instruction stringlengths 12 30k |
|---|
Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, suppose that we want to know the probability that a fish is red. Suppose that the person with partial information knows 1/3 of all fish are red, but we know that the particular species is actually red 2/3 of the... |
I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter w, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of the corresponding matrix.
For small values of w, the corresponding matrix is small and I can use the so-called power ... |
If $h(x_t)$ is a decreasing function such that $h(0)=1$. Why is it that $h(x_t)=1_{\{t<\infty\}}$?; if at all its true. If not what can I say along these line (ie using indicator function). |
I would like to learn Graph Theory from the beginning. It seems to me that one does not need to be familiar with many abstract type subjects to be able to understand the more basic concepts of graphs.
1. Which subjects should one know prior to learn Graph Theory at the introductory level?
2. And which book or lec... |
If $f$ is a positive function, the intuitive interpretation of the Riemann integral
$\int_a^b f(x) dx$
is the area under the curve $f$ between $a$ and $b$.
Suppose $f$ and $g$ are smooth positive functions. Is it correct to interpret the Riemann-Stieltjes integral
$\int_a^b f(x) d g(x)$ as the volume under ... |
Is this interpretation of Stieltjes integration correct? |
If $\varphi:U\subset \mathbb{R}^n \to \mathbb{R}^m$ is $C^1$, let $\mathrm{T}\varphi:\mathrm{T}U \to \mathrm{T}R^m$ be its tangent map. The inverse function theorem tells us that if $\ker(\mathrm{T}\varphi(x))$ is zero, $\varphi$ is injective in some neighborhood of $x$. If the kernel is non-zero, what can we say about... |
If $h(x_{t\wedge \tau})$ is a submartingale such that $h(0)=1$ and $\tau$ is a stopping time. Why is it that $h(x_\tau)=1_{\{\tau<\infty\}}$?. Also $h(x)$ is decreasing. |
If you look at [Wolfram Alpha][1]
![alt text][2]
or this Wikipedia page [List of countries by median age][3]
![alt text][4]
Clearly [**median**][5] seems to be the statistic of choice when it comes to ages. I am not able to explain to myself why [**arithmetic mean**][6] would be a worse statistic. Why is it... |
Why is "median age" a better statistic than "mean age"? |
If you look at [Wolfram Alpha][1]
![alt text][2]
or this Wikipedia page [List of countries by median age][3]
![alt text][4]
Clearly [**median**][5] seems to be the statistic of choice when it comes to ages.
I am not able to explain to myself why [**arithmetic mean**][6] would be a worse statistic. Why is... |
How to solve this trigonometric inequality: $k*sin(\frac {\Pi} {2}*x)*cos(\frac {\Pi} {2}*x)>0$? |
Why is median age a better statistic than mean age? |
How to solve this trigonometric inequality: $k*sin(\frac {\pi x} {2})*cos(\frac {\pi x} {2})>0$? |
How to solve this trigonometric inequality: $k \sin(\frac {\pi x} {2})\cos(\frac {\pi x} {2})>0$? |
How to solve this trigonometric inequality: $k*sin(\frac {\pi x} {2})*cos(\frac {\pi x} {2})>0$? |
How to solve this trigonometric inequality: $k \sin(\frac {\pi x} {2})\cos(\frac {\pi x} {2})>0$? |
I'm a noob, and I'm not a mathematician (Although I will be a Math major next semester). My question is:
I have 68 maps I would like to use as posters on my wall at home. They are all rectangles, and all different dimensions. Is there an algorithm (or better yet, a website or other simple computer program I can u... |
Proposition: For a set $X$ and its power set $P(X)$, any function $f\colon P(X)\to X$ has at least two sets $A\neq B\subseteq X$ such that $f(A)=f(B)$.
I can see how this would be true if $X$ is a finite set, since $|P(X)|\gt |X|$, so by the pidgeonhole principle, at least two of the elements in $P(X)$ would have t... |
The following probability question appeared in an [earlier thread][1]:
> I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The claim was that it is not actually a mathematical problem and it is only a language problem.
