Numerical constraints and non-spatial open boundary conditions for the Wigner equation

We discuss boundary value problems for the characteristic stationary von Neumann equation (stationary sigma equation) and the stationary Wigner equation in a single spatial dimension. The two equations are related by a Fourier transform in the non-spatial coordinate. In general, a solution to the characteristic equation does not produce a corresponding Wigner solution as the Fourier transform will not exist. Solution of the stationary Wigner equation on a shifted k-grid gives unphysical results. Results showing a negative differential resistance in IV-curves of resonant tunneling diodes using Frensley’s method are a numerical artefact from using upwinding on a coarse grid. We introduce the integro-differential sigma equation which avoids distributional parts at $$k=0$$ k = 0 in the Wigner transform. The Wigner equation for $$k=0$$ k = 0 represents an algebraic constraint needed to avoid poles in the solution at $$k=0$$ k = 0 . We impose the inverse Fourier transform of the integrability constraint in the integro-differential sigma equation. After a cutoff, we find that this gives fully homogeneous boundary conditions in the non-spatial coordinate which is overdetermined. Employing an absorbing potential layer double homogeneous boundary conditions are naturally fulfilled. Simulation results for resonant tunneling diodes from solving the constrained sigma equation in the least squares sense with an absorbing potential reproduce results from the quantum transmitting boundary with high accuracy. We discuss the zero bias case where also good agreement is found. In conclusion, we argue that properly formulated open boundary conditions have to be imposed on non-spatial boundaries in the sigma equation both in the stationary and the transient case. When solving the Wigner equation, an absorbing potential layer has to be employed.

Mechanical and mathematical modeling of seismic shear vibrations of a glacial massif

В статье впервые в мире разработаны теоретические положения сдвиговых сейсмических колебаний ледникового массива. Актуальность представленных научных разработок в приложении к инженерной сейсмологии и гляциологии обусловлено тем, что в недавнее время в различных регионах нашей планеты имели место внезапные срывы с гор грандиозных масс льда, что приводило к образованию мощных гляциальных селевых потоков. Эти потоки уничтожали населенные пункты и народохозяйственные объекты с многочисленными жертвами. Все мы помним катастрофический сход ледника Колка в Геналдонском ущелье в 2002г., унесшего 125 человеческих жизней. Причиной срыва ледяных масс со своих подстилающих поверхностей примерзаний является динамическое воздействие, в качестве которого мы рассматриваем землетрясение. Цель исследования. На основе современных научных методов механики сплошных сред проведение механико-математического компьютерного моделирования колебательного процесса в ледниковом массиве, когда колебание спровоцировано гармонической сейсмической волной, упавшей на подстилающую поверхность примерзания массива. В рамках выполненного моделирования содержится постановка и решение соответствующей начально-краевой задачи. Начальными данными являются как физико-механические характеристики льда, его плотность, модуль сдвига, коэффициент внутреннего (вязкого) сопротивления, так и геометрические размеры и непризматическая конфигурация массива. Искомыми величинами в поставленной начально-краевой задаче являются перемещения и напряжения, как в самом теле массива, так и на подстилающей поверхности примерзания. Методы исследования. Составленная модель представляет собой начально-краевую задачу математической физики для дифференциального уравнения гиперболического типа, в котором один коэффициент является комплексной величиной, названной комплексным модулем сдвига согласно с гипотезой Е.С. Сорокина, а другой коэффициент является переменной величиной, зависящей от пространственной координаты. Эти два особых фактора создают трудности в аналитическом способе решения начально-краевых задач. В представленной работе найден путь решения поставленной задачи в частном случае – при экспоненциальной зависимости переменного коэффициента от пространственной координаты. Результаты работы. Получена совокупность расчётных формул для вычисления напряжений и деформаций в ледниковом массиве. Доказано утверждение о том, что низкобалльная сейсмическая околорезонансная волна может отколоть ледниковый массив от подстилающей поверхности примерзания, что приведет к образованию гляциального селевого потока Theoretical studies of seismic oscillations of the glacial massif are an urgent task in the field of engineering seismology and glaciology. This statement is confirmed if we recall the case of the sudden catastrophic collapse of the Kolka glacier in 2002, which claimed the lives of 125 human lives. Aim. Conducting a mechanical and mathematical simulation of the oscillatory process in a glacial massif, when the oscillation is triggered by a harmonic seismic wave that has fallen on the underlying surface of the frozen massif. Formulation and solution of the initial boundary value problem for calculating stresses and deformations in a glacial massif. Methods. The compiled model represents an initial boundary value problem of mathematical physics for a hyperbolic differential equation, in which one coefficient is a complex quantity called the complex shift modulus according to the hypothesis of E.S. Sorokin, and the other coefficient is a variable value depending on the spatial coordinate. These two special factors create difficulties in the analytical way of solving initial-boundary value problems. In the present paper, we find a way to solve the problem in the special case - with an exponential dependence of the variable coefficient on the spatial coordinate. Results. A set of calculation formulas for calculating stresses and deformations in the glacial massif is obtained. It is proved that a low-point seismic near-resonant wave can break off the glacial massif from the underlying freezing surface, which will lead to the formation of a glacial mudflow.

