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Pathology in Complex Analysis
Clash Royale CLAN TAG#URR8PPP
up vote
19
down vote
favorite
Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the reals. ~
Charles Pugh
People often like to talk about elegant "miracles" in Complex Analysis. However, what's are "pathological" objects/properties in Complex Analysis?
EDIT (09/13/18): Also posted as
https://math.stackexchange.com/questions/2912320/most-pathological-object-in-complex-analysis
EDIT: Changed the wording of the question.
cv.complex-variables
 |Â
show 7 more comments
up vote
19
down vote
favorite
Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the reals. ~
Charles Pugh
People often like to talk about elegant "miracles" in Complex Analysis. However, what's are "pathological" objects/properties in Complex Analysis?
EDIT (09/13/18): Also posted as
https://math.stackexchange.com/questions/2912320/most-pathological-object-in-complex-analysis
EDIT: Changed the wording of the question.
cv.complex-variables
3
First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries?
– Alex M.
21 hours ago
3
Charles Chapman Pugh
– Robert Israel
21 hours ago
15
@AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time.
– M. Winter
20 hours ago
4
This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it?
– arsmath
20 hours ago
4
@j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation.
– Zachary Selk
19 hours ago
 |Â
show 7 more comments
up vote
19
down vote
favorite
up vote
19
down vote
favorite
Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the reals. ~
Charles Pugh
People often like to talk about elegant "miracles" in Complex Analysis. However, what's are "pathological" objects/properties in Complex Analysis?
EDIT (09/13/18): Also posted as
https://math.stackexchange.com/questions/2912320/most-pathological-object-in-complex-analysis
EDIT: Changed the wording of the question.
cv.complex-variables
Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the reals. ~
Charles Pugh
People often like to talk about elegant "miracles" in Complex Analysis. However, what's are "pathological" objects/properties in Complex Analysis?
EDIT (09/13/18): Also posted as
https://math.stackexchange.com/questions/2912320/most-pathological-object-in-complex-analysis
EDIT: Changed the wording of the question.
cv.complex-variables
cv.complex-variables
edited 5 hours ago
community wiki
3 revs, 2 users 81%
Qi Zhu
3
First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries?
– Alex M.
21 hours ago
3
Charles Chapman Pugh
– Robert Israel
21 hours ago
15
@AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time.
– M. Winter
20 hours ago
4
This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it?
– arsmath
20 hours ago
4
@j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation.
– Zachary Selk
19 hours ago
 |Â
show 7 more comments
3
First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries?
– Alex M.
21 hours ago
3
Charles Chapman Pugh
– Robert Israel
21 hours ago
15
@AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time.
– M. Winter
20 hours ago
4
This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it?
– arsmath
20 hours ago
4
@j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation.
– Zachary Selk
19 hours ago
3
3
First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries?
– Alex M.
21 hours ago
First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries?
– Alex M.
21 hours ago
3
3
Charles Chapman Pugh
– Robert Israel
21 hours ago
Charles Chapman Pugh
– Robert Israel
21 hours ago
15
15
@AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time.
– M. Winter
20 hours ago
@AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time.
– M. Winter
20 hours ago
4
4
This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it?
– arsmath
20 hours ago
This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it?
– arsmath
20 hours ago
4
4
@j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation.
– Zachary Selk
19 hours ago
@j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation.
– Zachary Selk
19 hours ago
 |Â
show 7 more comments
6 Answers
6
active
oldest
votes
up vote
21
down vote
I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance).
add a comment |Â
up vote
17
down vote
I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $mathbb D$, $g$ or $g-f$ has a zero in $mathbb D$.
This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638
add a comment |Â
up vote
11
down vote
In an old MO question of mine, I had wondered the following (I'm quoting my question):
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$.
Is f then necessarily holomorphic?
The answer turns out to be no.
add a comment |Â
up vote
10
down vote
A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.
A simple example is the series function:
$$f(z) = sum_n=0^infty z^2^n$$
This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.
1
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
add a comment |Â
up vote
6
down vote
The rigidity of complex domains in higher dimension For example the unit ball in $mathbbC^2$ is not holomorphic equivalent to the unit cube.
add a comment |Â
up vote
4
down vote
I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'
For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions.
