Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding angles from some other angles related to incircle Let $ABC$ be a triangle and $O$ the center of its enscribed circle.
Let $M = BO \cap AC$ and $N=CO \cap AB$ such that $\measuredangle NMB = 30°, \measuredangle MNC = 50°$.
Find $\angle ABC, \angle BCA$ and $\angle CAB$.
I also posted this here at the Art of Pro... | At first, I took $O$ to denote the center of the incircle inscribed into the triangle. Only a comment below clarified that you were actually taling about the center of the circumcircle circumscribed around the triangle. Therefore, I have two solutions, one for each interpretation.
Circumcircle
Angles without a proof
I ... | {
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"timestamp": "2023-03-29T00:00:00",
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Stone Weierstrass on noncompact subset I would like to ask whether we can create a function $f:\mathbb{R}\rightarrow\mathbb{R}$ which is continuous on $\mathbb{R}$ but $f$ is not the pointwise limit of any sequence of polynomial function $\{p_n(x)\}_{n\in\mathbb{N}}$.
Thank you for all helping.
| By Stone-Weierstass applied to $f$ restricted to interval $[-n,n]$ there is a polynomial $p_n$ such that $\sup_{x\in [-n,n]} |f(x)-p_n(x)|< \frac{1}{n}$. The sequence $(p_n)$ converges pointwise to $f$.
On the other hand, there is no sequence of polynomials converging uniformly to $f(x)=\sin x$, since polynomials are u... | {
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Closed subsets $A,B\subset\mathbb{R}^2$ so that $A+B$ is not closed I am looking for closed subsets $A,B\subset\mathbb{R}^2$ so that $A+B$ is not closed.
I define
$A+B=\{a+b:a\in A,b\in B\}$
I thought of this example, but it is only in $\mathbb{R}$. Take:
$A=\{\frac{1}{n}:n\in\mathbb{Z^+}\}\cup\{0\}$ and $B=\mathbb{Z}$... | Let $A=\mathbb{Z}$ and $B=p\mathbb{Z}:=\{pn: n\in\mathbb{Z}\}$ where $p$ is any irrational number.
So $A$ and $B$ are two closed subsets of $\mathbb{R}$,
but, $A+B :=\{m+pn: m,n\in\mathbb{Z}\}$ is not closed in $\mathbb{R}$.
| {
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Centroids of triangle On the outside of triangle ABC construct equilateral triangles $ABC_1,BCA_1, CAB_1$, and inside of ABC, construct equilateral triangles $ABC_2,BCA_2, CAB_2$. Let $G_1,G_2,G_3$ $G_3,G_4,G_6$be respectively the centroids of triangles $ABC_1,BCA_1, CAB_1$, $ABC_2,BCA_2, CAB_2$.
Prove that the centroi... | The easiest way to get at the result is to apply an affine transformation to the original triangle so as to make it equilateral. The constructed "interior" triangles will all then coincide with the original triangle, and the conclusion can be easily drawn using basic geometry.
| {
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A question about the definition of fibre bundle The canonical definition of fibre bundle is the following:
Let $B,X,F$ be three topological spaces and $\pi:X\rightarrow B$ a continuous surjective map; then $(X,F,B,\pi)$ is a fibre bundle on $B$ if for all $b\in B$ exist an open neighbourhood of $U$ of $b$ and a homeo... | If $\phi_U$ is a homeomorphism, then $\phi_U$ restricted to $\pi^{-1}(p)$ will also be a homeomorphism. But this is just a map from $\pi^{-1}(p)$ to $p\times F$ and $p\times F$ is homeomorphic to $F$.
| {
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How do I prove this using the definition of a derivative? If $f$ is a differentiable function and $g(x)=xf(x)$, use the definition of a derivative to show that $g'(x)=xf'(x)+f(x)$.
| Just do it: set up the difference quotient and take the limit as $h\to 0$. I’ll get you started:
$$\begin{align*}
g'(x)&=\lim_{h\to 0}\frac{g(x+h)-g(x)}h\\
&=\lim_{h\to 0}\frac{(x+h)f(x+h)-xf(x)}h\\
&=\lim_{h\to 0}\frac{x\big(f(x+h)-f(x)\big)+hf(x+h)}h\\
&=\lim_{h\to 0}\frac{x\big(f(x+h)-f(x)\big)}h+\lim_{h\to 0}\frac{... | {
"language": "en",
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Prove $ax - x\log(x)$ is convex? How do you prove a function like $ax - x\log(x)$ is convex? The definition doesn't seem to work easily due to the non-linearity of the log function.
Any ideas?
| A function $f(x) \in C^2(\Omega)$ is convex if its second derivative is non-negative.
$$f(x) = x \log(x) \implies f'(x) = x \cdot \dfrac1x + \log(x) \implies f''(x) = \dfrac1x > 0$$
EDIT
If $f(x) = ax - x\log(x)$, then
$f''(x) = - \dfrac1x$ and hence the function is concave.
| {
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A continuous function with measurable domain.
Let $D$ and $E$ be measurable sets and $f$ a function with domain $D\cup E$. We proved that $f$ is measurable on $D \cup E$ if and only if its restrictions to $D$ and $E$ are measurable. Is the same true if measurable is replaced by continuous.
I wrote the question stra... | That's wrong. Let $f\colon[0,1] \to \mathbb R$ be given by
\[
x\mapsto \begin{cases} 1 & x = 0\\ 0 & x > 0
\end{cases} \]
Then $f$ is not continuous, but it is, restricted to $D := \{0\}$ and $E := (0,1]$, both of which are measurable.
Let me add something concerning your edit: As the example above shows, for just m... | {
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Number of songs sung. There were 750 people when the first song was sung. After each song, 50 people are leaving the hall. How many songs are sung to make them zero?
The answer is 16, I am unable to understand it. I am getting 15 as the answer. Please explain.
| Listen to the question carefully: "There were $750$ people when the first song was sung"
First song was already sung $+1$;
remaining people is $750$.
$750/50=15$;
Answer is $15+1=16$
| {
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"question_score": "2",
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"answer_id": 2
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Write down a proof for $\bot\Rightarrow q$ in proposition calculus I am given the hint in the question that I will need to use the axiom $(((s\Rightarrow \bot)\Rightarrow \bot)\Rightarrow s)$.
The axioms I am using are $$(s\Rightarrow (t \Rightarrow s)) \\((s\Rightarrow(t\Rightarrow u))\Rightarrow((s\Rightarrow t)\Righ... | Using your first two axioms you can prove the deduction theorem. So to prove $\vdash \bot \Rightarrow q$, it's enough to prove $\bot \vdash q$. The hint suggests using the third axiom. With that, you can show that $((q \Rightarrow \bot)\Rightarrow \bot)\vdash q$. So you're done if you can prove $\bot \vdash ((q \Righta... | {
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Find the value of f(343, 56)? I have got a problem and I am unable to think how to proceed.
$a$ and $b$ are natural numbers. Let $f(a, b)$ be the number of cells that the line joining $(a, b)$ to $(0, 0)$ cuts in the region $0 ≤ x ≤ a$ and $0 ≤ y ≤ b$. For example $f(1, 1)$ is $1$ because the line joining $(1, 1)$ and... | A start: Note that $343=7\cdot 49$, and $56=7\cdot 8$. First find $f(49,8)$.
More: If $a$ and $b$ are relatively prime, draw an $a\times b$ chessboard, and think of the chessboard squares you travel through as you go from the beginning to the end.
| {
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Show that $f$ has at most one fixed point Let $f\colon\mathbb{R}\to\mathbb{R}$ be a differentiable function. $x\in\mathbb{R}$ is a fixed point of $f$ if $f(x)=x$. Show that if $f'(t)\neq 1\;\forall\;t\in\mathbb{R}$, then $f$ has at most one fixed point.
My biggest problem with this is that it doesn't seem to be true. ... | This problem is straight out of baby Rudin.
Assume by contradiction that $f$ has more than one fixed point. Select any two distinct fixed points, say, $x$ and $y$.
Then, $f(x) = x$ and $f(y) = y$. By the Mean Value Theorem, there exists some $\alpha \in (x,y)$ such that $f'(\alpha) = \frac{f(x)-f(y)}{x-y} = \frac{x-y}{... | {
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Reference for topology and fiber bundle I am looking for an introductory book that explains the relations of topology and bundles.
I know a basic topology and algebraic topology. But I don't know much about bundles. I want a book that
*
*explains the definition of bundles carefully and give some intuition on bundle... | Steenrod's "The Topology Of Fibre Bundles" is a classic. It isn't particularly modern but it does the basics very well.
