Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Probability that two random permutations of an $n$-set commute? From this MathOverflow question:
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. -- Benjamin Steinberg
Unfortunately, it's not well known to me. Can ... | It's a simple matter of combining two other well-known facts:
*
*In any finite group $G$, the probability that $a,b$ commute, with $(a,b)\in G\times G$ chosen uniformly at random, is $k/G$, where $k$ is the number of conjugacy classes of $G$.
*Conjugacy classes in $S_n$ correspond to cycle types, which correspond t... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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question on limits and their calculation In taking each of the limits
$$\lim_{x\to -\infty}\frac{x+2}{\sqrt {x^2-x+2}}\quad \text{ and } \quad \lim_{x\to \infty}\frac{x+2}{\sqrt {x^2-x+2}},$$
I find that both give the value $1$, although it should in fact be getting $-1$ and $1$, respectively.
This however doesn't show... | Your fundamental problem arises regarding the issue of signs when dealing with $x\to -\infty$ and it can be handled most easily (without applying too much thought and in almost mechanical fashion) by putting $x=-t$ and then letting $t \to\infty$. Thus we have $$\lim_{x\to -\infty}\frac{x+2}{\sqrt{x^{2}-x+2}}=\lim_{t \t... | {
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If there exists an integrable function that is not zero a.e., then the measure is $\sigma$-finite Suppose $f\in L^1(\Omega,\mathcal{A},\mu)$ and $f(x)\neq 0$ for almost every $x\in \Omega$. How to prove $\mu$ is $\sigma-$finite?
I only got that $\Omega=\cup_{n=1}^\infty \{x\in \Omega:|f(x)|\geq \frac{1}{n}\}\cup \{{ x\... | As you observed, in the decomposition
$$\Omega=\bigcup_{n=1}^\infty \left\{x\in \Omega:|f(x)|\geq \frac{1}{n}\right\}\cup \{{ x\in \Omega: f(x)=0\}}$$
every set on the right has finite measure. Therefore, the measure is $\sigma$-finite.
If needed, one can also be more precise and say: "$\mu$ is $\sigma$-finite on $\... | {
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For the polynomial For the polynomial, -2 is a zero. $h(x)= x^3+8x^2+14x+4$. Express $h(x)$ as a product of linear factors.
Can someone please explain and help me solve?
| Hint: $a$ is a root of the polynomial $f(x)$ if and only if the polynomial $x-a$ divides $f(x)$. So if you divide $h(x)$ by $x+2$ you get a polynomial of degree $2$. Do you know how to find the roots of a polynomial of degree $2$?
| {
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Idempotent entire complex function problem Problem statement
Find all the entire functions $f:\mathbb C \to \mathbb C$, that satisfy $f(f(z))=f(z)$ for all $z \in \mathbb C$.
I have no idea how to attack this problem, I would appreciate hints and suggestions.
| I think this could work: assuming $f(z)$ is not constant, and considering the points where $f'(z) \neq 0$ , we have : $$f'(f(z))f'(z)=f'(z) $$, so that $f'(f(z))=1$. Since the range of $f(z)$ *, since the zeros of an analytic point are isolated) contains a limit point in $\mathbb C$, we can use the identity theorem on ... | {
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What's the intuition behind the 2D rotation matrix? Can anyone offer an intuitive proof of why the 2D rotation matrix works?
http://en.wikipedia.org/wiki/Rotation_matrix
I've tried to derive it using polar coordinates to no avail.
| If I rotate $(1,0)^T$ by an angle of $\theta$ counterclockwise, it should end up at $(\cos\theta,\sin\theta)^T$. This will be the first column in the rotation matrix.
If I rotate $(0,1)^T$ by an angle of $\theta$ counterclockwise, it should end up at $(-\sin\theta,\cos\theta)^T$. This will be the second column in the r... | {
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Application Closed Graph Theorem to Cauchy problem Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms.
Consider
$$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f
$$
with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\
a_i\in C^0([a,b]),w_i\in\mathbb{R},t_0\in [a,b]$$ Then, let $T:E\to F$,... | Consider the map $D\colon F\to E$ given by
$$D(u) = \left(u^{(n)} + \sum_{i=0}^{n-1} a_i\cdot u^{(i)}, (u^{(i)}(t_0))\right).$$
It is elementary to verify that $D$ is continuous, and then you can note that $D = T^{-1}$ to conclude.
Alternatively, if you want to directly use the closed graph theorem, consider sequences ... | {
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Basic induction proof methods so we're looking to prove $P(n)$ that
$$1^2+2^3+\cdots+n^3 = (n(n+1)/2)^2$$
I know the basis step for $p(1)$ holds.
We're going to assume $P(k)$
$$1^3+2^3+\cdots+k^3=(k(k+1)/2)^2$$
And we're looking to prove $P(k+1)$
What I've discerned from the internet is that I should be looking to add... | What you need to show is that $S(k-1)+k^3=S(k)$, i.e.
$$\frac{(k-1)^2k^2}4+k^3=\frac{k^2(k+1)^2}4.$$
Simplifying by $\frac{k^2}4$, you get
$$(k-1)^2+4k=(k+1)^2.$$
QED.
| {
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Induction hypothesis misunderstanding and the fundamental theorem of arithmetic. The fundamental theorem of arithmetic is made of two parts:
*
*The existence part:
There exist primes
such that for any natural number $j$, we can write $j$ as a product of prime numbers.
*
*The uniqueness part:
That we can write a... | Prime numbers are the atoms of divisibility: you cannot divide them any more.
Thus they form a set of building blocks for the natural numbers: any natural number can be built from these atoms.
The building rule for the construction is the reverse of division, i.e. multiplication.
The proof is "proof by destruction" ... | {
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"timestamp": "2023-03-29T00:00:00",
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functions satisfying $f(x - y) = f(x) f(y) - f(a - x) f(a + y)$ and $f(0)=1$ A real valued function $f$ satisfies the functional equation
$$f(x - y) = f(x) f(y) - f(a - x) f(a + y) \tag 1 \label 1$$
Where $a$ is a given constant and $f(0) = 1$. Prove that $f(2a - x) = -f(x)$, and find all functions which satisfy the gi... | It seems that if $a \ne 0$, then $f(x) = \cos\frac{\pi }{2a}x$, and if $a = 0$, there is no solution!
Sketch of proof: The case $a=0$ is obvious. So let that $a\ne0$. With a change of variable the equation can be change to
$$
f(x+y) = f(x)f(y)-f(\pi/2-x)f(\pi/2-y).
$$
Let $g(x)=f(x)+i f(\pi/2 -x)$. Then one can easily ... | {
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Optimize matrix arrangement Let's imagine I have a Matrix $\textbf{C}$ whose construction depends on several parameters (and constraints).
I'm interested in maximizing a value $K$ calculated as:
$K=\frac{-1}{C_{1,1}^{-1}}$
where $C_{1,1}^{-1}$ is the first element of the inverse matrix.
I know that we should never cal... | There is a good chapter in Wilf's book "Mathematical Methods for Digital Computers" p 78. But it is a Monte Carlo approach so therefore it has limited accuracy.
They talk about it here:
https://mathoverflow.net/questions/61813/how-to-find-one-column-or-one-entry-of-the-matrix-inversion
| {
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Is $n! + 1$ often a prime? Related to another question (If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$), I wonder: How often is $n!+1$ a prime?
There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not. Does anybody know more about it?
| Such numbers are called factorial primes. There is only a limited number of known such numbers.
The largest factorial primes were discovered only recently. From an announcement of an organization called PrimeGrid PRPNet:
On 30 August 2013, PrimeGrid’s PRPNet found the 2nd largest known
Factorial prime: $$147855!-1$$ T... | {
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sequential continuity vs. continuity A short and hopefully simple question for someone with more experience in topology:
If a topology is induced by a mode of convergence and in fact nothing more is known apriori, whether this topology is metrizable, first-countable or anything else? Is continuity then equivalent to ... | There are sequential spaces, these are topological spaces such that a set $A$ is closed if $A$ is sequentially closed, meaning $A$ contains the limits of all sequences in $A$. One can say that a sequential space has the final topology with respect to all continuous maps from $\hat{\Bbb N}$, the one-point-compactificati... | {
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What are these symbols used for? I do not understand these symbols.
