Strict Approximation Research Articles

On the properties of the set where a generalized function of bounded variation takes an infinite value

We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim, we prove a new result on strict approximation of the sets of finite perimeter from the outside with open sets.

An Analytic Approximate Solution of the SIR Model

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure exact (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms; moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the exact equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations; the easy and practical formulation of the final approximate solutions; the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called basic reproduction ratio, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

Combined Kinetic-Hydrodynamic Model of Polyatomic Gas Flow

A mathematical model of the flow of a polyatomic gas containing a combination of the Navier-Stokes-Fourier (NSF) model and the model kinetic equation of polyatomic gases is presented. At the heart of the composed components is a unified physical model, as a result of which the NSF model is a strict first approximation of the model kinetic equation. The model allows calculations of flow fields in a wide range of Knudsen numbers (Kn), as well as fields containing regions of a high degree of dynamic nonequilibrium. The boundary conditions on a solid surface are set at the kinetic level, which allows, in particular, to formulate the boundary conditions on the surfaces absorbing or emitting gas. The composed model was tested. The example of the problem of the shock wave profile shows that up to Mach numbers M ≈ 2 the combined model gives smooth solutions even in those cases where the sewing point is in a high gradient region. For the Couette flow, smooth solutions are obtained at M = 5 and Kn = 0.2.

Quantitive Evalution of Involvement of Countries of the World in the International Open Access Movement

The article presents the developed method for the quantitative evaluation of involvement of countries of the world in the international open access movement. It consists in the identification of eight country open access indicators initially connected with the open access initiatives and instruments, their weighing, normalization and aggregation in the form of a weighted average value. In a second more strict approximation the number of indicators has been reduced up to six for the account of discarding the data duplicated in ROAR and Open DOAR. Budapest initiative and Berlin declaration were considered as the ОА-initiatives, and data from the international registers DOAJ, SHERPA/RoMEO, ROAR MAP and the Webometrics ОА-repositories ranking was considered as the instruments. The calculation is done on the basis of the developed method for 133 countries.

Quantitative Evaluation of Involvement of Counties in the International Open Access Movement

The present paper investigated a developed method for the quantitative evaluation of involvement of countries in the international open access movement. It identified eight country open access indices which were initially connected with open access initiatives and instruments, their weighing, normalization and aggregation in a weighted average value. In a second more strict approximation, the number of indices was reduced up to six for the account of discarding duplicated data in ROAR and Open DOAR. Budapest initiative and Berlin declaration were considered as ОА-initiatives; and data of the international registers, DOAJ, SHERPA/RoMEO, ROAR MAP and the Webometrics ОА-repositories ranking, was considered as the tools. The calculation was done on the basis of a developed method for 133 countries.

Accurately identifying mode indices of Laguerre–Gaussian beams via weak measurements

We propose an efficient method to accurately identify two mode indices (p and l) of Laguerre–Gaussian (LG) beams simultaneously with weak value measurement (WVM). Concise relations between the spatial displacements from the WVM and two mode indices of LG beams have been obtained. Compared with the traditional measurements, the advantage of using our method is that LG beams with large p and l can be identified precisely. A joint spatial WVM with a looser approximate condition has been presented, which realizes better identification than the individual spatial WVM with correction terms or strict approximation. Furthermore, the signal-to-noise ratio (SNR) of the spatial WVM is also discussed.

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Strict interior approximation of sets of finite perimeter and functions of bounded variation

It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate anopensetΩ\Omegaof finite perimeter inRn\mathbb {R}^nstrictlyfrom within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the(n−1)(n{-}1)-dimensional Hausdorff measure of the topological boundary∂Ω\partial \Omegaequals the perimeter ofΩ\Omega. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation ofBVBV-functions from a prescribed Dirichlet class.