----------
If one wanted to restate this prob... |
Boy Born on a Tuesday - is it just a language trick? |
The following probability question appeared in an [earlier thread][1]:
> I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The claim was that it is not actually a mathematical problem and it is only a language problem.
----------
If one wanted to restate this prob... |
There are even trickier aspects to this question. For example, what is the strategy of the guy telling you about his family? If he always mentions a boy first and not a daughter, we get one probability; if he talks about the sex of the first born child, we get a different probability. Your calculation makes a choice in... |
There are even trickier aspects to this question. For example, what is the strategy of the guy telling you about his family? If he always mentions a boy first and not a daughter, we get one probability; if he talks about the sex of the first born child, we get a different probability. Your calculation makes a choice in... |
The following probability question appeared in an [earlier thread][1]:
> I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The claim was that it is not actually a mathematical problem and it is only a language problem.
----------
If one wanted to restate this prob... |
I've been struggling with the following problem I found for a while now:
Suppose $(T,\preceq)$ is a partially ordered subset of $(U,\preceq)$ and $S\subseteq T$. If $\inf_T S$ and $u=\inf_U S$ both exist, and there exists a subset $S'\subseteq T$ such that $u=\sup_U S'$, then $\inf_U S=\inf_T S$.
First I noted t... |
For some sets $S\subseteq T\subseteq U$, when is $\inf_T S=\inf_U S$? |
What is the precise difference or relation between these terms in logic:
Inference, Reasoning, Deduction, and Induction? |
Difference or relation between Inference, Reasoning, Deduction, and Induction? |
I guess the following two versions of the experiment provide two different answers:
1. Dave has two children. Is atleast one of them a boy who is born on Tuesday? Dave answers Yes.
2. Dave has two children. I ask him to first choose and fix one child at random, and tell me if it is a boy who was born on Tuesday. ... |
We know that if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function, then $f$ carries connected sets to connected sets and compact sets to compact sets. That is if $A \subset \mathbb{R}$ is connected then $f(A)$ is connected, and if $A$ is compact then $f(A)$ is compact.
**Question:** Suppose $f: \mathbb{R} \to... |
We know that if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function, then $f$ carries connected sets to connected sets and compact sets to compact sets. That is if $A \subset \mathbb{R}$ is connected then $f(A)$ is connected, and if $A$ is compact then $f(A)$ is compact.
**Question:** Suppose $f: \mathbb{R} \to... |
In a proof or deduction system, there are some axioms and some inference rules. To have a small proof system, axioms are defined for the primitives (for example for negation and implication in propositional logic). Now, we want to have a proof system that proves all provable formulae including the formulae contain non-... |
Can we define definitions as axioms in logic? |
I'm not a mathematician and for a long time I've struggled with the question if numbers exist.
Mathematics is a language. Numbers are part of its vocabulary, like "apple" is part of the English vocabulary. Just like you can build a sentence with the word "apple" you can build equations with numbers, and just lik... |
One need only consult the history of algebra to find many informal
discussions on the existence and consistence of complex numbers.
Any *informal* attempt to justify the existence of $\mathbb C$
will face the same obstacles that existed in earlier times. Namely, the lack of any rigorous (set-theoretic) foundation... |
**REMARK** Such results on continuous induction are folklore dating back over a century - much older than the references in the Kalantari paper referenced Pete L. Clark. I once had a large file of papers on this topic but, alas, it is long lost. However I can easily recall one [old paper \[1\]][1] because the author ... |
In classical real analysis I've only seen absolute continuity defined for functions on compact interval $[a,b]$, where the two equivalent definitions are: $f:[a,b]\rightarrow\mathbb{R}$ is AC if
(1) Given $\epsilon > 0$ there is a $\delta > 0$ such that $\sum_{i=1}^n |f(y_i)-f(x_i)|< \epsilon$ for every finite coll... |
Absolute continuity on an open interval of the real line? |
What's the difference between a bar chart and a histogram? |
**Question**
Do you know any application or in which field this trajectories can be applied?
