Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
For what functions $f(x)$ is $f(x)f(y)$ convex? For which functions $f\colon [0,1] \to [0,1]$ is the function $g(x,y)=f(x)f(y)$ convex over $(x,y) \in [0,1]\times [0,1]$ ? Is there a nice characterization of such functions $f$?
The obvious examples are exponentials of the form $e^{ax+b}$ and their convex combinations. ... | Suppose $f$ is $C^2$. First of all, because $g$ is convex in each variable, it follows that $f$ is convex, and hence $f''\geq0$. I did not initially have Slowsolver's insight that log convexity would be a criterion to look for, but naïvely checking for positivity of the Hessian of $g$ leads to the inequalities
$$f''(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/15707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
} |
Stochastic integral and Stieltjes integral My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here is some background to make this clearer.
Suppose $f\colon \Re \mapsto \Re $ ... | First write
$$
R_n - L_n = \sum\limits_{i = 1}^n {[W(t_i ) - W(t_{i - 1} )]^2 },
$$
then consider Quadratic variation of Brownian motion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/15749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Simplicity of $A_n$ I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (at least my head). In a sense, I still do not have the feeling that I know why they are simple and why should it be 5 and not any other number (perh... | The one I like most goes roughly as follows (my reference is in French [Daniel Perrin, Cours d'algèbre], but maybe it's the one in Jacobson's Basic Algebra I) :
*
*you prove that A(5) is simple by considering the cardinal of the conjugacy classes and seeing that nothing can be a nontrivial normal subgroup (because ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/15773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 5,
"answer_id": 2
} |
How does (wikipedia's axiomatization of) intuitionistic logic prove $p \rightarrow p$? I'm looking at Wikipedia's article on Intuitionist logic, and I can't figure out how it would prove $(p \rightarrow p)$.
Does it prove $(p \rightarrow p)$? If yes, how?
If no, is there a correct (or reasonable, if there is no "corre... | *
*$p→(p→p)$ (THEN-1)
*$p→((p→p)→p)$ (THEN-1)
*$(p→((p→p)→p))→((p→(p→p))→(p→p))$ (THEN-2)
*$(p→(p→p))→(p→p)$ (MP from lines 2,3)
*$(p→p)$ (MP from lines 1,4)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/15844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Is long division the most optimal "effective procedure" for doing division problems? (Motivation: I am going to be working with a high school student next week on long division, which is a subject I strongly dislike.)
Consider: $\frac{1110}{56}=19\frac{46}{56}$.
This is really a super easy problem, since once you real... | The OP asks
Are there other distinct effective procedures for doing division
problems besides long division?
You can certainly tweak the long division algorithm to make it better. Here, we will describe one such 'tweak improvement', motivated by the OP's sample problem.
Problem: Divide $1,110$ by $56$.
$560 \times ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/15881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 2
} |
Does the sum of reciprocals of primes converge? Is this series known to converge, and if so, what does it converge to (if known)?
Where $p_n$ is prime number $n$, and $p_1 = 2$,
$$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
| No, it does not converge. See this: Proof of divergence of sum of reciprocals of primes.
In fact it is known that $$\sum_{p \le x} \frac{1}{p} = \log \log x + A + \mathcal{O}(\frac{1}{\log^2 x})$$
Related: Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/15946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
Path for learning linear algebra and multivariable calculus I'll be finishing Calculus by Spivak somewhat soon, and want to continue into linear algebra and multivariable calculus afterwards. My current plan for learning the two subjects is just to read and work through Apostol volume II; is this a good idea, or would ... | I LOVED Advanced Calculus: A Differential Forms Approach by Edwards. Great book.. It's great to move to the next level in geometry.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 3
} |
Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$? Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$?
Where $\pi(n)$ is the prime counting function. Using PMT it seems asymptotically $\ p_n^{\pi(n)} \leq x^n$ where $e \leq x$
| Yes,
Asymptotically you have
$$(p_n)^{\pi(n)} \leq n^{n/\log n} = e^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
How do I write a log likelihood function when I have 2 mean values for my pdf? I have been given the following pdf :
fT (t; B, C) = ( exp(-t/C) - exp(-t/B) ) / ( C - B ) , (t>0)
where the overall mean is B+C.
I am unsure as to how to write the log likelihood function of B and C.
The next part of the Q asks me to deri... | By definition, the log-likelihood is given by
$$
\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = \sum\limits_{i = 1}^n {\ln f(x_i|B,C)}.
$$
Thus, in our example,
$$
\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = \sum\limits_{i = 1}^n {\ln \bigg[\frac{{e^{ - x_i /C} - e^{ - x_i /B} }}{{C - B}}\bigg]} .
$$
EDIT: In view of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Replacing $\text{Expression}<\epsilon$ by $\text{Expression} \leq \epsilon$ As an exercise I was doing a proof about equicontinuity of a certain function. I noticed that I am always choosing the limits in a way that I finally get:
$\text{Expression} < \epsilon$
However it wouldn't hurt showing that
$\text{Expression}... | Suppose for every $\epsilon >0$, there is an $N$ such that $n>N$ implies $x_n\leq\epsilon$. Let $k>0$. Then $\frac{k}{2}>0$, so there is an $M$ such that $n>M$ implies $x_n\leq\frac{k}{2}$ which implies that $x_n<k$. This corresponds to the original definition of continuity, doesn't it?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
What is the point of logarithms? How are they used? Why do you need logarithms?
In what situations do you use them?
| In the geometric view of real numbers there are two basic forms of "movements", namely (a) shifts: each point $x\in{\mathbb R}$ is shifted a given amount $a$ to the right and (b) scalings: all distances between points are enlarged by the same factor $b>0$. In some instances (e.g. sizes of adults) the first notion is ap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 5,
"answer_id": 3
} |
Finding Concavity in Curves Suppose
$$\frac{d^2y}{dx^2} = \frac{e^{-t}}{1-e^t}.$$
After finding the second derivative, how do I find concavity? It's much easier to solve for $t$ in a problem like $t(2-t) = 0$, but in this case, solving for $t$ seems more difficult. Does it have something to do with $e$? $e$ never becom... | From the t on the RHS I assume that this comes from a pair of parametric equations where you were given x(t) and y(t) and that you got the second derivative by calculating dm/dx=m'(t)/x'(t) where m(t)=dy/dx=y'(t)/x'(t).
In any case, you are right that the curve will be concave up at points where the second derivative ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sum of series $2^{2m}$ How to sum $2^{2m}$ where $m$ varies from $0$ to $n$?
| Note that you can get: $2^{2m}=4^m$, now
$$\displaystyle\sum_{m=0}^{n} 2^{2m} = \displaystyle\sum_{m=0}^{n} 4^{m}$$
The last is easy, (is geometric with $r=4$), so
$$\sum_{m=0}^{n} 2^{2m} = \sum_{m=0}^{n} 4^{m} = \frac{4-4^{n+1}}{1-4}+1=\frac{4^{n+1}-4}{3}+1=\frac{4^{n+1}-1}3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why do books titled "Abstract Algebra" mostly deal with groups/rings/fields? As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem to 'only' cover groups, rings and fields. W... | Historically, the first "modern algebra" textbook was van der Waerden's in 1930, which followed the groups/rings/fields model (in that order).
As far as I know, the first paper with nontrivial results on semigroups was published in 1928, and the first textbook on semigroups would have to wait until the 1960s.
