Examples where existence is harder than evaluationexistence of antiderivatives of nasty but elementary functionsAre some numbers more irrational than others?Evaluation of an $n$-dimensional integralEvaluation of the multiple integralExistence of an equivariant Morse functionExistence of a uniformly continuous function $g$ on $mathbbR$ where $f = g$ a.e.?Existence of local minimizerEvaluation of an interesting IntegralA question on existence of a Sobolev Hilbert space, where convergence implies uniform convergenceEvaluation of a double definite integral with a singularity
Examples where existence is harder than evaluation
existence of antiderivatives of nasty but elementary functionsAre some numbers more irrational than others?Evaluation of an $n$-dimensional integralEvaluation of the multiple integralExistence of an equivariant Morse functionExistence of a uniformly continuous function $g$ on $mathbbR$ where $f = g$ a.e.?Existence of local minimizerEvaluation of an interesting IntegralA question on existence of a Sobolev Hilbert space, where convergence implies uniform convergenceEvaluation of a double definite integral with a singularity
$begingroup$
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical:
$$ sqrt1+2sqrt1+3sqrt1+ldots.
qquad(*)
$$
Assuming the above is well-defined, there is a slick trick showing that it evaluates
to $3$.
But such careless assumptions can lead to trouble, as in the example of the expression:
$$ -5 + 2(-6 + 2(-7 + 2(-8 + ldots))).
qquad(**)
$$
Applying the identity $n = -(n + 2) + 2(n + 1)$
repeatedly for $n=3,4,5,ldots$, we get
beginalign
3 &= -5 + 2(4) \
&= -5 + 2(-6 + 2(5))\
&= -5 + 2(-6 + 2(-7 + 2(6))\
&= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\
&=ldots,
endalign
which would falsely suggest that $(**)$ evaluates to $3$.
What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?
real-analysis
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
$endgroup$
add a comment |
$begingroup$
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical:
$$ sqrt1+2sqrt1+3sqrt1+ldots.
qquad(*)
$$
Assuming the above is well-defined, there is a slick trick showing that it evaluates
to $3$.
But such careless assumptions can lead to trouble, as in the example of the expression:
$$ -5 + 2(-6 + 2(-7 + 2(-8 + ldots))).
qquad(**)
$$
Applying the identity $n = -(n + 2) + 2(n + 1)$
repeatedly for $n=3,4,5,ldots$, we get
beginalign
3 &= -5 + 2(4) \
&= -5 + 2(-6 + 2(5))\
&= -5 + 2(-6 + 2(-7 + 2(6))\
&= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\
&=ldots,
endalign
which would falsely suggest that $(**)$ evaluates to $3$.
What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?
real-analysis
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
$endgroup$
$begingroup$
Feel free to make an answer out of this.
$endgroup$
– Aryeh Kontorovich
1 hour ago
1
$begingroup$
Many recursions $x_n+1=f(x_n)$ ("find the limit") provide examples, though this is of course exactly the type of example that you give yourself.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
@ChristianRemling if you have any good ones, feel free to share!
$endgroup$
– Aryeh Kontorovich
1 hour ago
2
$begingroup$
This is most decidedly not a good one, but something like $x_n+1=x_n/2+1$ fits your description. Admittedly, both showing convergence and finding the limit are very easy, but if we compare the two, then finding the limit is much easier still.
$endgroup$
– Christian Remling
1 hour ago
add a comment |
$begingroup$
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical:
$$ sqrt1+2sqrt1+3sqrt1+ldots.
qquad(*)
$$
Assuming the above is well-defined, there is a slick trick showing that it evaluates
to $3$.
But such careless assumptions can lead to trouble, as in the example of the expression:
$$ -5 + 2(-6 + 2(-7 + 2(-8 + ldots))).
qquad(**)
$$
Applying the identity $n = -(n + 2) + 2(n + 1)$
repeatedly for $n=3,4,5,ldots$, we get
beginalign
3 &= -5 + 2(4) \
&= -5 + 2(-6 + 2(5))\
&= -5 + 2(-6 + 2(-7 + 2(6))\
&= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\
&=ldots,
endalign
which would falsely suggest that $(**)$ evaluates to $3$.
What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?
real-analysis
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
$endgroup$
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical:
$$ sqrt1+2sqrt1+3sqrt1+ldots.
qquad(*)
$$
Assuming the above is well-defined, there is a slick trick showing that it evaluates
to $3$.
But such careless assumptions can lead to trouble, as in the example of the expression:
$$ -5 + 2(-6 + 2(-7 + 2(-8 + ldots))).
qquad(**)
$$
Applying the identity $n = -(n + 2) + 2(n + 1)$
repeatedly for $n=3,4,5,ldots$, we get
beginalign
3 &= -5 + 2(4) \
&= -5 + 2(-6 + 2(5))\
&= -5 + 2(-6 + 2(-7 + 2(6))\
&= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\
&=ldots,
endalign
which would falsely suggest that $(**)$ evaluates to $3$.
What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?
real-analysis
real-analysis
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
asked 1 hour ago
Aryeh KontorovichAryeh Kontorovich
2,5481729
2,5481729
$begingroup$
Feel free to make an answer out of this.