*
*a.s.
*e.g.
*i.e.
*c.f.
*...
| See Wikepedia for a nice list of mathematical abbreviations.
Mathematically, a.s. is used to shorten "almost surely."
And a.e. is used to shorten "almost everywhere."
You might also want to consult the list of mathematical jargon, particularly if English isn't your native language, and even if it is!
| {
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Arithmetic Progression - two series I have two arithmetic progressions: $a, b, c, d$ and $w, x, y, z$
If the arithmetic progressions are merged together like this:
$aw, bx, cy, dz$, is it possible to find the sum of the series?
Let $a$ be the first term and $c$ be the last term of the series.
Let $n$ be the number of t... |
I have two arithmetic progressions: $a, b, c, d$ and $w, x, y, z$
If the arithmetic progressions are merged together like this:
$aw, bx, cy, dz$, is it possible to find the sum of the series?
The result is $$(b+c)(x+y)+5(c-b)(y-x),
$$
or, equivalently, $$2b(3x-2y)+2c(3y-2x),
$$
or, equivalently, $$2x(3b-2c)+2y(3c... | {
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"timestamp": "2023-03-29T00:00:00",
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Definition of cluster point I'm studying if the book Multidimensional Real Analysis by Duistermaat and the definition of cluster point is:
A point $a \in \mathbb{R}^n$ is said to be a cluster point of a subset $A$ if for every $\delta >0$ we have $B(a; \delta) \cap A \neq \emptyset$, where $B(a; \delta) = \{x \in \math... | Indeed the definitions aren't equivalent. I always saw the terms accumulation point (or limit point), and adherence point for those definitions, respectively. In simple terms, a point is adherent to a set if it is a limit point that is not isolated. My approach would be to follow the definition that each specific book ... | {
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$n$ players roll a die. For every pair rolling the same number, the group scores that number. Find the variance of the total score. This is problem 3.3.3.(b) in Probability and Random Processes by Grimmett and Stirzaker.
Here's my attempted solution:
We introduce the random variables $\{X_{ij}\}$, denoting the scores... | Your solution appears to be correct. If $n=3$, there are $216$ equally likely outcomes of the dice. They're easy to enumerate, and the variance of the scores comes out to be $\frac{1015}{144}$. This agrees with your answer, not the book's.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Quick floor function This isn't true, right?
$$k\left\lfloor\frac n {2k}\right\rfloor\leq \left\lfloor\frac n k\right\rfloor$$
$2<k\leq \left\lfloor\dfrac {n-1} 2\right\rfloor$, $n>4$, $k,n$ are coprime.
| Let $n=15$ and $k=4$. Not that $15>4$, $2<4<\left\lfloor\dfrac{15-1}{2}\right\rfloor=7$, and that $gcd(4,15)=1$
Now
$$k\left\lfloor\dfrac {2n} k\right\rfloor=4\left\lfloor\dfrac {30} 4\right\rfloor=28$$
and
$$\left\lfloor\dfrac n k\right\rfloor=\left\lfloor\dfrac {15}4\right\rfloor=3$$
Clearly $28>3$, so this provides... | {
"language": "en",
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What is the proper definition of cylinder sets? in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with $G_i:=\sigma(X_i,X_{i+1},...)$. The question I asked myself was what the proper definition of $\sigma... | The idea is about the same as for finite collections. We want to define $\sigma(X_{i},X_{i+1},\dots)$ so that it has just enough sets for each of the variables $X_i$, $X_{i+1}$, ... to be measurable. What do we need for this? For every open set $A\subset \mathbb R$ and every $j\ge i$ the set $X_{j}^{-1}(A)\phantom{}$ ... | {
"language": "en",
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Solve $y = 2 + \int^x_2 [t - ty(t) \,\, dt]$ While working on some differential equation problems, I got one of the following problems:
$$y = 2 + \int^x_2 [t - ty(t) \,\, dt]$$
I have no idea what an integral equation is however, the hint give was "Use an initial condition obtained from the integral equation".
I don't... | $$y = 2 + \int^x_2 [t - ty(t) \,\, dt]$$ will be a function of $x$ after the integration. Differentiating both sides, we will get$$ \frac{dy}{dx}=x-xy(x)$$(This is Newton-leibnitz rule). Also you have $y(2)=2$
| {
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upper bound of probability with s^2 + a^2 as denominator Let X be a discrete random variable with $E(X) = 0$ and $\sigma^{2}$ = var(X) < $\infty$. Show that $P(X$ $\geq$ $a)$ $\leq$ $\frac{\sigma^{2}}{(\sigma^{2}+a^{2})}$ , for all $a$ $\geq$ $0$.
Please help.
| For a constant $b>0,$ define the variable:
$$Z = (X+b)^2.$$
Then, with $E(X)=0$, we have
$$E(Z) = E(X^2)+b^2=\sigma^2 +b^2.$$
By Markov's inequality
$$P(X\geq a) \leq P[Z\geq (a+b)^2] \leq \frac{E(Z)}{(a+b)^2}=\frac{\sigma^2+b^2}{(a+b)^2}.$$
Choose $b = \sigma^2/a$, which minimizes the bound.
Then
$$P(X\geq a) \leq ... | {
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Is $||A||_F ||x||_2^2 \geq x^TAx$ Given a symmetric matrix $A$ and a vector $x$
Is $||A||_F ||x||_2^2 \geq x^TAx$?
If yes, how to show this?
| The quantity $\frac{x^T A x}{||x||_2^2}$ is the Rayleigh quotient of $A$ and its maximum value is the largest eigenvalue of $A$, $\lambda_{max}$.
Noting that $||A||_F = \sqrt{tr(A A^T)} = \sqrt{tr(A A)} = \sqrt{tr(A^2)} = \sqrt{\sum_i \lambda_i^2} \geq \sqrt{\lambda_{max}^2} = |\lambda_{max}| \geq \lambda_{max}$, we g... | {
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Show that Axioms 7, 8, and 9 hold.
I'm having trouble seeing how axiom 7 holds since ku makes the first element a 0 but not kv.. also I'm not sure what the m is in axiom 8 and 9..
| The definition of scalar multiplication given in the question is for a general scalar $k$ and vector $\vec u$. And in order to show that the axioms hold, you must use general values, not the specific values given in the question. So $k\vec u = (0, ku_2)$ and $k\vec v = (0, kv_2)$. Then according to the addition defined... | {
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Prove $(a,b,c)=((a,b),(a,c))$ The notation is for the greatest common divisor. I know that
$$(a,b,c)=((a,b),c)=((a,c),b)=(a,(b,c))$$
Suppose $g=(a,b,c)$. Then $g\mid a,b,c$. Also, $g\mid(a,b),c$ and $g\mid(a,c),b$. Thus there exist integers $k,m$ such that
$$(a,b)=gk, (a,c)=gm$$
Then
$$((a,b),(a,c))=(gk,gm)=g(k,m)... | You can use just the one identity you know along with symmetry, and nothing else, to simplify
$$ ((a,b),(a,c)) = ((a,b),a,c) = (((a,b),a), c) = ((a,a,b),c) = (((a,a),b),c) = ((a,a),b,c)$$
In fact, the associative identity for a binary operator $((a,b),c) = (a,(b,c))$ -- or in its more common expression for operators w... | {
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A prime ideal in the intersection of powers of another ideal
Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$?