There is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "79",
"answer_count": 8,
"answer_id": 3
} |
Example of a ring with $x^3=x$ for all $x$ A ring $R$ is a Boolean ring if $x^2=x$ for all $x\in R$. By Stone representation theorem
a Boolean ring is isomorphic to a subring of the ring of power set of some set.
My question is what is an example of a ring $R$ with $x^3=x$ for all $x\in R$ that is
not a Boolean ring?... | You can always pick a set $X$, consider the free $\mathbb Z$-algebra $A=\mathbb Z\langle X\rangle$, and divide by the ideal generated by all the elements $x^3-x$ for $x\in A$ to que a ring $B_X$, no?
(Since the quotient is going to be commutative, you can also start from the polynomial ring...)
To show that the resulti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
A converse of sorts to the intermediate value theorem, with an additional property I need to solve the following problem:
Suppose $f$ has the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists a value $d$ between $a$ and $b$ for which $f(d)=c$, and also has the additional property that $f^{-1}(a)$ i... | As noted in the comments, this is a slightly more general version of a problem in Rudin. I assume for simplicity that the dense set is $\mathbb{Q}$.
Solution: Fix $x_0\in\mathbb{R}$, and fix a sequence $\{x_n\}$ converging to $x_0$. By the sequential characterization of continuity, it suffices to show that $f(x_n)\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What is $\frac{d}{dx}\left(\frac{dx}{dt}\right)$? This question was inspired by the Lagrange equation, $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$. What happens if the partial derivatives are replaced by total derivatives, leading to a situation where a function's derivative wi... | Write $$\frac{d}{dx}\left(\frac{dx}{dt}\right)=\frac{d}{dx}\left(\frac{1}{\frac{dt}{dx}}\right)$$ and use the chain rule. $$\frac{d}{dx}\left(\frac{dx}{dt}\right)=\left(\frac{-1}{(\frac{dt}{dx})^2}\right)\frac{d^2t}{dx^2}=-\left (\frac{dx}{dt}\right )^2\frac{d^2t}{dx^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
A divisor on a smooth curve such that $\Omega$ minus it has no nonzero sections Let $X$ be a smooth projective curve of genus $g$. Let $\Omega$ be the sheaf of differentials. Mumford (in Abelian Varieties, sec. 2.6, in proving the theorem of the cube) asserts that there is an effective divisor of degree $g$ on $X$ such... | Let $\mathcal L$ be an invertible sheaf on the curve $X$, and suppose
that $H^0(X,\mathcal L)$ has dimension $d$. (In the case when $\mathcal L
= \Omega$, we have $d = g$.)
If we choose a closed point $x \in X$, then there is a map $H^0(X,\mathcal L) \to
\mathcal L_x/\mathfrak m_x \mathcal L_x$ given by mapping a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
An approximation of an integral Is there any good way to approximate following integral?
$$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$
$\mu$ is between $0$ and $0.25$, the problem is in $\sigma$ which is always positive, but it can be arbitrarily small.
I w... | If you write y=x^2 and pull the constants out you have $$\frac{1}{2\sqrt{2\pi}\sigma}\int_0^{0.25}\sqrt{y}\cdot \exp(-\frac{(y-\mu )^2}{2\sigma ^2})dy$$ If $\sigma$ is very small, the contribution will all come from a small area in $y$ around $\mu$. So you can set $\sqrt{y}=\sqrt{\mu}$ and use your error function tab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Proving $A \subset B \Rightarrow B' \subset A'$ Suppose that A is a subset of B. How can we show that B-complement is a subset of A-complement?
| Instead of using a proof by contradiction, $b\in B^c \implies b \notin A$ since $A \subset B$ hence $b \in A^c$ which implies the result.
The key here being that for all $x$, either $x\in X$ or $x\in X^c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Got to learn matlab I have this circuits and signals course where i have been asked to learn matlab all by myself and that its treated as a basic necessity
now i wanted some help as to where should i start from as i hardly have around less than a month before my practicals session start
should i go with video lectures/... | For the elementary syntax, you may look up introduction videos on youtube,
and start from simple task in this MIT introduction - problem set.
visit the Mathworks website god idea, it's got very good explanatory videos.
And dont foget 2 most important commands in matlab: "help" and "doc".
Matlab got very good documentat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Proving there is no natural number which is both even and odd I've run into a small problem while working through Enderton's Elements of Set Theory. I'm doing the following problem:
Call a natural number even if it has the form $2\cdot m$ for some $m$. Call it odd if it has the form $(2\cdot p)+1$ for some $p$. Show t... | Here is a complete proof. We assume that $n^+=m^+$ implies $n=m$ (*) and say that $n$ is even if it is $m+m$ for some natural number $m$ and odd if it is $(m+m)^+$ for some natural number $m$.
We know that $\phi=\phi+\phi$ so $\phi$ is even and $\phi\neq p^+$ for any $p$ and so is therefore not odd. Now assume that $k\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
$A,B \subset (X,d)$ and $A$ is open dense subset, $B$ is dense then is $A \cap B$ dense? I am trying to solve this problem, and i think i did something, but finally i couldn't get the conclusion. The question is:
*
*Let $(X,d)$ be a metric space and let $A,B \subset X$. If $A$ is an open dense subset, and $B$ is a d... | Let $(X,\tau)$ be a topological space. Let $A$ and $B$ be two dense subsets of $(X,\tau)$ with $A$ is an open set.
Let $G\in \tau$. Then since $A$ is dense in $(X,\tau)$, $G\cap A\neq\emptyset$.
Now since $B$ is dense in $(X,\tau)$, for open set $G\cap A$ which is a
nonempty set, $(G\cap A)\cap B\neq \emptyset$.
Hence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Finite Model Property on the First-Order Theory of Two Equivalence Relations I know that there is a first-order sentence $\varphi$ such that
*
*$\varphi$ is written in the vocabulary given by just two binary relation symbols $E_1$, $E_2$ (and hence, without the equality symbol*),
*$\varphi$ is satisfiable in a mo... | Let's call two elements $x$ and $y$ $E_1$-neighbors if $(x \neq y \wedge E_1(x, y))$, and $E_2$-neighbors if $(x \neq y \wedge E_2(x, y))$. Then the following assertions should suffice to force an infinite model:
*
*Every element has exactly one $E_1$-neighbor.
*There exists an element with no $E_2$-neighbor; ever... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Different ways to represent functions other than Laurent and Fourier series? In the book "A Course of modern analysis", examples of expanding functions in terms of inverse factorials was given, I am not sure in today's math what subject would that come under but besides the followings : power series ( Taylor Series, La... | You can add the following to your list of representations
*
*Continued fractions. See Jones & Thron or Lorentzen's books
*Integral representations (Mellin–Barnes, etc). See the ECM entry
*Exponentials. See Knoebel's paper Exponentials reiterated
*Nested radicals. See Schuske & Thron's Infinite radicals in the com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Torsion module over PID Suppose $p$ is irreducible and $M$ is a tosion module over a PID $R$ that can be
written as a direct sum of cyclic submodules with annihilators of the form $p^{a_1} | \cdots | p^{a_s}$ and $p^{a_i}|p^{a_i+1}$. Let now $N$ be a submodule
of $M$. How can i prove that $N$ can be written a direct ... | just look at the size of the subgroups annihilated by p^k for various k.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/17132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Apparent inconsistency between integral table and integration using trigonometric identity According to my textbook:
$$\int_{-L}^{L} \cos\frac{n \pi x}{L} \cos\frac{m \pi x}{L} dx =
\begin{cases} 0 & \mbox{if } n \neq m \\
L & \mbox{if } n = m \neq 0 \\
2L& \mbox{if } n = m = 0
\end{cas... | The integration is wrong if $n=m$ or if $n=-m$, because it is false that $\int \cos(0\pi x)\,dx = \frac{\sin(0\pi x)}{0}+C$. So the very use of the formula assumes that $|n|\neq|m|$.