On a perturbation approach to open mapping theorems

The open covering property, also known in the literature as metric regularity, is investigated for certain classes of set-valued maps in a Banach space setting. The focus of the present study is on a perturbation approach for deriving open covering criteria, which stems from a theorem due to Milyutin, and which is developed here by means of an abstract notion of first-order approximation for single-valued maps between normed spaces. As a result, a criterion is achieved for parametric set-valued maps in terms of open covering of their strict approximation. Two applications are presented: the first one relates to Robinson-type theorems in the context of quasidifferential analysis, whereas the second concerns the estimation of the distance from the solution set to a nonsmooth problem in parametric convex optimization.

Generalized Quantum Hydrodynamics and Principles of Non-Local Physics

This paper addresses the fundamental principles of generalized Boltzmann physical kinetics, as a part of non-local physics. It is shown that the theory of transport processes (including quantum mechanics) can be considered in the frame of unified theory based on the non-local physical description. Generalized Boltzmann physical kinetics leads to the strict approximation of non-local effects in space and time and after transmission to the local approximation leads to parameter of non-locality T, which on the quantum level corresponds to the uncertainty principle time-energy. Schrodinger equation is consequence of the Liouville equation as result of the local approximation of non-local equations. Generalized quantum hydrodynamics leads to Schrodinger equation as a deep particular case of the generalized Boltzmann physical kinetics and therefore of non-local hydrodynamics. Generalized quantum hydrodynamics can be used for solution of fundamental problems in nanoelectronics.

Abelian strict approximation in $AW^*$-algebras and Weyl-von Neumann type theorems

In this paper, for a C ∗ -algebra A with M = M(A) an AW ∗ -algebra, or equivalently, for an essential, norm-closed, two-sided ideal A of an AW ∗ -algebra M , we investigate the strict approximability of the elements of M from commutative C ∗ -subalgebras of A. In the relevant case of the norm-closed linear span A of all finite projections in a semi-finite AW ∗ -algebra M we shall give a complete description of the strict closure in M of any maximal abelian self-adjoint subalgebra (masa) of A. We shall see that the situation is completely different for discrete respectively continuous M : In the discrete case, for any masa C of A, the strict closure of C is equal to the relative commutant C ' M , while in the continuous case, under certain conditions concerning the center valued quasitrace of the finite reduced algebras of M (satisfied by all von Neumann algebras), C is already strictly closed. Thus in the continu- ous case no elements of M which are not already belonging to A can be strictly approximated from commutative C ∗ -subalgebras of A. In spite of this pathology of the strict topology in the case of the norm-closed linear span of all finite projections of a continuous semi-finite AW ∗ -algebra, we shall prove that in general situations including also this case, any normal y 2 M is equal modulo A to some x 2 M which belongs to an order theoretical closure of an appropriate commutative C ∗ -subalgebra of A. In other words, if we replace the strict topology with order theoretical approximation, Weyl-von Neumann-Berg-Sikonia type theo- rems will hold in substantially greater generality.

Optimization with generalized invexity

Constrained optimization is studied, with nonsmooth (Lipschitz) functions in abstract spaces and cone-constraints. Some more general Lagrangian necessary conditions are obtained, using strict minimum and approximation methods. These conditions are sufficient for a minimum under generalized invex assumptions. A characterization is obtained for generalized invexity, generalizing a known result for differentiable functions. Generalized invexity happens exactly when the generalized Wolfe and Lagrangian dual problems coincide.

Temporal relaxation of plasma electrons acted upon by direct current electric and magnetic fields

On the basis of the time-dependent electron Boltzmann equation the temporal relaxation of the electrons in the presence of electric and magnetic fields in weakly ionized, collision dominated plasmas has been studied. The relaxation process is treated by using a strict time-dependent two-term approximation of the velocity distribution function expansion in spherical harmonics. A new technique for solving the time-dependent electron kinetic equation in this two-term approximation for arbitrary angles between the electric and magnetic fields has been developed and the main aspects of the efficient solution method are presented. Using this new approach and starting from steady-state plasmas under the action of time-independent electric fields only, the impact of superimposed DC magnetic fields on the electron relaxation is analyzed with regard to the control of a neon plasma. The investigations reveal an important effect of the magnetic field on the temporal relaxation process. In particular, it has been found that the relaxation time of the electron component with respect to the establishment of steady-state can be enlarged by some orders of magnitude when increasing the magnetic field strength.

