Conditions for massiveness of the range of the derivation of a Banach algebra and associated differential operators

1987 ◽  
Vol 42 (2) ◽  
pp. 669-674 ◽  
Author(s):  
Yu. V. Turovskii ◽  
V. S. Shul'man
Keyword(s):  
Banach Algebra ◽  
Differential Operators ◽  
Associated Differential Operators

scholarly journals SMOOTH PROBABILITY MEASURES AND ASSOCIATED DIFFERENTIAL OPERATORS

1999 ◽  
Vol 02 (01) ◽  
pp. 51-78 ◽  
Author(s):  
O. G. SMOLYANOV ◽  
H. v. WEIZSÄCKER
Keyword(s):  
Differential Operators ◽  
Finite Dimensional ◽  
Differentiable Measure ◽  
Chaos Decomposition ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Associated Differential Operators ◽  
Locally Convex ◽  
Nonsmooth Mappings

We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the L2-space of a differentiable measure the analog of the classical concepts of gradient, divergence and Laplacian (which coincides with the Ornstein–Uhlenbeck operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite straight forward and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the procedure of quantization of anharmonic oscillators is discussed.

scholarly journals Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

2007 ◽  
Vol 143 (3) ◽  
pp. 703-729 ◽  
Author(s):  
LUIS J. ALÍAS ◽  
A. GERVASIO COLARES
Keyword(s):  
Mean Curvature ◽  
Differential Operators ◽  
Spacelike Hypersurface ◽  
Convergence Condition ◽  
Higher Order ◽  
Spacelike Hypersurfaces ◽  
Newton Transformations ◽  
Higher Order Mean Curvature ◽  
Associated Differential Operators

AbstractIn this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

scholarly journals The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows

2019 ◽  
Vol 81 (2) ◽  
pp. 990-1018 ◽  
Author(s):  
L. Beirão da Veiga ◽  
D. Mora ◽  
G. Vacca
Keyword(s):  
Degrees Of Freedom ◽  
Differential Operators ◽  
Complex Structure ◽  
Navier Stokes ◽  
Virtual Element Method ◽  
Element Space ◽  
Trilinear Form ◽  
Numerical Tests ◽  
Virtual Element ◽  
Associated Differential Operators

Abstract In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier–Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space $${\varPhi }_h \subset H^2({\varOmega })$$ Φ h ⊂ H 2 ( Ω ) and prove that the triad $$\{{\varPhi }_h, {\varvec{V}}_h, Q_h\}$$ { Φ h , V h , Q h } (with $${\varvec{V}}_h$$ V h and $$Q_h$$ Q h denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.

scholarly journals Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators

2020 ◽  
Vol 2020 ◽  
pp. 1-27
Author(s):  
Yoritaka Iwata
Keyword(s):  
Banach Algebra ◽  
Algebraic Structure ◽  
Differential Operators ◽  
Mathematical Physics ◽  
Rotation Group ◽  
Module Structure ◽  
Unbounded Operators ◽  
Infinitesimal Generators ◽  
Rigorous Formulation

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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