Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Limit of $\lim\limits_{n \rightarrow \infty}(\sqrt{x^8+4}-x^4)$ I have to determine the following:
$\lim\limits_{n \rightarrow \infty}(\sqrt{x^8+4}-x^4)$
$\lim\limits_{n \rightarrow \infty}(\sqrt{x^8+4}-x^4)=\lim\limits_{x \rightarrow \infty}(\sqrt{x^8(1+\frac{4}{x^8})}-x^4 = \lim\limits_{x \rightarrow \infty}(x^4\sqrt... | A short way to (non-rigorously) find the limit is to observe that for large $x$,
$$
\sqrt{x^8+4} \approx \sqrt{x^8}=x^4
$$
so that for large $x$ (especially in $\lim_{x \to \infty}$)
$$
\sqrt{x^8+4}-x^4 \approx x^4-x^4=0
$$
So the limit must be $0$.
| {
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Help using the C-S-B theorem Let $\Bbb R$ denote the set of real numbers. Let $H\subseteq\Bbb R$ and assume that there are real numbers $a,b$ with $a>b$ such that the open interval $(a,b)$ is a subset of $S$. Prove that the cardinality of $H$ equals $\mathfrak{c}$.
| $w\mapsto x+ \dfrac{y-x}{1+2^w}$ is an injective mapping in one direction.
(As $w\to\infty$, this function goes to $x$; as $w\to-\infty$, this function goes to $y$; for other values of $w$, it's between $x$ and $y$.)
$v\mapsto v$ is an injective mapping in the other direction.
| {
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Why is this combination of a covariant derivative and vector field a (1,1)-tensor? I have a question regarding something Penrose says in section 14.3 of The Road to Reality.
It says '...when $\nabla$ acts on a vector field $\xi$, the resulting quantity $\nabla \xi$ is a $(1,1)$-valent tensor.'
I understand that $\nabla... | A $(1,1)$-tensor can be thought of as a linear map that sends vectors to vectors; so given a vector $X$ based at $p$, $\nabla\xi(X)=\nabla_X \xi$ will be another vector based at $p$, which you should think of as the change in the vector field $\xi$ when you move a small amount in the direction $X$ starting from $p$.
| {
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How do you solve this exponential equation? $3(16)^x+2(81)^x=5(36)^x$
How do you change the bases to combine the terms? The correct answer should be 0 and 0.5.
Edit: So this equation can't be solved algebraically? I have to use creative logic to solve it?
| Note that
$$3(16)^x=3(4)^{2x}, 2(81)^x=2(9)^{2x}, \text{and}\; 5(36)^x=5(6)^{2x}$$
If $2x=1$, you get that $12+18=30$, a true statement. Thus, $x=\frac{1}{2}$.
The case where $x=0$ is trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/599127",
"timestamp": "2023-03-29T00:00:00",
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Distance from point in circle to edge of circle The situation is as follows:
I have a circle with a diameter of $20$ and a center at $(0,0)$.
A point $P$ inside that circle is at $(2,0)$.
How do I calculate the distance from $P$ to the edge of the circle for a given angle $\theta$?
| Let the centre of the circle be $O$, and let the point $(2,0)$ be $P$. Draw a line $PQ$ to the periphery of the circle, making an angle $\theta$ with the positive $x$-axis. We want to find the length of $PQ$.
Consider the triangle $OPQ$. We have $\angle OPQ=180^\circ-\theta$. By the Cosine Law, with $x=PQ$, we have
$$1... | {
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Continuity correction: Change P(2 ≤ x < 9) to continuous? Convert discrete probability into continuous probability using continuous correction:
attempt:
Discrete: P(2 ≤ x < 9)
therefore continuous should be
Continuous: P(1.5 < X < 8.5)
Is this right? or should it be P (1.5 < x < 9)?
| I will assume that your discrete random variable takes integer values.
I find it difficult to remember a bunch of rules, so I remember only one: That if $k$ is an integer, and we are approximating the discrete $X$ by a continuous $Y$, then $\Pr(X\le k)$ is often better approximated by $\Pr(Y\le k+0.5)$.
Now $\Pr(2\le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Where is $k$ algebraically closed used? Suppose $k$ is algebraically closed, $A$, $B$ are $k$-algebras and $A$ is an affine $k$-algebra. It is known that then $A\otimes_k B$ is a domain if $A$ and $B$ are domains. This can be found in Milne's Algebraic Geometry notes as Proposition 4.15(b). I do not see where the assum... | I like this question a lot. While he does give a counterexample when $k$ is not algebraically closed, it is hard to see in his proof where this property is used. Where Milne uses algebraic closure is in the lines
For each maximal ideal $\mathfrak{m}$ of $A$, we know $(\sum\overline{a}_ib_i)(\sum \overline{a}_i~'b_i')=... | {
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Proof of Proposition IV.3. 8 in Hartshorne Hartshorne book Proposition (IV.3. 8) is that
Let $X$ be a curve in $\mathbb{P}^3$, which is not contained in any plane.
where, curve means a complete, nonsingular curve over algebraically closed field $k$.
Suppose either
(a) every secant of $X$ is a multisecant. or
(b)... | Here is the proof of (1). If $P =R$, it is trivial. If $P \ne R$, since $\phi$ is inseparable, $\phi$ must be ramified at $P$, this implies the line $\overline {PR}$ must be $L_P$. Check Figure 13 on page 299 of Hartshorne's book.
| {
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Geometric Slerp - Calculating Points along an Arc I'm trying to understand how to use Geometric Slerp, as seen here.