Buoyancy-affected laminar film condensation

The flow and heat transfer in a laminar condensate flim on an isothermal vertical plate is modelled mathematically. The strict Boussinesq approximation is adopted to account for buoyancy due to local temperature variations within the film. A similarity transformation reduces the governing boundary-layer type equations to a coupled set of ordinary differential equations and the resulting three-parameter twopoint boundary value problem is solved numerically for Prandtl numbers,Pr, ranging from 0.001 to 1000 and Jakob numbers,Ja, between 0.0001 and 1.5. The principal effects of the favourable buoyancy are to reduce the thickness of the condensate film and increase the film velocity at the smooth liquid-vapour interface, whereas the friction and heat transfer at the plate are enhanced. In accordance with the classical Nusselt theory, it is found that the temperature varies nearly linearly across the film. The computed similarity profiles for velocity reveal, however, substantial departures from the parabolic distribution assumed in the simplified Nusselt analysis.

Strict Approximation of Matrices

This paper describes a mechanism that includes the well-known strict approximation of a real vector which can be applied in the case of spectral approximation to define a unique strict spectral approximant of a matrix. For this purpose a new ordering is introduced.

A strict best approximant by non-decreasing functions

A strict best approximant by non-decreasing functions

Convergence of ‖πapproximations to the strict approximation in Rn

Suppose V is a subspace of R n and assume that ‖∞-norm is obtained as a suitable limit of convex functions {π m }. Let x m be a best π m -approximation to x by elements of V. We prove that the sequence x m converges to the strict best approximation of x. As a crollary we obtain convergence of the best ‖πm -approximation.

Combined forced and natural convection in laminar film flow

A mathematical model for the flow and heat transfer in a gravity-driven liquid film is presented, in which the strict Boussinesq approximation is adopted to account for buoyancy. A similarity transformation reduces the governing equations to a coupled set of ordinary differential equations. The resulting two-parameter problem is solved numerically for Prandtl numbers ranging from 1 to 1000. Favourable buoyancy arises when the temperatureTw of the isothermal surface is lower than the temperatureT0 of the incoming fluid, and the principal effects of the aiding buoyancy are to increase the wall shear and heat transfer rate. For unfavourable buoyancy (Tw>T0), the buoyancy force and gravity act in opposite directions and the flow in the film boundary layer decelerates, whereas the friction and heat transfer are reduced. The observed effects of buoyancy diminish appreciably for higher Prandtl numbers.

Linear-Approximation in l∞n

The paper studies the approximation behavior of a linear subspace U in l ∞ n; i.e., in R n equipped with the maximum norm. As a principal tool the Plücker-Graßmann coordinates of U are used; they allow a classification of the index set {1, ..., n} through which we determine the extremal points of the intersection of the orthogonal complement U⊥ of U and the closed l 1 n -unit ball in R n, leading to the dual problem. As a consequence, we describe the metric complement U (0) of U and give a decomposition of R n \\ Uinto a finite set of pairwise disjoint convex cones on which the metric projection P U has some characteristic properties. In the Chebyshev case, e.g., the metric projection is linear on these cones and, consequently, globally lipschitz continuous. A refinement allows an analogous statement for the strict approximation, proving a conjecture of Wu Li. Besides the strict approximation, were studying continuous selections of P U with and without the Nulleigenschaft, and characterize those subspaces U which admit a linear selection.

Extended method of moving asymptotes based on second-order information

The well-known and successful method of moving asymptotes was mainly developed for sizing problems in structural optimization. Applied to general problems, e.g. shape optimal design, the method occasionally exhibits some deficiencies. To further generalize the method, a simple extension is presented with respect to strict convex approximation of the objective function, deterministic asymptote adaption, and consistent treatment of equality constraints. It is based on second-order information estimated by forward finite differences. It is shown that the method is identical with diagonal quasi Newton sequential quadratic programming, if upper and lower asymptotes tend to positive or negative infinity, respectively. Comparative numerical examples show the success of the proposed extensions for various kinds of nonlinear optimization problems.