Uniform Boundedness Property Research Articles

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

An adaptive neural design for planar rigid formation of three coleaders in unknown flowfields

AbstractThis article deals with the robust planar rigid formation control problem of three second‐order coleaders with unknown flowfields acting on the velocity and acceleration respectively. To yield the uniform boundedness property of the resulting system, an adaptive projection is introduced to design the novel adaptive neural control law. To avoid the derivatives of Gaussian functions of neural networks, dynamic surface is used. Simulation results are provided to illustrate the effectiveness of the proposed control law.

Harmonic analysis of little [formula omitted]-Legendre polynomials

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their L1-algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i.e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any L1-algebra of a locally compact group; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little q-Legendre polynomials yield the first example of a polynomial hypergroup whose L1-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.

A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and some uniform boundedness property of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

Generalized Bloch spaces, integral means of hyperbolic harmonic mappings in the unit ball

In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball Bn in Rn(n≥2). Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space B(Bn) and the generalized Bloch space L∞,ωBα,a0(Bn), respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's related ones in the setting of hyperbolic harmonic mappings. Finally, we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric.

A robust prolongation operator for non-nested finite element methods

The paper is to present a general framework for designing a prolongation operator for general nonnested finite element methods. The main idea is to take the inclusion interpolation in the unrefined region that consists of all unrefined elements, and the canonical interpolation of the quasi-interpolation in the quasi-strict refined region that consists of children elements which are not neighbors of the unrefined region, and define a careful mixture in the small layer between these two regions. This leads to a robust prolongation operator that has both local projection and uniform boundedness properties for most of nonnested finite element methods in literature.

Finite multiplicity theorems for induction and restriction

We find upper and lower bounds of the multiplicities of irreducible admissible representations π of a semisimple Lie group G occurring in the induced representations IndHGτ from irreducible representations τ of a closed subgroup H. As corollaries, we establish geometric criteria for finiteness of the dimension of HomG(π,IndHGτ) (induction) and of HomH(π|H,τ) (restriction) by means of the real flag variety G/P, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.

UBIQUITOUS PROPERTIES OF CERTAIN OPERATIONAL QUANTITIES OF LINEAR RELATIONS IN NORMED SPACES

Abstract Let X and Y be normed spaces and T a linear relation in X x Y. Certain known operational quantities of linear operators are extended to linear relations. Properties of T such as precompactness, strict singularity, upper semi-Fredholmness and partial continuity are defined and characterised in terms of the given quantities, and certain uniform boundedness properties of these quantities derived.

On uniform boundedness properties in exhausting additive set function spaces

SynopsisValdivia (1978) introduced the class of suprabarrelled spaces, and (1979) deduced some uniform boundedness properties for scalar valued exhausting additive set functions on a σ-algebra from the suprabarrelledness of certain spaces. In this paper, it is shown that those uniform boundedness properties hold for G-valued exhausting additive set functions, G being a commutative topological group, on a larger class of Boolean algebras. Such properties are proved in Valdivia (1979) by means of duality theory arguments and ‘sliding hump’ methods, whereas here they are derived from the Baire category theorem. This generalization enables us to find a wide class of compact topological spaces K such that the subspaces of C(K) which satisfy a mild property are suprabarrelled.