Husemoller's "Fibre Bundles" is a bit more modern and has a bit more of a physics-y outlook but still very much a book for mathematicians. I find it not as pleasant to read as Steenrod's book but i... | {
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If $\#A$ and $m$ are relatively prime, then $a\mapsto ma$ is automorphism? Is it true if $A$ is a finite, abelian group and $m$ is some integer relatively prime to the order of $A$, then the map $a\mapsto ma$ is an automorphism?
It's left as an exercise in some course notes, but I cannot verify it.
In particular it is ... | Hint: since $\gcd(m,\#A)=1$, you can find an integer $d$ such that $dm\equiv1\pmod{\#A}$. Show that the map $a \mapsto da$ is the inverse of $a \mapsto ma$.
| {
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How do we draw the number hierarchy from natural to complex in a Venn diagram? I want to make a Venn diagram that shows the complete number hierarchy from the smallest (natural number) to the largest (complex number). It must include natural, integer, rational, irrational, real and complex numbers.
How do we draw the ... | Emmad's second link is just perfect, IMHO. For something right in front of you, here's this:
| {
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Jordan Form of a matrix I'm trying to find a matrix $P$ such that $J=P^{-1}AP$, where $J$ is the Jordan Form of the matrix:
$$A=\begin{pmatrix}
-1&2&2\\
-3&4&3\\
1&-1&0
\end{pmatrix}
$$
The characteristic polynomial is: $p(\lambda)=(\lambda-1)^3$, and a eigenvector for $A-I$ is $\begin{pmatrix}
0 \\ 1 \\-1 \end{pmatrix... | Maybe your fault is, that there is a second eigenvector, the Jordan normal form is $$\begin{pmatrix}1&0&0\\0&1&1\\0&0&1\end{pmatrix}.$$
You can find a second eigenvector $w$ and a vector $z$ such that $(A-I)z=w$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Continuity of function which is Lipschitz with respect to each variables separately Let a function $f: I\times J \rightarrow \mathbb R$, where $I,J$ are intervals in $\mathbb R$, be Lipschitz with respect of each variable separately. Is it then $f$ continuous with respect of both variables?
Thanks
| It depends on what you mean with Lipschitz with respect to each variable separately.
Consider the function $f\colon\mathbb R^2\to \mathbb R$, $(x,y)\mapsto xy$.
Then for fixed $y$, the function $x\mapsto f(x,y)$ is Lipschitz (with Lipschitz constant $y$ depending on $y$) and similarly $y\mapsto f(x,y)$ for fixed $x$.
H... | {
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Let $G=(V,E)$ be a connected graph with $|E|=17$ and for all vertices $\deg(v)>3$. What is the maximum value of $|V|$? Let $G=(V,E)$ be a connected graph with $|E|=17$ and for all vertices $\deg(v)>3$. What is the maximum value of $|V|$? (What is the maximum possible number of vertices?)
| HINT: Suppose that $V=\{v_1,\dots,v_n\}$. Then $$\sum_{k=1}^n\deg(v_k)=34\;;\tag{1}$$ why?
If $\deg(v_k)\ge 4$ for $k=1,\dots,n$, then $$\sum_{k=1}^n\deg(v_k)\ge\sum_{k=1}^n4\;.\tag{2}$$ Now combine $(1)$ and $(2)$.
| {
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showing $\nexists\;\beta\in\mathbb N:\alpha<\beta<\alpha+1$ I want to prove that $\nexists\; \beta\in\mathbb N$ such that $\alpha<\beta<\alpha+1$ for all $\alpha\in\mathbb N$. I just want to use the Peano axioms and $+$ and $\cdot$
If $\alpha<\beta$ then there is a $\gamma\in\mathbb N$ such that $\beta=\alpha+\gamma$.
... | I'm not sure if the following is allowed but:
Inserting the first equation in the second we get:
$\alpha+1=\alpha+\gamma+\delta$.
Now we can substract $\alpha$ at both sides to get $1=\gamma+\delta$, which is a contradiction.
| {
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How do you find the vertex of a (Bézier) quadratic curve? Before I elaborate, I do not mean a quadratic function! I mean a quadratic curve as seen here.
With these curves, you are given 3 points: the starting point, the control point, and the ending point. I need to know how to find the vertex of the curve. Also, I am ... | A nice question! And it has a nice answer. I arrived at it by a series of hand-drawn sketches and scribbled calculations, so I don't have time right now to present the derivation. But here is the answer:
We are given three point $P_0$, $P_1$, and $P_2$ (the start-, control-, and end-points). The Bézier curve for these ... | {
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Proof needed for a function $x^2$ for irrationals and $x$ for rationals How can I prove that the function $f:[0,1] \rightarrow \mathbb{R}$, defined as
$$ f(x) = \left\{\begin{array}{l l} x &\text{if }x \in \mathbb{Q} \\ x^2 & \text{if } x \notin \mathbb{Q}
\end{array} \right. $$
is continuous on $0$ and $1... | Hint: Choose a sequence of rationals converging to an irrational and vice-versa and recall
that continuity also implies sequential continuity, to conclude what you want.
Move your mouse over the gray area for a complete solution.
Consider $a \in [0,1] \backslash \mathbb{Q}$. For this $a$, choose a sequence of rational... | {
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A question about infinite utility streams At the end of Diamond's Evaluation of Infinite Utility Streams he proves a theorem (which he doesn't give a name to, but it's at the very end of the article). There is a step in which he jumps from $(u,0)_{rep}\succ (0,u)_{rep}$ to $(u,0)\succ_t (0,u)$, and I don't understand w... | Step: $(u,0)_{rep}\succ (0,u)_{rep}\implies (u,0)\succ_2(0,u)$
Proof: Suppose not. Since $\succeq_2$ is complete, we would have $(0,u)\succeq_2 (u,0)$ otherwise, and hence by A2 $(0,u)_{rep}\succeq (u,0)_{rep}$. This cannot be.
I don't see how the rest of Diamond's proof works out though. $(u,0,U)\succeq(0,u,U)$ follow... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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All possible combinations of x letters (what is this called in mathematics) Firstly, thank you for looking at my question.
I would like to know what this kind of problem is called in mathematics:
Given a set of letters, find all possible 'words' you can make with those letters. For example for 'abc' the solution would ... | All possible outcomes of a probability problem are called the "sample space". Is that what you were looking for?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is intersection of two independent set probability a multiplication process? Why is the probability of intersection of two independent sets $A$ and $B$, a multiplication of their respective probabilities i.e. Why is
$$\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)?$$
this question is about the intuition b... | Perhaps, one way of looking at it is the fact that intersection means to add conditions. In that sense, for every individual satisfying the condition A you have to compute how many satisfy condition B, thus the multiplication issue. Note that satisfying two conditions is scarcer than satisfying just one, so the "weight... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/217815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 10,
"answer_id": 3
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Benford's law with random integers I tried testing random integers for compliance with Benford's law, which they are apparently supposed to do. However, when I try doing this with Python,
map(lambda x:str(x)[0], [random.randint(0, 10000) for a in range(100000)]).count('1')
I get approximately equal frequencies for al... | Benford's law applies only to distributions that are scale-invariant and thus applies approximately to many real-life data sources, especially when we measure with arbitrary units: If the leading-digit distribution of a sample is essentially the same whether we measure in inches or centimeters, this is only possible if... | {
"language": "en",
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Triple integration in cylindrical coordinates Determine the value of $ \int_{0}^{2} \int_{0}^{\sqrt{2x - x^2}} \int_{0}^{1} z \sqrt{x^2 +y^2} dz\,dy\,dx $
My attempt: So in cylindrical coordinates, the integrand is simply $ \rho$.
