scholarly journals Applications of Nachbin's Theorem concerning Dense Subalgebras of Differentiable Functions

2018 ◽  
Vol 19 (3) ◽  
pp. 465
Author(s):  
Marcia Sayuri Kashimoto
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Differentiable Functions ◽  
Real Functions

In this paper, we give some applications of  Nachbin's Theorem  to approximation and interpolation in the the space of all k times continuously differentiable real functions on  any open subset of the  Euclidean space.

An asymptotic analysis for Hamilton–Jacobi equations with large Hamiltonian drift terms

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Taiga Kumagai
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Asymptotic Analysis ◽  
Open Subset ◽  
Dimensional Euclidean Space ◽  
Asymptotic Behavior Of Solutions ◽  
Two Dimensional ◽  
Jacobi Equations ◽  
Drift Terms

AbstractWe investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by

Most Infinitely Differentiable Functions are Nowhere Analytic

1973 ◽  
Vol 16 (4) ◽  
pp. 597-598 ◽  
Author(s):  
R. B. Darst
Keyword(s):  
Open Subset ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
First Category ◽  
Infinitely Differentiable Functions

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

Continuity of solutions of Schrödinger equations

1991 ◽  
Vol 110 (3) ◽  
pp. 581-597
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Total Variation ◽  
Open Subset ◽  
Radon Measure ◽  
Dimensional Euclidean Space ◽  
Schrödinger Equations ◽  
Kato Class ◽  
Continuity Of Solutions ◽  
Image Position ◽  
Newtonian Kernel

We denote by N(x, y) the Newtonian kernel on the d-dimensional Euclidean space (where d ≥ 2) so that N(x, y) = log|x–y|-1 for d = 2 and N(x, y) = |x−y|2−d for d ≥ 3. A signed Radon measure μ on an open subset Ω in d is said to be of Kato class iffor every y in Ω. where |μ| is the total variation measure of μ.

UNCERTAINTY AND IMPRECISION IN ANALYTICAL CONTEXT: FUZZY LIMITS AND FUZZY DERIVATIVES

2001 ◽  
Vol 09 (05) ◽  
pp. 563-585
Author(s):  
MARK BURGIN
Keyword(s):  
Incomplete Information ◽  
Differential Calculus ◽  
Classical Case ◽  
Open Problems ◽  
Weak Derivative ◽  
The Third ◽  
Differentiable Functions ◽  
Fourth Part ◽  
Real Functions ◽  
Derivatives Of

The main goal of the present paper is to extend such classical constructions as limits and derivatives making them appropriate for management of imprecise, vague, uncertain, and incomplete information. In the second part of the paper, going after introduction, elements of the theory of fuzzy limits are presented. The third part is devoted to the construction of fuzzy derivatives of real functions. Two kinds of fuzzy derivatives are introduced: weak and strong ones. It is necessary to remark that the strong fuzzy derivatives are similar to ordinary derivatives of real functions being their fuzzy extensions. The weak fuzzy derivatives generate a new concept of a weak derivative even in a classical case of exact limits. In the fourth part fuzzy differentiable functions are studied. Different properties of such functions are obtained. Some of them are the same or at least similar to the properties of the differentiable functions while other properties differ in many aspects from those of the standard differentiable functions. Many classical results are obtained as direct corollaries of propositions for fuzzy derivatives, which are proved in this paper. Some of the classical results are extended and completed. The fifth part of the paper contains several interpretations of fuzzy derivatives aiming at application of fuzzy differential calculus to solving practical problems. At the end, some open problems are formulated.

Estimates for the constant in two nonlinear Korn inequalities

2020 ◽  
pp. 108128652095770
Author(s):  
Maria Malin ◽  
Cristinel Mardare
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Rigid Motion ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Square Roots ◽  
Korn Inequality ◽  
New Inequalities

A nonlinear Korn inequality estimates the distance between two immersions from an open subset of [Formula: see text] into the Euclidean space [Formula: see text], [Formula: see text], in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in [Formula: see text]. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.

Topologizing Different Classes of Real Functions

1994 ◽  
Vol 46 (06) ◽  
pp. 1188-1207 ◽  
Author(s):  
Krzysztof Ciesielski
Keyword(s):  
Continuum Hypothesis ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Linear Functions ◽  
Hausdorff Topology ◽  
Completely Regular ◽  
Differentiable Functions ◽  
Real Functions ◽  
Baire One ◽  
Number Of Classes

AbstractThe purpose of this paper is to examine which classesof functions fromcan be topologized in a sense that there exist topologies τ1and τ2onandrespectively, such thatis equal to the class C(τ1, τ2) of all continuous functions. We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functionsfor which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈f ≠ g. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T onsuch that the class L of all linear functions g(x) = ax + b coincides with. This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the classof all polynomials, the classof all analytic functions and the class of all harmonic functions.We will also prove that several other classes of real functions cannot be topologized. This includes the classes of C∞functions, differentiable functions, Darboux functions and derivatives.

scholarly journals On Mappings Which Commute with Convolution

1971 ◽  
Vol 12 (1) ◽  
pp. 122-128
Author(s):  
S. R. Harasymiv
Keyword(s):  
Euclidean Space ◽  
Compact Support ◽  
Strong Topology ◽  
Dimensional Euclidean Space ◽  
Schwartz Distribution ◽  
Differentiable Functions

The symbolDwill be written for the space of indefinitely differentiable functions on the n-dimensional Euclidean spaceRnwhich have compact support andDapos; will denote the space of Schwartz distribution onRn, the topological dual ofD. Except where contrary is explicitly stated, it will be assumed thatD′ is equipped with the strong topology β (D′,D) induced byD.

scholarly journals Inner differentiability and differential forms on tangentially locally linearly independent sets

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 365-372
Author(s):  
Aneta Velkoska ◽  
Zoran Misajleski
Keyword(s):  
Euclidean Space ◽  
Differential Forms ◽  
Differentiable Manifold ◽  
Independent Sets ◽  
Natural Isomorphism ◽  
De Rham Cohomology ◽  
Linearly Independent ◽  
Singular Cohomology ◽  
Real Functions ◽  
De Rham

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

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