instruction stringlengths 12 30k |
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What are valent vertices? |
One direction is straightforward, so I'll only give the proof in the other direction.
Let $x \in \dot{\bar{C}}$. Then there exists an open ball $B_x$ containing $x$ such that $B_x \subset \bar{C}$.
Suppose $x \not\in \dot{C}$. But $x \in \bar{C}$, which means that $x$ is on the boundary of $C$. Therefore, $... |
How to solve \int_0^{\pi} \sin{2x}\sin{x} dx ?
|
How to solve \int_0^{\pi} \sin{2x}\sin{x} dx ? |
**Question 1:** Does such a ring can be found?
Note: The definition of a noetherian _topological space_ is similar to that in rings or sets. Every descending chain of closed subsets stops after a finite number of steps
(**Question 2:** is this equivalent to saying that every _descending_ chain of _opens_ ... |
How to solve $ \int_0^{\pi} \sin{2x}\sin{x} dx $? |
How to solve $\int_0^{\pi} \sin{2x}\sin{x} dx$ ?
|
Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just got stuck on another statement the author considered obvious. Could someone please give a proof that uses little more tha... |
Have a look here:<br>
http://www.wolframalpha.com/input/?i=integrate+sin%282x%29+sin%28x%29+from+0+to+Pi
...and for all the steps of the derivation (click on "show steps"):<br>
http://www.wolframalpha.com/input/?i=integrate+sin%282x%29+sin%28x%29 |
Suppose there are 9 events, that have a probability of 10%, 20%, 30%, ..., 90% of being a success.
How would I find the probability of exactly n number of these events succeeding?
For n = 1, I'm thinking it is something like
(first succeeds) -> (1-0.9) * 0.8 * 0.7 * 0.6 ... 0.1
(second succeeds) -> 0.... |
A quantile function Q is defined in terms of its distribution function F as:
> $Q(p) = \inf \\{ x\in R : p \le F(x) \\}, p \in (0,1).$
I was wondering:
(1) if the definition of Q can be extended to $[0,1]$ by letting $ x \in R \cup \\{\infty, -\infty \\}$? if not, why?
(2) if it is correct that a distributi... |
A quantile function Q is defined in terms of its distribution function F as:
> $Q(p) = \inf \\{ x\in R : p \le F(x) \\}, p \in (0,1).$
I was wondering:
(1) if the definition of Q can be extended to $[0,1]$ by letting $ x \in R \cup \\{\infty, -\infty \\}$? if not, why?
(2) if it is correct that a distributi... |
If the non-noetheriannes of the ring is hidden inside the nilradical, then $\mathrm{Spec}$ won't see it.
Let $k$ be any ring, and let $V$ be a free $k$-module of infinite rank. Consider $R=k\oplus V$, and turn it into a ring by defining $$(a,v)\dot(b,w)=(ab,aw+bv).$$ (Representation-people call this a *trivial exte... |
If the non-noetheriannes of the ring is hidden inside the nilradical, then $\mathrm{Spec}$ won't see it.
Let $k$ be any ring, and let $V$ be a free $k$-module of infinite rank. Consider $R=k\oplus V$, and turn it into a ring by defining $$(a,v)\dot(b,w)=(ab,aw+bv).$$ (Representation-people call this a *trivial exte... |
I've heard conflicting rumors about this site... (1) that your question will be quickly closed if it's not deemed up-to-snuff by the standards of professional mathematicians; (2) (from the page header that greeted me) that the site is for mathematicians *and those studying math at any level.* I'm banking on the latter,... |
variant on Sierpinski carpet; term for lack of empty regions? |
I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give me a good explanation. I can't seem to grasp other than the fact that it is just a particular integral of two functions. ... |
Meaning of convolution? |
I've heard conflicting rumors about this site... (1) that your question will be quickly closed if it's not deemed up-to-snuff by the standards of professional mathematicians; (2) (from the page header that greeted me) that the site is for mathematicians *and those studying math at any level.* I'm banking on the latter,... |
variant on Sierpinski carpet: classification? |
Have a look here:<br>
http://answers.yahoo.com/question/index?qid=20070125163821AA5hyRX
...and lots of good answers here:<br>
http://mathoverflow.net/questions/5892/what-is-convolution-intuitively |
I'm trying to solve this non-linear differential equation using substitution y/x to t. However, I can't solve this equation.
xy' = (y/x)^3 + y
How to solve this equation and what's the general solution?