*Comment*
If you know any book or any site that can show me the applications, please let me know. Thanks for your time |
**REMARK** Such results on continuous induction are folklore dating back over a century - much older than the references in the Kalantari paper referenced Pete L. Clark. I once had a large file of papers on this topic but, alas, it is long lost. However I can easily recall one [old paper \[1\]][1] because the author ... |
Such results on continuous induction are folklore dating back over a century - much older than the references in the Kalantari paper referenced by Pete L. Clark. I once had a large file of papers on this topic but, alas, it's long lost. But I can still recall one [old paper \[1\]][1] because the author was the world-... |
To prove the Cayleigh-Hamilton theorem in linear algebra, my professor said that a lemma was necessary:
> **Lemma**: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{k}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$, then $P(A) = 0$
The theorem (which says that if $f$ is an ... |
Is this proof really necessary? |
The question, as currently posted (even after I edited it) is a mess. Are $g(x)$ and $h(x)$ meant to be *constant* functions? Was that supposed to be $g(4) = \sqrt{4} + 8$ and $h(4)=(\sqrt{4}+8)^4$, or was it supposed to be $g(x)=\sqrt{x}+8$ and $h(x)=(\sqrt{x}+8)^4$? Something else?
In any case: a composition is a ... |
To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary:
> **Lemma**: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{k}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$, then $P(A) = 0$
The theorem (which says that if $f$ is an en... |
To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary:
> **Lemma**: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{k}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$, then $P(A) = 0$
The theorem (which says that if $f$ is an en... |
I've been reading Ravi Vakil's notes on algebraic geometry [(notes][1]) and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal properties (page 24).
Can anybody either explain the connection or provide some kind of 'roadmap' of the materi... |
I've been reading Ravi Vakil's notes on algebraic geometry [notes][1] and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal properties (page 24).
Can anybody either explain the connection or provide some kind of 'roadmap' of the material... |
From any textbook on fourier analysis:
"It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = \int_{-\pi}^{\pi}f(z)g(x-z)\;dz = (g \ast f)(x),$$ by using the substitution $z = x-y.$"
<blockquote>
I don't doubt that t... |
Is the proof of this lemma really necessary? |
From f(0)=0, set y=0 to get f(ax)=f(x) so that f(a(x+y))=f(x+y) and the function is additive.
This is too easy to be on rwbarton's list, and in fact I cannot find this problem in the file that you linked. Which problem number is it in the list? |
$f(0+0)=f(0) + f(0)$ implies $f(0)=0$. Set $y=0$ to get $f(\alpha x)=f(x)$ so that $f(a(x+y))=f(x+y)$ and the function is additive. Under any regularity assumption, $f$ is linear with $\alpha = 1$, or zero, so it is impossible to exhibit any nontrivial solutions.
This is too easy to be on rwbarton's list, and in fac... |
$f(0+0)=f(0) + f(0)$ implies $f(0)=0$. Set $y=0$ to get $f(\alpha x)=f(x)$ so that $f(\alpha (x+y))=f(x+y)$ and the function is additive. Under any regularity assumption, $f$ is linear with $\alpha = 1$, or zero, so it is impossible to exhibit any nontrivial solutions.
This is too easy to be on rwbarton's list, and ... |
$f(0+0)=f(0) + f(0)$ implies $f(0)=0$. Set $y=0$ to get $f(\alpha x)=f(x)$ so that $f(\alpha (x+y))=f(x+y)$ and the function is additive. If $\alpha \neq 1$ this means $f((\alpha - 1)x)=0$ and hence $f=0$.
This is too easy to be on rwbarton's list, and in fact I cannot find this problem in the file that you linked.... |
I've been reading Ravi Vakil's notes on algebraic geometry [notes][1] and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal properties (page 24).
Can anybody either explain the connection or provide some kind of 'roadmap' of the material... |
This is a simple terminology question. Let $S$ be a (let's say smooth)
surface in $\mathbb{R}^3$, and $p$ a point
on $S$. Suppose the principle curvatures $\kappa_1$ and $\kappa_2$ at $p$ are both negative.