My try is to choose the prime ideal generated by $x$. I wanted to show that $x$ can not b... | By Krull's Intersection Theorem we have $⋂_{n≥0}(x,y)^n=0$. I leave you the pleasure to draw the conclusion.
| {
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implication versus conjunction correctness in FOL? I've just started learning FOL and I'm really confused about whether to use conjunction or implications. For example, if I want to represent
some students who answer the easiest question do not answer the most difficult
I came up with several solutions that seem equiv... | It depends on the structure on which you evaluate your formulae. For simplicity I would introduce $3$ predicates $\mathsf{student}, \mathsf{solve\_easy}, \mathsf{solve\_hard}$. (The parameterized solve works but I think it is a little confusing)
If the universe of your structures contains both students and non-students... | {
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Sequence problem I have a calculus final two days from now and we have a test example. There's a sequence question I can't seem to solve and hope someone here will be able to help.
With $a_1$ not given, what are the possible values of it so that the sequence $a_{n+1}=\sqrt{3+a_n}$ will converge. If it does, what is the... | As @evinda rightly noticed, the limit must be $\frac12(1+\sqrt{13})$. For $a_1$ we can take an arbitrary number in $[-3,\infty)$. Note that $a_2$ will be nonnegative in any case and finally note that if $a_k\in [0,\frac12(1+\sqrt{13})]$, then $a_k\le a_{k+1}=\sqrt{a_k+3}\le \frac12(1+\sqrt{13})$ while if $a_k>\frac12(1... | {
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I'm missing the right substitute $\sqrt3\cos x=1-\sin x$ Please show me how to solve the following equation for $x$.
I've tried multiple substitutes but can't seem to find the right one.
$$\sqrt3\cos x=1-\sin x$$
| Let $f(x) = \sqrt{3}\cos x + \sin x$.
Then $f(x) = 2 ({\sqrt{3} \over 2}\cos x + {1 \over 2} \sin x) = 2 (\sin { \pi \over 3} \cos x + \cos { \pi \over 3} \sin x) = 2 \sin (x+{ \pi \over 3})$, so to solve $f(x) = 1$, we need to find $x$ such that $\sin (x+{ \pi \over 3}) = {1 \over 2}$.
Since $\sin^{-1} (\{ {1 \over 2}... | {
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Prove that in each year, the 13th day of some month occurs on a Friday Prove that in each year, the 13th day of some month occurs on a Friday.
No clue... please help!
| In fact, every year will contain a Friday the 13-th between March and October (so leap years don't enter into it).
If March 13 is assigned $0 \pmod 7$, then the other moduli occur as indicated below:
$$(\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\overbrace{31}^{\text{March}}}_{3 \pmod 7},\o... | {
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Is there a closed form solution to $e^{-x/b}(a+x) = e^{x/b}(a-x)$? I have the following equation
$$e^{-x/b}(a+x) = e^{x/b}(a-x)$$
where $b > 0$, and $a > 0$
I need to solve for $x$. I can do it numerically, but would prefer if there was a closed form solution.
It seems to me that there likely is no closed form solution... |
Is there a closed form solution to this equation ?
No. Not even one in terms of Lambert's W function.
| {
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Integral test for convergence: $\sum _1^\infty \frac{e^{1/n}}{n^2}$ Integral test for convergence: $$\sum _1^\infty \frac{e^{1/n}}{n^2}$$
I tried approaching this as an IBP but I haven't been able to sort the solution. Can this be made into a improper integral? and if so could someone show me the process?
| If you really want to do this with the integral test, we first need to realize that the function $\dfrac{e^{1/x}}{x^2}$ is decreasing (which it is, as it has negative derivative) and is positive (which is pretty clear).
Then we may use the integral test. We consider the integral
$$\int_1^\infty \dfrac{e^{1/x}}{x^2} \ma... | {
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$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant. Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that
$$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
| Hint: for a given $h$ one has
$${f(x + h) - f(x) \over h} = {f(x + h) - f(x + h/2) \over h} + {f(x + h/2) - f(x + h/4) \over h} + ....$$
$$= {1 \over 2}{f(x + h) - f(x + h/2) \over h/2} + {1 \over 4}{f(x + h/2) - f(x + h/4) \over h/4 } + ....$$
You actually need continuity of $f(x)$ already for the above. Now take lim... | {
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what is a smart way to find $\int \frac{\arctan\left(x\right)}{x^{2}}\,{\rm d}x$ I tried integration by parts, which gets very lengthy due to partial fractions.
Is there an alternative
| Put $\tan^{ -1}x = y$
Then it becomes $\tan y$ = $x$.
Then differentiate w.r.t $y$ then $dx = \sec^2y dy$.
Then finally,
$$I = \int y \csc^2 y dy$$
Then finally apply integration by parts. It will be little easier than directly applying Integration by parts.
| {
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What is physical interpretation of dot product? Consider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$.
When we take their dot product we get a real number.
How is that number related to the vectors?
Is there any way we can visualize it?
| Temporarily imagine that $V_2$ is of unit length. Then, $V_1 \cdot V_2$ is the projection of the vector $V_1$ onto the vector $V_2$. Picture here. Now we let $V_2$ have its original length and to do so we multiply the result of the dot product by the new length of $V_2$. (This has the effect of making it not matter... | {
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Decomposition of a function into positive and negative parts and its integrability 1)Is it true that any function can be decomposed as a difference of its positive and its negative part as $f=f^{+}-f^{-}$ or that function should belong to $\mathcal{L}^{1}(\mathbb{R})$.
Also if that function doesn't belong to $\mathcal{... | 1) Just define $f^{+}\left(x\right)=\max\left\{ 0,f\left(x\right)\right\} $
and $f^{-}\left(x\right)=\max\left\{ 0,-f\left(x\right)\right\} $.
Then $f^{+}$ and $f^{-}$ are nonnegative functions with $\left|f\right|=f^{+}+f^{-}$
and $f=f^{+}-f^{-}$.
This is true for any function $f$.
2) $\int f\left(x\right)dx=0$ can o... | {
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Finding double root of $x^5-x+\alpha$ Given the polynomial
$$x^5-x+\alpha$$
Find a value of $\alpha>0$ for which the above polynomial has a double root.
Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm looking for $\alpha$ when the 2 points in the plot meet.
Also, this is not homework
| As Daniel pointed out, since $p(x)$ will have a double root, $p'(x)$ must have the same root as well.
Also, by using Descartes rules of signs,
$$p(x) = x^5 -x +\alpha$$
$$p(-x) = -x^5 +x +\alpha$$
Therefore, p(x) has either 2 or 0 positive roots, 1 negative root, and either 2 or 4 complex root.
Since we are assumed the... | {
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How to read this matrix notation Excuse me for this basic question, but when reading some mathematic books I have encountered the following matrix:
W = 2diag([1 1 0,01])
Could anybody explain to me how can I read this? Is it just a diagonal matrix multiplied by 2?
| My guess would be $\texttt{2diag([1 1 0,01])}=\begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0,02 \end{bmatrix}.$
| {
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Hyperplanes and Support Vector Machines I have the following question regarding support vector machines: So we are given a set of
training points $\{x_i\}$ and a set of binary labels $\{y_i\}$.
Now usually the hyperplane classifying the points is defined as:
$ w \cdot x + b = 0 $
First question: Here $x$ does not denot... | For the first equation, $w\cdot x+b=0$, $w$ is the direction normal (orthogonal/perpendicular) to the hyperplane. You are correct that the $x$ (satisfying this equation) are the points on the hyperplane.
In the second equation, the $x$ are the training (or test) data. These should not lie on the hyperplane, where $f(x)... | {
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Evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$. How do I evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$.
I have tried to use binomial expansion and combine factorials, but I have gotten nowhere. I don't really know how to start this problem.
The answer is ${n+... | Suppose we want to pick $n$ children from a group of $n$ boys and $m$ girls. Then we can pick $n$ boys and $0$ girls, or $n - 1$ boys and $1 $ girl, or $n - 2$ boys and $2$ girls, ... There are
$$\sum_{r = 0}^n\binom{n}{n - r}\binom{m}{r} = \sum_{r = 0}^n\binom{n}{r}\binom{m}{r}$$
ways to do this.