Thanks.
|
How to solve this non-linear differential equation? |
I've heard conflicting rumors about this site... (1) that your question will be quickly closed if it's not deemed up-to-snuff by the standards of professional mathematicians; (2) (from the page header that greeted me) that the site is for mathematicians *and those studying math at any level.* I'm banking on the latter,... |
Choose numbers from 1 to 2n uniformly at random. How many numbers must be chosen, on average, before at least n have been picked?
This is similar to the coupon-collector problem, but looking for only partial completion.
Note: Choosing an appropriate meaning of 'random' is part of the question. |
How long to choose n out of 2n numbers? |
I've heard conflicting rumors about this site... (1) that your question will be quickly closed if it's not deemed up-to-snuff by the standards of professional mathematicians; (2) (from the page header that greeted me) that the site is for mathematicians *and those studying math at any level.* I'm banking on the latter,... |
The limit can be found. What are we saying when we assert that $lim_{t\to\infty} f(t) = \infty$, with $f:\mathbb{R} \to \mathbb{R}$? What does it mean?
It means, *by definition*, that given $M >0$ there exists $\varepsilon >0$ such that if $t>\varepsilon$ then $f(t) > M$. Simply this. In (not so rigorous) words, for... |
Here is my solution. Is this right?
$\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y\frac{y}{x}=u,y = \text{ux},\text{y$\prime $} = \text{u$\prime $x} + ux^2\text{u$\prime $} = u^3\int \frac{1}{u^3} \, du =\int \frac{1}{x^2} \, dx-\frac{1}{2u^2} = -\frac{1}{x} + cu^{2 }= \frac{1}{\frac{2}{x}+c}y^{2 }= \frac{x^2}{... |
I'm trying to solve this non-linear differential equation using substitution $\dfrac{y}{x}$ to $t$. However, I can't solve this equation.
$$
xy' = \dfrac{y}{x}^3 + y \ .
$$
How to solve this equation and what's the general solution?
Thanks.
|
Here is my solution. Is this right?
$$
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y
\frac{y}{x}=u,y = \text{ux},\text{y$\prime $} = \text{u$\prime $x} + ux^2\text{u$\prime $} = u^3\int \frac{1}{u^3} \, du =\int \frac{1}{x^2} \, dx-\frac{1}{2u^2} = -\frac{1}{x} + cu^{2 }= \frac{1}{\frac{2}{x}+c}y^{2 }= \frac{... |
If the non-noetheriannes of the ring is hidden inside the nilradical, then $\mathrm{Spec}$ won't see it.
Let $k$ be any ring, and let $V$ be a free $k$-module of infinite rank. Consider $R=k\oplus V$, and turn it into a ring by defining $$(a,v)\cdot(b,w)=(ab,aw+bv).$$ (Representation-people call this a *trivial ext... |
Here is my solution. Is this right?
$$
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y
$$
$$
\frac{y}{x}=u,
$$
y = \text{ux},
$$
$$
\text{y$\prime $} = \text{u$\prime $x} + ux^2\text{u$\prime $} = u^3\int \frac{1}{u^3} \,
$$
$$
du =\int \frac{1}{x^2} \,
$$
$$
dx-\frac{1}{2u^2} = -\frac{1}{x} + cu... |
I've heard conflicting rumors about this site... (1) that your question will be quickly closed if it's not deemed up-to-snuff by the standards of professional mathematicians; (2) (from the page header that greeted me) that the site is for mathematicians *and those studying math at any level.* I'm banking on the latter,... |
Here is my solution. Is this right?
\begin{align}
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y \\\\
\frac{y}{x}=u,
\end{align} |
Here is my solution. Is this right?
\begin{align}
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y \\\\ \\\\
\frac{y}{x}=u, \\\\
y = \text{ux}, \\\\
\text{y$\prime $} = \text{u$\prime $x} + u \\\\ \\\\
x^2\text{u$\prime $} = u^3 \\\\
\int \frac{1}{u^3} \, du =\int \frac{1}{x^2} \, dx \\\\
-\frac{1}{2u^2} = ... |
I'm trying to solve this non-linear differential equation using substitution $\dfrac{y}{x}$ to $t$. However, I can't solve this equation.
$$
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y
$$
How to solve this equation and what's the general solution?