$\sqrt{2x-x^2} $ is a circle of centre (1,0) in the xy plane. So $ x^2 + y^2 = 2x => \rho... | As joriki noted completely; your integral would be $$ \int_{0}^{\frac{\pi}{2}} \int_{0}^{2\cos\theta} \int_{0}^{1} z\,\,\rho^2\,dz\,d\rho\,d\theta=\bigg(\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\cos\theta}\rho^2\,d\rho\,d\theta \bigg)\times \int_{0}^{1} z\ dz\\\ =\bigg(\int_{0}^{\frac{\pi}{2}} \frac{\rho^3}{3}\bigg|_0^{2\c... | {
"language": "en",
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Constructing Riemann surfaces using the covering spaces In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow:
A polynomail-like map of degree d is a triple $(U,U',f)$ where $U$ and $U'$ are open subse... | I will give an informal answer. Your covering space is just a collection of holed spaces right? The equivalence relation just projects the holed space down onto a space isomorphic to $U'-L$. It does this by pasting the image and the preimage of $f$ together. (In an informal sense with each iteration the covering spac... | {
"language": "en",
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Relations between p norms The $p$-norm on $\mathbb R^n$ is given by $\|x\|_{p}=\big(\sum_{k=1}^n
|x_{k}|^p\big)^{1/p}$. For $0 < p < q$ it can be shown that $\|x\|_p\geq\|x\|_q$ (1, 2). It appears that in $\mathbb{R}^n$ a number of opposite inequalities can also be obtained. In fact, since all norms in a finite-dimensi... | *
*Using Cauchy–Schwarz inequality we get for all $x\in\mathbb{R}^n$
$$
\Vert x\Vert_1=
\sum\limits_{i=1}^n|x_i|=
\sum\limits_{i=1}^n|x_i|\cdot 1\leq
\left(\sum\limits_{i=1}^n|x_i|^2\right)^{1/2}\left(\sum\limits_{i=1}^n 1^2\right)^{1/2}=
\sqrt{n}\Vert x\Vert_2
$$
*Such a bound does exist. Recall Hölder's inequality ... | {
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$X_n\overset{\mathcal{D}}{\rightarrow}X$, $Y_n\overset{\mathbb{P}}{\rightarrow}Y \implies X_n\cdot Y_n\overset{\mathcal{D}}{\rightarrow}X\cdot Y\ ?$ The title says it. I know that if limiting variable $Y$ is constant a.s. (so that $\mathbb{P}(Y=c)=1)$ then the convergence in probability is equivalent to the convergence... | Let $Y$ represent a fair coin with sides valued $0$ (zero) and $1$ (one). Set $Y_n = Y$, $X = Y$, $X_n = 1-Y$. The premise is fulfilled, but $X_n\cdot Y_n = 0\overset{\mathcal{D}}{\nrightarrow}Y = X\cdot Y$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate congruences with non-prime divisor with Fermat's Little Theorem I can evaluate $ 17^{2012}\bmod13$ with Fermat's little theorem because $13$ is a prime number. (Fermat's Little theorem says $a^{p-1}\bmod p\equiv1$.)
But what if when I need to evaluate for example $12^{1729}\bmod 36$? in this case, $36$ is not ... | Your example is slightly trivial, because already $12^2\equiv0\bmod36$. If the base and modulus were coprime, you could use Euler's theorem. In cases in between, where the base contains some but not all factors of the modulus, you can reduce by the common factors and then apply Euler's theorem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Showing $\mathbb{Z}_6$ is an injective module over itself I want to show that $\mathbb{Z_{6}}$ is an injective module over itself. I was thinking in using Baer's criterion but not sure how to apply it. So it suffices to look at non-trivial ideals, the non-trivial ideals of $\mathbb{Z_{6}}$ are:
(1) $I=\{0,3\}$
(2) $J=... | I found a solution for the 'general' case:
Let I be a ideal of $\mathbb{Z}/n\mathbb{Z}$ then we know that $I=\langle \overline{k} \rangle$ for some $k$ such that $k\mid n$.
If $f:I\rightarrow \mathbb{Z}/n \mathbb{Z}$ is a $\mathbb{Z}/n \mathbb{Z}$-morphism then $im f\subset I$. To show this we note that if $\overline{x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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About the homework of differentiation This problem is not solved.
$$
\begin{align}
f(x) &=\log\ \sqrt{\frac{1+\sqrt{2}x +x^2}{1-\sqrt{2}x +x^2}}+\tan^{-1}\left(\frac{\sqrt{2}x}{1-x^2}\right) \cr
\frac{df}{dx}&=\mathord?
\end{align}
$$
| The answer is $$\frac{2\sqrt{2}}{1+x^4}$$
Hints: $$\frac{d\left(\log(1+x^2\pm\sqrt{2}x\right)}{dx}=\frac{2x\pm\sqrt{2}}{1+x^2\pm\sqrt{2}x}$$ and $$\left(1+x^2+\sqrt{2}x\right)\left(1+x^2-\sqrt{2}x\right)=\left(1+x^2\right)^2-\left(\sqrt{2}x\right)^2$$
Similarly,
$$\frac{d\left(\tan^{-1}\left(\frac{\sqrt{2}x}{1-x^2}\ri... | {
"language": "en",
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How does $\sqrt{|e^{-y}\cos x + ie^{-y}\sin x|}= e^{-y}$ How does $\sqrt{|e^{-y}\cos x + ie^{-y}\sin x|} = e^{-y}$ which is less than $1$?
This is a step from a question I am doing but I am not sure how the square root equaled & $e^{-y}$
| I'd start by using properties of the modulus:
$$|e^{-y} \cos x + i e^{-y} \sin x|=|e^{-y}||\cos x + i \sin x|=e^{-y}|e^{ix}|=e^{-y}$$
| {
"language": "en",
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Linear algebra: finding a Tikhonov regularizer matrix A more general soft constraint is the Tikhonov regularization constraint
$$
\mathbf{w}^\text{T}\Gamma^\text{T}\Gamma\mathbf{w} \leq C
$$
which can capture relationships among the $w_i$ (the matrix $\Gamma$ is the Tikhonov regularizer).
(a) What should $\Gamma$ be to... | (a) You are right, in order to obtain $\mathbf{w}^T\Gamma^T \Gamma \mathbf{w}=\sum_{q=0}^Q w_q^2$, you should use $\Gamma=I$, where $I$ is the identity matrix.
(b) Be careful with your dimensions, $\mathbf{w}\mathbf{w}$ is not defined. You are trying to multiply a $Q\times 1$ matrix by a $Q\times 1$ matrix, which is no... | {
"language": "en",
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Why the spectral theorem is named "spectral theorem"? "If $V$ is a complex inner product space and $T\in \mathcal{L}(V)$. Then $V$ has an orthonormal basis Consisting of eigenvectors of T if and only if $T$ is normal".
I know that the set of orthonormal vectors is called the "spectrum" and I guess that's where the na... | I think the top voted answer is very helpful:
Since the theory is about eigenvalues of linear operators, and Heisenberg and other physicists related the spectral lines seen with prisms
It might be worth explaining a bit more precisely how the theory is related to optics, especially for people not familiar with how a ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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harmonic conjugates and cauchy riemann eqns I'm trying to find function $v(x,y)$ such that the pair $(u,v)$ satisfies the Cauchy-Riemann equations for the following functions $u(x,y)$:
a) $u = \log(x^2+y^2)$
$$
u_x = v_y \Rightarrow \frac{2x}{x^2+y^2} = v_y \Rightarrow v = \frac{2xy}{x^2+y^2}?
$$
b) $u = \sin x \cosh ... | Here's how to find the corresponding imaginary parts by educated guessing. Maybe not the most systematic method, but maybe it improves your intuition about how these things behave.
For the first, remember that $e^{a + ib} = e^a(\cos b + i\sin b)$ (assuming $a,b \in \mathbb{R}$). In other words, $e^z$ sort of maps from ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Ratio of Boys and Girls In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same)
| Obviously the ratio of boys to girls could be any rational number, or infinite. If you either fix the number of families or, more generally, specify a probability distribution over the number of families, then B/G is a random variable with infinite expected value (because there's always some non-zero chance that G=0)... | {
"language": "en",
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Smooth function on $\mathbb R$ whose small increments are not controlled by the first derivative at infinity I need some help in finding a (as simple as possible) smooth function $f:\mathbb R \rightarrow \mathbb R$ which does NOT satisfy the following:
There exist a constant $C>0$, a compact $K\subset\mathbb R$ and $h_... | Try e.g. $f(x) = \cos(x)$. All you need is that $f$ is not constant on any interval and $f'$ has arbitrarily large zeros.
EDIT: With the new condition, take $$f(x) = \int_0^x (1 + t^2 \cos^2(t^2))\ dt = x+\frac{x \sin \left( 2\,{x}^{2} \right)}{8}-\frac{\sqrt {\pi }}{16}{\rm
FresnelS} \left( 2\,{\frac {x}{\sqrt {\pi ... | {
"language": "en",
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"source": "stackexchange",
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Question about exponential functions (easy one) Can someone please explain me, why when we are looking at the function $f(x)=a^x $ , we should remember that $1 \neq a >0 $ ? (And not saying that we can't put an x that satisfies:
$ 0 < x < 1 $ ?