But if $|n|\neq|m|$, then your formula does say that the integral should be $0$, the same thing you get after the substitution. What make... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How many possible combinations are there in Hua Rong Dao? How many possible combinations are there in Hua Rong Dao?
Hua Rong Dao is a Chinese sliding puzzle game, also called Daughter in a Box in Japan. You can see a picture here and an explanation here .
The puzzle is a $4 \times 5$ grid with these pieces
*
*$2 \ti... | A straightforward search yields the figure of $4392$ for all but the $1 \times 1$ stones. The former fill $14$ out of $20$ squares, so there are $\binom{6}{4} = 15$ possibilities to place the latter. In total, we get $$4392 \times 15 = 65880.$$ These can all be generated, and one can in principle calculate the number o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
If $4^x + 4^{-x} = 34$, then $2^x + 2^{-x}$ is equal to...? I am having trouble with this:
If $4^x + 4^{-x} = 34$, then what is $2^x + 2^{-x}$ equal to?
I managed to find $4^x$ and it is:
$$4^x = 17 \pm 12\sqrt{2}$$
so that means that $2^x$ is:
$$2^x = \pm \sqrt{17 \pm 12\sqrt{2}}.$$
Correct answer is 6 and I am not... | You haven't done anything wrong! To complete your answer, one way you can see the answer is $6$ is to guess that
$$17 + 12 \sqrt{2} = (a + b\sqrt{2})^2$$
Giving us
$$17 = a^2 + 2b^2, \ \ ab = 6$$
Giving us
$$a = 3, \ \ b = 2$$
Thus $$ \sqrt{17 + 12 \sqrt{2}} = 3 + 2\sqrt{2}$$
which gives $$2^x + 2^{-x} = 6$$
(And sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Frequency Swept sine wave -- chirp I am experiencing what I think is really simple confusion.
Take $y(t) = \sin(2 \cdot \pi \cdot t \cdot\omega(t))$
and $\omega(t) = a \cdot t+b$ for $t \in [0,p)$ and let $\omega(t)$ have a periodic extension with period $p$. The values $a,b,p$ are parameters
$\omega(t)$ looks like the... | It seems my comment has answered the question, so I suppose I should turn it into an actual answer.
For a chirp signal of the form $\sin(2\pi\phi(t))$, the instantaneous frequency $\omega(t)$ is the time derivative of the instantaneous phase $\phi(t)$. So for given $\omega(t)$, your signal should be $\sin\left(2\pi\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Division of Other curves than circles The coordinates of an arc of a circle of length $\frac{2pi}{p}$ are an algebraic number, and when $p$ is a Fermat prime you can find it in terms of square roots.
Gauss said that the method applied to a lot more curves than the circle. Will you please tell if you know any worked exa... | To flesh out the comment I gave: Prasolov and Solovyev mention an example due to Euler and Serret: consider the plane curve with complex parametrization
$$z=\frac{(t-a)^{n+2}}{(t-\bar{a})^n (t+i)^2}$$
where $a=\frac{\sqrt{n(n+2)}}{n+1}-\frac{i}{n+1}$ and $n$ is a positive rational number.
The arclength function for thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Creating a parametrized ellipse at an angle I'm creating a computer program where I need to calculate the parametrized circumference of an ellipse, like this:
x = r1 * cos(t), y = r2 * sin(t)
Now, say I want this parametrized ellipse to be tilted at an arbitrary angle. How do I go about this? Any obvious simplificatio... | If you want to rotate $\theta$ radians, you should use $$t\mapsto \left(
\begin{array}{c}
a \cos (t) \cos (\theta )-b \sin (t) \sin
(\theta ) \\
a \cos (t) \sin (\theta )+b \sin (t) \cos
(\theta )
\end{array}
\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/17465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Time until x successes, given p(failure)? I hope this is the right place for help on this question! I expect this should be easy for this audience (and, no, this isn't homework).
I have a task that takes $X$ seconds to complete (say, moving a rock up a hill). However, sometimes the task is restarted from the beginning ... | Let $T$ be the requested time, and $R$ a random variable measuring the time until the next restart. We have $$\mathbb{E}[T] = \Pr[R > X] \cdot X + \Pr[R \leq X]( \mathbb{E}[T] + \mathbb{E}[R|R\leq X]).$$ Simplifying, we get $$\mathbb{E}[T] = X + \frac{\Pr[R \leq X]}{\Pr[R > X]} \mathbb{E}[R|R\leq X].$$
Edit: Here's a m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Going back from a correlation matrix to the original matrix I have N sensors which are been sampled M times, so I have an N by M readout matrix. If I want to know the relation and dependencies of these sensors simplest thing is to do a Pearson's correlation which gives me an N by N correlation matrix.
Now let's say if... | You could make your "10%" implementation faster by using gradient descent
Here's an example of doing it for covariance matrix because it's easier.
You have $k\times k$ covariance matrix C and you want to get $k \times n$ observation matrix $A$ that produces it.
The task is to find $A$ such that
$$A A' = n^2 C$$
You cou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
How do you integrate a Bessel function? I don't want to memorize answers or use a computer, is this possible? I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various Schaum's) f... | At the very least, $\int u J_{2n}(u)\mathrm du$ for integer $n$ is expressible in terms of Bessel functions with some rational function factors.
To integrate $u J_0(u)$ for instance, start with the Maclaurin series:
$$u J_0(u)=u\sum_{k=0}^\infty \frac{(-u^2/4)^k}{(k!)^2}$$
and integrate termwise
$$\int u J_0(u)\mathrm ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Integrate Form $du / (a^2 + u^2)^{3/2}$ How does one integrate
$$\int \dfrac{du}{(a^2 + u^2)^{3/2}}\ ?$$
The table of integrals here: http://teachers.sduhsd.k12.ca.us/abrown/classes/CalculusC/IntegralTablesStewart.pdf
Gives it as: $$\frac{u}{a^2 ( a^2 + u^2)^{1/2}}\ .$$
I'm getting back into calculus and very rusty. ... | A trigonometric substitution does indeed work.
We want to express $(a^2 + u^2)^{3/2}$ as something without square roots. We want to use some form of the Pythagorean trigonometric identity $\sin^2 x + \cos^2 x = 1$. Multiplying each side by $\frac{a^2}{\cos^2 x}$, we get $a^2 \tan^2 x + a^2 = a^2 \sec^2 x$, which is in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
"answer_id": 1
} |
How does teacher get first step? Below are the steps the teacher took to solve:
$y = \sqrt{3}\sin x + \cos x$ find min and max on $[0, 2\pi)$
Step 1: = $2\sin(x + \pi/6))$
How does the teacher get this first step?
Note: No calculus please.
| picakhu's answer is the simplest way to see how it works having already arrived at $y=2\sin(x+\frac{\pi}{6})$ (use the identity there to expand this form). In general, given $a\sin x+b\cos x$ (let's say for $a,b>0$), it is possible to arrive at a similar equivalent form:
$$\begin{align}
a\sin x+b\cos x
&=a\left(\sin x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Why a complete graph has $\frac{n(n-1)}{2}$ edges? I'm studying graphs in algorithm and complexity,
(but I'm not very good at math) as in title:
Why a complete graph has $\frac{n(n-1)}{2}$ edges?