Measurement of Human Semicircular Canal Spatial Attitude

Located deep in the temporal bone, the semicircular canal is a subtle structure that requires a spatial coordinate system for measurement and observation. In this study, 55 semicircular canal and eyeball models were obtained by segmentation of MRI data. The spatial coordinate system was established by taking the top of the common crus and the bottom of the eyeball as the horizontal plane. First, the plane equation was established according to the centerline of the semicircular canals. Then, according to the parameters of the plane equation, the plane normal vectors were obtained. Finally, the average unit normal vector of each semicircular canal plane was obtained by calculating the average value of the vectors. The standard normal vectors of the and left posterior semicircular canal, superior semicircular canal and lateral semicircular canal were [−0.651, 0.702, 0.287], [0.749, 0.577, 0.324], [−0.017, −0.299, 0.954], [0.660, 0.702, 0.266], [−0.739, 0.588, 0.329], [0.025, −0.279, 0.960]. The different angles for the different ways of calculating the standard normal vectors of the right and left posterior semicircular canal, superior semicircular canal and lateral semicircular canal were 0.011, 0.028, 0.008, 0.011, 0.024, and 0.006 degrees. The technology for measuring the semicircular canal spatial attitudes in this study are reliable, and the measurement results can guide vestibular function examinations and help with guiding the diagnosis and treatment of BPPV.

Development and Application of Fast Simulation Based on the PSS Pressure as a Spatial Coordinate

Recent studies have shown the utility of the Fast Marching Method and the Diffusive Time of Flight for the rapid simulation and analysis of Unconventional reservoirs, where the time scale for pressure transients are long and field developments are dominated by single well performance. We show that similar fast simulation and multi-well modeling approaches can be developed utilizing the PSS pressure as a spatial coordinate, providing an extension to both Conventional and Unconventional reservoir analysis. We reformulate the multi-dimensional multi-phase flow equations using the PSS pressure drop as a spatial coordinate. Properties are obtained by coarsening and upscaling a fine scale 3D reservoir model, and are then used to obtain fast single well simulation models. We also develop new 1D solutions to the Eikonal equation that are aligned with the PSS discretization, which better represent superposition and finite sized boundary effects than the original 3D Eikonal equation. These solutions allow the use of superposition to extend the single well results to multiple wells. The new solutions to the Eikonal equation more accurately represent multi-fracture interference for a horizontal MTFW well, the effects of strong heterogeneity, and finite reservoir extent than those obtained by the Fast Marching Method. The new methodologies are validated against a series of increasingly heterogeneous synthetic examples, with vertical and horizontal wells. We find that the results are systematically more accurate than those based upon the Diffusive Time of Flight, especially as the wells are placed closer to the reservoir boundary or as heterogeneity increases. The approach is applied to the Brugge benchmark study. We consider the history matching stage of the study and utilize the multi-well fast modeling approach to determine the rank quality of the 100+ static realizations provided in the benchmark dataset against historical data. The multi-well calculation uses superposition to obtain a direct calculation of the interaction of the rates and pressures of the wells without the need to explicitly solve flow equations within the reservoir model. The ranked realizations are then compared against full field simulation to demonstrate the significant reduction in simulation cost and the corresponding ability to explore the subsurface uncertainty more extensively. We demonstrate two completely new methods for rapid reservoir analysis, based upon the use of the PSS pressure as a spatial coordinate. The first approach demonstrates the utility of rapid single well flow simulation, with improved accuracy compared to the use of the Diffusive Time of Flight. We are also able to reformulate and solve the Eikonal equation in these coordinates, giving a rapid analytic method of transient flow analysis for both single and multi-well modeling.