I am imagining $p$ sitting at the bottom of a dent in the surface.
Is there an accepted term to describe su... |
Terminology for point in dent in surface? |
As the Rotman's, or Voloch's, constructions of Dubuque's answers show, you *can't* recover *exactly* "the usual definition" of the localization of a ring from its universal property. That is, a universal property won't tell you, in general, how to construct an object that verifies it (as you can see, there are at least... |
I fear I over simplified the following problem:
For any partially ordered set $(A,\leq)$, let $A^* = A- ${$ maxA,minA $} if $maxA$ and $minA$ exist. Show the inclusion $(A^*,\leq)\hookrightarrow (A,\leq)$ is sup-continuous.
So I took any set $B\subseteq A^*$ such that $supB$ exists. Then $\iota(B)=B$, and $\iota(... |
I am learning some graph theory myself as an independent study in college.
I started with a very simple, but informative text, Introductory Graph Theory by Chatrand. It is a Dover book, and can be bought for very cheap on Amazon.
I then went to my university library and took out Modern Graph Theory by Bollobas. It ... |
My goal here is to make my own custom ["polyhedral map"][1] of Earth. If you print out something from the ["Map Fold-outs"][2] page, you will have something almost exactly like what I'm trying to make.
I have 2 triangles. One is a spherical triangle drawn on a 3D globe. By definition, each edge of a spherical triang... |
How do I map a spherical triangle to a plane triangle? |
To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary:
> **Lemma**: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{K}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$, then $P(A) = 0$
The theorem (which says that if $f$ is an en... |
As the Rotman's, or Voloch's, constructions of Dubuque's answers show, you *can't* recover *exactly* "the usual definition" of the localization of a ring from its universal property. That is, a universal property won't tell you, in general, how to construct an object that verifies it (as you can see, there are at least... |
I'm working my way through Niven's Introduction to Number Theory, and the wording of the following problem is making me unsure of my answer:
Show that there is a one-to-one correspondence between twin primes and numbers $n$ such that $n^2-1$ has just four positive divisors.
I felt an obvious bijection would be $... |
In [Wikipedia it is stated that the volume of the parallelepiped][1] given its edge lengths $a,b,c$, and the internal angles between the edges $\alpha ,\beta ,\gamma $ is given by
$V=abc\sqrt{1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha \cos
^{2}\beta \cos ^{2}\gamma }\qquad$(*).
I was not able to der... |
In [Wikipedia it is stated that the volume of the parallelepiped][1] given its edge lengths $a,b,c$, and the internal angles between the edges $\alpha ,\beta ,\gamma $ is:
$V=abc\sqrt{1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha \cos
^{2}\beta \cos ^{2}\gamma }\qquad$(*).
I was not able to derive it ... |
In [Wikipedia it is stated that the volume of the parallelepiped][1] given its edge lengths $a,b,c$, and the internal angles between the edges $\alpha ,\beta ,\gamma $ is:
$V=abc\sqrt{1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha
- \cos^{2}\beta - \cos ^{2}\gamma }\qquad$(*).
I was not able to derive ... |
Perhaps what you mean is how differential equations can be applied to find othogonal trajectories?
http://www.sosmath.com/diffeq/first/orthogonal/orthogonal.html |
The Fourier transform can be defined on $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$, and we can extend this to $X:=L^2(\mathbb{R}^n)$ by a density argument.
Now, by Plancherel we know that $\|\widehat{f}\|_2 = \|f\|_2$, so the Fourier transform is an isometry on this space.
My question now is, what is a theorem th... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
I got this interesting question in my mind:
How do we prove that if $a \in \mathbb{N}$ then $\sqrt{a}$ is an integer or an irrational number?
Can we extend this result? That is, can it be shown that if $a,b \in \mathbb{N}$, then $a^{1/b}$ is an integer or an irrational? |
Let $\mathcal{M}$ be an n-dimensional manifold endowed with an affine connection $\nabla$. Let $\gamma_1:[a,b]\rightarrow M$ and $\gamma_2:[c,d]\rightarrow \mathcal{M}$ be two curves with the same initial and final points, that is,
$p=\gamma_1(a)=\gamma_2(c), q=\gamma_1(b)=\gamma_2(d)$. Take $X\in T_p\mathcal{M}$. Pa... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
This is not intended as an answer, but clearly is too long for comments.