And how this is related with combinatorics?
| A complete graph has an edge between any two vertices. You can get an edge by picking any two vertices.
So if there are $n$ vertices, there are $n$ choose $2$ = ${n \choose 2} = n(n-1)/2$ edges.
Does that help?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/17747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
"answer_count": 4,
"answer_id": 2
} |
Coefficients for Taylor series of given rational function Looking at an earlier post Finding the power series of a rational function, I am trying to get a closed formula for the n'th coefficient in the Taylor series of the rational function (1-x)/(1-2x-x^3). Is it possible to use any of the tricks in that post to not o... | Denoting the coefficients of $T(x)$ as $t_i$ (so $T(x) = t_0 + t_1 x + ...$), consider the coefficient of $x^a$ in $(1 - 2x - x^3) T(x)$. It shouldn't be hard to show that it's $t_a - 2t_{a-1} - t_{a-3}$.
So we have $t_0 - 2t_{-1} - t_{-3} = 1$, $t_1 - 2t_0 - t_{-2} = -1$, and $a > 1 \Rightarrow t_a - 2t_{a-1} - t_{a-3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Unbounded subset of $\mathbb{R}$ with positive Lebesgue outer measure The set of rational numbers $\mathbb{Q}$ is an unbounded subset of $\mathbb{R}$ with Lebesgue outer measure zero. In addition, $\mathbb{R}$ is an unbounded subset of itself with Lebesgue outer measure $+\infty$. Therefore the following question came ... | I guess you mean with positive and finite outer measure. An easy example would be something like $[0,1]\cup\mathbb{Q}$. But perhaps you also want to have nonzero measure outside of each bounded interval? In that case, consider $[0,1/2]\cup[1,1+1/4]\cup[2,2+1/8]\cup[3,3+1/16]\cup\cdots$. If you want the set to have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Why do we use the smash product in the category of based topological spaces? I was telling someone about the smash product and he asked whether it was the categorical product in the category of based spaces and I immediately said yes, but after a moment we realized that that wasn't right. Rather, the categorical produ... | From nLab:
The smash product is the tensor product in the closed monoidal category of pointed sets. That is,
$$\operatorname{Fun}_*(A\wedge B,C)\cong \operatorname{Fun}_*(A,\operatorname{Fun}_*(B,C))$$
Here, $\operatorname{Fun}_*(A,B)$ is the set of basepoint-preserving functions from $A$ to $B$, itself made into ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/17955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 0
} |
Prove A = (A\B) ∪ (A ∩ B) I have to demonstrate this formulae:
Prove $A = (A\setminus B) ∪ (A ∩ B)$
But it seems to me that it is false.
$(A\setminus B) ∪ (A ∩ B)$
*
*$X \in A\setminus B \implies { x ∈ A \text{ and } x ∉ B }$
or
*
*$X ∈ A ∩ B \implies { ... | To show that two sets are equal, you show they have the same elements.
Suppose first $x\in A$. There are two cases: Either $x\in B$, or $x\notin B$. In the first case, $x\in A$ and $x\in B$, so $x\in A\cap B$ (by definition of intersection). In the second case, $x\in A$ and $x\notin B$, so $x\in A\setminus B$ (again, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 0
} |
Simpler solution to this geometry/trig problem? i had a geometry/trignometry problem come up at work today, and i've been out of school too long: i've lost my tools.
i'm starting with a rectangle of known width (w) and height (h). For graphical simplification i can convert it into a right-angle triangle:
i'm trying to... | Parametrize the line from $(w,0)$ to $(0,h)$ by $(w,0) + t(-w, h)$. Then you are searching for the point $(x,y)$ on the line such that $(x,y)\cdot (-w,h) = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
Diophantine equations solved using algebraic numbers? On Mathworld it says some diophantine equations can be solved using algebraic numbers. I know one example which is factor $x^2 + y^2$ in $\mathbb{Z}[\sqrt{-1}]$ to find the Pythagorean triples.
I would be very interested in finding some examples of harder equations ... | Fermat's Last Theorem was originally "proved" by using some elementary arguments and the "fact" that the ring of integers of $\mathbb{Q}(\zeta_p)$ is a UFD (which it is not), where $\zeta_p$ is a primitive $p$th root of unity. These arguments do hold when $\mathcal{O}_p$ is a UFD though; for a thorough account of the a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 4
} |
What are the sample spaces when talking about continuous random variables? When talking about continuous random variables, with a particular probability distribution, what are the underlying sample spaces?
Additionally, why these sample spaces are omitted oftentimes, and one simply says r.v. $X$ follows a uniform dist... | the sample space are the numbers that your random variable X can take. If you have a uniform distribution on the interval (0,1) then the value that your r.v. can take is any thing from O to 1. If your the r.v is outside this interval then your pdf is zero. So that means that the universal space are the all the real num... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 3
} |
Map of Mathematical Logic My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a mathematician.
Can anyone suggest a study-map of Mathematical Logic.
such as
(1) Learn The follo... | I recommend Teach Yourself Logic by Peter Smith as an extremely helpful resource. In addition to detailed textbook suggestions, it has descriptions of the basic topics and the key points that you want to learn regardless what book you learn them from.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
$n$th derivative of $e^{1/x}$ I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula
$$\frac{\mathrm d^n}{\mathrm dx^n}f(x)=(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 n+k}$$
I ... | We may obtain a recursive formula, as follows:
\begin{align}
f\left( t \right) &= e^{1/t} \\
f'\left( t \right) &= - \frac{1}{{t^2 }}f\left( t \right) \\
f''\left( t \right) &= - \frac{1}{{t^2 }}f'\left( t \right) + f\left( t \right)\frac{2}{{t^3 }} \\
&= - \frac{1}{{t^2 }}\left\{ {\frac{1}{{t^2 }}f\left( t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 6,
"answer_id": 0
} |
Integral Representations of Hermite Polynomial? One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a homework problem that he is having trouble with.)
http://functions.wolfram.c... | \begin{align*}
H_n(x)
&=(-)^n\mathrm{e}^{x^2}\partial_x^n\mathrm{e}^{-x^2} \\
&=(-1)^n\mathrm{e}^{x^2}\partial_x^n\mathrm{e}^{-x^2}\Big(\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{-(t\pm \mathrm{i}x)^2}\mathrm{d}t\Big) \\
&=(-1)^n\mathrm{e}^{x^2}\frac{1}{\sqrt{\pi}}\partial_x^n\int_{-\infty}^{\infty}\mathrm{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Spread evenly $x$ black balls among a total of $2^n$ balls Suppose you want to line up $2^n$ balls of which $x$ are black the rest are white. Find a general method to do this so that the black balls are as dispersed as possible, assuming that the pattern will repeat itself ad infinitum. The solution can be in closed ... | Let $\theta = x/2^n$. For each $m$, put a (black) ball at $$\min \{ k \in \mathbb{N} : k\theta \geq m \}.$$ In other words, look at the sequence $\lfloor k \theta \rfloor$, and put a ball in each position where the sequence increases.
