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In the general framework of [[Generalized function algebras|generalized function algebras]] developed by E.E. Rosinger [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], a distinguished role is played by ideals in the sequence algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301602.png" /> an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301603.png" />, which are defined by vanishing properties. Given a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301604.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301605.png" />, stable under finite unions, one considers the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301606.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301607.png" /> determined by those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301608.png" /> for which there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r1301609.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016011.png" /> vanishes near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016012.png" /> eventually, that is, there are a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016013.png" /> and a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016017.png" />. The nowhere-dense generalized function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016018.png" /> is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016019.png" /> is the class of nowhere-dense, closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016020.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016021.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016022.png" /> via the constant imbedding. It has two distinguishing features. First, the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016023.png" /> forms a [[Flabby sheaf|flabby sheaf]], and in a certain sense the smallest flabby sheaf containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016024.png" />, see [[#References|[a2]]]. Secondly, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016025.png" /> of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016026.png" /> can be imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016027.png" />.
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In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016028.png" /> (no growth restrictions near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016029.png" />) can be interpreted as global solutions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016030.png" /> by means of a suitable regularization method. The space of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016031.png" /> (cf. also [[Generalized functions, space of|Generalized functions, space of]]) is imbedded in any algebra of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016032.png" /> by a general procedure [[#References|[a4]]] using an algebraic basis.
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Further generalizations of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016033.png" /> to include larger exceptional sets as well as applications to non-smooth differential geometry can be found in [[#References|[a1]]]; non-linear Lie group actions on generalized functions using the framework of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016034.png" /> are studied in [[#References|[a7]]].
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In the general framework of [[Generalized function algebras|generalized function algebras]] developed by E.E. Rosinger [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], a distinguished role is played by ideals in the sequence algebra $\mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} }$, $\Omega$ an open subset of ${\bf R} ^ { n }$, which are defined by vanishing properties. Given a family $S$ of subsets of $\Omega$, stable under finite unions, one considers the ideal $\mathcal{I} _ { S }$ in $\mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} }$ determined by those $( u_j )_{ j \in \mathbf{N}}$ for which there is a $\Gamma \in S$ such that for all $x \in \Omega \backslash \Gamma$, $( u_j )_{ j \in \mathbf{N}}$ vanishes near $x$ eventually, that is, there are a $j _0$ and a neighbourhood $V \subset \Omega \backslash \Gamma$ of $x$ such that $u _ { j } | _ { V } \equiv 0$ for $j \geq j_0$. The nowhere-dense generalized function algebra $\mathcal{R} _ { \text{nd} } ( \Omega ) = \mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} } / \mathcal{I} _ { \text{nd} }$ is obtained when $S$ is the class of nowhere-dense, closed subsets of $\Omega$. The space $\mathcal{R} _ { \text{nd} } ( \Omega )$ contains $\mathcal{C} ^ { \infty } ( \Omega )$ via the constant imbedding. It has two distinguishing features. First, the family $\{ \mathcal{R} _ { \text{nd} } ( \Omega ) : \Omega \, \text{open} \}$ forms a [[Flabby sheaf|flabby sheaf]], and in a certain sense the smallest flabby sheaf containing $\mathcal{C} ^ { \infty } ( \Omega )$, see [[#References|[a2]]]. Secondly, the algebra $C _ { \text{nd} } ^ { \infty } ( \Omega )$ of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of $\Omega$ can be imbedded into $\mathcal{R} _ { \text{nd} } ( \Omega )$.
  
