetric tensor. This system is studied for plane wave solutions. In massive case, we have found 5 independent solutions. Separately, we have examined the case of a massless field. There are found 6 independent solutions, 4 special combinations of them are identified with the gauge solutions in accordance with Pauli - Fierz approach. Finally we have found two solutions which do not contain any gauge components

Spin 3/2 Particle in External Magnetic Field, the Method of Projective Operators, Algebraic Approach

In the present paper, an algebraic method for solving the system of equations describing the spin 3/2 particle in presence of the uniform magnetic field has been elaborated. The method is based on decomposition of 16-components wave function with transformation properties of vector-bispinor in the sum of four constitutes, which are determined by four projective operators. With the use of formalism of elements of complete matrix algebra, the system is transformed to the form, in which only projective constituents Ψ±1/2(x), Ψ±3/2(x) enter. This system of equations is transformed to cylindric coordinates. On the wave functions three operators are diagonalized: the energy, the third projection of linear momentum and the third projection of the total angular momentum. After separating the variables, we derive 4 linked subsystems of equations for 16-component functions Ψ±1/2(r), Ψ±3/2(r). After performing needed calculations, the problem reduces to independent second order equations for 4 primary functions. This equations are solved in terms of confluent hypergeometric functions, 4 different energy spectra are found

Безмассовая частица со спином 2: цилиндрическая симметрия, проективные операторы, калибровочные степени свободы

In the present paper, we have developed the theory of a massless spin 2 particle. We apply the matrix equation in Minkowski space-time, specifying it in cylindrical coordinates t r z φ and tetrad. By diagonalizing energy operators, the third projection of total angular momentum, and the third projection of linear momentum, we derive the system of 39 differen - tial equations in a polar coordinate r. In order to resolve this system, we apply the Fedorov–Gronskiy method based on the projective operator method. In accordance with this method, the dependence of all 39 functions is determined only by five different functions of the polar variable r that in the considered case are expressed in terms of Bessel functions. We find the explicit form of six independent solutions of the basic matrix equation. In order to eliminate gauge degrees of freedom, we use the general structure of gauge solutions according to the Pauli–Fierz approach, when the gauge solutions for the spin 2 field are constructed on the basis of the exact solution for a massless spin 1 field (in Bessel functions as well). In this way, we find the explicit form of two independent gauge solutions for the spin 2 field. In the end, we derive the explicit form of two gauge- free solutions for the massless spin 2 field, as should be expected by physical reason

Безмассовая частица со спином 2: цилиндрическая симметрия, проективные операторы, калибровочные степени свободы

In the present paper, we have developed the theory of a massless spin 2 particle. We apply the matrix equation in Minkowski space-time, specifying it in cylindrical coordinates t, r, φ, z and tetrad. By diagonalizing energy operators, the third projection of total angular momentum, and the third projection of linear momentum, we derive the system of 39 differential equations in a polar coordinate r. In order to resolve this system, we apply the Fedorov–Gronskiy method based on the projective operator method. In accordance with this method, the dependence of all 39 functions is determined only by five different functions of the polar variable r that in the considered case are expressed in terms of Bessel functions. We find the explicit form of six independent solutions of the basic matrix equation. In order to eliminate gauge degrees of freedom, we use the general structure of gauge solutions according to the Pauli-Fierz approach, when the gauge solutions for the spin 2 field are constructed on the basis of the exact solution for a massless spin 1 field (in Bessel functions as well). In this way, we find the explicit form of two independent gauge solutions for the spin 2 field. In the end, we derive the explicit form of two gauge-free solutions for the massless spin 2 field, as should be expected by physical reason.Исследуется безмассовая частица со спином 2, при этом применяется матричное уравнение в цилиндрической тетраде пространства Минковского. На решениях диагонализируются операторы энергии, третьей проекции полного момента и третьей проекции импульса; после разделения переменных получена система из 39 уравнений по полярной координате r. Для нахождения решений этих уравнений используется метод Федорова–Гронского, основанный на теории проективных операторов. В соответствии с этим 39 функций выражаются через пять основных функций от переменной r, строящихся в терминах функций Бесселя. Найден явный вид шести независимых решений. Чтобы исключить калибровочные степени свободы, используется явный вид четырех калибровочных решений, строящихся согласно теории Паули–Фирца на основе точных решений уравнения для безмассовой частицы со спином 1 в цилиндрических координатах. После исключения из шести решений четырех калибровочных найдены явные выражения для двух независимых решений, не содержащих калибровочных степеней свободы