Another thing is related to The_Sympathizer's answer: Any open set in $mathbbC$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
add a comment |Â
6 Answers
6
active
oldest
votes
6 Answers
6
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
21
down vote
I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance).
add a comment |Â
up vote
21
down vote
I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance).
add a comment |Â
up vote
21
down vote
up vote
21
down vote
I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance).
I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance).
answered 20 hours ago
community wiki
Francesco Polizzi
add a comment |Â
add a comment |Â
up vote
17
down vote
I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $mathbb D$, $g$ or $g-f$ has a zero in $mathbb D$.
This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638
add a comment |Â
up vote
17
down vote
I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $mathbb D$, $g$ or $g-f$ has a zero in $mathbb D$.
This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638
add a comment |Â
up vote
17
down vote
up vote
17
down vote
I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $mathbb D$, $g$ or $g-f$ has a zero in $mathbb D$.
This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638
I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $mathbb D$, $g$ or $g-f$ has a zero in $mathbb D$.
This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638
answered 21 hours ago
community wiki
Robert Israel
add a comment |Â
add a comment |Â
up vote
11
down vote
In an old MO question of mine, I had wondered the following (I'm quoting my question):
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$.
Is f then necessarily holomorphic?
The answer turns out to be no.
add a comment |Â
up vote
11
down vote
In an old MO question of mine, I had wondered the following (I'm quoting my question):
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$.
Is f then necessarily holomorphic?
The answer turns out to be no.
add a comment |Â
up vote
11
down vote
up vote
11
down vote
In an old MO question of mine, I had wondered the following (I'm quoting my question):
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$.
Is f then necessarily holomorphic?
The answer turns out to be no.
In an old MO question of mine, I had wondered the following (I'm quoting my question):
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$.
Is f then necessarily holomorphic?
The answer turns out to be no.
edited 11 hours ago
community wiki
André Henriques
add a comment |Â
add a comment |Â
up vote
10
down vote
A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.
A simple example is the series function:
$$f(z) = sum_n=0^infty z^2^n$$
This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.
1
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
add a comment |Â
up vote
10
down vote
A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.
A simple example is the series function:
$$f(z) = sum_n=0^infty z^2^n$$
This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.
1
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
add a comment |Â
up vote
10
down vote
up vote
10
down vote
A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.
A simple example is the series function:
$$f(z) = sum_n=0^infty z^2^n$$
This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.
A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.
A simple example is the series function:
$$f(z) = sum_n=0^infty z^2^n$$
This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.
answered 11 hours ago
community wiki
The_Sympathizer
1
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
add a comment |Â
1
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
1
1
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.
– Robert Furber
2 hours ago
add a comment |Â
up vote
6
down vote
The rigidity of complex domains in higher dimension For example the unit ball in $mathbbC^2$ is not holomorphic equivalent to the unit cube.
add a comment |Â
up vote
6
down vote
The rigidity of complex domains in higher dimension For example the unit ball in $mathbbC^2$ is not holomorphic equivalent to the unit cube.
add a comment |Â
up vote
6
down vote
up vote
6
down vote
The rigidity of complex domains in higher dimension For example the unit ball in $mathbbC^2$ is not holomorphic equivalent to the unit cube.
The rigidity of complex domains in higher dimension For example the unit ball in $mathbbC^2$ is not holomorphic equivalent to the unit cube.
answered 6 hours ago
community wiki
Ali Taghavi
add a comment |Â
add a comment |Â
up vote
4
down vote
I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'
For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions.
Another thing is related to The_Sympathizer's answer: Any open set in $mathbbC$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
add a comment |Â
up vote
4
down vote
I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'
For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions.
Another thing is related to The_Sympathizer's answer: Any open set in $mathbbC$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
add a comment |Â
up vote
4
down vote
up vote
4
down vote
I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'
For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions.
Another thing is related to The_Sympathizer's answer: Any open set in $mathbbC$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.
I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'
For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions.
Another thing is related to The_Sympathizer's answer: Any open set in $mathbbC$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.
answered 7 hours ago
community wiki
James Hanson
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
add a comment |Â
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?
– Robert Furber
2 hours ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
Yeah, those are the example I know of.
– James Hanson
4 mins ago
add a comment |Â
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3
First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries?
– Alex M.
21 hours ago
3
Charles Chapman Pugh
– Robert Israel
21 hours ago
15
@AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time.
– M. Winter
20 hours ago
4
This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it?
– arsmath
20 hours ago
4
@j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation.
– Zachary Selk
19 hours ago