But we can also look ... | {
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"question_score": "12",
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Closed form of $\displaystyle\sum_{n=1}^\infty x^n\ln(n)$ Is there a closed form of this :
$$\sum_{n=1}^\infty x^n\ln(n),$$
where $|x|<1$.
Thanks in advance.
| By definition, $\displaystyle\sum_{n=1}^\infty\frac{x^n}{n^a}=\text{Li}_a(x)$. Now, differentiate both sides with regard to a, and then let $a=0$.
| {
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What is the algorithm for the "Shorten" command in Maple? There is a package in Maple called "PolynomialTools". That has a command "Shorten". Does anybody know on what algorithm this is based. The maple manual does not explain much.
Example:
with(PolynomialTools):
Shorten(x^2+x+1,x);
| Here is what the "shorten" command does to a polynomial $f(x)
$\bullet$ make a substitution to remove the $x^(n-1)$ term
$\bullet$ scale $x$ by some rational number,
$\bullet$ if $\mbox{deg } (f,x)=2$ then square-free factor the discriminant
I was hoping that through some substitutions in $x$ it would find a polynomi... | {
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If a uniquness for all functions exist shouldn't there be uniquness to recursion? What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example.
Let's make a table of values for $^{x}2$
(0,1)
(1,2)
(2,4)
(3,16)
(4,65536)
The recursive relation is ... | Any relation that produces this sequence can of course be 'ultimately' simplified to $a_{n+1}=2^{a_n}$ just because this is how consecutive elements in the sequence are related. You do not however specify which identities are 'allowed' in a simplification. There are analytic functions that vanish on all integers, $\si... | {
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Surjective Homomorphism $D_{12}$ I'm trying to find all groups $H$ up to isomorphism such that there is a surjective homomorphism from $D_{12}$ onto H.
The possible $H$ are the factor groups $D_{12}/N$ where $N$ is normal in $G$.
I'm stuck at the possibility the size of the Normal Subgroup is $4$. This implies that th... | No, it doesn't exist. Take an homomorphism $\phi:D_{12}\to C_3$: every reflection has order 2, hence its image in $C_3$ must be the identity. Hence we have at least seven elements in the kernel (six reflections and the identity), and since the number of elements in the kernel must divide 12, it must be 12. Another way ... | {
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If $K = \frac{2}{1}\times \frac{4}{3}\times \cdots \times \frac{100}{99}.$ Then value of $\lfloor K \rfloor$
Let $K = \frac{2}{1}\times \frac{4}{3}\times \frac{6}{5}\times \frac{8}{7}\times \cdots \times \frac{100}{99}.$ Then what is the value of $\lfloor K \rfloor$, where $\lfloor x \rfloor$ is the floor function?
M... | If we make the problem more general and write $$\displaystyle K_n = \frac{2}{1}\times \frac{4}{3}\times \frac{6}{5}\times \frac{8}{7}\times \cdots \times \frac{2n}{2n-1}$$ the numerator is $2^n \Gamma (n+1)$ and the denominator is $\frac{2^n \Gamma \left(n+\frac{1}{2}\right)}{\sqrt{\pi }}$. So, $$K_n=\frac{\sqrt{\pi } ... | {
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Initial value of Newton Raphson Method I am currently studying Newton-Raphson Method. I feel that I understand the concept of it.
Somehow, I am facing some question in my head about how to actually apply it.
The questions that I have are below
- How should I decide the first initial value?
- How should I find all the r... | It will depend on the application. In most practical problems, you are likely to have some idea of the order of magnitude of the solution you expect to find. You take the initial value to be the best guess you have available.
If you're lucky, Newton-Raphson might still work even if this initial guess is quite far fro... | {
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What's the property of $g$ necessary and sufficient to commute with $\sup$? I asked myself the following question:
Does it hold that $\left ( \sup_x |f(x)|\right)^2 = \sup_x |f(x)|^2$.
The answer in this case is: yes. Then I went on to wonder what the defining property of square is that makes it commute with $\sup$. My... | The property does not depend on $f$ much, the only input we get from $f$ is the set of all of its values, i.e., the range. Let's denote this set by $E$ and forget about $f$. What properties of $g$ ensure $g(\sup E) = \sup g(E)$, you ask?
Trying two-point sets $E=\{a,b\}$, we discover that $g$ needs to be increasing (n... | {
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Evaluate the sum ${n \choose 1} + 3{n \choose 3} +5{n \choose 5} + 7{n \choose 7}...$ in closed form How do I evaluate the sum:$${n \choose 1} + 3{n \choose 3} +5{n \choose 5} + 7{n \choose 7} ...$$in closed form?
I don't really know how to start and approach this question. Any help is greatly appreciated.
| Hint: We have $\binom{n}{2k-1}=\frac{n}{2k-1}\binom{n-1}{2k-2}$.
Note that $\binom{a}{b}=0$ if $b\gt a$.
| {
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For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$ Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that
$$B(K,\varepsilon) ... | What you can do is to show that $\Gamma := \rm{dist}(\cdot, U^c)$ is a continuous (even Lipschitz-continuous) map.
Then $\Gamma$ attains its minimum $\gamma$ on the compact set $K$. We have $\gamma > 0$, because for each $x \in K \subset U$, we have $B_\varepsilon (x) \subset U$ for some $\varepsilon > 0$ and thus $\Ga... | {
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Polynomials vanishing on subsets of $\mathbb{R}^2$ Let $\mathcal{S}\subset\mathbb{R}^2$ such that every point in the real plane is at most at distance $1$ from a point in $\mathcal{S}$. Is it true that if $P\in\mathbb{R}[X,Y]$ is a polynomial that vanishes on $\mathcal{S}$, then $P=0$?
| Following the suggestions in comments, let's expand $P$ into a sum of homogeneous polynomials: $P=P_0+\dots+P_n$, with each $P_k$ homogeneous of degree $k$, and $P_n$ is not identically zero. Pick a closed ball $B\subset \mathbb R^n$ on which $P_n$ is nonzero (this is possible because the zero set of $P_n$ has empty in... | {
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Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50$. I'm confused about the following question in my math textbook.
Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50.$
The or is what gets me. For or to be true don't we need only on... | The problem is with the connective 'or'. It is not (always) exclusive. Suppose you took $x=12$, then to satisfy the inequality $x+y\geq 100$ you need $y \geq 88$, which does satisfy $y \geq 50$.
Hint: Prove the contrapositive.
Edit: Answering the comment below, no, because $x+y=72 < 100$. Your hypothesis is that $x+y \... | {
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Integral dependence and field extension Let $R$ be a domain (commutative with unity). $k$ is field algebraically dependent on $k_0$. $A$ is some ideal of $R \otimes_{k_0} k$ and $A_0$ = $A \cap R$.
How to prove that $(R \otimes_{k_0} k)/A$ is integrally dependent on $R/A_0$
| First of all $R$ must be a $k_0$-algebra in order to define $R \otimes_{k_0} k$.
Then we can see a general frame of this problem which is easily proven:
If $R\subset S$ is an integral extension and $J\subset S$ an ideal, then $R/J\cap R\subset S/J$ is also an integral extension.
In your case take $S=R \otimes_{k_0} ... | {
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discrete mathematics and proofs Let $a$ and $b$ be in the universe of all integers, so that $2a + 3b$ is a multiple of $17$.
Prove that $17$ divides $9a + 5b$.
In my textbook they do $17|(2a+3b) \implies 17|(-4)(2a+3b)$.
They do this with the theorem of $a|b \implies a|bx$.
However, I don't know how the book got $x=-4... | The author of the book "cheated" here.
We know: if $17$ divides $2a+3b$, then $17$ divides $k(2a+3b)$ for any integer $k$.