U.S. stock market interaction network as learned by the Boltzmann Machine

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented analysis shows that binarization preserves market correlation structure. Properties of distributions of external fields and couplings as well as industry sector clustering structure are studied for different historical dates and moving window sizes. We found that a heavy positive tail in the distribution of couplings is responsible for the sparse market clustering structure. We also show that discrepancies between the model parameters might be used as a precursor of financial instabilities.Comment: 15 pages, 17 figures, 1 tabl

Living with joint hypermobility syndrome: patient experiences of diagnosis, referral and self-care.

BACKGROUND: Musculoskeletal problems are common reasons for seeking primary health care. It has been suggested that many people with 'everyday' non-inflammatory musculoskeletal problems may have undiagnosed joint hypermobility syndrome (JHS), a complex multi-systemic condition. JHS is characterized by joint laxity, pain, fatigue and a wide range of other symptoms. Physiotherapy is usually the preferred treatment option for JHS, although diagnosis can be difficult. The lived experience of those with JHS requires investigation. OBJECTIVE: The aim of the study was to examine patients' lived experience of JHS, their views and experiences of JHS diagnosis and management. METHODS: Focus groups in four locations in the UK were convened, involving 25 participants with a prior diagnosis of JHS. The focus groups were audio recorded, fully transcribed and analysed using the constant comparative method to inductively derive a thematic account of the data. RESULTS: Pain, fatigue, proprioception difficulties and repeated cycles of injury were among the most challenging features of living with JHS. Participants perceived a lack of awareness of JHS from health professionals and more widely in society and described how diagnosis and access to appropriate health-care services was often slow and convoluted. Education for patients and health professionals was considered to be essential. CONCLUSIONS: Timely diagnosis, raising awareness and access to health professionals who understand JHS may be particularly instrumental in helping to ameliorate symptoms and help patients to self-manage their condition. Physiotherapists and other health professionals should receive training to provide biopsychosocial support for people with this condition

Water-stable zirconium-based metal-organic framework material with high-surface area and gas-storage capacities.

We designed, synthesized, and characterized a new Zr-based metal-organic framework material, NU-1100, with a pore volume of 1.53 ccg(-1) and Brunauer-Emmett-Teller (BET) surface area of 4020 m(2) g(-1) ; to our knowledge, currently the highest published for Zr-based MOFs. CH4 /CO2 /H2 adsorption isotherms were obtained over a broad range of pressures and temperatures and are in excellent agreement with the computational predictions. The total hydrogen adsorption at 65 bar and 77 K is 0.092 g g(-1) , which corresponds to 43 g L(-1) . The volumetric and gravimetric methane-storage capacities at 65 bar and 298 K are approximately 180 vSTP /v and 0.27 g g(-1) , respectively.OKF, JTH and RQS thank DOE ARPA-E and the Stanford Global Climate and Energy Project for support of work relevant to methane and CO2, respectively. TY acknowledges support by the U. S. Department of Energy through BES Grant No. DE-FG02-08ER46522. WB acknowledges support from the Foundation for Polish Science through the “Kolumb” Program. DFJ acknowledges the Royal Society (UK) for a University Research Fellowship. This material is based upon work supported by the National Science Foundation (grant CHE-1048773).This is the accepted manuscript. The final version is available as 'Water-Stable Zirconium-Based Metal–Organic Framework Material with High-Surface Area and Gas-Storage Capacities' from Wiley at http://onlinelibrary.wiley.com/doi/10.1002/chem.201402895/abstract