The equivalence of the norms of deviations of the desired density of a body from operators such as finite density transformation with specially constructed elements and the Radon transformation from it is stated. It is shown how computer science, previously established in the theory of computational (numerical) diameter, immediately leads to nontrivial results in computed tomography.
A new geometric condition necessary for regularity of a curved three-web is found. A class of three-webs from circles generalizing the regular three-web of Blaschke from three elliptic pencils of circles with pairwise coinciding vertices is considered, and it is shown that only webs equivalent to the Blaschke web are regular in this class.
In the present paper the lattice optimal cubature formulas are constructed by the variational method in the Sobolev space. In addition, the square of the norm of the error functional of the constructed lattice optimal cubature formulas in the conjugate Sobolev space is explicitly calculated.
A new symmetric linear variational statement is proposed for the problem of eigenvibrations of a plate with an attached oscillator. The existence of the sequence of positive eigenvalues of finite multiplicity with the limit point at infinity and the corresponding complete orthonormal system of eigenvectors is established. A new symmetric scheme of the finite element method with Hermite finite elements is stated. Error estimates consistent with the solution smoothness for the approximate eigenvalues and approximate eigenvectors are proved. The results of numerical experiments illustrating the influence of the smoothness of the solution on the computation accuracy are presented.
In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for a generalized Cauchy–Riemann equation with supersingular coefficients. To solve the problem, it was necessary to derive a structural formula for the general solution to this equation and to investigate the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index due to the power-order vorticity point.
The Lebesgue constant of the classical Fourier operator is uniformly approximated by a logarithmic-fractional-rational function depending on three parameters; they are defined using the specific properties of logarithmic and rational approximations. A rigorous study of the corresponding residual term having an indefinite (nonmonotonic) behavior has been carried out. The obtained approximation results strengthen the known results by more than two orders of magnitude.
This paper presents a calculation method and algorithm, as well as numerical results of studying chemically reacting turbulent jets based on three-dimensional parabolic systems of Navier–Stokes equations for multicomponent gas mixtures. Continuity equations are used to calculate the mass imbalance when solving with constant pressure, and with variable pressures, with the equations of motion and continuity. Diffusion combustion of a propane–butane mixture flowing from a square-shaped nozzle in a submerged flow of an air oxidizer is numerically studied. Pressure variability significantly affects the velocity (temperature) profiles in the initial sections of the jet, and, when moving away from the nozzle exit, the pressure effect can be considered imperceptible, but the plume length is longer than that at constant pressure, but it does not significantly affect the shape of the plume. The saddle-shaped behavior of the longitudinal velocity in the direction of the major axis is numerically obtained for large initial values of the turbulence kinetic energy of the main jet. The presented method allows studying nonreacting and reactive turbulent jets flowing from a rectangular nozzle.
The author has previously put forward a hypothesis that in n-dimensional Euclidean space, curves, any two oriented arcs of which are similar, are rectilinear. The author also proved this statement for dimensions of n = 2 and n = 3. In a space of arbitrary dimension, this hypothesis was confirmed in the class of rectifiable curves. In this study, the author provides a complete solution to this problem that is even stronger: (a) a curve in E n, where any two oriented arcs starting at an common nonfixed point are similar is rectilinear; (b) if a curve in E n has a half-tangent at its boundary point and any two of its oriented arcs emanating from this point are similar, then the curve is rectilinear; (c) if a curve in E n has a tangent at an inner point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in E2 and E3 are given, that are not rectilinear, although their arcs having a common startpoint are similar, and a complete description of such curves in E2 is given. Research methods are topological and set-theoretic using the apparatus of functional equations.
This paper considers the inverse problem of determining the time-dependent coefficient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag–Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.
Let (tau ) be a faithful normal semifinite trace on a von Neumann algebra (mathcal{M}). The block projection operator ({{mathcal{P}}_{n}}) ((n geqslant 2)) in the *-algebra (S(mathcal{M},tau )) of all (tau )-measurable operators is investigated. It has been shown that (A leqslant n{{mathcal{P}}_{n}}(A)) for any operator (A in S{{(mathcal{M},tau )}^{ + }}). If (A in S{{(mathcal{M},tau )}^{ + }}) is invertible in (S(mathcal{M},tau )), then ({{mathcal{P}}_{n}}(A)) is invertible in (S(mathcal{M},tau )). Let (A = Atext{*} in S(mathcal{M},tau )). Then, (i) if ({{mathcal{P}}_{n}}(A) leqslant A) (or if ({{mathcal{P}}_{n}}(A) geqslant A)), then ({{mathcal{P}}_{n}}(A) = A), (ii) ({{mathcal{P}}_{n}}(A) = A) if and only if ({{P}_{k}}A = A{{P}_{k}}) for all (k = 1, ldots ,n); and (iii) if (A,{{mathcal{P}}_{n}}(A) in mathcal{M}) are projections, then ({{mathcal{P}}_{n}}(A) = A). Four corollaries have been obtained. One example presented in paper (A. Bikchentaev and F. Sukochev, “Inequalities for the Block Projection Operators,” J. Funct. Anal. 280 (7), 108851 (2021)) has been refined and strengthened.
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