Any understandable explanation will be great!
Thanks !
| Besides to @André's theoretical answer(+1); imagine what would be happen if $a<0$? For example if $a=-4$, what would you do with $(-4)^{\frac{m}{n}}$ wherein $n$ is even and $m$ is odd? Would we have a real number? How many fractions of this type are there in $\mathbb R$? I see the close story can be when $a=1$.
| {
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Continued Fractions Approximation I have come across continued fractions approximation but I am unsure what the steps are.
For example How would you express the following rational function in continued-fraction form:
$${x^2+3x+2 \over x^2-x+1}$$
| This might be what you are looking for:
$$
\begin{align}
\frac{x^2+3x+2}{x^2-x+1}
&=1+\cfrac{4x+1}{x^2-x+1}\\[4pt]
&=1+\cfrac1{\frac14x-\frac5{16}+\cfrac{\frac{21}{16}}{4x+1}}\\[4pt]
&=1+\cfrac1{\frac14x-\frac5{16}+\cfrac1{\frac{64}{21}x+\frac{16}{21}}}
\end{align}
$$
At each stage, we are doing a polynomial division i... | {
"language": "en",
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Showing that $f$ is differentiable at $x=0$, using the mean value theorem The following exercise is again about the Mean Value Theorem :)
Let $f : [0,1] \rightarrow \mathbb{R}$ be continuous and differentiable on $(0,1)$.
Assume that $$ \lim_{x\rightarrow 0^+} f'(x)= \lambda.$$
Show that $f$ is differentiable (from ... | Being differentiable at $0$ from the right means $\displaystyle{\lim_{x \to 0^+} \frac{f(x)-f(0)}{x-\alpha}}$ exist. Its value is $f'_+(0)$.
From the Mean Value Theorem, for each $x>0$ there exists a $y$ with $0<y<x$, such that $$f'(y)=\lambda=\frac{f(x)-f(0)}{x} = \frac{f(x)-f(0)}{x-0}$$
This implies that:$$\display... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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functional analysis complementary subspace Let $Y$ and $Z$ be closed subspaces in a Banach space $X$. Show that each $x \in X$ has a unique decomposition $x = y + z$, $y\in Y$, $z\in Z$ iff $Y + Z = X$ and $Y\cap Z = \{0\}$. Show in this case that there is a constant $\alpha>0$ such that $ǁyǁ + ǁzǁ \leq\alphaǁxǁ$ for e... | Hint: Assume that $Y\cap Z=\{0\}$. If $z+y=x=z'+y'$ s.t. $z,z'\in Z$, $y,y'\in Y$ then $z-z'=y'-y$.
For the other direction, if $y\in Y\cap Z$ then $0+0=y+(-y)$.
| {
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Calculus 1- Find directly the derivative of a function f. The following limit represents the derivative of a function $f$ at a point $a$. Evaluate the limit.
$$\lim\limits_{h \to 0 } \frac{\sin^2\left(\frac\pi 4+h \right)-\frac 1 2} h$$
| Let $f(x)=\sin^2x$. We have, $f^{\prime}(x)=2\sin x\cos x$. In the other hand
\begin{equation}
\begin{array}{lll}
\lim_{h\rightarrow 0}\frac{\sin^2\left(\frac{\pi}{4}+h\right)-\frac{1}{2}}{h}&=&\lim_{h\rightarrow 0}\frac{f\left(\frac{\pi}{4}+h\right)-f\left(\frac{\pi}{4}\right)}{h}\\
&=&f^{\prime}(\pi/2)\\
&=&2\sin(\pi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Discrete Subgroups of $\mbox{Isom}(X)$ and orbits Let $X$ be a metric space, and let $G$ be a discrete subgroup of $\mbox{Isom}(X)$ in the compact-open topology. Fix $x \in X$. If $X$ is a proper metric space, it's not hard to show using Arzela-Ascoli that $Gx$ is discrete. However, is there an easy example of a metric... | Let $X = \mathbb{R}^2$ with the following metric: $$d((x_1,y_1),(x_2,y_2)) = \begin{cases} |y_1-y_2| & \text{if $x_1=x_2$,} \\ |y_1|+|y_2|+|x_1-x_2| & \text{if $x_1\ne x_2$}. \end{cases}$$ I don't know if there is a name for this metric, it is the length of the shortest path if we only allow arbitrary vertical segments... | {
"language": "en",
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Simple Tensor Product Question about Well-definedness If I want to define a homomorphism, $f$, from $A\otimes_R B$ into some $R$ module $M$. If I defined it on simple tensors $a\otimes b$ what are the conditions I need to check to make this is well defined.
Does it suffice to check that $f(r(a\otimes b))=f((ra)\otimes ... | You simply need to define an $R$-bilinear map $\tilde{f} : A \times B \rightarrow M$. The universal property of the tensor product then induces an $R$-module homomorphism $f : A \otimes_R B \rightarrow M$. To check that $\tilde{f}$ is $R$-bilinear, you must show:
(1) $\tilde{f}(ra,b) = \tilde{f}(a,rb) = r\tilde{f}(a,... | {
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If a function is bounded almost everywhere, then globally bounded? Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is bounded almost everywhere. Then is it bounded?
If so, what is the main idea or method in tis proof, and can I generalize this for upto what?
| Let $f$ map the irrationals to $0$ and the rationals to themselves. Since the rational numbers form a countable set, it has measure $0$. Then f is bounded almost everywhere but is not bounded.
More generally, on any infinite set, one can define a function that is bounded almost everywhere but is not bounded. Simply tak... | {
"language": "en",
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Show that $(f_n)$ is equicontinuous, given uniform convergence
Let $f_n: [a,b] \rightarrow, n \in \mathbb{N}$, be a sequence of functions converging uniformly to $f: [a,b] \rightarrow \mathbb{R}$ on $[a,b]$. Suppose that each $f_n$ is continuous on [a,b] and differentiable on (a,b), and that the sequence of derivative... | Hint
$$ |f_n(x)-f_n(y)|=|(f_n(x)-f(x))+(f(x)-f(y))+(f(y)-f_n(y))| $$
$$ \leq |f_n(x)-f(x)|+|f(x)-f(y)|+|f(y)-f_n(y)| \,.$$
Now, use the assumptions you have been given.
| {
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Show that f is uniformly continuous and that $f_n$ is equicontinuous
$f_n: A \rightarrow \mathbb{R}$,$n \in \mathbb{N}$ is a sequence of functions defined on $A \subseteq \mathbb{R} $. Suppose that $(f_n)$ converges uniformly to $f: A \rightarrow \mathbb{R}$, and that each $f_n$ is uniformly continuous on $A$.
1.) Ca... | For 1, let $\epsilon >0$. Then pick $n$ such that $|f_n(x) - f(x)| < \epsilon/3$ on $A$. By uniform continuity of $f_n$, there exists a $\delta$ such that $|x-y| < \delta \Longrightarrow |f_n(x)-f_n(y)| < \epsilon/3$. Now if $|x-y| < \delta$,
$$ |f(x)-f(y)| = |f(x)-f_n(x)+f_n(x)-f_n(y)+f_n(y) -f(y)| \leq $$
$$|f(x)-f_n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Continuous extension of a real function Related;
Open set in $\mathbb{R}$ is a union of at most countable collection of disjoint segments
This is the theorem i need to prove;
"Let $E(\subset \mathbb{R})$ be closed subset and $f:E\rightarrow \mathbb{R}$ be a contiuous function. Then there exists a continuous function $g... | This is a special case of the Tietze extension theorem. This is a standard result whose proof can be found in any decent topology text. A rather different proof can be found here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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prove an analytic function has at least n zeros I`m confused about this problem:
Let G be a bounded region in C whose boundary consists of n circles.
Suppose that f is a non-constant function analytic on G: Show that
if absolute value of f(z) = 1 for all z in the boundary of G then f has at least n zeros (counting mul... | I don't know what tools you have at your disposal, but this follows from some basic topology. The assumptions imply that $f$ is a proper map from the region $G$ to the unit disk $\mathbb{D}$. As such the map has a topological degree, and since the preimage of the boundary $|z|=1$ contains $n$ components, this degree ha... | {
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How many ways can the letters of the word TOMORROW be arranged if the Os can't be together? How many ways can the letters of the word TOMORROW be arranged if the Os cant be together?