The author, aiming to write an interesting problem, would have chosen $k$ so that $(2k,3k) \text{ mod } 17$ didn't look like an obvious multiple of $(2,3)$ modulo $17$. So the author of the questi... | {
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Can there be more than one power series expansion for a function. I guess the answer is NO, for polynomials.
I know that there are more than one series expansion for every function.
But I am talking about power series here.
All Ideas are appreciated
| Two different power series (around the same point) cannot converge to the same function.
If the power series both have positive radius of convergence, and their $n$th coefficients differ, then the $n$th derivatives of the functions they define also differs, so they cannot be the same function.
| {
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If a non-decreasing function $f: \mathbb{R}\rightarrow (0,+\infty)$ satisfies $\lim\inf (f(n+1)-f(n))>0$, then $\lim \sup \frac{f(x)}{x}>0$ Prove if a non-decreasing function $f: \mathbb{R}\rightarrow (0,+\infty)$ satisfies $\lim \inf_{n\rightarrow \infty} (f(n+1)-f(n))>0$, then $\lim \sup_{x\rightarrow \infty} \frac{f... | You don't know $f$ to be differentiable. But if indeed it were: if for all $x > N$, you have $f'(x)<c$, then $f(n+1)-f(n)<c$ for all $n> N$ by mean value theorem, and you can derive a contradiction with the lim inf being positive.
Try something else for the general case: if the lim inf is $c$ then for $n\geqslant N$, $... | {
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How to prove Godunova's inequality?
Let $\phi$ be a positive and convex function on $(0,\infty)$. Then
$$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$
The application of this inequality is this :
$(1)$ Hardy's inequality.
With $\phi(u)=u^p$, we ... | This is just an explicit execution (as a community wiki answer) of the hint given in the comments:
By Jensen's inequality, we get (because $(0,x)$ with the measure $\frac{dt}{x}$ is a probability space)
\begin{eqnarray*}
\int_{0}^{\infty}\phi\left(\int_{0}^{x}g\left(t\right)\frac{dt}{x}\right)\frac{dx}{x} & \leq & \int... | {
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Derivative and integral of the abs function I would like to ask about how to find the derivative of the absolute value function for example :
$\dfrac{d}{dx}|x-3|$
My try:$$
f(x)=|x-3|\\
f(x) =
\begin{cases}
x-3, & \text{if }x \geq3 \\
3-x, & \text{if }x \leq3
\end{cases}
$$
So:
$$f'(x) =
\begin{cases}
1, & \text{if }x ... | 1) Differentiation:
Define the signum function
$$\mathop{sgn}{(x)}=
\begin{cases}
-1 \quad \text{if } x<0 \\
+1 \quad \text{if } x>0 \\
0 \quad \text{if } x=0 \\
\end{cases}$$
Claim: $$ \frac{d |x|}{dx} = \mathop{sgn}(x), x\neq 0$$
Proof: Use the definition of the absolute value function and observe the left and rig... | {
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How to find $\frac{f(z)}{z-a}$ I hope that you can help me to find some residues. I know two ways to find the residue in a value $a \in \mathbb{C}$:
*
*Straight forward calculation: $ \int_{C(a,\epsilon)^+} f(z) dz$
*Rewriting a function end using the equality $\frac 1{2 \pi i}Res_{z=a}\frac{f(z)}{z-a}\ = \ f(a)$
... | If you have a function holomorphic on some annulus we have a Laurent expansion
$$f(z) = \sum_{n= \infty}^{\infty} a_n (z-z_0)^n$$
where
$$a_n = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{(z-z_0)^{n+1}}dz$$
where $\gamma$ is some closed curve in your annulus.
We have the the residue at $z_0$ is $Res_{z=z_0} f(z) = a_{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/857341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Partial differentiation in transformed coordinates Following lecture notes from MIT it says that, given some variable $A = A(x, y, z(x, y, r, t), t)$ where $r$ is a transformed vertical coordinate
$\left. \frac{\partial A}{\partial x} \right|_r = \left. \frac{\partial A}{\partial x} \right|_z + \frac{\partial A}{\parti... | I think that the confusion comes from transforming from $(x,z) \rightarrow (x,r)$. These 2 $x$s are not the same in terms of partial derivatives because, the first assumes that $z$ remains constant and the second assumes that $r$ remains constant. So if instead we were to start with:
$$A(x(x^\prime), z(x^\prime,r))$$
T... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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What is the intutive explanation of why the notation of matrices is as it is? If I want to solve a system of linear equations, like
2x-y=1
x+2y=4
Then the matrix notation for the same would be:
$$ \begin{bmatrix} 2 & -1 \\ 1 & 2 \\ \end{bmatrix} \begin{bmatrix} X\\ Y\\ \end{bmatrix} = \begin{bmatrix} 1\\ 2\\ \end{bma... | As in my post here, when matrix theory was developed, this notation was not used. Instead, it looked more like
$$ (X,Y,Z)= \left( \begin{array}{ccc}
a & b & c \\
a' & b' & c' \\
a'' & b'' & c'' \end{array} \right)(x,y,z)$$
Which represented the set of linear functions $(ax + by + cz, a'z + b'y + c'z, a''z + b''y + c''z... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Boundary under transformation of a closed curve from $R^2\to R^3$
Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where
$R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected
region. We call the boundary of the surface (which we assume to be a
regular closed curve) created by this map... | As I was out walking, I answered my own question.
Our mapping $\phi$ must divide the surface it is mapping onto in two sections (asymmetrical or otherwise---that is, in some sense, in order to 'unfold' it), each of which has a specific orientation. As such an orientation must have a smooth change, and must change at th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
finding explicit formula through substitution method The question ask us to guess an explicit formula for the sequence
$$t_k = t_{k-1} + 3k + 1 ,$$ for all integers $k$ greater than or equal to 1 and
$t_0 = 0$
Can someone help me with this? Because I am not really familiar with substitution method.
Thanks in advance.
| Hint
Another solution : define $y_k=t_k-t_{k-1}=3k+1$. Adding all terms together, since they telescope,
$$y_k=t_k-t_0=\sum_{i=1}^{k}(3i+1)=3\sum_{i=1}^{k}i+\sum_{i=1}^{k}1=???$$
I am sure that you can take from here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Another formula for number of onto function. Let A and B be two sets.
$A=\{1,2,\dots m\}$
$B=\{1,2,\dots n\}$
We have to find the number of onto functions from A to B
In the following link , the approach of the answer was applying Inclusion Exclusion to count the complement. Can't we use it directly?
Number of onto fu... | A function is onto iff every element of the codomain has nontrivial fiber.
So you need to compute $|\bigcap J_i|$, not $|\bigcup J_i|$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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} |
Is $\cup_{k=1}^\infty (r_k-\frac{1}{k}, r_k+\frac{1}{k}) = \mathbb{R}$? Let $r_k$ be the rational numbers in $\mathbb{R}$.
(1).Is $\cup_{k=1}^\infty (r_k-\frac{1}{k^2}, r_k+\frac{1}{k^2}) = \mathbb{R}$?
(2).Is $\cup_{k=1}^\infty (r_k-\frac{1}{k}, r_k+\frac{1}{k}) = \mathbb{R}$?
(1).Because $m(\mathbb{R})=+\infty, \sum... | I think (2) depends on how you enumerate the rationals. For example lets say you dont want $e$ in the image. Then enumerate the rationals so that if $q$ is a rational and
$e-q \sim \frac{1}{n}$ the make sure that if $r_k=q$ we have $k > n$.
(better is given in Ayman's answer and in the comments afterwards).
Conversel... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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$(18B^2/(A^2-9B^2)) - (A/A+3B) + 2$ Simplify:
$$ \frac{18B²}{A²-9B²} - \frac{A}{A+3B} + 2$$
If the notation doesn't work like I wrote it above it's; Simplify: 18B^2/A^2-9B^2 - A/A+3B + 2.