Having looked at the following equation:
How can P0 and P1 be calculated in order to using this equation? Aren't P0 and P1 represented by 2 numbers? The 2 numbers being x and y coordinates? or have I mi... | Your equation is a vector equation. So, yes, $P_0$ and $P_1$ are 2D points. You multiply these points by scalars, and then add together.
| {
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Is the matrix corresponding to an equivalence relation positive semidefinite? Let $|X| < \infty$ and $(X,R)$ be an equivalence relation. Define the $|X| \times |X|$ matrix $A$ by
$$(A)_{ij} = \begin{cases}1 & (i,j) \in R,\\0 & \text{ otherwise}.\end{cases}$$
Is this matrix positive semidefinite? Is there a simple way t... | Notice that $A$ is equal to $I$ plus the adjacency matrix of a graph consisting of a disjoint union of cliques. The eigenvalues of a clique $K_n$ are well-known to be $n-1$ and $-1$ and the spectrum of a disjoint union of graphs is the union of the spectra of the connected components. It follows that $A$ is positive se... | {
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Equivalent Conditions of Projection Map I got a problem in doing the following:
Let $A_1,\dots,A_k$ be linear operators on a vector space $V$ with dimension $n<+\infty$ such that
$$A_1+\cdots+A_k=I.$$
Prove that the following conditions are equivalent:
1) the operator $A_i$ are projections, i.e. $A_i^2=A_i$.
2) $A_iA_j... | Let $V_k$ be the range of $A_k$. Suppose (3) holds. Then the map $(V_1\oplus \dots\oplus V_k)\to V$ given by $(x_1,\dots,x_k)\mapsto x_1+\dots +x_k$ is a surjective map between spaces of the same dimension; thus, the map is an isomorphism.
So, every element of $ V$ has a unique representation as a sum of elements of... | {
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"timestamp": "2023-03-29T00:00:00",
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Any two norms on finite dimensional space are equivalent Any two norms on a finite dimensional linear space are equivalent.
Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for all $x$. Define $||\cdot||''=\sum |x_i|\cdot||e_i||$ (*). This is a ... | This lecture note answers this question quite well.
https://math.mit.edu/~stevenj/18.335/norm-equivalence.pdf
| {
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How to calculate the partial derivative of matrices' product Let $U = \frac{1}{2}u^TKu$,
then $\frac{\partial U}{\partial u} = Ku$.
How could I get this answer? Is there any book explains how to calculate derivative of matrices?
| By definition
$$
U=u^TKu=\frac{1}{2}\sum_{j=1}^n\sum_{i=1}^n K_{ij}u_iu_j.
$$
Differentiating with respect to the $l$-th element of $u$ we have
$$
2\frac{\partial U}{\partial u_l}=\sum_{j=1}^n K_{lj}u_j+\sum_{i=1}^n K_{il}u_i
$$
for all $l=1,\,\ldots,n$ and consequently
$$
\frac{\partial U}{\partial u}=\frac{1}{2}(Ku+K... | {
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Connected Subsets If C is a collection of connected subsets of M, all having a point in
common, prove that union of C is connected.
I know a set is connected if it is not disconnected. Also, from the above, I know the intersection of all subsets C is nonempty. I am not sure where to go from there.
| Hint:Let $C=(C_i)_{i\in I}$.Let $A=\cup_{i\in I} C_i$.
Now,(by contrdiction) suppose that $A$ is not connected.Then there is a function $g:A\to ${$0,1$} which is continuous and onto. Let $x_0\in \cap_{i\in I} C_i$. Then $x_0\in A$. Suppose that $g(x_0)=0$. Because $A$ is disconnected,there is a $x_1\in A:g(x_1)=1$.Also... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/599973",
"timestamp": "2023-03-29T00:00:00",
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How to determine the matrix of adjoint representation of Lie algebra? My questions will concern two pages:
http://mathworld.wolfram.com/AdjointRepresentation.html
and
http://mathworld.wolfram.com/KillingForm.html
In the first page, we know the basis of four matrix $\{e_1,e_2,e_3,e_4\}$, and my try to find their adjoint... | $\newcommand{\ad}{\operatorname{ad}}$
Answer to Q1:
You shouldn't bother too much with this, it's just a matter of notation. Anyway, I think there's a mistake in their $\ad(Y)$ in the sense that, if they want to be coherent with the first page, they should have your $\ad(Y)$ and not the transpose of it.
Answer to Q2:
T... | {
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For all but finitely many $n \in \mathbb N$ In my book I have the following theorem:
A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in \mathbb N$.
Can anyone clarify what the phrase, "All but finitely many", means?
| In your sentence, "but" (is not a conjunction, but a preposition and) means "except". At first, I didn't know that the word "but" has two meanings and thus I had exactly the same doubt as you.
Notice that the bold word "but" above (is a conjunction and) doesn't have the same meaning as in your sentence.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Deriving the formula of a summation Derive the formula for
$$
\sum_{k=1}^n k^2
$$
The solution's that I was given has $k^3 + (k-1)^3$ as the first step but doesn't say how it got to that. Any help?
| Hint:
For nonnegative integers $n,r$ with $r\leq n$ it is surprisingly
easy to prove by induction that
$\sum_{k=r}^{n}\binom{k}{r}=\binom{n+1}{r+1}$
This result allows you to find formulas for $\sum_{k=1}^{n}k^{r}$
for $r=1,2,3,\ldots$
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that if $(a,b)=1$, $a\mid c$ and $b\mid c$, then $a\cdot b\mid c$
Show that if $(a,b)=1$, $a\mid c$ and $b\mid c$, then $a\cdot b\mid c$.
Tried
$c=a\cdot k$ and $c=b\cdot j$ with $k,j\in\mathbb{N}$ then $a\cdot b\mid c^2=c\cdot c$.
| $c=a.k=b.j$
But $(a,b)=1$, and $a$ divides $b.j$, so $a$ divides $j$. Hence $a.b$ divides $c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/600401",
"timestamp": "2023-03-29T00:00:00",
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How to prove $\sum_{n=0}^\infty \left(\frac{(2n)!}{(n!)^2}\right)^3\cdot \frac{42n+5}{2^{12n+4}}=\frac1\pi$? In an article about $\pi$ in a popular science magazine I found this equation printed in light grey in the background of the main body of the article:
$$
\color{black}{
\sum_{n=0}^\infty \left(\frac{(2n)!}{(n!)^... | $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displaystyle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/600483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Prove the lecturer is a liar... I was given this puzzle:
At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are leaving in random order, the probability of that is precisely 1/3." Show the lecturer is l... | Let $a$ = the number of women, $b$ = the number of men, and $n = a + b$ be the total number of attendees.
