Can the number of solutions to a system of PDEs be bounded using the characteristic variety?
$begingroup$
I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate question). In that question, Robert Bryant noted that the characteristic variety consists of three points. Concretely, I'm wondering if this means that there are at most three solutions to this system of equations (perhaps modulo some affine transformations).
More generally, does the number of points in the characteristic variety bound the number of real analytic solutions for a given system of PDEs?
My understanding is that the characteristic variety gives something like the formal power series solutions to the system, so there shouldn't be more real analytic solutions than that. However, my knowledge of $D$-modules is really lacking, so I was wondering if anyone could either correct me on this or else point me to a good reference to learn more. I've been trying to read through Bryant's Exterior Differential Systems, and I apologize if this question is obvious for those who understand the theory.
reference-request dg.differential-geometry ap.analysis-of-pdes differential-equations
edited 12 hours ago
Ali Taghavi
18252085
asked 12 hours ago
Gabe KGabe K
898416
$endgroup$
add a comment |
$begingroup$
I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate question). In that question, Robert Bryant noted that the characteristic variety consists of three points. Concretely, I'm wondering if this means that there are at most three solutions to this system of equations (perhaps modulo some affine transformations).
More generally, does the number of points in the characteristic variety bound the number of real analytic solutions for a given system of PDEs?
My understanding is that the characteristic variety gives something like the formal power series solutions to the system, so there shouldn't be more real analytic solutions than that. However, my knowledge of $D$-modules is really lacking, so I was wondering if anyone could either correct me on this or else point me to a good reference to learn more. I've been trying to read through Bryant's Exterior Differential Systems, and I apologize if this question is obvious for those who understand the theory.
reference-request dg.differential-geometry ap.analysis-of-pdes differential-equations
edited 12 hours ago
Ali Taghavi
18252085
asked 12 hours ago
Gabe KGabe K
898416
$endgroup$
2
$begingroup$
You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution").
$endgroup$
– Deane Yang
11 hours ago
2
$begingroup$
Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty.
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
What about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
1
$begingroup$
@AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers.
$endgroup$
– Deane Yang
9 hours ago
$begingroup$
@DeaneYang What do you mean "Count using functions rather than numbers"? Do you mean a module consideration rather than vector space consideration?
$endgroup$
– Ali Taghavi
3 hours ago
add a comment |
$begingroup$
I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate question). In that question, Robert Bryant noted that the characteristic variety consists of three points. Concretely, I'm wondering if this means that there are at most three solutions to this system of equations (perhaps modulo some affine transformations).
More generally, does the number of points in the characteristic variety bound the number of real analytic solutions for a given system of PDEs?
My understanding is that the characteristic variety gives something like the formal power series solutions to the system, so there shouldn't be more real analytic solutions than that. However, my knowledge of $D$-modules is really lacking, so I was wondering if anyone could either correct me on this or else point me to a good reference to learn more. I've been trying to read through Bryant's Exterior Differential Systems, and I apologize if this question is obvious for those who understand the theory.
reference-request dg.differential-geometry ap.analysis-of-pdes differential-equations
edited 12 hours ago
Ali Taghavi
18252085
asked 12 hours ago
Gabe KGabe K
898416
$endgroup$
I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate question). In that question, Robert Bryant noted that the characteristic variety consists of three points. Concretely, I'm wondering if this means that there are at most three solutions to this system of equations (perhaps modulo some affine transformations).
More generally, does the number of points in the characteristic variety bound the number of real analytic solutions for a given system of PDEs?
My understanding is that the characteristic variety gives something like the formal power series solutions to the system, so there shouldn't be more real analytic solutions than that. However, my knowledge of $D$-modules is really lacking, so I was wondering if anyone could either correct me on this or else point me to a good reference to learn more. I've been trying to read through Bryant's Exterior Differential Systems, and I apologize if this question is obvious for those who understand the theory.
reference-request dg.differential-geometry ap.analysis-of-pdes differential-equations
reference-request dg.differential-geometry ap.analysis-of-pdes differential-equations
edited 12 hours ago
Ali Taghavi
18252085
asked 12 hours ago
Gabe KGabe K
898416
edited 12 hours ago
Ali Taghavi
18252085
asked 12 hours ago
Gabe KGabe K
898416
edited 12 hours ago
Ali Taghavi
18252085
edited 12 hours ago
Ali Taghavi
18252085
edited 12 hours ago
Ali Taghavi
18252085
18252085
asked 12 hours ago
Gabe KGabe K
898416
asked 12 hours ago
Gabe KGabe K
898416
asked 12 hours ago
Gabe KGabe K
898416
898416
2
$begingroup$
You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution").
