One Singularly Perturbed Problem of Turbulent Gas Dynamics

The problem of the interaction of a Prandtl–Mayer wave with a shear layer is solved using the small parameter method for the case where the flow vorticity in the shear layer is small. A direct expansion is constructed and its inadequacy at large distances from the vortex layer is proved. The strained coordinate method is used to obtain a uniformly adequate expansion. It is shown that for certain velocity distributions in the shear layer, the characteristics in the reflected simple wave resulting from the interaction intersect each other and a shock arises in the flow. There coordinates of the shock origin and the function describing the shock shape are obtained.     

A comparison principle for functions of a uniformly random subspace

This note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the Stiefel manifold in terms of the expected norm of a standard Gaussian matrix with the same dimensions. A related comparison holds for any convex function of a random matrix drawn from the Stiefel manifold. For certain norms, a reversed inequality is also valid.     

Calculation of thermal stresses in profiled single crystals taking into account the temperature dependence of thermal expansion

Approximate expressions for stress tensor components in cylindrical and ribbon crystals are obtained taking into account the temperature dependence of the thermal expansion coefficient. These expressions are used for estimating the effect of the thermal expansion nonlinearity factor on the stress level. It is shown that the stresses emerging in linear temperature fields due to this factor are comparable with critical stresses for defect formation.     

Shape of the surface of a vertical liquid bridge between two parallel solid planes taking into account the gravity force for small Bond numbers

A variational formulation has given the problem of the shape of the lateral surface of a small vertical liquid θ bridge between two parallel solid surfaces that take into account the gravity force in the axisymmetric case. An algorithm has been constructed for the iterative solution of the problem for small Bond numbers. The dependence of the number of solutions on the liquid bridge height has been analyzed.     

The Problem of the Shape of a Vertical Liquid Bridge between Convex Surfaces Taking the Gravity Force into Account

Technical Physics - The solution to the problem of the shape of the lateral surface of a vertical 3D catenoidal liquid bridge of small volume between two arbitrary convex solid surfaces in the...     

Calculation of the thermal effect of an electron probe on a sample of GaN

Stationary temperature fields due to the interaction of an electron probe with a GaN sample are examined. In order to calculate the density of generated heat, the process of electron energy loss is modeled by the Monte Carlo method. The heat generation region is assumed to have the shape of a half-ellipsoid. In the case of uniform heat generation in the ellipsoid, an analytical solution to the heat conduction problem is found and expressed in terms of elementary functions. It is shown that the maximum heating temperature and the temperature field distribution depend only slightly on the shape of the heat generation region. An approximation of the density of heat sources by a uniform distribution over a hemisphere of radius equal to the ultimate range of electrons leads to a considerably underestimated maximum heating temperature. An expression is derived for determining the characteristic size of the heat generation region in GaN; this expression allows one to calculate the maximum heat temperature with an accuracy of 3% in a wide range of electron beam energies.     

On the Linear Independence of Spikes and Sines

The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof depends on an extrapolation argument of Bourgain and Tzafriri.     

Directly detected Mn MRI: Application to phantoms for human hyperpolarized C MRI development

In this work we demonstrate for the first time directly detected manganese-55 (⁵⁵Mn) magnetic resonance imaging (MRI) using a clinical 3 T MRI scanner designed for human hyperpolarized ¹³C clinical studies with no additional hardware modifications. Due to the similar frequency of the ⁵⁵Mn and ¹³C resonances, the use of aqueous permanganate for large, signal-dense, and cost-effective “¹³C” MRI phantoms was investigated, addressing the clear need for new phantoms for these studies. Due to 100% natural abundance, higher intrinsic sensitivity, and favorable relaxation properties, ⁵⁵Mn MRI of aqueous permanganate demonstrates dramatically increased sensitivity over typical ¹³C phantom MRI, at greatly reduced cost as compared with large ¹³C-enriched phantoms. A large sensitivity advantage (22-fold) was demonstrated. A cylindrical phantom (d = 8 cm) containing concentrated aqueous sodium permanganate (2.7 M) was scanned rapidly by ⁵⁵Mn MRI in a human head coil tuned for ¹³C, using a balanced steady state free precession acquisition. The requisite penetration of radiofrequency magnetic fields into concentrated permanganate was investigated by experiments and high frequency electromagnetic simulations, and found to be sufficient for ⁵⁵Mn MRI with reasonably sized phantoms. A sub-second slice-selective acquisition yielded mean image signal-to-noise ratio of ~ 60 at 0.5 cm³ spatial resolution, distributed with minimum central signal ~ 40% of the maximum edge signal. We anticipate that permanganate phantoms will be very useful for testing HP ¹³C coils and methods designed for human studies.     

Matrix Concentration for Products

Foundations of Computational Mathematics - This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize...     

Separation of angular and energy relaxations of nonequilibrium electrons in a solid

We demonstrate that the collision integral of the kinetic equation for the interaction of hot electrons with phonons can be split into substantially different parts that correspond to elastic and inelastic collisions. In particular, this applies to electrons with energies of about 1 eV that propagate in semiconductors. The difference in the characteristic energy and momentum relaxation times makes it possible to separate the angular and energy relaxation processes. If the differential cross section of elastic scattering depends, not on the scattering angle, but on the directions of incident and scattered electrons (which is observed, e.g., for the interaction of an electron with piezoelectric lattice vibrations in A^IIIB^V compounds), the Laplacian in the equation that describes the spatial and energy distributions of electrons can be replaced by an elliptical operator; i.e., the electron diffusion turns out to be anisotropic.