The probability that the first 3 students to leave are all female is $\frac{a}{n} \cdot \frac{a-1}{n-1} \cdot \frac{a-2}{n-2}$. Setting this expression equal to $\frac{1}{3}$ and cross-multiplying gives $3a(a-1)(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/600595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Change $y= (1/2)x +1$ into standard form and get the answer $x-2y=-2$ I know the answer to the problem because I can check the answers in the back of the book but when I do the work myself I get
$$
-\frac{1}{2}x +y= 1
$$
when I attempt to change it into standard. I need a step by step explanation on how the book got
... | For standard form your x-coefficient needs to be positive and all coefficients must be integers, so multiply the entire expression by -2
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Complete representative set of squares modulo $15$.
$2.\,\,$Do the following computations.
$\text{(a)}$ Solve the equation $x^2\equiv 1\mod15$
Solution: We only need to choose a complete representative set modulo $15$ and verify the equation over such a set. In the following table, we choose the representative set
... | If you really wanted to, you could choose the set $\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14\}$, but notice that $8 \equiv -7 \mod 15$, $9 \equiv -6 \mod 15$, etc. So then it is more convenient to choose the representative set $\{0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7\}$. This cuts down the amount of work from $... | {
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Let $T:\mathbb{R}^{p} \rightarrow \mathbb{R}$ linear and $K=\left \{ \overline{x}\in \mathbb{R}^{p}:||x||_{2}\leq1 \right \}$. Show that T(K)=[-M,M] Let $T:\mathbb{R}^{p} \rightarrow \mathbb{R}$ linear (i.e. $T(x)=a_{1}x_{1}+\dots +a_{p}x_{p} $ ) and $K=\left \{ \overline{x}\in \mathbb{R}^{p}:||x||_{2}\leq1 \right \}$... | To find $M$:
$Max$ $T(x)$
s.t. $||x||_{2}=1$
$\mathcal{L}=a_{1}x_{1}+ \cdots + a_{p}x_{p}-\lambda[(x_{1}^{2}+ \cdots + x_{p}^{2} )^{\frac{1}{2}}-1]$
F.o.c.
$x_{i}:\text{ } \frac{a_{i}}{\lambda}$
$\lambda: \text{ }(x_{1}^{2}+ \cdots + x_{p}^{2} )^{\frac{1}{2}}=1$
$\therefore x^{*}=T(\overline{e}_{1}+ \cdots +... | {
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A diffusion partial differential equation, or Sturm-Liouville eigenvalue ODE What is the analytical solution for the following diffusion partial differential equation (initial value problem)?
$$\frac{\partial f}{\partial t} = (ax^2+b)\frac{\partial f}{\partial x}+\frac{\partial^2 f}{\partial x^2},$$
where $a$ and $b$ a... | Using Maple we get the solution is: $$f(x,t)=F1(x)\cdot F2(t)$$ where $F1$ and $F2$ are functions such that $$F1_{xx}=c_1\cdot F1-(a\cdot x^2-b)F1_x \quad\text{ and } $$$$F2_t=c_1\cdot F2, $$ where $c_1$ is an arbitrary constant. The ODE for $F1$ has an "explicit" solution in terms of the Heun Triconfluent function (ve... | {
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Prove a set is closed
Suppose $f\colon \mathbb R \to \mathbb R$ is a continuous function and $K$ is a closed subset of $\mathbb R$. Prove that the set $A = \{x \in \mathbb R : f(x) \in k\}$ is also closed.
Could someone show me direction as I am lost?
| The definition of a continuous function is : A function that inverse image of open (closed) sets are open (closed).
Thus your set $A=f^{-1}(K)$ is closed.
| {
"language": "en",
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$X$ is homeomorphic to $X\times X$ (TIFR GS $2014$) Question is :
Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:
*
*$X$ is not connected.
*$X$ is not compact
*$X$ is not homeomorphic to a subset of $\mathbb{R}$
*None of the abo... | The Cantor set is a counter-example to the second and third statement. Note that the Cantor set is homeomorphic to $\{0,1\}^{\mathbb N}$, hence it is homeomorphic to the product with itself.
An infinite set with the smallest topology (exactly two open sets) is a counter-example to the first statement. Martini gives a b... | {
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When does a Eigendecomposition result in a Q with det(Q)=1? With Eigendecomposition I can decompose a symmetric real matrix $A$ into $Q\Lambda Q^T$, where $Q$ is orthogonal. If $det(Q)=1$, $Q$ is a rotation matrix and if $det(Q)=-1$, $Q$ is a rotation matrix with reflection.
Is there a way to know beforehand if $det(... | You can always arrange that $\det Q = 1$. If $q_1, \ldots, q_n$ is your eigenbasis (the columns of $Q$), note that
$\det(q_1, \ldots, q_n) = -\det(q_2,q_1,\ldots, q_n)$ and $(q_2,q_1, \ldots, q_n)$ is also an orthonormal eigenbasis.
| {
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Question on probability in hashing Consider a hash table with n buckets, where external (overflow) chaining is used
to resolve collisions. The hash function is such that the probability that a key
value is hashed to a particular bucket is
1/n.