Mathematical Model of Human Semicircular Canal Spatial Attitude

Located deep in the temporal bone, the semicircular canal is a subtle structure that requires a spatial coordinate system for measurement and observation. In this study,Fifty-five semicircular canal and eyeball models were obtained by segmentation of MRI data. The spatial coordinate system was established by taking the top of the common crus and the bottom of eyeball as the horizontal plane. Firstly, the plane equation is calculated according to the centerline of the semicircular canals. Then, according to the parameters of the plane equation, the plane normal vectors are obtained. Finally, the average unit normal vector of each semicircular canal plane can be obtained by calculating the average value of the vectors. It is more intuitive and accurate to calculate the average normal vector of semicircular canal plane with the vector average method, which is different from the angular average method in different degrees. The mathematical model of semicircular canal spatial attitude established in this study is more reliable, which can guide the vestibular function examination, and also help guide the diagnosis and treatment of BPPV.

Păstoritul de pe Valea Argeșului – coordonate spațiale

"Shepherding on Argeș Valley – spatial coordinates This paper presents the essential elements that define the spatial coordinate characteristic of shepherding which is practised in the sub-Carpathian villages on the Argeș Valley, following the documentary attestations of the villages, the genesis and evolution of the village boundaries, the evolution of land ownership and the right to use lands (in condominium or individually). Also within the spatial coordinate, there are researched the traditional ways in which the potential of the rural land fund is highlighted, as a result of its geomorphological, pedological, climatic qualities, etc., the study taking into consideration all the areal types with pastoral potential, not only those on the administrative territory of the villages, but also those in the mountain area destined for summer grazing. If for the presentation of origins, attestations, borderline fixation of the villages from the studied areal, documents, monographic studies and other categories of specialized works have been used, for the understanding of the manner of pastoral, individual and especially collective exploitation of the land fund by the village communities, of valorizing the fodder qualities of the different types of surfaces, it was necessary to carry out field research in the three targeted villages: Albeștii de Argeș, Corbeni and Arefu, all in the Argeș County. The visit of sheepfolds from Lespezi, Lipitoarea, Ciocanu, Podeanu, Oticu, in the alpine hollow of the Făgăraș Mountains, were necessary both for studying some elements related to the spatial coordinate (sheepfold location, daily travel routes, travel routes from the village hearth to the mountains, etc.) as well as for conducting interviews, based on an elaborate questionnaire, interviews generating unique and extremely useful information also for researching the other coordinates of the pastoral system practiced in the area. Keywords: shepherding, Argeș Valley, spatial coordinate, pastoral system, field research "

Super Yang-Mills theories with inhomogeneous mass deformations

We construct the 4-dimensional $$ \mathcal{N}=\frac{1}{2} $$ N = 1 2 and $$ \mathcal{N} $$ N = 1 inhomogeneously mass-deformed super Yang-Mills theories from the $$ \mathcal{N} $$ N = 1* and $$ \mathcal{N} $$ N = 2* theories, respectively, and analyse their supersymmetric vacua. The inhomogeneity is attributed to the dependence of background fluxes in the type IIB supergravity on a single spatial coordinate. This gives rise to inhomogeneous mass functions in the $$ \mathcal{N} $$ N = 4 super Yang-Mills theory which describes the dynamics of D3-branes. The Killing spinor equations for those inhomogeneous theories lead to the supersymmetric vacuum equation and a boundary condition. We investigate two types of solutions in the $$ \mathcal{N}=\frac{1}{2} $$ N = 1 2 theory, corresponding to the cases of asymptotically constant mass functions and periodic mass functions. For the former case, the boundary condition gives a relation between the parameters of two possibly distinct vacua at the asymptotic boundaries. Brane interpretations for corresponding vacuum solutions in type IIB supergravity are also discussed. For the latter case, we obtain explicit forms of the periodic vacuum solutions.