*
*I made denominator common by expanding A²-9B²: (A+3B)(A-3B)
So the A after the minus should be still multiplied by (A-3B)
Thi... | $$\frac{18b^2}{a^2-9b^2}+2=\frac{2a^2}{a^2-9b^2}$$
As $a^2-9b^2=(a)^2-(3b)^2=(a+3b)(a-3b),$
$$\frac{18b^2}{a^2-9b^2}+2-\frac a{a+3b}=\frac{2a^2}{a^2-9b^2}-\frac a{a+3b}$$
$$=a\left(\frac{2a}{a^2-9b^2}-\frac1{a+3b}\right)$$
$$=a\cdot\frac{2a-(a-3b)}{(a+3b)(a-3b)}=a\cdot\frac1{a-3b}$$ assuming $a+3b\ne0$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Proving Induction $(1\cdot2\cdot3)+(2\cdot3\cdot4)+...+k(k+1)(k+2)=k(k+1)(k+2)(k+3)/4$ I need a little help with the algebra portion of the proof by induction.
Here's what I have:
Basis Step: $P(1)=1(1+1)(1+2)=6=1(1+1)(1+2)(1+3)/4=6$ - Proven
Induction Step:
$$(1\cdot2\cdot3)+(2\cdot3\cdot4)+...+k(k+1)(k+2)+(k+1)(k+2... | Easier: multiply out the brackets to get $k^3 + 3 k^2 +2k$ and then prove induction or perturbation (this is better!) for each sum. Then combine them back to get the answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/858036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Let $N$ and $M$ be two digit numbers. Then the digits of $M^2$ are those of $N^2$, but reversed.
Let $N$ be a two digit number and let $M$ be the number formed from $M$ by reversing $N$'s digits. The digits of $M^2$ are precisely those of $N^2$, but reversed.
$Proof$:
Since $N$ is a two digit number, we can write $N... | As Greg's examples and other comments point out this, can only be true if $a^2,2ab$ and $b^2$ are all less than $10$. Otherwise there is a carryover that spoils it, as your example shows...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Solve $a$ and $b$ for centre of mass in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Given ellipse:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
What length do $a$ and $b$ have to be so the centre of mass is $S(4;2)$?
I've tried steps to solve the equation to $$y=b\sqrt{1-\frac{x^2}{a^2}}$$ and integrate
$$A=b\int_0^a{\sqrt{1-\frac{x... | If you knew the location of the center of mass (COM) for a quarter circle, it'd be easy: you'd just find the scaling-transform that mapped that point to $(4, 2)$. By symmetry, the COM for the quarter-circle must be at some point $(s, s)$ along the line $y = x$. But I cannot see any way, other than actually doing the in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Proving that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12)\pi)$ I want to prove that $\tan(x) = x$ has exactly one solution per interval $((n-\frac12)\pi, (n+\frac12))$.
My attempt:
$\tan(x)$ is $\pi$-harmonic, and has a range of $(-\infty, \infty)$ for each interval $(\frac\pi2n, \f... | No, this argument is not sufficient. A function can be strictly increasing and still meet a linear function more than once - for example, $e^x$ meets $x + 2$ twice. The equation $\tan x = 3x$ has three solutions in the interval about $0$.
A hint towards a correct argument: Suppose that we have $\tan(x_1) = x_1$ and $\t... | {
"language": "en",
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Calculate the distance from a point to a line Por favor, alguém me ajude com essa questão de Geometria:
Please, can someone help me with this geometry question?
Given the point $A(3,4,-2)$ and the line $$r:\left\{\begin{array}{l} x = 1 + t \\ y = 2 - t \\ z = 4 + 2t \end{array} \right.$$
compute the distance from $A$... | Consider the vector $\vec{PA}=(-2,-2,-6)$ and the vector that gives the direction of the line $\vec{v}=(1,-1,2).$ These two vectors form a parallelogram and the height of this parallelogram is the distance between the point and the line (since distance is realized in the direction perperdicular to the line through $A$)... | {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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Counting Number of even and distinct digits The Question was: The number of even four-digit decimal numbers with no digit repeated.
So the first digit cannot be 0 so there are 9 ways to choose a digit. Then for the 3rd, 2nd and 1st digits there would be respectively 9 ways (adding back the zero as an option), 8 ways, ... | $4 \cdot 9 \cdot 8 \cdot 7=2016$
Since the last digti you have only $\ 2,\ 4,\ 6,\ 8\ $ four numbers to choose.
| {
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"timestamp": "2023-03-29T00:00:00",
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Trouble finding the derivative of $\frac{4}{\sqrt{1-x}}$ I've been trying to figure out how to differentiate this expression, apparently I don't know my differentiation rules as much as I thought. I've been trying to use Wolfram Alpha as a guide but I'm at a loss.
I need to differentiate $$\frac{4}{\sqrt{1-x}}$$
I firs... | In this case (like many others), despite of you are working with a quotient, the quotient rule is not needed because you can rewrite your function in a convenient way as you can see below.
$$\begin{align*}\frac{d}{dx}\left[\frac{4}{\sqrt{1-x}}\right]&=4\frac{d}{dx}\left[\frac{1}{\sqrt{1-x}}\right]&\text{ (basic rule)}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the gradient of $\frac{x}{x-y}$ It seems simple on the face of it, but I cannot figure out how to actually do this. I know that you have to find the partial with respect to $x$ and also with respect to $y$, but that's where I get lost.
| $$\left.\begin{array}{rcl}\frac{\partial}{\partial x}\left( \frac{x}{x-y}\right) = \frac{(x-y)-x}{(x-y)^2} &=&\frac{-y}{(x-y)^2}\\
\frac{\partial}{\partial y} \left(\frac{x}{x-y}\right) &=& \frac{x}{(x-y)^2}\end{array}\right\} \Longrightarrow \nabla \left(\frac{x}{x-y}\right) = \frac{1}{(x-y)^2}\begin{pmatrix}- y \\ x\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate $\sum_{k=1}^n \frac 1 {(k+1)(k+2)}$ I have homework questions to calculate infinity sum, and when I write it into wolfram, it knows to calculate partial sum...
So... How can I calculate this:
$$\sum_{k=1}^n \frac 1 {(k+1)(k+2)}$$
| We can use integrals to calculate this sum:
$$
\sum_{k=1}^{n}\dfrac{1}{(k+1)(k+2)} = \sum_{k=1}^{n}\biggl(\dfrac{1}{k+1} - \dfrac{1}{k+2}\biggr) = \sum_{k=1}^{n}\biggl(\int_{0}^{1}x^kdx - \int_{0}^{1}x^{k+1}dx \biggr)
$$
$$
=\sum_{k=1}^{n}\int_{0}^{1}x^k(1 - x)dx = \int_{0}^{1}(1 - x)\sum_{k=1}^{n}x^kdx = \int_{0}^{1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 6,
"answer_id": 5
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A triangle with integer co ordinates and integer sides Is there a triangle with integer sides as well as integer co ordinates when none of the angles is $90$? I tried to solve the general case but I am stuck with it.
Update:
Let the Triangle be $T$ whose vertics are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ such that $x_i\neq x_... | The triangle with vertexes $(-3,0),(3,0),(0,4)$
Note 1: Consider any Pythagorician triple $(a,b,c)$, then $a,b,c \in \mathbb{N}$ and $a^2+b^2=c^2$. Now consider the triangle with vertexes of coordinates $(0,0),(b,0),(0,a)$. Finally to avoid the right angle consider the triangle with vertexes $(-b,0),(b,0),(0,a)$. Clear... | {
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Finding domain of $f \circ g$ I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not.