I know TOMORROW can be arranged in $\frac{8!}{3!2!} = 3360$ ways. But how many ways can it be arranged if the Os can't be together? And w... | First, you have to remove the permutations like this TOMOORRW and TMOOORRW, so see OO as an element, then we have $3360-\frac{7!}{2!}$.
EDITED
Now, Notice you remove words like this TOMOORRW and TMOOORRW with OO and OOO, but you remove a little bit more you want, why? can you see this? how to fix this problem?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determinant of rank-one perturbations of (invertible) matrices I read something that suggests that if $I$ is the $n$-by-$n$ identity matrix, $v$ is an $n$-dimensional real column vector with $\|v\| = 1$ (standard Euclidean norm), and $t > 0$, then
$$\det(I + t v v^T) = 1 + t$$
Can anyone prove this or provide a referen... | I solved it. The determinant of $I+tvv^T$ is the product of its eigenvalues. $v$ is an eigenvector with eigenvalue $1+t$. $I+tvv^T$ is real and symmetric, so it has a basis of real mutually orthogonal eigenvectors, one of which is $v$. If $w$ is another one, then $(I+tvv^T)w=w$, so all the other eigenvalues are $1$.
I ... | {
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Prove that if X and Y are Normal and independent random variables, X+Y and X−Y are independent Prove that if X and Y are Normal and independent random variables, X+Y and X−Y are independent. Note that X and Y also have the same mean and standard deviation.
Note that this is a duplicate of Prove that if $X$ and $Y$ are ... | Define $U = X + Y, V = X - Y$. Then, $X = (U + V)/2, Y = (U - V)/2$. Find the Jacobian $J$ for the transformation.
Then, $f_{U,V}(u,v)=f_{X}(x=(u+v)/2)f_{Y}(y=(u-v)/2)|J|$.
You will find that $f_{U,V}(u,v)$ factors into a function of $u$ alone and a function of $v$ alone. Thus, by the Factorization thm, $U$ and $V$ are... | {
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A collection of Isomorphic Groups
So the answer is
Questions
1) How exactly is $<\pi>$ isomorphic to the other integer groups? I mean $\pi$ itself isn't even an integer.
2)What is exactly is the key saying for the single element sets? Are they trying to say they are isomorphic to themselves?
3) How exactly is $\{\mat... | 1) Notice that $\langle \pi \rangle=\{\pi^n \mid n \in \mathbb Z\}$ and we also have $\pi^n\pi^m=\pi^{n+m}$ what group does this remind you of?
2) The single element sets are not isomorphic to any of the other groups. As an aside it's not particularly meaningful to say a group is isomorphic to itself. You generally spe... | {
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How to Decompose $\mathbb{N}$ like this?
Possible Duplicate:
Partitioning an infinite set
Partition of N into infinite number of infinite disjoint sets?
Is it possible to find a family of sets $X_{i}$, $i\in\mathbb{N}$, such that:
*
*$\forall i$, $X_i$ is infinite,
*$X_i\cap X_j=\emptyset$ for $i\neq j$,
*$\m... | Let $p_n$ be the $n$-th prime number. That is $p_1=2; p_2=3; p_3=5; p_4=7$ and so on.
For $n>0$ let $X_n=\{(p_n)^k\mid k\in\mathbb N\setminus\{0\}\}$.
For $X_0$ take all the rest of the numbers available, namely $k\in X_0$ if and only if $k$ can be divided by two distinct prime numbers, or if $k=1$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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Weakly convex functions are convex Let us consider a continuous function $f \colon \mathbb{R} \to \mathbb{R}$. Let us call $f$ weakly convex if
$$
\int_{-\infty}^{+\infty}f(x)[\varphi(x+h)+\varphi(x-h)-2\varphi(x)]dx\geq 0 \tag{1}
$$
for all $h \in \mathbb{R}$ and all $\varphi \in C_0^\infty(\mathbb{R})$ with $\varphi ... | By a change of variables (translation-invariance of Lebesgue measure) the given inequality can be equivalently rewritten as
$$
\int [f(x+h)+f(x-h)-2f(x)]\varphi(x)\,dx \geq 0 \qquad \text{for all }0 \leq \varphi \in C^{\infty}_0(\mathbb{R})\text{ and all }h \gt 0.
$$
If $f$ were not midpoint convex then there would be... | {
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} |
Question about how to implicitly differentiate composite functions I have a question. How in general would one differentiate a composite function like $F(x,y,z)=2x^2-yz+xz^2$ where $x=2\sin t$ , $y=t^2-t+1$ , and $z = 3e^{-1}$ ? I want to find the value of $\frac{dF}{dt}$ evaluated at $t=0$ and I don't know how. Can so... | $$\frac{\partial F}{\partial t}=\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial y}\frac{dy}{dt}+\frac{\partial F}{\partial z}\frac{dz}{dt}=$$
$$=(4x+z^2)\cdot 2\cos t-z(2t-1)+(2xz-y)\cdot 0$$
You'll now to substitute:
$$t=0\Longrightarrow\,x=0\,,\,y=1\,,\,z=3\,e^{-1}$$
The final result is, if I'm ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/220258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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An inequality for all natural numbers Prove, using the principle of induction, that for all $n \in \mathbb{N}$, we have have the following inequality:
$$1+\frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n} \leq 2\sqrt n$$
| Suppose
$1+\frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n} \leq 2\sqrt n$
and
$1+\frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt {n+1}} > 2\sqrt {n+1}$
(i.e., that the induction hypothesis is false).
Subtracting these,
$\frac{1}{\sqrt {n+1}} > 2\sqrt {n+1} - 2\sqrt n
= 2(\sqrt {n+1} - \sqrt n)\frac{\sqrt {n+1} +\sqrt n}{\sqrt {n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/220323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Minimum Number of Nodes for Full Binary Tree with Level $\lambda$ If the level ($\lambda$) of a full binary tree at zero is just a root node, than I know that I can get the maximum possible number of nodes (N) for a full binary tree using the following:
N = $2^{\lambda+1}$- 1
Is the minimum possible number of nodes the... | Sorry to say but what all of you are discussing. AFAIK for full binary tree nodes = [(2^(h+1)) - 1] (fixed).
*
*For strict binary tree, max node = [2^h + 1] and min node = [2h + 1].
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Show $x^2 +xy-y^2 = 0$ is only true when $x$ & $y$ are zero. Show that it is impossible to find non-zero integers $x$ and $y$ satisfying $x^2 +xy-y^2 = 0$.
| The quadratic form factors into a product of lines $$0 = x^2 + xy - y^2 = -(y-\tfrac{1-\sqrt{5}}{2}x)(y-\tfrac{1+\sqrt{5}}{2}x),$$ equality holds if either
*
*$y=\tfrac{1-\sqrt{5}}{2}x$
*$y=\tfrac{1+\sqrt{5}}{2}x$
but this can't happen for $x,y$ integers unless they're both zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proving the 3-dimensional representation of S3 is reducible The 3-dimensional representation of the group S3 can be constructed by introducing a vector $(a,b,c)$ and permute its component by matrix multiplication.
For example, the representation for the operation $(23):(a,b,c)\rightarrow(a,c,b)$ is
$
D(23)=\left(\begin... | Here's another way to prove it's reducible, although it may depend on stuff you haven't learned yet. The order of the group is the sum of the squares of the degrees of the irreducible representations. So a group of order 6 can't have an irreducible representation of degree 3; $3^2\gt6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Orders of the Normal Subgroups of $A_4$
Prove that $A_4$ has no normal subgroup of order $3.$
This is how I started:
Assume that $A_4$ has a normal subgroup of order $3$, for example $K$.
I take the Quotient Group $A_4/K$ with $4$ distinct cosets, each of order $3$.
But I want to prove that these distinct cosets wil... | In fact, one can show that all normal subgroups of $A_4$ are $1$, $K_4$ (Klein four group) and $A_4$. Note that two permutations of $S_n$ are conjugate iff they have the same type. So we can write down (by some easy calculations) all conjugate classes of $A_4$ are the following $4 $ classes:
*
*type $1^4$: {(1)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/220659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Finding the limit of this function as n tends to infinity... $$\lim_{n\rightarrow\infty}\frac{n}{3}\left[\ln\left(e-\frac{3}{n}\right)t-1\right]$$
I'm having little trouble figuring this out.