The hash table is initially empty and K
distinct values are inserted in the ... | (a) The probability that bucket 1 is empty after ONE insertion is $(n-1)/n$. That's the probability that the first item didn't hash to bucket 1. The event that it's empty after TWO insertions is defined by "first item missed bucket 1" AND "2nd item missed bucket one". With this, you can (I hope) compute the probability... | {
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Weierstrass $\wp$ function question Given the Weierstrass $\wp$ function with periods $1,\tau$ and $\wp(z) := \sum\limits_{n^2+m^2\ne 0} \frac{1}{(z+m+n\tau)^2}-\frac{1}{(m+n\tau)^2}$, I am trying to show $\wp = (\pi^2 \sum\limits^\infty_{n=-\infty} \frac{1}{\sin^2(\pi(z+n\tau))})+K$ for some constant $K$. Note I am n... | The sum
$$\sum_{m,n\in\mathbb{Z}} \frac{1}{(z+m+n\tau)^2}$$
does not converge absolutely, so working with that is not easy, you have to explicitly prescribe the order of summation to get a well-defined sum, and need to justify each manipulation of the sum accordingly. That can be done here, but I think it's easier to p... | {
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Explain complex numbers My cousin asked me if I could provide him with a practical example with complex numbers. I found it hard to do, so does anyone have a easy practical example with the use of complex numbers?
I tried to show him that complex numbers is needed to solve $x^2 = -1$, but he was not impressed.
| Prelude
You mentioned in your comments that he is 13 years old.
I'm only a couple years older than that, and don't have any knowledge of practical uses.
Short Answer
However, I can tell you what imaginary numbers are used for (more generically): to describe numbers that aren't real.
I think it is best described with a... | {
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Begging the question in Rudin? I read this in Theorem 2.35 of Baby Rudin:
Corollary. In the context of metric spaces) If $F$ is closed and $K$ is compact then $F \cap K$ is compact.
Proof. Because intersections of closed sets are closed and because compact subsets of metric spaces are closed, so is $F \cap K$; since $... | I have a Second Edition (1964) of Rudin in which the proof is given this way:
Theorems $2.26(b)$ and $2.34$ show that $F\cap K$ is closed; since
$F\cap K \subset K$, Theorem $2.35$ shows that $F\cap K$ is compact.
Theorem $2.26(b)$ says that intersections of closed sets are closed, $2.34$ says that compact subsets ... | {
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What can be said about a function that is odd (or even) with respect to two distinct points? This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that
$$
(1): \quad f(x) = -f(-x... | No. There's really not much else to say. You get a periodic function that's got a certain symmetry (on some period, it's "even", and on a period offset from this by a half-period, it's also even) from your conditions. But if I give you a periodic function satisfying this "double evenness" property on some period, then ... | {
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Proof that there is an order relation For an arbitrary set M there is a relation $R \subseteq 2^M \times 2^M$ about
$$ A \mathrel R B \Leftrightarrow A \cup \{x\} = B$$ The join is a disjoint join. There are not more details what is $x$.
Show that $R^*$, the reflexive and transitive hull of R is a order relation.
So I ... | Since the union is specified be a disjoint union, $R$ is not itself reflexive: you cannot choose $x\in A$, since in that case $A$ and $\{x\}$ are not disjoint. In fact $A\,R\,B$ if and only if $B$ is obtained from $A$ by adding one extra element of $M$ that was not in $A$.
I would begin by taking the reflexive closure ... | {
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Why these are equivalent? Situation: operator theory, spectrum of a operator.
We consider this as definition:
$\lambda$ is a eigenvalue if $\lambda x=Tx$ for some $x\ne 0$
but I see someone saying this:
$\lambda x-Tx=0\not \Rightarrow x=0 $ so $\lambda $ is a eigenvalue.
I cannot see why the latest sentence impli... | The statement "$\lambda x = Tx$ for some nonzero $x$" is the same as "$\lambda x - Tx = 0$ for some nonzero $x$." So if $\lambda x - Tx$ doesn't imply $x = 0$, then there's a nonzero $x$ satisfying the equation, so you're back to the first statement.
It might help to see that both formulations are equivalent to the thi... | {
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Is $\sqrt x \sin\frac{1}{x}$ continuous at $0$? If it is not defined, does it count as continuous? Is $\sqrt x \sin\frac{1}{x}$ continuous at $0$?
I found the limit of the function which is $0$, but the function is not defined at $0$. Is it continuous then?
| If the function is undefined, it cannot be continuous. However, if the limit exists, you can define $g(x)$ to be $\sqrt{x} \sin(1/x)$ for $x \neq 0$ and let $g(0)=0$. Then $g$ would be continuous (provided you took the limit correctly).
| {
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Minimal time to ride all ski slopes
Suppose we want to know what the minimum time is to ride all ski slopes on a mountain. We know the time it takes to ride a slope, and we know the time it takes to take a ski lift to get from one ski station to another, given that we have to end up where we started.
This screems Mi... | As far as I can tell, your problem is a lightly disguised version of Traveling Salesman (note that since you have to ski all the slopes, for algorithmic purposes the skiing time is irrelevant, all that matters is the time to travel between slopes.)
| {
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Why is the Cech nerve $C(U)$ of a surjective map $U\to X$ weakly equivalent to $X$? Let $f:U\to X$ be a surjective map of sets and
$$
...U\times_XU\times_XU \substack{\textstyle\rightarrow\\[-0.6ex]
\textstyle\rightarrow \\[-0.6ex]
\textstyle\rightarrow}
U\times_X\times U \s... | Show that $f': C(U)\to X$ is an acyclic fibration. Using the fact that $C(U)$ is a groupoid, it suffices to verify two conditions.
| {
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Another question on continuous functions and Cauchy's Integral formula. Let $C(z,r)$ denote the circle centered at $z$ with radius $r$. Let $f$ be a continuous function defined on a domain $D$. For $n=1,2$ and each $z \in D$ let
$A_n(z)=\lim_{r\to 0} \frac{1}{2\pi ir^n} \int_{C(z,r)} f(\zeta) d\zeta$ if the limit exi... | Hint: Consider
$$f(z) := \frac{\overline{z}}{|z|^{\frac{1}{2}}}$$
| {
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Cohomology groups for the following pair $(X,A)$ Let $X=S^1\times D^2$, and let $A=\{(z^k,z)\mid z\in S^1\}\subset X$. Calculate the groups and homomorphisms in the cohomology of the exact sequence of the pair $(X,A)$.