I can't stop thinking about this problem and I really want to know the solution. Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>... |
Any linear map between two finite-dimensional vector spaces can be represented as a matrix under the bases of the two spaces.
But if one or all of the vector spaces is infinite dimensional, is the linear map still represented as a matrix under their bases?
If there is matrix of infinite dimension, what is it use... |
This is not intended as an answer, but clearly is too long for comments.
Here are some approaches I have been playing around with; I would appreciate any comments, advice, or suggestions.
Clearly the condition s<sub>i</sub> = s<sub>j</sub> + s<sub>k</sub> is the key thing about this problem. It seems like in ord... |
An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it.
Is there a similar intuition behind generalized eigenvectors? |
<b>Bounty update</b>: this can be solved by change of basis, but I'm intrigued by David's solution relying on Fourier Transform of Dirac Delta function, so the bounty is for whoever finds a way to fix his solution to give the right result.
Suppose I have a non-negative real-valued function over $d$-dimensional real ... |
Consider first eigenvectors associated to $0$; these are elements of the nullspace. Now, it is possible for a vector to not lie in the nullspace, but for its image to lie in the nullspace (for example, consider the linear transformation $T\colon \mathbb{R}^2\to\mathbb{R}^2$ given by $T(x,y) = (y,0)$; then $T(0,1) = (1,... |
To prove the <a href="http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem">Cayley-Hamilton</a> theorem in linear algebra, my professor said that a lemma was necessary:
> **Lemma**: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{K}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b... |
What is a conditional double limit and how to compute it for $\lim_{(x,y)\rightarrow\infty}\frac{2k-1}{k-1}+\frac{x}{y}$? |
What is a conditional double limit and how to compute it? |
I take it there isn't a classification of finite complete groups yet. Has someone put together a classification of small complete groups? I.e. $S_4$, $\text{Aut}(G)$ for $G$ simple, $\text{Hol}(C_p)$ for $p$ odd prime are complete. The smallest complete group not of one of these forms is $H$ of order $\left|H\right|$, ... |
What is a conditional double limit and how to compute it for $\lim_{(x,y)\rightarrow\infty,x\leq y}\frac{2x-1}{x-1}+\frac{x}{y}$? |
Here's a nicer approach ...
![alt text][1]
Given two segments emanating from a point and ending at points of tangency with a circle, we know that those segments are congruent. In the figure, the three vertices of the right triangle give rise to three fundamental lengths; note that, since the "green" angle is a ri... |
Here's a nicer approach ...
![alt text][1]
Given two segments emanating from a point and ending at points of tangency with a circle, we know that those segments are congruent. In the figure, the three vertices of the right triangle give rise to three fundamental lengths; note that, since the "green" angle is a ri... |
In a inner product space with inner product <,> and real or complex line as its base field, for each point x in the space, is <x,> continuous function on the second argument, and is <,x> continous function on the first argument? "continuous" is defined respect to the topology induced by the inner product.
Thanks and... |
Is Inner product continous when one arg is fixed? |
In a inner product space with inner product <,> and real or complex line as its base field, for each point x in the space, is < x, > continuous function on the second argument, and is < ,x> continous function on the first argument? "continuous" is defined respect to the topology induced by the inner product.
Thanks ... |
In a inner product space with inner product $\langle\ ,\ \rangle$ and real or complex line as its base field, for each point $x$ in the space, is $\langle x,-\rangle$ continuous function on the second argument, and is $\langle - ,x\rangle$ continous function on the first argument? "Continuous" is defined respect to the... |
An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it.
Is there a similar intuition behind generalized eigenvectors?
EDIT: By generalized eigenvectors I'm referring to vectors in the kernel of $(T-\lambda I)^k$ for some $k$. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.