If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$
$ $
Then $f \circ g = x$
$ $
I think domain of $f \circ g $ is $\mathbb{R} - \left\{0\right\}$
$ $
But m... | One sensible way to resolve this issue is to understand what the equation $f \circ g(x)=x$ does and does not say. It does not say "the function $f \circ g(x)$ is the same as the function $x$". By being careful about domains, what this equation does say is that "the function $f \circ g(x)$ is the same as the restriction... | {
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"source": "stackexchange",
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TT* + I is invertible I've the following exercise which I can't solve:
Prove that:
$$ AA^* + I $$
is invertible for all Matrix $ A $ in finite-dimensional field $V$ with inner product. $ A^* $ is the adjoint operator.
Any help will be appreciated.
| Assume that there's a $v\in V$ such that $AA^*v=-v$. Can you use the definition of the adjoint to conclude that $v=0$?
| {
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Zermelo–Fraenkel set theory the natural numbers defines $1$ as $1 = \{\{\}\}$ but this does not seem right If 1 can be defined as the set that contains only the empty set then what of sets which contain one thing such as the set of people who are me.
number 1 does not just mean $1$ nothing, it means $1$ something. The ... | $\{\{\}\}$ does contain one thing. The thing that it contains is $\{\}$, which is something.
| {
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"url": "https://math.stackexchange.com/questions/859195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solution of $\exp(z)=z$ in $\Bbb{C}$. I have posted a related question here. I thinkg this one is more interesting:
What about the solution of $\exp(z)=z$ in $\Bbb{C}$?
My try :
$z \mapsto e^z - z$ is entire non-constant.
Perhaps $z \mapsto e^z - z$ can be developed in Weierstrass product.
Also any numerically appro... | If
$$z = e^z$$
then
$$-ze^{-z} = -1$$
so
$$-z = W(-1)$$
and thus
$$z = - W(-1),$$
where $W$ is any branch of the Lambert W function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/859274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Derivative of $\frac1{1-x}$ Why is this not correct:
$$
\frac{1}{1-x}= (1-x)^{-1}
$$
now use chain-rule which gives: $(1-x)^{-2}$ times derivative of $(1-x)$ which is $-1$
so
$$
-1\cdot (1-x)^{-2}= \frac{-1}{(1-x)^2}
$$
why is this incorrect?
Because if I use quotient rule on $1/(1-x)$ I get
$$
\frac{0 \cdot (1-x) - 1... | $$\frac d{dx} (1-x)^{-1} = -1\cdot (1 - x)^{-2} \cdot \underbrace{\frac{d}{dx}(1-x)}_{\large =\,-1}$$
$$ = -1\cdot -1\cdot (1 - x)^{-2}= \frac{1}{(1-x)^2}$$
| {
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"url": "https://math.stackexchange.com/questions/859438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$? Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own.
Here's the question again in case someone has their own proof (which may ... | We shall first consider how many different ordered bases $\Bbb F_q^n$ has.
Recall that $|GL_n(\Bbb F_q)|=(q^n-1)(q^n-q)\cdots(q^n-q^{n-1})$. Each element of $GL_n(\Bbb F_q)$ represents a linear map that carries the standard (ordered) basis $\{e_1, e_2, \ldots, e_n\}$ to another ordered basis. We can establish a bijecti... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Positive integral everywhere implies positive function a.e I would like to get feedback on my demonstration of this simple statement :
Let $f$ be an integrable function on the measure space $(X,S,\mu)$.
\begin{align}
\text{If }\int_E f \, d\mu \geq 0\text{ for all }E\in S\text{ then }f \geq 0\text{ a.e.}
\end{align}
I ... | $D$ is certainly measurable. Just replace your strict inequality with $\leq$, and you are good to go.
| {
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"timestamp": "2023-03-29T00:00:00",
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Express the length a, b, c, and d in the figure in terms of the trigonometric ratios of θ. Problem
Express the length a, b, c, and d in the figure in terms of the trigonometric ratios of $θ$. (See the image below)
Progress
I can figure out $c$ usng the pythagorean theorem. $a^2+b^2=c^2$ which would be $2$. Is that cor... | Compute $\sin(\theta)$, $\cos(\theta)$ and $\tan(\theta)$ using your picture. What do you see?
| {
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"url": "https://math.stackexchange.com/questions/859698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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correlation estimator variance Consider I have realisations of two random variables $X$ and $Y$ and I estimate their correlation thanks to the classic formula :
$$\rho=\frac{\sum_{i=1}^{n}{x_iy_i}-\sum_{i=1}^{n}x_i\sum_{i=1}^{n}y_i}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2\sum_{i=1}^{n}(y_i-\bar{y})^2}}$$
1- What is the var... | The variance of the sample correlation is not an easy question; nor is an easy general answer available. If you refer to:
*
*Stuart and Ord (1994), Kendall's Advanced Theory of Statistics, volume 1 - Distribution Theory, sixth edition, Edward Arnold
... an approximation is provide at eqn (10.17) (based on what is ... | {
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What do you get when you differentiate a $e^{f(x)}$-like function I need help with exponential functions. I know that the derivative of $e^x$ is $e^x$, but wolfram alpha shows a different answer to my function below. If you, for example, take the derivative of $e^{-2x}$ do you get $-2e^{-2x}$ or $e^{-2x}$?
| You have to use the chain rule here. Writing $f(x) = e^x$ and $g(x) = -2x$ we have $h(x) := f(g(x)) = e^{-2x}$, hence by the chain rule
$$ h'(x) = f'(g(x))g'(x) $$
Now $f'(x) = e^x$, hence $f'(g(x)) = e^{-2x}$, and $g'(x) = -2$, this gives
$$ h'(x) = f'(g(x))g'(x) = e^{-2x} \cdot (-2) = -2e^{-2x} $$
| {
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"url": "https://math.stackexchange.com/questions/859936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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For which values of $a, b$ does the system of equations not have any solutions? I am trying to solve the following problem:
For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution?
$$ \left[\begin{array}{ccc|c}
1&-2&3&-4\\
2&1&1&2\\
1&a&2&-b
... | By row reduction the system becomes (if I didnt make a mistake (highly likely))
$$\begin{pmatrix}
1&0&1&0\\
0&1&-1&2\\
0&0&a+1&-b-2a\\
\end{pmatrix}$$
In order for the rank to be less than $3$ we need that $a+1=0$, so $a=-1$ for no solution we then need $-b-2a \neq 0$ so $b\neq 2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/860018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Value of $\int_{0}^{1}\dfrac{\log x}{1-x}dx$. What is my wrong step? I would like to evaluate the value of this integral:
$$I=\int_{0}^{1}\dfrac{\log x}{1-x}dx.$$
On one hand, I proceed using integration by parts as follows:
$$I=\int_{0}^{1}f(x)g'(x)dx,$$
where $f(x)=\log x$ and $g'(x)=\dfrac{1}{1-x}$. From this, I c... | Don't use integration by parts straight away. Instead, expand the denominator: $\sum_{k=0}^{\infty} x^k$ because the bounds on $x$ are strictly between $0$ and $1$. After this, interchange integration and summation (due to uniform convergence and you'll get integrals of the form $\int_{0}^{1} x^k \log x dx$. Now use in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $ This integral looks simple, but it appears that its not so.
All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
| Let $x=\sinh^2 u$. (This is the same transformation as CountIblis used, but I'll employ it slightly differently.) Observe that $dx=2\cosh u \sinh u \, du$ and $$x+x^2=\sinh^2 u+\sinh^4 u=\sinh^2 u(1+\sinh^2 u)=\sinh^2 \cosh^2 u$$ since $\cosh^2 u-\sinh^2 u=1$.
Therefore \begin{align}
\int_0^1 x^2 \sqrt{x+x^2}\,dx
&=\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Example of a bijection from the set of real numbers to a subset of irrationals I need an example of a bijection from the set of real numbers to a subset of the irrationals.