I did try to differentiate it about 3 times and ended up with something like this
$$f'''(n) = \frac{1}{3} \left(\frac{1}{e - \... | Your limit is equal to
$$\lim_{n\to +\infty}\frac{n}{3}\,\log\left(1-\frac{3}{en}\right),$$
but since:
$$\lim_{x\to 0}\frac{\log(1-x)}{x}=-1,$$
(by squeezing, by convexity or by De l'Hopital rule) you have:
$$\lim_{n\to +\infty}\frac{n}{3}\,\log\left(1-\frac{3}{en}\right)=-\frac{1}{e}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $ A$ is open and $ B$ is closed, is $B\setminus A$ open or closed or neither? If $ A$ is open and $ B$ is closed, is $B\setminus A$ open or closed or neither?
I think it is closed, is that right? How can I prove it?
| Yes, if $A$ is open and $B$ is closed, then $B\setminus A$ is closed. To prove it, just note that $X\setminus A$ is closed (where $X$ is the whole space), and $B\setminus A=B\cap(X\setminus A)$, so $B\setminus A$ is the intersection of two closed sets and is therefore closed.
Alternatively, you can observe that $X\setm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/220792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Is every subset of a metric space a metric subspace? Is every subset of a metric space a metric subspace? A simple proof does justify that all are subspaces, still, wanted to know if I missed something.
| Let $(x,p)$ be metric space and letting $Y$ be a non empty subset of $X$. Define the function $§$ on $Y$. $Y$ by $§(x,y)=p(x,y)$ for all $x,y$ in $Y$.Then $(Y,§)$ is also a metric space called a subspace of the metric space $(x,p)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 2
} |
Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$ Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$. Please show me the way you find it.
The answer in my textbook is $f(x)=\frac{1+x^2+x^4}{x\cdot \sqrt{1-x^2}}$
| $$f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2} = \left(x+\frac{1}{x}\right)^2-2$$
Let $x+\frac{1}{x}=z$.
Then we get, $$f(z)=z^2-2.$$
Hence we put x on the place of z. And we get
$f(x)=x^2-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How do i solve this double integral $$\int_0^1\int_{-\pi}^\pi x\sqrt{1-x^2\sin^2(y)}\mathrm{d}y\mathrm{d}x$$
How do I solve this question here?
| Switch the order of integration.
Integrating first over $x$, we obtain ($u=1-x^2 \sin^2y$)
$$\int_0^1\!dx\, x \sqrt{1-x^2 \sin^2 y}
= \frac{1}{2 \sin^2 y} \int_{\cos^2y}^1\!du\,\sqrt{u}
= \frac{1}{3\sin^2 y} ( 1 - |\cos^3 y|).$$
What is missing is the integral over $y$. Using one of the standard method to integral rati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Lie Groups question from Brian Hall's Lie Groups, Lie Algebras and their representations. In page 60 of Hall's textbook, ex. 8 assignment (c), he asks me to prove that if $A$ is a unipotent matrix then $\exp(\log A))=A$.
In the hint he gives to show that for $A(t)=I+t(A-I)$ we get
$$\exp(\log A(t)) = A(t) , \ t<<1$$
I... | If $A$ is unipotent, then $A - I$ is nilpotent, meaning that $(A-I)^n = 0$ for all sufficiently large $n$. This will turn your power series into a polynomial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/221072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A tough differential calculus problem This is a question I've had a lot of trouble with. I HAVE solved it, however, with a lot of trouble and with an extremely ugly calculation. So I want to ask you guys (who are probably more 'mathematically-minded' so to say) how you would solve this. Keep in mind that you shouldn't ... | By "touches" I assume you mean that the line is tangent to the graph of $f_p$. You can try implicit differentiation. Start with
$$ y = \frac{9\sqrt{x^2 + p}}{x^2 + 2}. $$
Multiply by $x^2 + 2$ to get
$$ y(x^2 + 2) = 9\sqrt{x^2 + p}. $$
Squaring, you get
$$ y^2 (x^2 + 2)^2 = 81 (x^2 + p). $$
Differentiate both sides im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Do these $\delta-\epsilon$ proofs work? I'm new to $\delta, \epsilon$ proofs and not sure if I've got the hand of them quite yet.
$$ \lim_{x\to -2} (2x^2+5x+3)= 1 $$
$|2x^2 + 5x + 3 - 1| < \epsilon$
$|(2x + 1)(x + 2)| < \epsilon$
$|(2x + 4 - 3)(x + 2)| < \epsilon$
$|(2(x+2)^2 -3(x + 2)| \leq 2|x+2|^2 +3|x + 2| < \epsil... | $$ \lim_{x\to -2} (2x^2+5x+3)= 1 $$
Finding $\delta$: $|2x^2 + 5x + 3 - 1| < \epsilon$
$|(2x + 1)(x + 2)| < \epsilon$
$|x - (-2)| < \delta $, pick $\delta = 3$
$|x+2| < 3 \Rightarrow -5 < x < 1 \Rightarrow -9 < 2x + 1 < 3$
This implies $|2x + 1||x + 2| < 3 \cdot |x + 2| < \epsilon \Rightarrow |x+2| < \frac{\epsilon}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $1-x/3\le\frac{\sin x}x\le1.1-x/4, \forall x\in(0,\pi]$ Prove that
$$1- \frac{x}{3} \le \frac{\sin x}x \le 1.1 - \frac{x}{4}, \quad \forall x\in(0,\pi].$$
| From the concavity of $f(x)=\cos x$ over $[0,\pi/2]$, we have:
$$ \forall x\in[0,\pi/2],\quad \cos x\geq 1-\frac{2}{\pi}x, $$
from which
$$ \forall x\in[0,\pi/2],\quad \sin x\geq \frac{1}{\pi}x(\pi-x) = x-\frac{1}{\pi}x^2$$
follows, by integration. Now, both the RHS and the LHS are symmetric wrt $x=\frac{\pi}{2}$, so w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question about complex numbers (what's wrong with my reasoning)? Can someone point out the flaw here?
$$e^{-3\pi i/4} = e^{5\pi i/4}$$
So raising to $\frac{1}{2}$, we should get
$$e^{-3\pi i/8} = e^{5\pi i/8}$$
but this is false.
| Paraphrase using $e^0=1$ and $e^{\pi i}=-1$. We can write
$$
e^{-3\pi i/4}\;1^2=e^{-3\pi i/4}\;(-1)^2
$$
Raising to the $\frac12$ power yields
$$
e^{-3\pi i/8}\;1=e^{-3\pi i/8}\;(-1)
$$
The problem is that without proper restrictions (e.g. branch cuts), the square root is not well-defined on $\mathbb{C}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/221386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Does the category framework permit new logics? It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't have to be {0,1}.
But I wonder, aren't all the multivalued logics also part... | No, you can't model all multivalued logics in set theory. Set theory models classical propositional logic, but it does not model a logic where say the principle of contradiction fails and its negation fail also. All formal theorems of any multivalued logic exist within classical logic in the sense that if A comes as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Why is every representation of $\textrm{GL}_n(\Bbb{C})$ completely determined by its character? I know that every (Lie group) representation of $\textrm{GL}_n(\Bbb{C})$ is completely reducible; this I believe comes from the fact that every representation of the maximal compact subgroup $\textrm{U}(n)$ is completely red... | This boils down to facts about the representation theory of compact groups: there every complex representation is determined by its character. Now a representation of $GL(n,\mathbb C)$ is determines by a representation of its Lie algebra. But this is the complexification of $u(n)$ and complex representations of a Li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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If every proper quotient is finite, then $G\cong\mathbb Z$ Here is my problem:
Let $G$ is an infinite abelian group. Prove that if every proper quotient is finite, then $G\cong\mathbb Z$.
And here is my incompleted approach:
I know that the quotient subgroup $\frac{G}{tG}$ wherein $tG$ is torsion subgroup of $G$ is a... | Let $a_0\in G$ be a nonzero element.
Then $\langle a\rangle\cong \mathbb Z$ as $G$ is torsion-free (which you have shown).
Now $Q_0=G/\langle a_0\rangle$ is a finite abelian group.
If $Q_0\cong 1$, we are done.
Otherwise, select $a_1\in G\setminus\langle a_0\rangle$. Then let $Q_1=G/\langle a_0, a_1\rangle$, etc.