For example, if $n=3$ and $x = 3$ then $\theta = 3/8$ and the sequence of floors is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Balancing an acid-base chemical reaction I tried balancing this chemical equation
$$\mathrm{Al(OH)_3 + H_2SO_4 \to Al_2(SO_4)_3 + H_2O}$$
with a system of equations, but the answer doesn't seem to map well. I get a negative coefficient which is prohibitive in this equations. How do I interpret the answer?
| Let $x$ be the number of $\mathrm{Al(OH)_3}$; $y$ the number of $\mathrm{H_2SO_4}$; $z$ the number of $\mathrm{Al_2(SO_4)_3}$, and $w$ the number of $\mathrm{H_2O}$. Looking at the number of $\mathrm{Al}$, you get $x = 2z$. Looking at $\mathrm{O}$, you get $3x + 4y = 12z + w$. Looking at $\mathrm{H}$ you get $3x + 2y =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How can I solve for a single variable which occurs in multiple trigonometric functions in an equation? This is a pretty dumb question, but it's been a while since I had to do math like this and it's escaping me at the moment (actually, I'm not sure I ever knew how to do this. I remember the basic trigonometric identiti... | Hint:
Can you solve $$p = \frac{q\sin 2\theta + r}{s\cos 2\theta + t}$$
Ok, more details.
$$a = \frac{\sin \theta \cos \theta}{\cos^2 \theta} - \frac{b}{\cos^2 \theta} = \frac{\sin 2 \theta }{2\cos^2 \theta} - \frac{b}{\cos^2 \theta} $$
$$ = \frac{\sin 2\theta - 2b}{2cos^2 \theta} = \frac{ \sin 2\theta - 2b}{\cos 2\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
decompose some polynomials [ In first, I say "I'm sorry!", because I am not a Englishman and I don't know your language terms very well. ]
OK, I have some polynomials (like $a^2 +2ab +b^2$ ). And I can't decompress these (for example $a^2 +2ab +b^2 = (a+b)^2$).
Can you help me? (if you can, please write the name or for... | For
1) $(a^2-b^2)x^2+2(ad-bc)x+d^2-c^2$
think about rearranging
$$(a^2-b^2)x^2+2(ad-bc)x+d^2-c^2=a^2x^2+2adx+d^2-(b^2x^2+2bcx+c^2)$$
The same idea can be applied to all your questions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Golden Number Theory The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers $\mathbb{Z}[\varphi]$ used in number theory though. If anyone happens to know some equations we can apply this in and h... | You would probably solve the Mordell equation $y^2=x^3+5$ by working in that field.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "61",
"answer_count": 4,
"answer_id": 0
} |
Convolution of two Gaussians is a Gaussian I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian.
| *
*the Fourier transform (FT) of a Gaussian is also a Gaussian
*The convolution in frequency domain (FT domain) transforms into a simple product
*then taking the FT of 2 Gaussians individually, then making the product you get a (scaled) Gaussian and finally taking the inverse FT you get the Gaussian
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 0
} |
Proof of a combination identity:$\sum \limits_{j=0}^n{(-1)^j{{n}\choose{j}}\left(1-\frac{j}{n}\right)^n}=\frac{n!}{n^n}$ I want to ask if there is a slick way to prove:
$$\sum_{j=0}^n{(-1)^j{{n}\choose{j}}\left(1-\frac{j}{n}\right)^n}=\frac{n!}{n^n}$$
Edit:
I know Yuval has given a proof, but that one is not direct. I ... | This is inclusion-exclusion for counting the number of onto maps from $n$ to $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 1
} |
If $\sum^n_{i=1}{f(a_i)} \leq \sum^n_{i=1}{f(b_i)}$. Then $\sum^n_{i=1}{\int_0^{a_i}{f(t)dt}} \leq \sum^n_{i=1}{\int_0^{b_i}{f(t)dt}}$? I have been puzzling over this for a few days now. (It's not homework.) Suppose $f$ is a positive, non-decreasing, continuous, integrable function. Suppose there are two finite sequen... | The answer is no, even for convex functions. Let $f(t)=(t)^+$ (the positive part of $t$). Then $f$ is convex, the first inequality means that $a_1+\cdots+a_n\le b_1+\cdots+b_n$, the second inequality means that $a_1^2+\cdots+a_n^2\le b_1^2+\cdots+b_n^2$. As soon as $n\ge2$, the former cannot imply the latter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Help me spot the error? I have a determinant to expand which is
$$\triangle =
\begin{bmatrix}
p& 1 & \frac{-q}{2}{}\\
1& 2 &-q \\
2& 2 & 3
\end{bmatrix} = 0 $$
But when I am expanding the determinant along the first row such as $ p(6+2q) - (3+2q) = 0 $ but when I am trying to expand along first column I am gett... | Seems you like you forgot the third column ($\frac{-q}{2}$) when expanding using the first row.
In the first row expansion, just using the first and second columns gives you $p(6+q) - (3 +2q)$. The third column gives an extra $q$, after adding which it matches the second expression you got, using the first column.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Property of a Cissoid? I didn't think it was possible to have a finite area circumscribing an infinite volume but on page 89 of Nonplussed! by Havil (accessible for me at Google Books) it is claimed that such is the goblet-shaped solid generated by revolving the cissoid y$^2$ = x$^3$/(1-x) about the positive y-axis bet... | The interpretation is mangled. The volume is finite but the surface area infinite, much like Gabriel's Horn. The idea of the quote is much as in Gabriel's Horn: since the volume is finite, you can imagine "filling it up" with a finite amount of paint. But the surface area is infinite, which suggests that you are "paint... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Finding the area of a quadrilateral I have a quadrilateral whose four sides are given
$2341276$, $34374833$, $18278172$, $17267343$ units.
How can I find out its area? What would be the area?
| I can't comment (not enough reputation), but as this sort of a partial answer anyway, I'll just post it as an answer. Rahula Narain's comment is correct that the area depends on more than just the sidelengths. But assuming the sidelengths are listed in the question in order around the polygon as usual (in fact, assumin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Primitive integer solutions to $2x^2+xy+3y^2=z^3$? The class number of $\mathbb{Q}(\sqrt{-23})$ is $3$, and the form
$$2x^2 + xy + 3y^2 = z^3$$
is one of the two reduced non-principal forms with discriminant $-23$.
There are the obvious non-primitive solutions $(a(2a^2+ab+3b^2),b(2a^2+ab+3b^2), (2a^2+ab+3b^2))$.
I'm pr... | If $4|y$, then $2|z$, which in turn, by reducing mod $4$, $2|x$, contradicting primitivity. Multiply by $2$ and factor the equation over $\mathbb{Q}(\sqrt{-23})$:
$$(\frac{4x+y+\sqrt{-23}y}{2})(\frac{4x+y-\sqrt{-23}y}{2}) = 2z^3$$
Note that both fractions are integral. The gcd of the two factors divides $\sqrt{-23}y$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Sigma algebra/Borel sigma algebra proof problem In my stochastics class we were given the following problem ($\mathcal{B}(\mathbb{R})$ stands for the Borel $\sigma$-algebra on the real line and $\mathbb{R}$ stands for the real numbers):
Let $f : \Omega → \mathbb{R}$ be a function. Let $F = \{ A \subset \Omega : \text{ ... | You have to show that $F$ is (i) non-empty, (ii) closed under complementation, and (iii) closed under countable unions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/19005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is there a problem in defining a complex number by $ z = x+iy$? The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, which satisfy the following operations $+$ and $\cdot$:... | There is no "explicit" problem, but if you are going to define them as formal symbols, then you need to distinguish between the + in the symbol $a$+$bi$, the $+$ operation from $\mathbb{R}$, and the sum operation that you will be defining later until you show that they can be "confused"/identified with one another.
Tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Non-squarefree version of Pell's equation? Suppose I wanted to solve (for x and y) an equation of the form $x^2-dp^2y^2=1$ where d is squarefree and p is a prime. Of course I could simply generate the solutions to the Pell equation $x^2-dy^2=1$ and check if the value of y was divisible by $p^2,$ but that would be slow... | A different and important way to view this equation is still as a norm equation, only in a non-maximal order of the quadratic field - namely, we are looking at the ring $\mathbb{Z}[p\sqrt{d}]$.
Since we are still in the realm of algebra, we can prove results algebraically. For example:
Proposition. Let $p$ be a prime t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
Algorithm to compute Gamma function The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to arbitrary precision.
Is there a good algorithm to compute approximations of the Gamma ... | Someone asked a similar question yesterday. I thought of replacing $e^{-t}$ by a series.
$$\Gamma (z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt \approx \sum_{j=0}^{a} \frac{(-1)^j b^{j+z}}{(j + z) j !} . \text{Choose } a > b ,$$ but as J. M. points out, I should have checked this a bit better. Take great care in the choice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 5,
"answer_id": 1
} |
$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$ I was working out some problems. This is giving me trouble.
*
*If $p$ is a prime number of the form $4n+1$ then how do i show that:
$$ \sum\limits_{i=1}^{p-1} \Biggl( \biggl\... | Here are some more detailed hints.
Consider the value of $\lfloor 2x \rfloor - 2 \lfloor x \rfloor$ where $x=n+ \delta$ for
$ n \in \mathbb{Z}$ and $0 \le \delta < 1/2.$
Suppose $p$ is a prime number of the form $4n+1$ and $a$ is a
quadratic residue modulo $p$ then why is $(p-a)$ also a quadratic residue?
What does thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
Suggestions for topics in a public talk about art and mathematics I've been giving a public talk about Art and Mathematics for a few years now as part of my University's outreach program. Audience members are usually well-educated but may not have much knowledge of math. I've been concentrating on explaining Richard Ta... | I heard Thomas Banchoff give a nice talk about Salvador Dali's work a few years ago. Apparently they were even friends.
Here's a link to a lecture by Banchoff on Dali.
It looks like Banchoff wrote a paper in Spanish Catalan on Dali, too:
"La Quarta Dimensio i Salvador Dali," Breu Viatge al mon de la Mathematica, 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 4
} |
If $A$ is an $n \times n$ matrix such that $A^2=0$, is $A+I_{n}$ invertible? If $A$ is an $n \times n$ matrix such that $A^2=0$, is $A+I_{n}$ invertible?
This question yielded two different proofs from my professors, which managed to get conflicting results (true and false). Could you please weigh in and explain what's... | I suggest thinking of the problem in terms of eigenvalues. Try proving the following:
If $A$ is an $n \times n$ matrix (over any field) which is nilpotent -- i.e., $A^k = 0$ for some positive integer $k$, then $-1$ is not an eigenvalue of $A$ (or equivalently, $1$ is not an eigenvalue of $-A$).
If you can prove this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 0
} |
extension/"globalization" of inverse function theorem I am curious as to what changes do we need to make to the hypotheses of the
inverse function theorem in order to be able to find the global differentiable inverse to a differentiable function. We obviously need $f$ to be a bijection, and $f'$ to be non-zero. Is this... | There is a theorem ("Introduction to Smooth Manifolds," Lee, Thm 7.15) for differentiable manifolds which says that:
If $F: M \to N$ is a differentiable bijective map of constant rank, then $F$ is a diffeomorphism -- so in particular $F^{-1}$ is differentiable.
Here, the rank of differentiable map $F\colon M \to N$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Filtration in the Serre SS I knew this at one point, and in fact it is embarassing that I have forgotten it. I am wondering what filtartion of the total space of a fibration we use to get the Serre SS. I feel very comfortable with the Serre SS, I am just essentially looking for a one line answer. I checked have Hatche... | Yup. I win the bet :)
See J. M. McCleary's User's guide to spectral sequences, Chapter 5.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/19645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Transformation T is... "onto"? I thought you have to say a mapping is onto something... like, you don't say, "the book is on the top of"...
Our book starts out by saying "a mapping is said to be onto R^m", but thereafter, it just says "the mapping is onto", without saying onto what. Is that simply the author's version ... | You do indeed hear these terms in relation to functions.
One-to-one means the same as injective.
Onto means the same as surjective.
One-to-one and onto means bijective.
A function can be just one of them or all three of them.
To answer your specific question, onto means each value of the codomain is mapped to by a memb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Combinatorial interpretations of elementary symmetric polynomials? I have some questions as to some good combinatorial interpretations for the sum of elementary symmetric polynomials. I know that for example, for n =2 we have that:
$e_0 = 1$
$e_1 = x_1+x_2$
$e_2 = x_1x_2$
And each of these can clearly been seen as the... | Here are two specific interesting cases to go with Mariano Suárez-Alvarez's general explanation. On $n$ variables, $e_k(1,1,\ldots,1) = \binom{n}{k}$ and $e_k(1,2,\ldots,n) = \left[ n+1 \atop n-k+1 \right]$, where the latter is a Stirling number of the first kind. (See Comtet's Advanced Combinatorics, pp. 213-214.) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Does this expression represent the largest real number? I'm not very good at this, so hopefully I'm not making a silly mistake here...
Assuming that $\infty$ is larger than any real number, we can then assume that:
$\dfrac{1}{\infty}$ is the smallest possible positive real number.
It then stands to reason that anything... | Since $\infty$ is not a real number, you cannot assume that $\dfrac{1}{\infty}$ is a meaningful statement. It is not a real number.
You might want to investigate non-standard, hyperreal and surreal numbers and infinitesimals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/19790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Finding number of roots of a polynomial in the unit disk I would like to know how to find the number of (complex) roots of the polynomal $f(z) = z^4+3z^2+z+1$ inside the unit disk. The usual way to solve such a problem, via Rouché's theorem does not work, at least not in an "obvious way".
Any ideas?
Thanks!
edit: here ... | This one is slightly tricky, but you can apply Rouché directly.
Let $g(z) = 3z^2 + 1$. Note that $|g(z)| \geq 2$ for $|z| = 1$ with equality only for $z = \pm i$ (because $g$ maps the unit circle onto the circle with radius $3$ centered at $1$).
On the other hand for all $|z| = 1$ we have the estimate $h(z) = |f(z) - g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to deduce trigonometric formulae like $2 \cos(\theta)^{2}=\cos(2\theta) +1$? Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)... | In my experience, almost all trigonometric identities can be obtained by knowing a few values of $\sin x$ and $\cos x$, that $\sin x$ is odd and $\cos x$ is even, and the addition formulas:
\begin{align*}
\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta,\\
\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 0
} |
If $b$ is the largest square divisor of $a$ and $a^2|c$ then $a|b$? I think this is false, a counter example could be:
$c = 100,$
$b = 10,$
$a = 5$
But the book answer is true :( ! Did I misunderstand the problem or the book's answer was wrong?
Thanks,
Chan
| With or without your edit, b does not divide a. I suspect the question you want is
If b is the largest square divisor of c (not a) and a^2|c then a|b?
Then the answer would be true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/19979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Special arrows for notation of morphisms I've stumbled upon the definition of exact sequence, particularly on Wikipedia, and noted the use of $\hookrightarrow$ to denote a monomorphism and $\twoheadrightarrow$ for epimorphisms.