I tried something like
$f(x)=x+\sqrt{2}$,
but where should I map $-\sqrt{2}$?
| Let $f(x) = \dfrac{\arctan x}{\pi}$, so $f^{-1}(x) = \tan \pi x$. $f$ maps $\mathbb{R}$ to $(-\dfrac12,\dfrac12)$.
$$g(x) = \begin{cases}
x \in \mathbb{Q} & x + \sqrt{5}\\
x \notin \mathbb{Q} & x\\
\end{cases}$$
$$g^{-1}(x) = \begin{cases}
x > 1 & x - \sqrt{5}\\
x \le 1 & x\\
\end{cases}$$
So $g \circ f$ maps from $\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Evaluate the integral $\int_{-2}^{2} \frac{1+x^2}{1+2^x}dx$ My friend asked me ot evaluate the integral:
$$\int_{-2}^{2} \frac{1+x^2}{1+2^x}dx$$
And he gave me the hint: substitute $u = -x$. And so I did that, but I can't seem to get any farther than that. Could someone please provide some hints and help as to how to e... | Substituting yields $$\int_{2}^{-2}-\frac{1+x^2}{1+2^{-x}},$$ which we can add to the original integral to get $$\int_{-2}^{2} \frac{1+x^2}{1+2^x}dx + \int_{2}^{-2}-\frac{1+x^2}{1+2^{-x}} = \int_{-2}^{2}\frac{1+x^2}{1+2^x} + \frac{1+x^2}{1+2^{-x}} = \int_{-2}^{2}1+x^2 = \frac{28}{3},$$ so our original integral is half ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Basis in the vector space of all polynomials Let $V$ vector space of all polynomials $p(t) = a_0 + a_1t + \cdots + a_nt^n$,$\forall n \in\mathbb{N}$ and $a_0,\ldots,a_n \in\mathbb{R}$. How can I prove that $ \gamma = \{1,t,t^2,\ldots\}$ is a basis of $V$, and use it to find a linear transformation $T:V \rightarrow V$... | You seem to have a fundamental misunderstanding about this question.
$T$ is a transformation from the set of polynomials on $t$ to the set of polynomials on $t$. So, the input to $T$ should be a polynomial, and the output should be some other polynomial. Two common linear transformations are differentiation and integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Has the Gödel sentence been explicitly produced? I do not pretend to know much about mathematical logic. But my curiosity was piqued when I read Hofstadter's Gödel, Escher, Bach, which tries to explain the proof of Gödel's first incompleteness theorem by using an invented formal system called TNT--Typographical Number... | I have once played around with this stuff myself and obtained this example of such a sentence. The long thing at the end of that page (there are in fact two long things, just minor variants if I recall it correctly).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/860603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $? I would be interest to show :
if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $ ?
my second question that's make me a problem is that :
what is :$ f^{-1}(\pi) $ ?
I would be interest for any replies or any comments .
| There is something questionable in the wording (about the bounds of the integral). So, two interpretations are presented below :
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/860690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
How to show that $ \sum_{n = 0}^{\infty} \frac {1}{n!} = e$?
How to show that $\sum\limits_{n = 0}^{\infty} \frac {1}{n!} = e$ where $e = \lim\limits_{n\to\infty} \left({1 + \frac 1 n}\right)^n$?
I'm guessing this can be done using the Squeeze Theorem by applying the AM-GM inequality. But I can only get the lower bo... | Let $a_n = \left(1+ 1/n\right)^n$. By the binomial theorem,
$$a_n = 1 + 1 + \frac1{2!}\left(1- \frac1{n}\right)+ \ldots +\frac1{n!}\left(1- \frac1{n}\right)\ldots\left(1- \frac{n-1}{n}\right)\leq \sum_{k=0}^{\infty}\frac1{k!}=e,$$
The sequence $a_n$ is increasing and bounded, so it converges.
Hence, $$\lim_{n\rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 0
} |
Distributing Set Intersections Over an Intersection I was working through some examples, and found this to be true:
$(A \cap B) \cap (B \cap C) = A \cap B \cap C $
$(A \cap B) \cap(A \cap C) = A \cap B \cap C$
$(A \cap B) \cap(A \cap C) \cap (B \cap C) = A \cap B \cap C$
Is there some proof, or rule for these statemen... | Of course you can prove them, starting from the definition of intersection of sets :
$x \in A \cap B$ iff $x \in A$ and $x \in B$
and the "basic fact" that equality between sets amounts to mutual inclusion : $A = B$ iff $A \subseteq B$ and $B \subseteq A$.
In turn, to prove inclusion you have to use its definition :
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Continous surjective map from $S^1$ to $S^n$ Is there any continous surjective map from $S^1$ or $[0,1]$ onto $S^n$, for some $n\geq 2$.
Thank you.
| Yes, there is. Start with a space filling curve $\gamma \colon [0,1] \to [0,1]^2$. Induction gives you a continuous onto $\gamma_n \colon [0,1] \to [0,1]^{n}$. Identifying the boundary to one point is a quotient map $\pi \colon [0,1]^n \to S^n$, together with $\gamma_n$, we have $\pi \circ\gamma_n \colon [0,1]\to S^n$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Some "Product" of Positive Definite Matrices I could remember that if $A,B$ are two positive definite matrices, then $(a_{ij}b_{ij})$ is positive definite also. But I could not see how to prove it then.
| Edit: Following the remark of user126154, I suppose here that the two matrix $A$ and $B$ are symmetric.
Let $x=(x_1,\cdots,x_n)\in \mathbb{R}^n=E$ and put $q_A(x)=\sum_{i,j}a_{i,j}x_ix_j$. As $A$ is positive definite, there exists $n$ independant linear forms $\displaystyle T_k(x)=\sum_{l=1}^n \alpha_{k,l}x_l$ such tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$A+A^2B+B=0$ implies $A^2+I$ invertible? Let $A$ and $B$ be two square matrices over a field such that $A+A^2B+B=0$.
Is it true that $A^2+I$ is always invertible ?
| We have $A+(A^2+I)B=0$. We multiply by $A$:
$$A^2+(A^2+I)BA=0$$
We add $I$:
$$I+A^{2}+(A^2+I)BA=I=(A^2+I)(I+BA)$$
Hence $A^2+I$ is invertible, and its inverse is $I+BA$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/861181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Can any collection of open sets in $\mathbb{R}$ be covered by a countable subcollection? Let $A$ be a collection of open sets in $\mathbb{R}$. is there a countable subcollection $G_i$ of $A$ such that $$\cup_{G\in A} G=\cup_{i=1}^\infty G_i$$
I guess there must be such subcollection, but I don't know how to establish i... | For each open interval $(a,b)$ with rational endpoints, if there is some $G\in A$ with $(a,b)\subseteq G$, then pick one such $G$. As there are only countably many rational intervals, you'll pick only countably many $G$'s. I claim that their union equals the union of all the original $G$'s. To see this, consider any $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Harmonic functions and polar differential forms Given a harmonic function $u$, its differential and conjugate differential are
$$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy,\qquad ^{*}du = -\frac{\partial u}{\partial y}dx + \frac{\partial u}{\partial x}dy.$$ We also know that Laplace's equati... |
Also, he states that the form of $∗du$ given holds for a circle $|z|=r$, if that makes a difference.
Oh yes, it does make a difference. This is why context matters. A differential form is a device that eats vectors and produces numbers. For example, $du$ is the form that takes a vector $\vec a$ and returns the direc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Independence of $X$ and $2X$ Are these two random variables independent? Unfortunately, I don't know probability theory enough to answer this question. I know for a fact that if $X$ and $Y$ are independent random variables and $g$,$h$ are measurable functions, then $g(X)$ and $h(Y)$ are independent as well. However, I ... | The formula for the mgf of a sum $X+Y$ is correct when $X$ and $Y$ are independent.
Apart from a few degenerate examples, $X$ and $2X$ are not independent.
For instance, toss a fair coin, and let $X=1$ if we get a head, and $X=0$ otherwise.
Then $\Pr(X=1\cap 2X=0)=0$. But $\Pr(X=1)\nee 0$, and $\Pr(2X=0)\ne 0$, so
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
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