$endgroup$
– Aryeh Kontorovich
1 hour ago
1
$begingroup$
Many recursions $x_n+1=f(x_n)$ ("find the limit") provide examples, though this is of course exactly the type of example that you give yourself.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
@ChristianRemling if you have any good ones, feel free to share!
$endgroup$
– Aryeh Kontorovich
1 hour ago
2
$begingroup$
This is most decidedly not a good one, but something like $x_n+1=x_n/2+1$ fits your description. Admittedly, both showing convergence and finding the limit are very easy, but if we compare the two, then finding the limit is much easier still.
$endgroup$
– Christian Remling
1 hour ago
add a comment |
$begingroup$
Feel free to make an answer out of this.
$endgroup$
– Aryeh Kontorovich
1 hour ago
1
$begingroup$
Many recursions $x_n+1=f(x_n)$ ("find the limit") provide examples, though this is of course exactly the type of example that you give yourself.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
@ChristianRemling if you have any good ones, feel free to share!
$endgroup$
– Aryeh Kontorovich
1 hour ago
2
$begingroup$
This is most decidedly not a good one, but something like $x_n+1=x_n/2+1$ fits your description. Admittedly, both showing convergence and finding the limit are very easy, but if we compare the two, then finding the limit is much easier still.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
Feel free to make an answer out of this.
$endgroup$
– Aryeh Kontorovich
1 hour ago
$begingroup$
Feel free to make an answer out of this.
$endgroup$
– Aryeh Kontorovich
1 hour ago
1
1
$begingroup$
Many recursions $x_n+1=f(x_n)$ ("find the limit") provide examples, though this is of course exactly the type of example that you give yourself.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
Many recursions $x_n+1=f(x_n)$ ("find the limit") provide examples, though this is of course exactly the type of example that you give yourself.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
@ChristianRemling if you have any good ones, feel free to share!
$endgroup$
– Aryeh Kontorovich
1 hour ago
$begingroup$
@ChristianRemling if you have any good ones, feel free to share!
$endgroup$
– Aryeh Kontorovich
1 hour ago
2
2
$begingroup$
This is most decidedly not a good one, but something like $x_n+1=x_n/2+1$ fits your description. Admittedly, both showing convergence and finding the limit are very easy, but if we compare the two, then finding the limit is much easier still.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
This is most decidedly not a good one, but something like $x_n+1=x_n/2+1$ fits your description. Admittedly, both showing convergence and finding the limit are very easy, but if we compare the two, then finding the limit is much easier still.
$endgroup$
– Christian Remling
1 hour ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Brownian motion is an example of this phenomenon in probability.
I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$. (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on.
But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923).
(From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.)
(The historical notes are from Pitman and Yor, Guide to Brownian Motion, which see for more references.)
$endgroup$
add a comment |
Your Answer
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%2f330977%2fexamples-where-existence-is-harder-than-evaluation%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
$begingroup$
Brownian motion is an example of this phenomenon in probability.
I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$. (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on.
But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923).
(From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.)
(The historical notes are from Pitman and Yor, Guide to Brownian Motion, which see for more references.)
$endgroup$
add a comment |
$begingroup$
Brownian motion is an example of this phenomenon in probability.
I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$. (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on.
But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923).
(From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.)
(The historical notes are from Pitman and Yor, Guide to Brownian Motion, which see for more references.)
$endgroup$
add a comment |
$begingroup$
Brownian motion is an example of this phenomenon in probability.
I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$. (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on.
But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923).
(From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.)
(The historical notes are from Pitman and Yor, Guide to Brownian Motion, which see for more references.)
$endgroup$
Brownian motion is an example of this phenomenon in probability.
I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian with variance proportional to $t$. (It seems that Bachelier may have also done it independently in 1900.) These properties uniquely define Brownian motion (up to scaling), and so any question you may have about Brownian motion can in principle be deduced from these axioms. For instance, you can compute its quadratic variation, and show that it is a Markov process and a martingale, and define and compute stochastic integrals, and so on.
But proving that there actually exists a process with these properties is harder. Historically, it took another 18 years or so before this was done (by Wiener in 1923).
(From Wiener's point of view, the object in question is a measure on the Banach space $C([0,1])$; the aforementioned properties tell us the finite-dimensional projections of this measure, which would uniquely determine it; but it is not trivial to prove the existence of a measure with those projections.)
(The historical notes are from Pitman and Yor, Guide to Brownian Motion, which see for more references.)
answered 1 hour ago
community wiki
Nate Eldredge
add a comment |
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%2f330977%2fexamples-where-existence-is-harder-than-evaluation%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
$begingroup$
Feel free to make an answer out of this.
$endgroup$
– Aryeh Kontorovich
1 hour ago
1
$begingroup$
Many recursions $x_n+1=f(x_n)$ ("find the limit") provide examples, though this is of course exactly the type of example that you give yourself.
$endgroup$
– Christian Remling
1 hour ago
$begingroup$
@ChristianRemling if you have any good ones, feel free to share!
$endgroup$
– Aryeh Kontorovich
1 hour ago
2
$begingroup$
This is most decidedly not a good one, but something like $x_n+1=x_n/2+1$ fits your description. Admittedly, both showing convergence and finding the limit are very easy, but if we compare the two, then finding the limit is much easier still.
$endgroup$
– Christian Remling
1 hour ago