I was wondering whether this notation was widely used, or if it is common to define a morphi... | Some people use those notations, some don't. Using $\hookrightarrow$ to mean that the map is a mono is not a great idea, in my opinion, and I much prefer $\rightarrowtail$ and use the former only to denote inclusions. Even when using that notation, I would say things like «consider the epimorphism $f:X\twoheadrightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
"answer_id": 1
} |
Probability of "clock patience" going out Here is a question I've often wondered about, but have never figured out a satisfactory answer for. Here are the rules for the solitaire game "clock patience." Deal out 12 piles of 4 cards each with an extra 4 card draw pile. (From a standard 52 card deck.) Turn over the first ... | The name clock patience (solitaire in the US) is appropriate, not just because of the shape of the layout but because it is about cycles in the permutation. As you start with the King pile, a cycle ends when you find a King. If four cycles (one for each King) include all 52 cards, you win. You lose if the bottom car... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Best way to exactly solve a linear system (300x300) with integer coefficients I want to solve system of linear equations of size 300x300 (300 variables & 300 equations) exactly (not with floating point, aka dgesl, but with fractions in answer). All coefficients in this system are integer (e.g. 32 bit), there is only 1 ... | This kind of thing can be done by symbolic computation packages like Maple, Mathematica, Maxima, etc.
If you need a subroutine to do this that you'll call from a larger program, then the answer will depend a lot on the programming language that you're using and what kinds of licensing you're willing to work with.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 1
} |
Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series) I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, if we let $S_n$ act on $(\mathbb... | I share the thoughts above on the Yang-Baxter equation. My viewpoint on this is perhaps more algebraic: in light of the quantum group theory, the Yang R-matrix, as well as its trigonometric and elliptic counterparts, are indeed somewhat miraculous objects. Since no classification of solution of the Yang-Baxter equation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
Probability of Sum of Different Sized Dice I am working on a project that needs to be able to calculate the probability of rolling a given value $k$ given a set of dice, not necessarily all the same size. So for instance, what is the distribution of rolls of a D2 and a D6?
An equivalent question, if this is any easier... | Suppose we have dice $Da$ and $Db$, with $a \le b$. Then there are three cases:
*
*If $2 \le n \le a$, the probability of throwing $n$ is $\frac{n-1}{ab}$.
*If $a+1 \le n \le b$, the probability of throwing $n$ is $\frac{1}{b}$.
*If $b+1 \le n \le a+b$, the probability of throwing $n$ is $\frac{a+b+1-n}{ab}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Integrals as Probabilities Firstly, I'm not a mathematician as will become evident in a quick moment. I was pondering some maths the other day and had an interesting thought: If you encased an integrable function over some range in a primitive with an easily computable area, the probability that a random point within s... | No, this is a very good observation! It is the basis of the modern definition of probability, where all probabilities are essentially defined as integrals. Your particular observation about $\pi$ being closely related to the probability that a point lands in a circle is also very good, and actually leads to a probabili... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
How to know that $a^3+b^3 = (a+b)(a^2-ab+b^2)$ Is there a way of go from $a^3+b^3$ to $(a+b)(a^2-ab+b^2)$ other than know the property by heart?
| Keeping $a$ fixed and treating $a^3 + b^3$ as a polynomial in $b$, you should immediately notice that $-a$ will be a root of that polynomial. This tells you that you can divide it by $a + b$. Then you apply long division as mentioned in one of the comments to get the other factor.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 0
} |
Converting recursive function to closed form My professor gave us a puzzle problem that we discussed in class that I could elaborate on if requested. But I interpreted the puzzle and formed a recursive function to model it which is as follows:
$$f(n) = \frac{n f(n-1)}{n - 1} + .01 \hspace{1cm} \textrm{where } f(1) = .0... | If you define $g(n) = \frac{f(n)}{n}$ then your expression is $g(n) = g(n-1) + \frac{0.01}{n}$ and $g(0) = 0$. So $g(n) = 0.01\sum_{i=1}^n \frac{1}{n} = 0.01 h(n)$ where $h(n)$ is the $n^{\text{th}}$ harmonic number. Therefore $f(n) = 0.01n \cdot h(n)$. I think this is about as closed of a form as you're going to ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to find the sum of the following series How can I find the sum of the following series?
$$
\sum_{n=0}^{+\infty}\frac{n^2}{2^n}
$$
I know that it converges, and Wolfram Alpha tells me that its sum is 6 .
Which technique should I use to prove that the sum is 6?
| It is equal to $f(x)=\sum_{n \geq 0} n^2 x^n$ evaluated at $x=1/2$.
To compute this function of $x$, write $n^2 = (n+1)(n+2)-3(n+1)+1$, so that $f(x)=a(x)+b(x)+c(x)$ with:
$a(x)= \sum_{n \geq 0} (n+1)(n+2) x^n = \frac{d^2}{dx^2} \left( \sum_{n \geq 0} x^n\right) = \frac{2}{(1-x)^3}$
$b(x)=\sum_{n \geq 0} 3 (n+1) x^n = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
a coin toss probability question I keep tossing a fair coin until I see HTHT appears. What is the probability that I have already observed the sequence TTH before I stop?
Edit:
Okay. This problem is the same as before. I am trying to think of the following. Could anyone please tell me if this is also the same as before... | By the standard techniques explained here in a similar context, one can show that the first time $T_A$ when the word A=HTHT is produced has generating function
$$
E(s^{T_A})=\frac{s^4}{a(s)},\qquad a(s)=(2-s)(8-4s-s^3)-2s^3,
$$
and that the first time $T_B$ when the word B=TTH is produced has generating function
$$
E(s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to map discrete numbers into a fixed domain? I have several numbers, say 3, 1, 4, 8, 5, and I wanted them to be mapped into a fixed domain [0.5, 3]. In this case, 1 should be mapped as 0.5 and 8 is 3. Then the rest numbers should be scaled down to their correspondences.
So, my question is what should I do to deal w... | Suppose you have $I=[a,b]$ and you want to map it to $J=[c,d]$ such that there is a constant C and for every $x,y \in I$ and their images $f(x),f(y)\in J$ you have $(x-y)C = (f(x)-f(y))$ (I hope this is what you mean by "scaled down...").
You also want that $a\mapsto c$ and $b\mapsto d$ (in your case $1\mapsto 0.5,\; 8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are there any series whose convergence is unknown? Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will eventually be shown to converge or diverge?
EDIT: People were kind enough to ... | As kind of a joke answer, but technically correct, and motivated by Chandru's deleted reply,
$$\sum_{n=0}^\infty \sin(2\pi n!\,x)$$
where $x$ is the Euler-Mascheroni constant, or almost any other number whose rationality has not been settled. (If $x$ is rational, the series converges. The implication does not go the ot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "191",
"answer_count": 8,
"answer_id": 3
} |
Motivating differentiable manifolds I'm reading lectures for the first time starting next week ^_^ The subject is calculus on manifolds I was told that the students I'll be lecturing are not motivated (at all), so I need to kick off the series with an impressive demonstration of what an cool subject it will be that I'l... | Personally I always loved to link maths and physics.
For instance, why not introduce hysteresis? Some hysteresis phenomenons can be modelled using manifolds I believe.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Maximize the product of two integer variables with given sum Let's say we have a value $K$. I want to find the value where
$A+B = K$ (all $A,B,K$ are integers) where $AB$ is the highest possible value.
