On some discrete subgroups of the Lorentz group

Some discrete subgroups of the Lorentz group are found using Fedorov's parametrization by means of complex vector-parameter. It is shown that the discrete subgroup of the Lorentz group, which have not fixed points, are contained in boosts along a spatial direction for time-like and space-like vectors and are discrete subgroups of the group SO(1,1), whereas discrete subgroups of isotropic vector are subgroups of SO(1,1)\times E(1,1).Comment: 9 page

Dynamical relations in the system of two objects with internal degrees of freedom

A system of $N$ interacting objects with internal degrees of freedom is considered. Derivation of system of equations for the description of two interacting objects with spin is given. Relations between the parameters describing subsystems and the parameters describing the system as a whole are obtained. In particular, relation between energies of subsystems and energy of system is found, on the basis of which the assumption is made that energy of subsystem should be oscillatory functions of time, so that interaction between objects is reduced to permanent energy exchang

Impact of PVT Properties of the Fluid on the LBM Scheme Within the Scale Integration for Shale Reservoirs

Modelling of the performance of shale gas reservoirs is known for the presence of multiple scales. The latter includes pore-scale, fracture scale and field scale. The nature of flow-mechanisms at various scales is different. Therefore, separate treatment of the physical processes is required. On the other hand, an integrated approach is highly beneficial for practical implementation. One of the candidates for seamless integration concerned is the Lattice-Boltzmann Method. The latter fact together with the demands of the industry provides the major motivation for the present work. In this study the novel Lattice-Boltzmann Model for pore-scale simulations has been introduced. The major advantage of the approach concerned is that the mathematical formulation of the model has a high degree of self-consistency. The latter means that it does not have an artificially introduced terms like pseudo-potentials, which are common for conventional Lattice-Boltzmann schemes. Despite the advantages of the approach in terms of mathematical formulation, there exist certain limitations because of the issues with numerical stability. One of the most important results of the present work is that the issues concerned can not be resolved by the reasonable increase of the number of lattice vectors in the model. The limitations involved make the scheme impractical for fieldscale simulations. Therefore, an alternative formulation of Lattice-Boltzmann method for reservoir modelling is required. In the present work, a novel pseudo-potential model for field-scale simulations has been introduced. The model concerned demonstrates a reasonable agreement with the analytical techniques in the case of steady-state flow. However, further investigation shows significant deviations because of the numerical diffusion. Moreover, it has been shown that significant numerical diffusion is a feature of the majority of the existent pseudo-potential models. The numerical effect concerned is critically important in the case of the multiphase flow, because it can lead to non-physical solutions. In order to resolve the problem concerned a novel Lattice-Boltzmann Scheme has been introduced. The scheme demonstrates reasonable agreement with analytical methods and with simulations performed with trusted programs for reservoir modelling. Finally, the major contribution of the present work includes the development of selfconsistence approach for simulations at pore-scale, the proof of fundamental limitations of the model introduced, observation of numerical diffusion in pseudo-potential Lattice- Boltzmann Methods, and the solution of the latter issue through the development of the novel Lattice-Boltzmann scheme for field-scale simulations

Treatment and Companion Diagnostics of Lower Back Pain Using Self-Controlled Energo-Neuroadaptive Regulator (SCENAR) and Passive Microwave Radiometry (MWR)

Evaluation of the effectiveness of treatment of nonspecific lower back pain (LBP) is currently largely based on the patient’s subjective feelings. The purpose of this study was to use passive microwave radiometry (MWR) as a tool for assessing the effectiveness of various treatment methods in patients with acute and subacute nonspecific LBP. Patients with a pain assessment on a visual analogue scale (VAS) of 6 to 10 points were divided into two groups: Group I included patients with pharmacological, syndrome-oriented treatment (n = 30, age 54.9 ± 2.3 years); Group II included a combination of pharmacotherapy with self-controlled energy-neuroadaptive regulation (SCENAR) (n = 25, age 52.8 ± 2.5 years). The analysis showed that the addition of SCENAR therapy (Group II) significantly potentiated the analgesic effect at the stages of treatment, and after 3 weeks, this had increased by more than two times, by 1.3 points on the VAS. There was also a significant decrease in the maximum internal temperature and normalization of the gradient of internal and skin temperatures, and a decrease in thermo-asymmetry, as assessed by temperature fields. Thermal asymmetry visualization allows the identification of the area of pathological muscle spasm and/or inflammation in the projection of the vertebral-motor segment for the possible targeted use of treatment methods such as percutaneous electro neurostimulation, massage, manual therapy, diagnostic and treatment blocks, etc. The MWR method also avoids unnecessary radiation exposure

Generalized boosts with shell structure of the parameter space

A modification of boost transformation in arbitrary pseudo-Euclidean space is suggested, which in the case of the Minkowski space admits the existence of inertial reference frames moving with velocities taking values in a certain bounded interval. The velocity space may be partitioned by hypersurfaces \beta^2=\beta^2_k=const, k=1,2,3,..., into a finite or countable number of domains (shells), each of which has own class of inertial "reference frames" and the velocity composition law. These shells are in one-to-one correspondence. A set of mappings of shells to each other forms the group, isomorphic to permutation group in the case of finite number of shells, or the group of integers in the case of countable number of shells in the velocity spaceComment: 6 page