I know that theorically one has $$0\rightarrow C_n(A)\rightarrow C_n(X)\rightarrow C_n(X,A)\rightarr... | As both $X$ and $A$ are homotopic to $\mathbb S^1$, $H_1(X) = H_1(A) = \mathbb Z$ and all other homology groups vanish. The long exact sequence is
$$0 \to H_2(X, A) \to \mathbb Z \overset{f}{\to} \mathbb Z \to H_1(X, A)\to \mathbb Z \overset{g}{\to} \mathbb Z \to H_0(X, A)\to 0,$$
where the first two $\mathbb Z$'s corr... | {
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Describe units and maximal ideals in this ring
If $p$ is a fixed prime integer, let $R$ be the set of all rational numbers that can be written in a form $\frac{a}{b}$ with $b$ not divisible by $p$. I need to describe all the units in $R$ and all maximal ideals in $R$.
$\mathbb{Z} \subset R$, because $n=\frac{n}{1}$ f... | You have an example of something called a local ring, this is a ring with a unique maximal ideal. You are taking the ring $\mathbb{Z}$ and you are localizing at $(p)$. (In short, localization a ring $R$ at a prime ideal $I$ is just inverting everything that is outside of $I$. The fact that $I$ is prime tells us that $R... | {
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how does $\sum_{n=0}^{\infty} (-1)^n \frac{1}{1+n}$ diverge? I thought that to prove an alternating series two tests needed to be proven
$$a_n \ge a_{n+1}$$
which is true and
$$ \lim_{n\to\infty} b_n = 0 \ \ \ \ \ \ \text{which} \ \ \ \ \ \ \lim_{n\to\infty}\frac{1}{1+n}=0$$
yet sources (wolfram alpha) indicate tha... | In fact it converges. Let $b_n=(-1)^n$ and $a_n=\frac {1}{n+1}$. Then $a_n>0$ is decreasing that goes to $0$.Also $b_n$ has bounded partial sums because $\sum_{k=0}^{n} b_n\leq 1$.So using Dirichlet's Proposition we have that it converges. See here http://en.wikipedia.org/wiki/Dirichlet's_test
| {
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Compute $\lim_{x\to 0}\dfrac{\sqrt[m]{\cos x}-\sqrt[n]{\cos x}}{x^2}$ For two positive integers $m$ and $n$, compute
$$\lim_{x\to 0}\dfrac{\sqrt[m]{\cos x}-\sqrt[n]{\cos x}}{x^2}$$
Without loss of generality I consider $m>n$ and multiply the numerator with its conjugate. But what next? Cannot proceed further! Help pl... | Since $x$ is going to zero, expand $\cos x$ as a Taylor series (one term would be sufficient) and use the fact that, for small values of $y$, $(1-y)^a$ is close to $(1-a y)$ (this is also coming from a Taylor series). So, you will easily establish that
$\cos^{1/m}(x) = 1- \dfrac{x^2}{2m}$
Doing the same for $n$, you en... | {
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Topology - Projections I'm pretty sure I have this right, but want to double check and make sure.
Let $X_1$ = $X_2$ = $\mathbb{R}$ and let $p_1: X_1 \times X_2 \rightarrow X_1$ and $p_2: X_1 \times X_2 \rightarrow X_2$ be the projections. Let $A = {(x,y): 1 \le x \le 2, 3 \le y \le 3x}$. Find $p_1(A)$ and $p_2(A)$. ... | Your solution is correct.
Since $A=\{(x,y)\mid 1\le x\le2, 3\le y \le3x \}$, its projection $p_1(A)$ is a subset of $[1,2]$. On the other hand, for each $x\in[1,2]$ there is a $y\in[3,3x]$, so there is a point in $A$ which projects to $x$, thus $p_1(A)=[1,2]$
Since $3\le y\le3x\le6$, we have $p_2(A)\subseteq[3,6]$. And... | {
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How many different colourings are there of the cube using 3 different colours? A cube is called coloured if each of its faces is coloured by one of 3 given colours. Two colourings are considered to be the same if there is a rotation of the cube carrying one colouring to the other. How would you prove there are exactly ... | Yes your approach is correct, and do you see that you should use a group of order 24? Which one? See also here for the full answer. But first try it yourself!
| {
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Hamming distance for a linear code I want to solve this exercise:
"""
Prove the equality $d_{min}(D)=\min\{wt_H(z) | z \in D \} $ for a linear code D.
"""
$wt_H $ denotes the Hamming weight. What is $d_{min}$? The read that it is the mininmum distance of the error? What do I have to calculate then to get this value $d_... | Let $x,y\in D$ be such that $d_{min}(D)=d(x,y)$. Note that $d(a,b)=d(a-c,b-c)$ for all $a,b,c\in D$. Hence $d_{min}(D)=d(x,y)=d(x-y,0)=wt_H(x-y)$.
| {
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Given the probability that X > Y and X If $P[X>Y]=\frac{1}{2}$, and $[X<Y]=\frac{1}{2}$, then is $E[X]=E[Y]$?
How can I visualize this problem?
| Assume that $X$ is standard normal and consider
$$Y=2X+X^+.$$
Then $E[X]=0$, $[Y\gt X]=[X\gt0]$ hence $P[Y\gt X]=\frac12$, and $[Y\lt X]=[X\lt0]$ hence $P[Y\lt X]=\frac12$.
Furthermore, $E[Y]=2E[X]+E[X^+]=2\cdot0+\frac1{\sqrt{2\pi}}$. Thus, $E[Y]\ne E[X]$.
| {
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When can a metric space be embedded in the plane? It's easy to check if a graph can be embedded in the plane: just check for forbidden minors. Is it also easy to check if a "distance function" can be embedded?
Are there any necessary and sufficient conditions one can check?