I've found out that it's:
*
*$a = K/2$; $b = K/2$; when $K$ is even
*$a = (K-1)/2$; $b = ((K-1)/2)+1$; when $K$ i... | We know that $B = K-A$. So you want to maximize $AB = A(K-A)$. So we have:
$$AB = A(K-A)$$
$$ = AK-A^2$$
Find the critical points and use some derivative tests.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$ \mathbb{C} $ is not isomorphic to the endomorphism ring of its additive group Let $M$ denote the additive group of $ \mathbb{C} $. Why is $ \mathbb{C}$, as a ring, not isomorphic to $\mathrm{End}(M)$, where addition is defined pointwise, and multiplication as endomorphisms composition?
Thanks!
| Method 1: $\operatorname{End}(M)$ is not an integral domain.
Method 2: Count idempotents. There are only 2 idempotent elements of $\mathbb{C}$, but lots in $\operatorname{End}(M)$, including for example real and imaginary projections along with the identity and zero maps.
Method 3: Cardinality. Assuming a Hamel basis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Connected components of a fiber product of schemes The underlying set of the product $X \times Y$ of two schemes is by no means the set-theoretic product of the underlying sets of $X$ and $Y$.
Although I am happy with the abstract definition of fiber products of schemes, I'm not confident with some very basic questions... | "No" in both cases:
let $f\in K[X]$ be a separable irreducible polynomial of degree $d>1$ over some field $K$.
Let $L$ be the splitting field of $f$ over $K$.
Let $X:=\mathrm{Spec} (K[X]/fK[X])$, $Y:=\mathrm{Spec}(L)$ and $S:=\mathrm{Spec}(K)$.
Then $X$ is irreducible and $X\times_K Y=\mathrm{Spec}(L[X]/fL[X])$ consist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Graph for $f(x)=\sin x\cos x$ Okay, so in my math homework I'm supposed to draw a graph of the following function:
$$f(x)=\sin x \cos x.$$
I have the solution in the textbook, but I just can't figure out how they got to that. So, if someone could please post (a slightly more detailed) explanation for this, it would be ... | Here is a hint: Can you draw the graph of $\sin x$? What about $a\sin{(bx)}$? Then recall the identity $$\sin{x} \cos{x} = \frac{1}{2} \sin {2x}.$$
Maybe that helps.
Edit: This is just one way, there are many. As asked in the above comments, how are you supposed to solve it? What tools do you have your disposal?... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Uniform Boundedness/Hahn-Banach Question Let $X=C(S)$ where $S$ is compact. Suppose $T\subset S$ is a closed
subset such that for every $g\in C(T),$ there is an $f\in C(S)$
such that: $f\mid_{T}=g$. Show that there exists a constant $C>0$
such that every $g\in C(T)$ can be continuously extended to $f\in C(S)$
such that... | $C(S) \to C(T)$ is a surjective bounded linear map of Banach spaces (with sup norms), so there is a closed linear subspace $M \subset C(S)$ such that $C(S)/M \to T$ is bijective and bounded with the quotient norm. Inverse mapping theorem says that the inverse is a bounded linear map. The statement then follows.
By the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Showing $\frac{e^{ax}(b^2\cos(bx)+a^2\cos(bx))}{a^2+b^2}=e^{ax}\cos(bx).$ I've got:
$$\frac{e^{ax}(b^2\cos(bx)+a^2\cos(bx))}{a^2+b^2}.$$
Could someone show me how it simplifies to:
$e^{ax} \cos(bx)$?
It looks like the denominator is canceled by the terms that are being added, but then how do I get rid of one of the cos... | You use the distributive law, which says that $(X+Y)\cdot Z=(X\cdot Z)+(Y\cdot Z)$ for any $X$, $Y$, and $Z$. In your case, we have $X=b^2$, $Y=a^2$, and $Z=\cos(bx)$, and so
$$\frac{e^{ax}(b^2\cos(bx)+a^2\cos(bx))}{a^2+b^2}=\frac{e^{ax}((b^2+a^2)\cos(bx))}{(a^2+b^2)}=\frac{e^{ax}((a^2+b^2)\cos(bx))}{(a^2+b^2)}=e^{ax}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equation for $y'$ from $\frac{y'}{ [1+(y')^2]^{1/2}} = c$ In a book there is a derivation for y' that comes from
$$\frac{y'}{[1+(y')^2]^{1/2}} = c,$$
where $c$ is a constant. The result they had was
$$y' = \sqrt{\frac{c^2}{1-c^2}}.$$
How did they get this? I tried expanding the square, and other tricks, I cant seem to... | Square both sides, solve for ${y^{\prime}}^2$ and then take the square root.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$? The question contains 2 stages:
*
*Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$
This one is pretty clear by induction and by applying Hausdorff's formula.
*Prove ${\aleph_\omega} ^... | As you mention, the first equation is a consequence of Hausdorff's formula and induction.
For the second: Clearly the right hand side is at most the left hand side. Now: Either $2^{\aleph_1}\ge\aleph_\omega$, in which case in fact $2^{\aleph_1}\ge{\aleph_\omega}^{\aleph_1}$, and we are done, or $2^{\aleph_1}<\aleph_\om... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
Prove that an odd integer $n>1$ is prime if and only if it is not expressible as a sum of three or more consecutive integers. Prove that an odd integer $n>1$ is prime if and only if it is not expressible as a sum of three or more consecutive integers.
I can see how this works with various examples of the sum of three o... | First of all, you can assume you're adding only positive numbers; otherwise the question isn't correct as written.
Note that the sum of the numbers from $1$ to $n$ is ${\displaystyle {n^2 + n \over 2}}$. So the sum of the numbers from $m+1$ to $n$ is ${\displaystyle {n^2 + n \over 2} - {m^2 + m \over 2}
= {n^2 - m^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Physical meaning of the null space of a matrix What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.
E.g.: I think of the rank $r$ of a matrix as the minimum number of dimensions that a line... | Its the solution space of the matrix equation $AX=0$ . It includes the trivial solution vector $0$. If $A$ is row equivalent to the identity matrix, then the zero vector is the only element of the solution space. If it is not i.e. when the column space of $A$ is of a dimension less than the number of columns of A then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "136",
"answer_count": 10,
"answer_id": 6
} |
What is the simplification of $\frac{\sin^2 x}{(1+ \sin^2 x +\sin^4 x +\sin^6 x + \cdots)}$? What is the simplification of $$\frac{\sin^2 x}{(1+ \sin^2 x +\sin^4 x +\sin^6 x + \cdots)} \space \text{?}$$
| What does $1 + \sin^2 x + \sin^4 x + \sin^6 x + ....$ simplify to?
Or better, what does $1 + x^2 + x^4 + x^6 + ....$ simplify to?
Or better, what does $1 + x + x^2 + x^3 + ....$ simplify to?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/21182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Distribution of $\log X$ If $X$ has the density function
$$ f_\vartheta (x) = \Big \{ \begin{array}{cc}
(\vartheta - 1)x^{-\vartheta} & x \geq 1\\
0 & otherwise
\end{array}$$
How can I see that $\log X \sim Exp(\vartheta - 1)$?
I had the idea to look at $f(\log x)$ but I think that's n... | One way is to use the cdf technique. Let $Y = \log X$. Then
$$P(Y \leq y) = P(\log X \leq y) = P(X \leq e^y) = \int_1^{e^y} (\vartheta - 1)x^{-\vartheta} dx = \left.-x^{-\vartheta+1}\right|_1^{e^y} = 1-e^{-(\vartheta-1)y}.$$
Differentiating with respect to $y$ yields $(\vartheta-1) e^{-(\vartheta-1)y}$ as the pdf of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.