On the Possible Trajectories of Spinning Particles. I. Free Particles

By means of the method of moving Frenet-Serret frame the set of equations of motion is derived for spinning particle in an arbitrary external field, which is determined by potential depending from both position and the state of movement, as well as by two pseudo-vectors one of which is easily associated with external magnetic field, and another still remains undetermined. The equations give a possibility to describe the motion of both massive and massless particles with spin. All solutions of the equations of motion in the absence of external fields were found, and besides, we give more precise definition of a free object. It turns out that the massive particles always possess a longitudinal polarization. There are possible transversal motions of the following types: 1) oscillatory motion with proper frequency, 2) circular motion, and 3) complicated motion along rosette trajectories round the center of inertia with the velocity, varying in finite limits. Free massless particles can either fluctuate or move along complicated paths around fixed centers of balance, when the spin of the particles can have any direction.Comment: 16 pages, 4 figures; Extended report at 9-th International Conference Bolyai-Gauss-Lobachevsky: Non-Euclidean Geometry in Modern Physics, BGL-9, October 27-30, 2015, Minsk, Belaru

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Surrogate-modelling techniques including Polynomial Chaos Expansion (PCE) is commonly used for statistical estimation (aka. Uncertainty Quantification) of quantities of interests obtained from expensive computational models. PCE is a data-driven regression-based technique that relies on spectral polynomials as basis-functions. In this technique, the outputs of few numerical simulations are used to estimate the PCE coefficients within a regression framework combined with regularization techniques where the regularization parameters are estimated using standard cross-validation as applied in supervised machine learning methods. In the present work, we introduce an efficient method for estimating the PCE coefficients combining Elastic Net regularization with a data-driven feature ranking approach. Our goal is to increase the probability of identifying the most significant PCE components by assigning each of the PCE coefficients a numerical value reflecting the magnitude of the coefficient and its stability with respect to perturbations in the input data. In our evaluations, the proposed approach has shown high convergence rate for high-dimensional problems, where standard feature ranking might be challenging due to the curse of dimensionality. The presented method is implemented within a standard machine learning library (Scikit-learn) allowing for easy experimentation with various solvers and regularization techniques (e.g. Tikhonov, LASSO, LARS, Elastic Net) and enabling automatic cross-validation techniques using a widely used and well tested implementation. We present a set of numerical tests on standard analytical functions, a two-phase subsurface flow model and a simulation dataset for CO2 sequestration in a saline aquifer. For all test cases, the proposed approach resulted in a significant increase in PCE convergence rates.Comment: 28 pages, 7 figures, published in Journal of Computational Physics (2019

Impact of PVT Properties of the Fluid on the LBM Scheme Within the Scale Integration for Shale Reservoirs

Modelling of the performance of shale gas reservoirs is known for the presence of multiple scales. The latter includes pore-scale, fracture scale and field scale. The nature of flow-mechanisms at various scales is different. Therefore, separate treatment of the physical processes is required. On the other hand, an integrated approach is highly beneficial for practical implementation. One of the candidates for seamless integration concerned is the Lattice-Boltzmann Method. The latter fact together with the demands of the industry provides the major motivation for the present work. In this study the novel Lattice-Boltzmann Model for pore-scale simulations has been introduced. The major advantage of the approach concerned is that the mathematical formulation of the model has a high degree of self-consistency. The latter means that it does not have an artificially introduced terms like pseudo-potentials, which are common for conventional Lattice-Boltzmann schemes. Despite the advantages of the approach in terms of mathematical formulation, there exist certain limitations because of the issues with numerical stability. One of the most important results of the present work is that the issues concerned can not be resolved by the reasonable increase of the number of lattice vectors in the model. The limitations involved make the scheme impractical for fieldscale simulations. Therefore, an alternative formulation of Lattice-Boltzmann method for reservoir modelling is required. In the present work, a novel pseudo-potential model for field-scale simulations has been introduced. The model concerned demonstrates a reasonable agreement with the analytical techniques in the case of steady-state flow. However, further investigation shows significant deviations because of the numerical diffusion. Moreover, it has been shown that significant numerical diffusion is a feature of the majority of the existent pseudo-potential models. The numerical effect concerned is critically important in the case of the multiphase flow, because it can lead to non-physical solutions. In order to resolve the problem concerned a novel Lattice-Boltzmann Scheme has been introduced. The scheme demonstrates reasonable agreement with analytical methods and with simulations performed with trusted programs for reservoir modelling. Finally, the major contribution of the present work includes the development of selfconsistence approach for simulations at pore-scale, the proof of fundamental limitations of the model introduced, observation of numerical diffusion in pseudo-potential Lattice- Boltzmann Methods, and the solution of the latter issue through the development of the novel Lattice-Boltzmann scheme for field-scale simulations