I know there's a lot of research into approx... | I have the following
Conjecture. A metric space $(X,d)$ can be isometrically embedded in the plane $\mathbb R^2$ endowed with the standard metric $\rho$ iff each four-point subspace of $(X,d)$ can be isometrically embedded in the plane.
To build such an embedding $i$ we fix three different points $x$, $y$ and $z$ of $... | {
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Are there simple methods for calculating the determinant of symmetric matrices? I've seen that there are lots of exercises about determinants of symmetric matrices in my algebra books. Some are easy and others are a bit more twisted, but the basic problem is almost always the same. I have been trying to come up with a ... |
Edit (July 2021): As suggested in the comment, the answer here calculated the determinant of
$$\begin{pmatrix}
\ a & b & c \\
b & c & a \\
c & a & b \end{pmatrix},$$
instead of the one in the post.
Original answer:
do R1 --> R1+R2+R3
take out $(a+b+c)$
you will end up with
$$=(a+b+c)\begin{pmatrix}
\ 1 & 1 & 1 \\
b &... | {
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How to show that every $\alpha$-Hölder function, with $\alpha>1$, is constant? Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some $\alpha >1$
and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$.
I'm not sure which theorem should I look to prove this question. Can you gu... | Hint: Show that $f'(y)$ exists and is equal to $0$ for all $y$. Then as usual by the Mean Value Theorem our function is constant.
| {
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Radius of Convergence for analytic functions I know that the radius of convergence of any power series can be found by simply using the root test, ratio test etc.
I am confused as to how to find the radius of convergence for an analytic $f$ such as
$f(z)=\frac{4}{(z-1)(z+3)}$.
I can't imagine that I would have to fi... | It is very useful to remember that the radius of convergence of power series in the complex plane is basically the distance to nearest singularity of the function. Thus if a function has poles at $i$ and $-i$ and you do a power series expansion about the point $3+i$, then the radius of convergence will be $3$ since tha... | {
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Recursive Square Root Futility Closet This post on Futility Closet the other day: http://www.futilitycloset.com/2013/12/05/emptied-nest/
asked for the solution to this equation:
\begin{equation}\sqrt{x+\sqrt{x+\sqrt{x...}}} = 2\end{equation}
The problem can be described recursively as
\begin{equation} \sqrt{x + 2} = 2 ... | Some steps for further investigation.
*
*If $x > 0$ is fixed, show that the nested radical
$$
\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}
$$
converges to a positive number.
*Define the function
$$
f(x) = \sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}.
$$
Show that $f(x) > 1$ for all $x > 0$.
*Calculate
$$
\lim_{x\to 0^+} f(x).
$$
| {
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Uniform convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}}$ Prove that
$$ f(x) = \sum\limits_{n=1}^\infty \frac{(-1)^n}{ne^{nx}} = \sum\limits_{n=1}^\infty \frac{(-e^{-x})^n}{n}$$
is uniform convergent for $x \in [0,\infty)$.
Attempt:
At first, this looked a lot like the alternating series. At $x=0$, it is... | The series converges uniformly on $[1,\infty)$ by the Weierstrass M-Test (thanks to the exponential term). To prove it converges uniformly on $[0,1]$, use properties of convergent alternating series. For any convergent alternating series of reals with terms nonincreasing in absolute value, the absolute value of the d... | {
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Application of Fenchel Young- Inequality i'm stuck on the weak duality ineqiality.
For $X,Y$ euclidean spaces: $f: X\rightarrow (-\infty,\infty]$, $g: Y\rightarrow (-\infty,\infty]$ and $A:X\rightarrow Y$ linear bounded mapping.
I want to show that $\inf_{x\in X}\{f(x)+g(Ax)\}\geq \sup_{y \in Y}\{-f^{*}(A^{*}y)-g^{*}(-... | One approach is to reformulate the primal problem as
\begin{align*}
\operatorname*{minimize}_{x,y} & \quad f(x) + g(y) \\
\text{subject to} & \quad y = Ax.
\end{align*}
Now formulate the dual problem. The Lagrangian is
\begin{equation*}
L(x,y,z) = f(x) + g(y) + \langle z, y - Ax \rangle
\end{equation*}
and the dual f... | {
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Polynomial Interpolation and Error I have numerical analysis final coming up in a few weeks and I'm trying to tackle a practice exam.
Assuming $p(x)$ interpolates the function $f(x)$, find the polynomial $p(x)$ that satisfies the following conditions:
$$p(0) = 20, p(1) = 26, p'(1)=9, p(2) = 36, p'(2)=16$$.
I also have... | As asked by Neurax, i describe the way of getting the function without using matrix for the 5x5 system.
I keep the notation used on my previous answer and I use the conditions in the order they appear in the initial post.
Equations then write
p(0) = 20 gives a = 20
p(1) = 26 gives a + b + c + d + e = 26
p'(... | {
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Spring Calculation - find mass
A spring with an $-kg$ mass and a damping constant $9$ can be held stretched $2.5 \text{ meters}$ beyond its natural length by a force of $7.5 \text{ Newtons}$. If the spring is stretched $5 \text{ meters}$ beyond its natural length and then released with zero velocity, find the mass tha... | Critical damping occurs when $(\gamma^2)-4mk=0$
Therefore,
$$(9^2)-4m(7.5/2.5)=0$$
$$81=12m$$
$$m=6.75 kg$$
Bob Shannon also got the same answer, but I wanted to present it this way because I thought it might be more clear to some.
| {
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Why does Trapezoidal Rule have potential error greater than Midpoint? I can approximate the area beneath a curve using the Midpoint and Trapezoidal methods, with errors such that:
$Error_m \leq \frac{k(b-a)^3}{24n^2}$ and $Error_T \leq \frac{k(b-a)^3}{12n^2}$.
Doesn't this suggest that the Midpoint Method is twice as a... | On an interval where a function is concave-down, the Trapezoidal Rule will consistently underestimate the area under the curve. (And inversely, if the function is concave up, the Trapezoidal Rule will consistently overestimate the area.)