Thanks.
|
If the non-noetherianness of the ring is hidden inside the nilradical, then $\mathrm{Spec}$ won't see it.
Let $k$ be any ring, and let $V$ be a free $k$-module of infinite rank. Consider $R=k\oplus V$, and turn it into a ring by defining $$(a,v)\cdot(b,w)=(ab,aw+bv).$$ (Representation-people call this a *trivial ex... |
I wrote a bunch of stuff about the properties of this set assuming it existed, then I realized that it wasn't even obvious that the set was well-defined.
Here's the problem: Does the central point belong to the set? If it does, then it does not belong to the middle block, and vice versa, leading to a contradiction. ... |
I have a probability space $\mathcal{X} \times \mathcal{Y}$. $\mathcal{X}$ and $\mathcal{Y}$ could be infinite, but they are at worst case countable.
There is a matrix $A_{xy}$ ranging over $\mathcal{X}$ for rows $\mathcal{Y}$ for columns, which is either 0 or 1 for each $x,y$.
We have the information that
$P... |
can I bound the following probability? |
The pdf of the geomtric distribution is the following: $f(x) = (1-p)^{x}p$. Also $E[X] = \frac{1-p}{p}$ and $\text{Var}[X] = \frac{1-p}{p^2}$. The pdf of the negative binomial distribution is: $f(x) = \binom{r+x-1}{x}p^{r}(1-p)^{x}$. Also $E[X] = \frac{r(1-p)}{p}$ and $\text{Var}[X] = \frac{r(1-p)}{p^2}$. So is the geo... |
I wrote a bunch of stuff about the properties of this set assuming it existed, then I realized that it wasn't even obvious that the set was well-defined.
Here's the problem: Does the central point belong to the set? If it does, then it does not belong to the middle block, and vice versa, leading to a contradiction. ... |
In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be viewed as submanifolds (of the target manifold in a sigma model), possibl... |
What are D-branes (in a topological field theory)? |
I'm having trouble finding a good explanation of Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Lets take for example a function $f(x) = x^2$, how do we find intergral of $f(x)$ under $[0,1]$ using Lebesgue integral? |
I've heard conflicting rumors about this site... (1) that your question will be quickly closed if it's not deemed up-to-snuff by the standards of professional mathematicians; (2) (from the page header that greeted me) that the site is for mathematicians *and those studying math at any level.* I'm banking on the latter,... |
In [this answer on meta](http://meta.math.stackexchange.com/questions/901/request-for-reopening-equivalence-of-curved-and-straight-lines/903#903), Pete L. Clark said:
> I think the question concerns the idea that a differentiable curve becomes more and more like a straight line segment the closer one zooms in on its g... |
Is "locally linear" an appropriate description of a differentiable function? |
I was playing around with [Sierpinski carpets][1] (see [pretty GPU-produced picture here][2]), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell me an existing name for it, and help me determine where it fits into existing classifications of fractals.
... |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
Given that a population (eg. of bacteria) grows exponentially with the function
$$p 2^{\frac{t}{10}}$$
where $p$ is the initial population and $t$ is the time in minutes. Given that the population consumes (eg. food) at a steady rate, how would you calculate how much the population consumes?
Assume that each b... |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 years, each student has at some point been in a classroom with every other student?
More generally: Starting with an edg... |
What is the probability that every pair of students studies together at some point? |
I was playing around with [Sierpinski carpets][1] (see [pretty GPU-produced picture here][2]), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell me an existing name for it, and help me determine where it fits into existing classifications of fractals.