$endgroup$
– Deane Yang
11 hours ago
2
$begingroup$
Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty.
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
What about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
1
$begingroup$
@AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers.
$endgroup$
– Deane Yang
9 hours ago
$begingroup$
@DeaneYang What do you mean "Count using functions rather than numbers"? Do you mean a module consideration rather than vector space consideration?
$endgroup$
– Ali Taghavi
3 hours ago
add a comment |
2
$begingroup$
You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution").
$endgroup$
– Deane Yang
11 hours ago
2
$begingroup$
Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty.
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
What about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
1
$begingroup$
@AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers.
$endgroup$
– Deane Yang
9 hours ago
$begingroup$
@DeaneYang What do you mean "Count using functions rather than numbers"? Do you mean a module consideration rather than vector space consideration?
$endgroup$
– Ali Taghavi
3 hours ago
2
2
$begingroup$
You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution").
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution").
$endgroup$
– Deane Yang
11 hours ago
2
2
$begingroup$
Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty.
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty.
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
What about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
$begingroup$
What about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
1
1
$begingroup$
@AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers.
$endgroup$
– Deane Yang
9 hours ago
$begingroup$
@AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers.
$endgroup$
– Deane Yang
9 hours ago
$begingroup$
@DeaneYang What do you mean "Count using functions rather than numbers"? Do you mean a module consideration rather than vector space consideration?
$endgroup$
– Ali Taghavi
3 hours ago
$begingroup$
@DeaneYang What do you mean "Count using functions rather than numbers"? Do you mean a module consideration rather than vector space consideration?
$endgroup$
– Ali Taghavi
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The wave equation in the plane is $partial^2_x-partial^2_y=(partial_x+partial_y)(partial_x-partial_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.
Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $partial_x+partial_y, partial_x-partial_y$.
edited 11 hours ago
Joonas Ilmavirta
5,78352751
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
$endgroup$
1
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
3
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327234%2fcan-the-number-of-solutions-to-a-system-of-pdes-be-bounded-using-the-characteris%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The wave equation in the plane is $partial^2_x-partial^2_y=(partial_x+partial_y)(partial_x-partial_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.
Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $partial_x+partial_y, partial_x-partial_y$.
edited 11 hours ago
Joonas Ilmavirta
5,78352751
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
$endgroup$
1
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
3
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
add a comment |
$begingroup$
The wave equation in the plane is $partial^2_x-partial^2_y=(partial_x+partial_y)(partial_x-partial_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.
Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $partial_x+partial_y, partial_x-partial_y$.
edited 11 hours ago
Joonas Ilmavirta
5,78352751
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
$endgroup$
1
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
3
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
add a comment |
$begingroup$
The wave equation in the plane is $partial^2_x-partial^2_y=(partial_x+partial_y)(partial_x-partial_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.
Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $partial_x+partial_y, partial_x-partial_y$.
edited 11 hours ago
Joonas Ilmavirta
5,78352751
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
$endgroup$
The wave equation in the plane is $partial^2_x-partial^2_y=(partial_x+partial_y)(partial_x-partial_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.
Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $partial_x+partial_y, partial_x-partial_y$.
edited 11 hours ago
Joonas Ilmavirta
5,78352751
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
edited 11 hours ago
Joonas Ilmavirta
5,78352751
edited 11 hours ago
Joonas Ilmavirta
5,78352751
edited 11 hours ago
Joonas Ilmavirta
5,78352751
5,78352751
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
answered 12 hours ago
Ben McKayBen McKay
14.7k22961
14.7k22961
1
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
3
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
add a comment |
1
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
3
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
1
1
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
$begingroup$
Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution?
$endgroup$
– Gabe K
12 hours ago
3
3
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity.
$endgroup$
– Ben McKay
11 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
$begingroup$
@BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327234%2fcan-the-number-of-solutions-to-a-system-of-pdes-be-bounded-using-the-characteris%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
$begingroup$
You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution").
$endgroup$
– Deane Yang
11 hours ago
2
$begingroup$
Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty.
$endgroup$
– Deane Yang
11 hours ago
$begingroup$
What about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE?
$endgroup$
– Ali Taghavi
10 hours ago
1
$begingroup$
@AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers.
$endgroup$
– Deane Yang
9 hours ago
$begingroup$
@DeaneYang What do you mean "Count using functions rather than numbers"? Do you mean a module consideration rather than vector space consideration?
$endgroup$
– Ali Taghavi
3 hours ago