With the Midpoint Rule, each rectangle will sometimes overestimate and sometimes... | {
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Simultaneous irreducibility of minimal polynomials
Let $F$ be a field. Let $u,v$ be elements in an algebraic extension of $F$ with minimal polynomials $f$ and $g$ respectively. Prove that $g$ is irreducible over $F(u)$ if and only if $f$ is irreducible over $F(v)$.
I have only obtained that $f,g$ are irreducible ove... | $g$ is irreducible over $F(u)$ if and only if $[F(u,v):F(u)]=\deg g$ and $f$ is irreducible over $F(v)$ if and only if $[F(u,v):F(v)]=\deg f$. If consider $[F(u,v):F]$ you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/603904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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polynomial over a finite field Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$
Any ideas how to prove it?
| Take some element $\alpha_1\in F$
Then consider $f_1(x)=(x-\alpha_1)+1$.. What would be $f_1(\alpha_1)$?
Soon you will see that $f(\alpha_1)$ is non zero but may probably for some $\alpha_2$ we have $f_1(\alpha_2)=0$
Because of this i would now try to include $(x-\alpha_2)$ in $f_1(x)$ to make it
$f_2(x)=(x-\alpha_1)(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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idea for the completion of a metric space While doing the proof of the existence of completion of a metric space, usually books give an idea that the missing limit points are added into the space for obtaining the completion. But I do not understand from the proof where we are using this idea as we just make equivalenc... | For a metric space $\langle T, d\rangle$ to be complete, all Cauchy sequences must have a limit. So we add that limit by defining it to be an "abstract" object, which is defined by "any Cauchy sequence converging to it".
We have two cases:
*
*The Cauchy sequence already had a limit in $T$. In this case there is no n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/604070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Alternating sum of binomial coefficients $\sum(-1)^k{n\choose k}\frac{1}{k+1}$ I would appreciate if somebody could help me with the following problem
Q:Calculate the sum:
$$ \sum_{k=1}^n (-1)^k {n\choose k}\frac{1}{k+1} $$
| $$
\begin{align}
\sum_{k=1}^n(-1)^k\binom{n}{k}\frac1{k+1}
&=\sum_{k=1}^n(-1)^k\binom{n+1}{k+1}\frac1{n+1}\\
&=\frac1{n+1}\sum_{k=2}^{n+1}(-1)^{k-1}\binom{n+1}{k}\\
&=\frac1{n+1}\left(1-(n+1)+\sum_{k=0}^{n+1}(-1)^{k-1}\binom{n+1}{k}\right)\\
&=-\frac n{n+1}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/604173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to prove there exists a solution? Guillemin Pollack Prove there exists a complex number $z$ such that
$$
z^7+\cos(|z^2|)(1+93z^4)=0.
$$
(For heaven's sake don't try to compute it!)
| Although the answers above are correct ones, they fail to use $deg_2$ as the book of Guillemin & Pollack suggest. Heres an approach that use the notion of $deg_2$:
Let $f:\mathbb{C} \to \mathbb{C}$ be defined as
$$
f(z)=z^7+\cos(|z^2|)(1-93z^4).
$$
Consider the homotopy $F(z,t)=tf(z)+(1-t)z^7$ between $f(z)$ and $z^7$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$ for positive integers $x$ and $y$? Let $x$ and $y$ be positive integers. Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$?
I ran a program for $1\le{x,y}\le1\text{ }000\text{ }000$ and found no solution, so I believe there are none.
| The equation $x(x+1)(x+2) = 3y(y+1)(y+2)$ is equivalent to $\left(\frac{24}{3y-x+2}\right)^2 = \left(\frac{3y-9x-6}{3y-x+2}\right)^3-27\left(\frac{3y-9x-6}{3y-x+2}\right)+90$.
This is an elliptic curve of conductor $3888$. Cremona's table says its group of rational points is of rank $2$, and is generated by the obvious... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/604333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Integral involving a confluent hypergeometric function I have the following integral involving a confluent hypergeometric function:
$$\int_{0}^{\infty}x^3e^{-ax^2}{}_1F_1(1+n,1,bx^2)dx$$
where $a>b>0$ are real constants, and $n\geq 0$ is an integer.
Wolfram Mathematica returns the following solution: $\frac{a^{n-1}(a+b... | Let's start with the hypergeometric function. We have:
\begin{eqnarray}
F_{2,1}[1+n,1;b x^2] &=&
\sum\limits_{m=0}^\infty \frac{(1+n)^{(m)}}{m!} \cdot \frac{(b x^2)^m}{m!} \\
&=& \sum\limits_{m=0}^\infty \frac{(m+1)^{(n)}}{n!} \cdot \frac{(b x^2)^m}{m!} \\
&=&\left. \frac{1}{n!} \frac{d^n}{d t^n} \left( t^n \cdot e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/604502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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If $R[s], R[t]$ are finitely generated as $R$-modules, the so is $R[s + t]$.
Let $S \supset R$ as rings with $1 \in R$. Suppose that $s, t \in S$ and that the subrings $R[s], R[t]$ are finitely generated by $\{1, s, \dots, s^k\}$ and $\{1, t, \dots, t^m \}$. Then $R[s + t]$ is also finitely generated.
Let $g$ be a ... | Let $R[s]$ be fin-gen with monic $f$ and $R[t]$ fin-gen with monic $g$.
$(X + t)^k, k \geq 1$, is a polynomial in $R[t][X]$ . Since $s$ is integral over $R$ with $f$ it's also integral over $R[t]$ with $f$ since $R \subset R[t]$. Then $(X + t)^k = q(X) f(X) + r(X)$ for some $r = 0$ or $\deg r \lt \deg f$. This means... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Of any 52 integers, two can be found whose difference of squares is divisible by 100 Prove that of any 52 integers, two can always be found such that the difference of their squares is divisible by 100.