... |
You may want to consider the following sources:
- [Lebesgue's theory of integration: its origins and development - Thomas Hawkins]()
- [The calculus gallery: masterpieces from Newton to Lebesgue - William Dunham](http://books.google.com/books?id=QnXSqvTiEjYC&printsec=frontcover&dq=The+Calculus+Gallery+Masterpiece... |
I'm trying to remember a particular theorem of ZF but unfortunately my memory is quite incomplete.
* The theorem is of the form (some set operation) is either (expected answer) or the empty set.
* If the Axiom of Choice holds, (some set operation) is (expected answer).
* I believe that if (some set operation) is (... |
I have a polygon P, that may or may not be convex. Is there an algorithm that will enable me to find the collection of points A that are at a distance less than d from P?
Is A in turn always a polygon?
Does the solution change materially if we try to solve the problem on the surface of a sphere instead of on a Eucl... |
I have a text book which gives two diagrams of beads suspended from strings with different angles, and worked examples to calculate the tension in the strings.
The first bead is suspended by 2 strings attached to a rod, both strings at angles to the vertical (the bead is at the bottom, the rod is across the top). Th... |
The question [what-is-the-optimum-angle-of-projection-when-throwing-a-stone-off-a-cliff][1] was asked and answered a while back. This one has a much cleaner answer. Now you are on a uniform slope, a line through the origin that is not horizontal and you want to throw a stone as far as possible. As a function of the ... |
What is the optimal angle when throwing a stone on a slope? |
Here is my solution. Is this right?
\begin{align}
\text{xy$\prime $} = \left(\frac{y}{x}\right)^3+y \\\\ \\\\
\frac{y}{x}=u, \\\\
y = \text{ux}, \\\\
\text{y$\prime $} = \text{u$\prime $x} + u \\\\ \\\\
x^2\text{u$\prime $} = u^3 \\\\
\int \frac{1}{u^3} \, du =\int \frac{1}{x^2} \, dx \\\\
-\frac{1}{2u^2} = ... |
Denoted as $\zeta(s,a)$ for a > 0
> Where do I find topics on the Hurwitz zeta function for a < 0?
Any links or resources would be appreciated.
(Please dont mention wiki or mathworld)
Thanks |
I have a homework question "Show the following is true using theorems. State which theorem you use at each step." This is just one of many problems I have! So, if you can help me with this one problem than I can apply what I learn to finish the rest. I don't want a handout; I just don't know where to start or what to d... |
Number theory proving question? |
I have a homework question "Show the following is true using theorems. State which theorem you use at each step." This is just one of many problems I have! So, if you can help me with this one problem than I can apply what I learn to finish the rest. I don't want a handout; I just don't know where to start or what to d... |
I have a homework question "Show the following is true using theorems. State which theorem you use at each step." This is just one of many problems I have! So, if you can help me with this one problem than I can apply what I learn to finish the rest. I don't want a handout; I just don't know where to start or what to d... |
This one is a problem from a journal or from competitions at the level of the Putnam contest (see reference below).
Hint: $g(x) = x + Af(x)$ satisfies $g(f^n(x))=A^ng(x)$ when $A^2 = A + 1$; consider the cases $n \to \pm \infty$.
Source for a similar problem, with solution: http://books.google.com/books?id=-... |
I am trying to make a truth table from an SOP boolean algebra expression. I understand AND, OR, NOT truth tables. I just dont understand these types of tables and there outputs.
This is the expression
A'BD' + BCD + ABC' + AB'D = A'BD' + BCD + ABC' + AB'D + BC'D' + A'BC + ABD
I can use either side whicheve... |
I have a text book which gives two diagrams of beads suspended from strings with different angles, and worked examples to calculate the tension in the strings.
The first bead is suspended by 2 strings attached to a rod, both strings at angles to the vertical (the bead is at the bottom, the rod is across the top). Th... |
If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a constant, $\Gamma(.,.)$ denotes standard gamma probability density function and '$\sim$' represents 'distributed as'. |
what will be the distribution of ratio of correlated gamma distributed random variables? |
If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes standard gamma probability density function and '$\sim$' represents 'distributed as'. X and Y are independent random var... |
compute the fourier transform of f(t) = sin(t). Does that converge?
|
Compute the fourier transform of f(t) = sin(t)? |
Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true?