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Comparing Two Sums with Binomial Coefficients How do I use pascals identity:
$${2n\choose 2k}={2n-1\choose 2k}+{2n-1\choose 2k-1}$$ to prove that
$$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}=\displaystyle\sum_{k=0}^{2n-1}{2n-1\choose k}$$
for every positive integer $n$ ?
| Other than Pascal's identity, we just notice that the sums on the right are the same because they cover the same binomial coefficients (red=even, green=odd, and blue=both).
$$
\begin{align}
\sum_{k=0}^n\binom{2n}{2k}
&=\sum_{k=0}^n\color{#C00000}{\binom{2n-1}{2k}}+\color{#00A000}{\binom{2n-1}{2k-1}}\\
&=\sum_{k=0}^{2n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Laplace integral and leading order behavior Consider the integral:
$$
\int_0^{\pi/2}\sqrt{\sin t}e^{-x\sin^4 t} \, dt
$$
I'm trying to use Laplace's method to find its leading asymptotic behavior as $x\rightarrow\infty$, but I'm running into problems because the maximum of $\phi(t)$ (i.e. $-\sin^{4}t$) is $0$ and occu... | Plot integrand for few values of $x$:
It is apparent that the maximum shifts closer to the origin as $x$ grows.
Let's rewrite the integrand as follows:
$$
\int_0^{\pi/2} \sqrt{\sin(t)} \exp\left(-x \sin^4(t)\right) \mathrm{d}t = \int_0^{\pi/2} \exp\left(\frac{1}{2} \log(\sin(t))-x \sin^4(t)\right) \mathrm{d}t
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Check my workings: Show that $\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$ Let $f''$ be continuous on $\mathbb{R}$. Show that
$$\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$$
My workings
$$\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=\lim_{h\to0}\frac{f(x+h)-f(x)-[f(x)-f(x-h)]}{h^2}=\frac{\lim_{h\to0}\f... | Try applying L'Hospital's Rule to $h$, that is, differentiate with respect to $h$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/221905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 0
} |
Evaluate $\sum_{k=1}^\infty \frac{k^2}{(k-1)!}$.
Evaluate $\sum_{k=1}^\infty \frac{k^2}{(k-1)!}$
I sense the answer has some connection with $e$, but I don't know how it is. Please help. Thank you.
| For $\frac{P(n)}{(n-r)!},$ where $P(n)$ is a polynomial.
If the degree of $P(n)$ is $m>0,$ we can write $P(n)=A_0+A_1(n-r)+A_2(n-r)(n-r-1)+\cdots+A_m(n-r)(n-r-1)\cdots\{(n-r)-(m-1)\}$
Here $k^2=C+B(k-1)+A(k-1)(k-2)$
Putting $k=1$ in the above identity, $C=1$
$k=2,B+C=4\implies B=3$
$k=0\implies 2A-B+C=0,2A=B-C=3-1\imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/221951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
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Galois theory (Showing $G$ is not abelain) Suppose $G$ is the Galois group of an irreducible degree $5$ polynomial $f \in \mathbb{Q}[x]$ such that $|G| = 10$. Then $G$ is non-abelian.
Proof: Suppose $G$ is abelian. Let $M$ be the splitting field of $f$. Let $\theta$ be a root of $f$. Consider $\mathbb{Q}(\theta) \subs... | The only abelian group of order $10$ is cyclic. Since $G$ is a subgroup of $S_5$, it's enough to show that there's no element of order $10$ in $S_5$.
If you decompose a permutation in $S_5$ as a product of disjoint cycles, then the order is the LCM of the cycle lengths - and these can be any partition of $5$.
Since $5 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is this differe equation (and its solution) already known? While studying for an exam, I met the following nonlinear differential equation
$a\ddot{x}+b\dot{x}+c\sin x +d\cos x=k$
where $a,b,c,d,k$ are all real constants. My teacher says that this differential equation does not admit closed form solution, but on this I ... | $a\ddot{x}+b\dot{x}+c\sin x+d\cos x=k$
$a\dfrac{d^2x}{dt^2}+b\dfrac{dx}{dt}+c\sin x+d\cos x-k=0$
This belongs to an ODE of the form http://eqworld.ipmnet.ru/en/solutions/ode/ode0317.pdf
Let $\dfrac{dx}{dt}=u$ ,
Then $\dfrac{d^2x}{dt^2}=\dfrac{du}{dt}=\dfrac{du}{dx}\dfrac{dx}{dt}=u\dfrac{du}{dx}$
$\therefore au\dfrac{du... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find the volume of the solid under the plane $x + 2y - z = 0$ and above the region bounded by $y=x$ and $y = x^4$. Find the volume of the solid under the plane $x + 2y - z = 0$ and above the region bounded by $y = x$ and $y = x^4$.
$$
\int_0^1\int_{x^4}^x{x+2ydydx}\\
\int_0^1{x^2-x^8dx}\\
\frac{1}{3}-\frac{1}{9} = \fra... | I think it should be calculated as
\begin{eqnarray*}
V&=&\int_0^1\int_{x^4}^x\int_0^{x+2y}dzdydx\\
&=&\int_0^1\int_{x^4}^x(x+2y)dydx\\
&=&\int_0^1\left.\left(xy+y^2\right)\right|_{x^4}^xdx\\
&=&\int_0^1(2x^2-x^5-x^8)dx\\
&=&\left.(\frac{2}{3}x^3-\frac{1}{6}x^6-\frac{1}{9}x^9)\right|_0^1\\
&=&\frac{7}{18}
\end{eqnarray*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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problem on entire function Let $f$ be an entire function. For which of the following cases $f$ is not necessarily a constant
*
*$\operatorname{im}(f'(z))>0$ for all $z$
*$f'(0)=0$ and $|f'(z)|\leq3$ for all $z$
*$f(n)=3$ for all integer $n$
*$f(z) =i$ when $z=(1+\frac{k}{n})$ for every positive ... | You're right for $1$, where the $c$ you mention should have positive imaginary part. For example $f(z)=iz$ does the job.
For $2$, $f$ need not be identically $0$. $f(z)=2$ satisfies the requirements, for instance. $f$ does have to be constant, though, because if $f$ is entire, $f'$ is too, and so by Liouville's theorem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
convergent series, sequences? I want to construct a sequence of rational numbers whose sum converges to an irrational number and whose sum of absolute values converges to 1.
I can find/construct plenty of examples that has one or the other property, but I am having trouble find/construct one that has both these propert... | Find two irrational numbers $a > 0$ and $b < 0$ such that $a-b = 1$ but $a+b$ is irrational. Create a series with positive terms that sum to $a$ and another series with negative terms that sum to $b$. Combine the two series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/222268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
What is a rational $0$-cycle on an algebraic variety $X$ over $\mathbb{Q}$? I found the following assertion in a paper about Hilbert modular forms that I'm trying to read.
Let $X$ be an algebraic variety over $\mathbb{Q}$, and let $\Psi$ be a rational function on $X$ and $C = \sum n_P P$ be a rational $0$-cycle on $X$... | To say that $C = \sum n_P P$ is a rational $0$-cycle means that $n_P\in \mathbb Z$ and that $P\in X$ is a rational point i.e. a closed point with residue field $\kappa (P)=\mathbb Q$.
If the rational function $\Psi$ is defined at $P$ its value at $P$ is a rational number $\Psi(P) \in \mathbb Q$ and if $\Psi$ is defined... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Relationship between Legendre polynomials and Legendre functions of the second kind I'm taking an ODE course at the moment, and my instructor gave us the following problem:
Derive the following formula for Legendre functions $Q_n(x)$ of the second kind:
$$Q_n(x) = P_n(x) \int \frac{1}{[P_n(x)]^2 (1-x^2)}dx$$
where $P_... | The most general solution of Legendre equation is
$$y = A{P_n} + B{Q_n}.$$
Let $y(x) = A(x){P_n}(x)$. Then $y' = AP' + A'P$ and $y'' = AP'' + 2A'P' + A''P$. So
$$(1 - {x^2})(AP'' + 2A'P' + A''P) - 2x(AP' + A'P) + n(n + 1)AP = 0.$$
Note that
$$(1 - {x^2})(AP'') - 2x(AP') + n(n + 1)AP = 0$$
which means some terms in the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Compute $\lim_{n\to\infty}\int_0^n \left(1+\frac{x}{2n}\right)^ne^{-x}\,dx$. I'm trying to teach myself some analysis (I'm currently studying algebra), and I'm a bit stuck on this question. It's strange because of the $n$ appearing as a limit of integration; I want to apply something like LDCT (I guess), but it doesn't... | HINT Note that $$\left(1 + \dfrac{x}{2n} \right)^n < e^{x/2}$$ for all $n$. Hence, $$ \left(1 + \dfrac{x}{2n} \right)^n e^{-x} < e^{-x/2}$$
Your sequence $$f_n(x) = \begin{cases} \left(1 + \dfrac{x}{2n} \right)^n e^{-x} & x \in [0,n]\\ 0 & x > n\end{cases}$$ is dominated by $g(x) = e^{-x/2}$. Now apply LDCT.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/222485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Studying $ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $ I would like to find a simple equivalent of:
$$ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $$
We have:
$$ 0\leq u_{n}\leq \frac{1}{n!}\left(\frac{\pi}{2}\right)^n \rightarrow0$$
So $$ u_{n} \rightarrow 0$$
Clearly:
$$ u_{n} \sim \frac{1}{n!} \int... | The change of variable $x=\cos\left(\frac{\pi s}{2n}\right)$ yields
$$
u_n=\frac1{n!}\left(\frac\pi2\right)^{n+2}\frac1{n^2}v_n,
$$
with
$$
v_n=\int_0^n\left(1-\frac{s}n\right)^n\,\frac{2n}\pi \sin\left(\frac{\pi s}{2n}\right)\,\mathrm ds.
