Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

Uniform boundedness of Kantorovich operators in Morrey spaces

In this paper, the Kantorovich operators \(K_n, n\in \mathbb {N}\) are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference \(K_n(f)-f\) for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.     

多线性分数次积分算子的一致有界性

研究了Lebesgue空间上多线性分数次积分算子和相应的多线性分数次极大算子的有界性,利用多线性分数次积分转化为相对应的分数次积分的方法来讨论,得到算子TΩ,α,A1,A2和MΩ,α,A1,A2的(Lp,Lq)的一致L ipsch itz估计,获得一种简明的方法.     

Uniform ultimate boundedness of nonlinear nonstationary systems

A certain class of nonlinear, nonstationary systems of differential equations is studied. It is assumed that the right-hand sides of the equations under consideration are homogeneous functions of order smaller than one with respect to the phase variables. The purpose of this paper is to obtain sufficient conditions for the uniform ultimate boundedness of systems of this form. A method for constructing nonstationary Lyapunov functions is suggested and applied to prove that the asymptotic stability of the zero solution of the corresponding averaged system implies the uniform ultimate boundedness of the initial nonstationary system. Classes of perturbations that do not violate uniform ultimate boundedness, even in the case where the order of the perturbations exceeds the homogeneity order of the unperturbed equations, are described. Unlike in previous works, where the results are based on the averaging method, the presence of a small parameter on the right-hand sides of the equations under examination is not assumed. Dissipativity is ensured at the expense of homogeneity orders.     

Uniform boundedness of the inverse of a Jacobian matrix arising in regularized interior-point methods

This short communication analyses a boundedness property of the inverse of a Jacobian matrix that arises in regularized primal-dual interior-point methods for linear and nonlinear programming. This result should be a useful tool for the convergence analysis of these kinds of methods.     

Uniform Boundedness of Torsion Subgroups of Linear Groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.     

Rayleigh-Taylor instability of a compressible plasma in a horizontal magnetic field

A study of the Rayleigh instability of a compressible plasma, density stratified in horizontal planes and subjected to a horizontal magnetic field, is made. It is found that in the presence of the compressibility effects the magnetic field continues to affect the development of the Rayleigh instability even for disturbances that propagate across the magnetic field (taken perpendicular to gravity), contrary to the result for the case of an incompressible plasma. The special case of a plane interface separating two superposed uniform plasmas of different densities is treated as an example to illustrate the compressibility effects on the hydromagnetic Rayleigh instability. It is found that the compressibility effects have a stabilising nature.

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Zusammenfassung Es wird die Rayleigh-Instabilität eines kompressiblen Plasmas untersucht, mit Dichte-Schichtung in horizontalen Ebenen, in Gegenwart eines horizontalen Magnetfeldes. Es wird gezeigt, daß bei Kompressibilität das Magnetfeld auch die Instabilitätsentwicklung derjenigen Störungen beeinflußt, die sich quer zum Magnetfeld (welches senkrecht zur Gravitationsrichtung angenommen wurde) fortpflanzen; dies ist im kompressiblen Plasma nicht der Fall. Der Spezialfall einer ebenen Trennfläche zwischen zwei Plasmas mit verschiedenen gleichförmigen Dichten wird behandelt, um den Kompressibilitätseffekt auf die hydromagnetische Rayleigh-Instabilität zu zeigen; dieser erweist sich als stabilisierend.

     

Evolution of the magnetic field in plasma in a neighborhood of zero points

We consider a mathematical model of magnetic reconnection in a neighborhood of singular points in the framework of 3D magnetohydrodynamics. In the case of linear dependence of the magnetic field and the velocity on coordinates, we present self-similar solutions of the system of MHD equations. For an arbitrary initial magnetic configuration, we state a mixed problem, which is solved numerically. We describe a semi-implicit algorithm for the solution of the difference problem and present the results of several numerical experiments.     

Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion

Uniform boundedness and convergence of global solutions are proved for quasilinear parabolic systems with a single nonzero cross-diffusion in population dynamics. Gagliardo-Nirenberg type inequalities are used in the estimates of solutions in order to establish W¹₂-bounds uniform in time. By using the uniform bound, convergence of solutions are established for systems with large diffusion coefficients in the weak competition case.     

Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model

Using the energy estimate and Gagliardo–Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for a strongly coupled reaction–diffusion system are proved. This system is the Shigesada–Kawasaki–Teramoto three-species cooperating model with self- and cross-population pressure. Meanwhile, some criteria on the global asymptotic stability of the positive equilibrium point for the model are also given by Lyapunov function. As a by-product, we proved that only constant steady states exist if the diffusion coefficients are large enough.     

Uniform boundedness of the derivatives of meromorphic inner functions on the real line

Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions for them to have uniformly bounded derivative on the real line. This question was first studied by Louis de Branges in 1968 and was later revived by Anton Baranov in 2011.     

One dimensional flow of a plasma with an external magnetic field

Zusammenfassung Eine Lösung der Gleichungen der eindimensionalen Plasmaströmung in Anwesenheit eines externen magnetischen Feldes senkrecht zur Strömungsrischtung wird entwickelt im speziellen Fall: (i)P _r =3/4, (ii)c _p -constant und (iii)T=T(u).     

Collisionless diffusion of particles across a magnetic field at a plasma edge in the presence of turbulence

We study the collisionless diffusion of electrons and ions across a magnetic field at a plasma edge, in the turbulent field produced by a Kelvin–Helmholtz instability. Ions and electrons are described by gyro-kinetic equations which include the finite Larmor radius correction and the polarization drift. Characteristic methods associated with cubic spline interpolation in two dimensions (2D) are used to integrate the gyro-kinetic equations in a slab geometry.     

Uniform boundedness in time and large time behavior of weak solutions to a kind of dissipative system

This paper deals with the uniform boundedness (as well as the existence) and large time behavior of the weak entropy solutions to a kind of compressible Euler equation with dissipation effect. The existence and uniform boundedness in time of weak solutions are proved by using the Lax-Friedrichs scheme and compensate compactness. Time asymptotically, the density is showed to satisfy a kind of nonlinear Fokker-Planck equation and the momentum obeys to the Darcy’s law. As a by product, the exponentially decay rate is obtained.