Replacing the single differential algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016035.png" /> with chains of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016036.png" /> using the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016037.png" /> in the place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016038.png" /> at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016039.png" />-functions as well as the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016040.png" /> at each fixed level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130160/r13016041.png" />.
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In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets $\Gamma$ (no growth restrictions near $\Gamma$) can be interpreted as global solutions in $\mathcal{R} _ { \text{nd} } ( \Omega )$ by means of a suitable regularization method. The space of distributions $\mathcal{D} ^ { \prime } ( \Omega )$ (cf. also [[Generalized functions, space of|Generalized functions, space of]]) is imbedded in any algebra of the form $\mathcal{C} ^ { \infty } ( \Omega ) / \mathcal{I} _ { S }$ by a general procedure [[#References|[a4]]] using an algebraic basis.
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Further generalizations of the ideal $\mathcal{I} _ {\operatorname{nd} }$ to include larger exceptional sets as well as applications to non-smooth differential geometry can be found in [[#References|[a1]]]; non-linear Lie group actions on generalized functions using the framework of $\mathcal{R} _ { \text{nd} } ( \Omega )$ are studied in [[#References|[a7]]].
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Replacing the single differential algebra ${\cal R = C} ^ { \infty } ( \Omega ) / {\cal I} _ { S }$ with chains of algebras $\mathcal{R} ^ { \infty } \rightarrow \ldots \rightarrow \mathcal{R} ^ { m } \rightarrow \ldots \rightarrow \mathcal{R} ^ { 0 }$ using the spaces $\mathcal{C} ^ { m } ( \Omega )$ in the place of $\mathcal{C} ^ { \infty } ( \Omega )$ at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of $\mathcal{C} ^ { m }$-functions as well as the derivative of $\mathcal{C} ^ { m + 1 } \rightarrow \mathcal{C} ^ { m }$ at each fixed level $m$.
  
 
See also [[Generalized function algebras|Generalized function algebras]].
 
See also [[Generalized function algebras|Generalized function algebras]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Mallios, E.E. Rosinger, "Space-time foam dense singularities and de Rham cohomology" ''Acta Applic. Math.'' (to appear) {{MR|1847884}} {{ZBL|1005.46020}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Oberguggenberger, E.E. Rosinger, "Solution of continuous nonlinear PDEs through order completion" , North-Holland (1994) {{MR|1286940}} {{ZBL|0821.35001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.E. Rosinger, "Distributions and nonlinear partial differential equations" , Springer (1978) {{MR|0514014}} {{ZBL|0469.35001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) {{MR|0590891}} {{ZBL|0447.35001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.E. Rosinger, "Generalized solutions of nonlinear partial differential equations" , North-Holland (1987) {{MR|0918145}} {{ZBL|0635.46033}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.E. Rosinger, "Nonlinear partial differential equations, an algebraic view of generalized solutions" , North-Holland (1990) {{MR|}} {{ZBL|0717.35001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> E.E. Rosinger, "Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert's fifth problem" , Kluwer Acad. Publ. (1998)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> A. Mallios, E.E. Rosinger, "Space-time foam dense singularities and de Rham cohomology" ''Acta Applic. Math.'' (to appear) {{MR|1847884}} {{ZBL|1005.46020}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. Oberguggenberger, E.E. Rosinger, "Solution of continuous nonlinear PDEs through order completion" , North-Holland (1994) {{MR|1286940}} {{ZBL|0821.35001}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> E.E. Rosinger, "Distributions and nonlinear partial differential equations" , Springer (1978) {{MR|0514014}} {{ZBL|0469.35001}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) {{MR|0590891}} {{ZBL|0447.35001}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> E.E. Rosinger, "Generalized solutions of nonlinear partial differential equations" , North-Holland (1987) {{MR|0918145}} {{ZBL|0635.46033}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> E.E. Rosinger, "Nonlinear partial differential equations, an algebraic view of generalized solutions" , North-Holland (1990) {{MR|}} {{ZBL|0717.35001}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> E.E. Rosinger, "Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert's fifth problem" , Kluwer Acad. Publ. (1998)</td></tr></table>