**Definitions:**
1. $\angle ABC=90^{\circ}$
2. $\angle BCD=95^{\circ}$
3. $|AB|=|CD|$
4. $M:=$ the center of $BC$
5. $N:=$ the center of $AD$
6. $l:=$ a ... |
What is wrong in my proof that 90 = 95? Or is it correct? |
I was playing around with [Sierpinski carpets][1] (see [pretty GPU-produced picture here][2]), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell me an existing name for it, and help me determine where it fits into existing classifications of fractals.
... |
variant on Sierpinski carpet: classification and can it even exist? |
Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true?
**Definitions:**
1. $\angle ABC=90^{\circ}$
2. $\angle BCD=95^{\circ}$
3. $|AB|=|CD|$
4. $M:=$ the center of $BC$
5. $N:=$ the center of $AD$
6. $l:=$ a ... |
I have a question pertaining to linear separability with hyperplanes in a support vector machine.
According to [Wikipedia][1]:
> ...formally, a support vector machine
> constructs a hyperplane or set of
> hyperplanes in a high or infinite
> dimensional space, which can be used
> for classification, regression... |
Let $A$ be an abelian category. There are various types of full subcategories. I often wonder if it is assumed that these are nonempty, since in most proofs this is used implicitely, but also the empty case then works as a special case. To be sure, I ask you:
- $S \subseteq A$ is called thick iff $S$ is closed unde... |
What you seek is known as the _offset curve_ of a polygon. [Here][1] is the Wikipedia article.
This is closely related to the [Minkowski sum construction][2], in this case the Minkowski sum of a polygon and a disk. As others already indicated, the offset of a polygon is not in general
a polygon. There is a substan... |
Jessica: Preserving the order of all elements isn't sufficient; you have to show that it preserves the group operation.
Here's a hint though. $G^\prime$ is the set of symmetries of a square, right? Try labelling the square's corners. Then each symmetry is just a permutation of its corners... |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
Consider (non-trivial) functions that are their own nth derivatives. For instance
$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$
$\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$
$\frac{\mathrm{d}^3}{\mathrm{d}x^3} e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2}) = e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2})$
$\frac... |
I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic functional analysis so that I can be better prepared to take a graduate course in that material in my university. I plan to... |
I sincerely apologize if this question isn't appropriate here. Originally, I wanted to post this question on Overflow because I wanted to hear the opinions of mathematicians who are doing research, but it probably really is not appropriate to post there.
Long story short, I was looking for alternative ways to carry ... |
Kindle as a Tool for Mathematicians? |
What are good elementary examples for teaching/introducing/learning about Intuitionistic Logic or Heyting Algebras? |
Not to scare you, but list of requirements for a first course in functional analysis is rather long:
- Basic theorems of metric spaces including, but limited to:
- The Baire category theorem
- $\ell^p$ is complete
- [Arzelà-Ascoli](http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem) (how else ... |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
best book for topology? |
I am working with $G_{n,p}$ random graphs. Let $p$ (the probability of an edge) be $c/n$, so $c$ is the expected number of neighbors for a vertex when $n$ is large.
Now let $N$ be the number of components with more edges than vertices in the graph. I am trying to find the largest $c$ such that $\mathbf{E}(N)$ tends ... |
I have attempting to solve this using the infinite ramsey theorem, with colouring based on whether the sum of two vertices has an even or odd number of distinct prime factors.
This is leading to an infinite recursion.
Is this ok? At the end of all time I will be done. |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
I am working with $G_{n,p}$ random graphs. Let $p$ (the probability of an edge) be $c/n$, so $c$ is the expected number of neighbors for a vertex when $n$ is large.
Now let $N$ be the number of components with more edges than vertices in the graph. I am trying to find the largest $c$ such that $\mathbb{E}(N)$ tends ... |
Did you mean find, or did you mean prove one exists?
To prove it, you are on the right track.
Hint: Start with $\\{5n+1\\}$. |
Suppose one is given an arbitrary moment generating function $M_{X}(t)$. How would you determine $P(X=k)$ from this? We know that $M_{X}(t) = E[e^{tX}]$. |
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