$$
When $n\to\infty$, $\left(1-\frac{s}n\right)^n\mathbf 1_{0\leqslant s\leqslan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to show that this function is bijective
Possible Duplicate:
Proving the Cantor Pairing Function Bijective
Assume I define
$$ f: \mathbb N \times \mathbb N \to \mathbb N, (a,b) \mapsto a + \frac{(a + b ) ( a + b + 1)}{2} $$
How to show that this function is bijective? For injectivity I tried to show that if $f(a,... | The term $(a+b)(a+b+1)/2$ is the sum of the numbers from $1$ to $a+b$. For a fixed value of $s=a+b$, $a$ ranges from $0$ to $s$, so we need $s+1$ different results for these arguments. Now you can prove by induction that the range from $s(s+1)/2$ to $(s+1)(s+2)/2-1$ contains precisely the images of the pairs with sum $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Number of integral solutions for $|x | + | y | + | z | = 10$ How can I find the number of integral solution to the
equation
$|x | + | y | + | z | = 10.$
I am using the formula,
Number of integral solutions for $|x| +|y| +|z| = p$ is $(4P^2) +2 $, So the answer is 402.
But, I want to know, How we can find it without us... | (1) z=0, $40$ patterns
z=1, $36$
z=2, $32$
・・・
z=9,$4$
but z=10,$2$ the total is
$S=2(4+8+\dots+36)+40+2=402$
(2) Another counting
There are 8 areas by plus and minus of x,y,z, 40 patterns at x=0, 2 patterns at y=z=0, therefore
$40+8*10C2+2=402$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/222690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 8,
"answer_id": 6
} |
Absolute value and sign of an elasticity In my microeconomics book, I read that when we have $1+\dfrac{1}{\eta}$ where $\eta$ is an elasticity coefficient, we can write $1-\dfrac{
1}{|\eta|}$ "to avoid ambiguities stemming from the negative sign of the elasticity".
What does this mean? Is it always legitimate to perfor... | If the elasticity coefficient $\eta$ is negative, then $|\eta|=-\eta$. The ambiguity arises because some people may suppress the negative sign and write it as a positive number instead. In this case, using $1+\frac{1}{\eta}$ becomes ambiguous.
For example, if $\eta=-2$ but people write it as $\eta=2$, then $1+\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
existence of hyperbolic groups I perfectly understand that the Milnor-Schwarz lemma tells me that cocompact lattice in semisimple Lie groups of higher rank are not hyperbolic (in the sense of Gromov).
But do there exist noncocompact lattices in higher rank semisimple Lie groups which are hyperbolic? (wikipedia is sayin... | You asked a related question in another post, and you erased the question while I was posting an answer. So I post it here. The question was complementary to the one above so I think it's relevant to include the answer: why are non-uniform lattices in rank 1 symmetric spaces of noncompact type not hyperbolic except in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Simple matrix equation I believe I'm missing an important concept and I need your help.
I have the following question:
"If $A^2 - A = 0$ then $A = 0$ or $A = I$"
I know that the answer is FALSE (only because someone told me) but when I try to find out a concrete matrix which satisfies this equation (which isn't $0$ or ... | If for a polynomial $p$ and a matrix $A$ you have $p(A)=0$ then for every invertible matrix $W$ you have $$p(W^{-1}AW)=W^{-1}p(A)W=0 . $$
Here $p=x^2-x$, you can take $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$, $W$ any invertible matrix to make a lot of examples.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/222886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Understanding what this probability represents Say I want the probability that a five card poker hand contains exactly two kings. This would be
$$\frac{{4\choose 2}{48 \choose 3}}{52\choose 5}$$
Now if I drop the $48 \choose 3$, which represents the 3 non king cards, what can
the probability $\frac{4\choose 2}{52\choos... | It would be the probability of a hand containing exactly two kings and three specified non-kings, e.g. the ace, 2 and 3 of spades.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/222928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
modular multiplicative inverse I have a homework problem that I've attempted for days in vain... It's asking me to find an $n$ so that there is exactly one element of the complete residue system $\pmod n$ that is its own inverse apart from $1$ and $n-1$. It also asks me to construct an infinite sequence of $n's$ so th... | For the second: What about $3$ and $5 \pmod8$?
For the first: if $x^2\equiv 1$ then $(-x)^2\equiv 1$, too, so if there is only one (besides $\pm 1$), then $x\equiv -x \pmod{n}$, that is $2x=n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/222973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find all Laurent series of the form... Find all Laurent series of the form $\sum_{-\infty} ^{\infty} a_n $ for the function
$f(z)= \frac{z^2}{(1-z)^2(1+z)}$
There are a lot of problems similar to this. What are all the forms? I need to see this example to understand the idea.
| Here is related problem. First, convert the $f(z)$ to the form
$$f(z) = \frac{1}{4}\, \frac{1}{\left( 1+z \right)}+\frac{3}{4}\, \frac{1}{\left( -1+z \right) }+\frac{1}{2}\,\frac{1}{\left( -1+z \right)}$$
using partial fraction. Factoring out $z$ gives
$$ f(z)= \frac{1}{4z}\frac{1}{(1+\frac{1}{z})}- \frac{3}{4z}\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show there is no measure on $\mathbb{N}$ such that $\mu(\{0,k,2k,\ldots\})=\frac{1}{k}$ for all $k\ge 1$
For $k\ge1$, let $A_k=\{0,k,2k,\ldots\}.$ Show that there is no measure $\mu$ on $\mathbb{N}$ satisfying $\mu(A_k)=\frac{1}{k}$ for all $k\ge1$.
What I have done so far:
I am trying to apply Borel-Cantelli lemma (... | You are basically there. You proved in your last line that $\mu$-a.e. number is divisible by infinitely many primes, which is an obvious contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/223129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How Many Ways to Make a Pair Given Five Poker Cards I'm confused at the general method of solving this type of problem. The wikipedia page says that there are:
${13 \choose 1} {4 \choose 2} {12 \choose 3} {4 \choose 1}^{3}$ ways to select a pair when 5 cards are dealt. Can someone outline what each calculation means?... | $13\choose 1$ is the number of ways of choosing the denomination of the pair, whether it is a pair of kings, or a pair of threes, or whatever. Then there are $4\choose 2$ ways to choose suits for the two cards of the pair.
Then there are $12\choose 3 $ ways to choose the three different denominations of the remaining ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How many distinct functions can be defined from set A to B? In my discrete mathematics class our notes say that between set $A$ (having $6$ elements) and set $B$ (having $8$ elements), there are $8^6$ distinct functions that can be formed, in other words: $|B|^{|A|}$ distinct functions. But no explanation is offered an... | Let's say for concreteness that $A$ is the set $\{p,q,r,s,t,u\}$, and $B$ is a set with $8$ elements distinct from those of $A$. Let's try to define a function $f:A\to B$.
What is $f(p)$? It could be any element of $B$, so we have 8 choices.
What is $f(q)$? It could be any element of $B$, so we have 8 choices.
...
Wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 6,
"answer_id": 3
} |
Linear Algebra Fields question I have the following statement (which is false) and I'm trying to understand why, I can't find a concrete example.
If $a$ belongs to $Z_n$ and $a^2 = 1$ then $a=1$ or $a=-1$
Can someone give me a direction?
Guy
| The statement does hold when the structure is a field, i.e. when $n$ is prime. The statement is false for some composite $n$. For example, consider any $n$ of the form $x(x+2)$ for some $x\in\mathbb{N}^+$ then we will have
$$(x+1)^2 \equiv x^2 + 2x + 1 \equiv n + 1 \equiv 1 \pmod n$$
so that $\pm(x+1)$ is a solution in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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