Latest revision as of 16:56, 1 July 2020

In the general framework of generalized function algebras developed by E.E. Rosinger [a3], [a4], [a5], [a6], a distinguished role is played by ideals in the sequence algebra $\mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} }$, $\Omega$ an open subset of ${\bf R} ^ { n }$, which are defined by vanishing properties. Given a family $S$ of subsets of $\Omega$, stable under finite unions, one considers the ideal $\mathcal{I} _ { S }$ in $\mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} }$ determined by those $( u_j )_{ j \in \mathbf{N}}$ for which there is a $\Gamma \in S$ such that for all $x \in \Omega \backslash \Gamma$, $( u_j )_{ j \in \mathbf{N}}$ vanishes near $x$ eventually, that is, there are a $j _0$ and a neighbourhood $V \subset \Omega \backslash \Gamma$ of $x$ such that $u _ { j } | _ { V } \equiv 0$ for $j \geq j_0$. The nowhere-dense generalized function algebra $\mathcal{R} _ { \text{nd} } ( \Omega ) = \mathcal{C} ^ { \infty } ( \Omega ) ^ { \text{N} } / \mathcal{I} _ { \text{nd} }$ is obtained when $S$ is the class of nowhere-dense, closed subsets of $\Omega$. The space $\mathcal{R} _ { \text{nd} } ( \Omega )$ contains $\mathcal{C} ^ { \infty } ( \Omega )$ via the constant imbedding. It has two distinguishing features. First, the family $\{ \mathcal{R} _ { \text{nd} } ( \Omega ) : \Omega \, \text{open} \}$ forms a flabby sheaf, and in a certain sense the smallest flabby sheaf containing $\mathcal{C} ^ { \infty } ( \Omega )$, see [a2]. Secondly, the algebra $C _ { \text{nd} } ^ { \infty } ( \Omega )$ of (equivalence classes of) smooth functions defined off some nowhere-dense, closed subset of $\Omega$ can be imbedded into $\mathcal{R} _ { \text{nd} } ( \Omega )$.

In particular, solutions to partial differential equations defined piecewise off nowhere-dense closed sets $\Gamma$ (no growth restrictions near $\Gamma$) can be interpreted as global solutions in $\mathcal{R} _ { \text{nd} } ( \Omega )$ by means of a suitable regularization method. The space of distributions $\mathcal{D} ^ { \prime } ( \Omega )$ (cf. also Generalized functions, space of) is imbedded in any algebra of the form $\mathcal{C} ^ { \infty } ( \Omega ) / \mathcal{I} _ { S }$ by a general procedure [a4] using an algebraic basis.

Further generalizations of the ideal $\mathcal{I} _ {\operatorname{nd} }$ to include larger exceptional sets as well as applications to non-smooth differential geometry can be found in [a1]; non-linear Lie group actions on generalized functions using the framework of $\mathcal{R} _ { \text{nd} } ( \Omega )$ are studied in [a7].

Replacing the single differential algebra ${\cal R = C} ^ { \infty } ( \Omega ) / {\cal I} _ { S }$ with chains of algebras $\mathcal{R} ^ { \infty } \rightarrow \ldots \rightarrow \mathcal{R} ^ { m } \rightarrow \ldots \rightarrow \mathcal{R} ^ { 0 }$ using the spaces $\mathcal{C} ^ { m } ( \Omega )$ in the place of $\mathcal{C} ^ { \infty } ( \Omega )$ at each level, allows one to achieve consistency of the multiplication and derivation with the pointwise product of $\mathcal{C} ^ { m }$-functions as well as the derivative of $\mathcal{C} ^ { m + 1 } \rightarrow \mathcal{C} ^ { m }$ at each fixed level $m$.

See also Generalized function algebras.

References

[a1] A. Mallios, E.E. Rosinger, "Space-time foam dense singularities and de Rham cohomology" Acta Applic. Math. (to appear) MR1847884 Zbl 1005.46020
[a2] M. Oberguggenberger, E.E. Rosinger, "Solution of continuous nonlinear PDEs through order completion" , North-Holland (1994) MR1286940 Zbl 0821.35001
[a3] E.E. Rosinger, "Distributions and nonlinear partial differential equations" , Springer (1978) MR0514014 Zbl 0469.35001
[a4] E.E. Rosinger, "Nonlinear partial differential equations. Sequential and weak solutions" , North-Holland (1980) MR0590891 Zbl 0447.35001
[a5] E.E. Rosinger, "Generalized solutions of nonlinear partial differential equations" , North-Holland (1987) MR0918145 Zbl 0635.46033
[a6] E.E. Rosinger, "Nonlinear partial differential equations, an algebraic view of generalized solutions" , North-Holland (1990) Zbl 0717.35001
[a7] E.E. Rosinger, "Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert's fifth problem" , Kluwer Acad. Publ. (1998)
How to Cite This Entry:
Rosinger nowhere-dense generalized function algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rosinger_nowhere-dense_generalized_function_algebra&oldid=50121
This article was adapted from an original article by Michael Oberguggenberger (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article