I was thinking about using recurrence, but it seems like pigeonhole may also work. I don't know where to start.
| Look at your $52$ integers $\mod 100$. So, the difference of their squares resulting in division by $100$ can be given by $a^2=b^2(\mod 100)$. This will resolute in product of the difference of the numbers and sum of the numbers is divisible by $100$. since, any of $52$ integer numbers are asked, there can be no optim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/604635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 7,
"answer_id": 3
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Pullback of a vector bundle on Abelian variety via $(-1)$ Let $A$ be an abelian variety over some field $k$ and $(-1) : A \to A$ is the inverse map of $A$ as an algebraic group. If $V$ is a vector bundle over $A$ what is $(-1)^* V$? In other words, is there a way to describe $(-1)^* V$ in more standart functors? If it ... | For any morphism $f:X \to Y$ of varieties (or schemes), one has $f^*\mathcal O_Y = \mathcal O_X$.
A line bundle on an ellipitic curve of degree $d$ is of the form $\mathcal O(D)$ where $D = (d-1)O + P$, where $O$ is the origin and $P$ is a point on the curve. Applying $[-1]^*$ takes this to $\mathcal O(D')$, where ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/604744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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1000 Doors Homework Problem I am faced with the following problem as homework- a man has 1000 doors. he opens every door, and then he closes every second door. Then he works on every third door- if it's open, then he closes it. if it's closed, he opens it. Then he works on every fourth door, fifth door, and so on all ... | This is an absolute classic, in the first round he opens all of them, then he closes multiples of 2. Then he alters multiples of 3. So in round $j$ the door $a$ is opened or closed if and only if j divides it. How many times is each door altered think about its divisors.
full solution:
a door $j$ is altered the same... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sequence of distinct moments of $X_{n}$ converging to $1$ implies $X_{n}$ converges to $1$ Suppose for $0<\alpha<\beta$ and $X_{n}\geq0$ we have $EX_{n}^{\alpha},EX_{n}^{\beta}\to1$ as $n\to\infty$. Show that $X_{n}\to1$ in probability. In special cases this is pretty clear (for instance, assuming $\alpha\geq1$ and $... | Define $Y_n:=X_n^\alpha$ and $p:=\beta/\alpha\gt 1$. The assumptions give that the sequence $\{Y_n,n\geqslant 1\}$ is tight, so it's enough to prove that each subsequence converges in distribution to the constant $1$.
Take $\{Y_{n'}\}$ a subsequence which converges in distribution to $Y$·
Since $\{Y_n,n\geqslant 1\}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Another limit of summation Please help - how to solve this:
$$\lim_{n \to \infty}\frac1{n}\sum_{k=0}^{n}{\frac{1}{1+\frac{k^6}{n^3}}} $$
| It seems the following.
Put $$f(k,n)= \frac{1}{1+\frac{k^6}{n^3}}.$$ Then $$\sum_{k=0}^{n} f(k,n)\le \sum_{k\le n^{2/3}}f(k,n) + \sum_{k\ge n^{2/3}} f(k,n)
\le n^{2/3}+\frac{n}{1+n}.$$ Hence $$0\le \lim_{n \to \infty}\frac1{n}\sum_{k=0}^{n} f(k,n)\le \lim_{n \to \infty} \frac1{n}\left(n^{2/3}+\frac{n}{1+n}\right)=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/605106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the Area of a rectangle please help a pool 20 ft by 30 ft is going to have a deck x feet wide added all the way around the pool write an expression in simplified form for the area of the deck. I have tried doing this but have failed please help
| You are on the correct path cris.
The pool is still a rectangle, even with the new deck. Thus you are almost correct.
Each side will have $x$ feet added to it.
Thus it will be $(20+2x)(30+2x) = 4x^2+100x+600$
Try finding the answer to this:
If there is only 120 Feet of bamboo for the new deck,what is the largest ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/605181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative Counterexamples - Calculus I need counterexamples for the following (I guess these claims are not correct):
*
*If $ lim_{n\to \infty} n\cdot (f(\frac{1}{n}) - f(0) ) =0$ then $f$ is differentiable at $x=0$ and $f'(0)=0$ .
*If f is defined in a neighberhood of $a$ including $a$ and differentiable at a ne... | For 1, if $\lim_{n\to\infty}n(f(1/n)-f(0))=0$ then we have $\displaystyle\lim_{n\to\infty}\frac{f(\frac{1}{n})-f(0)}{\frac{1}{n}}=0$. It is common to assume that $n$ denotes a natural number but this was not indicated in the problem statement. So assuming the stronger statement (i.e. that the limit is taken for $n\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/605233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Index map defines a bijection to $\mathbb{Z}$? In the book "Spin Geometry" by Lawson and Michelsohn, page 201, proposition 7.1(chapter III), it asserts that the mapping which assign a Fredholm operator from one Hilbert space to another its index ($\dim\ker-\dim\text{coker} $) defines a bijection from the set of connec... | If $H$ has infinite dimension, the shift operator under a base is Fredholm with index $-1$. As $ind$ is a groups homomorphism between $\text{Fredholms}/\text{Compact}$ and $\mathbb{Z}$, and $-1$ is a generator of $\mathbb{Z}$, this proves that $ind$ is indeed surjective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/605421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Extended ideals If $R\subset S$ is a ring extension where $1\in R$ and $I$ is an ideal of $R$ is it true that $IS$, the subset of $S$ generated by $I$ is an ideal of $S$? Should we assume $R$ is commutative?
| It would be sensible to look at an example where $S$ is a simple ring to limit the number of ideals possible in $S$.
Take, for example, the $2\times 2$ upper triangular matrices $R=T_2(\Bbb R)\subseteq M_2(\Bbb R)=S$, and consider the ideal $I$ of $R$ of matrices of the form $\begin{bmatrix}0&x\\0&0\end{bmatrix}$.
Clea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Index of conjugate subgroup equals the index of subgroup Show that if $G$ is a group and $H$ is its subgroup then $[G:H]=[G:gHg^{-1}]$, $g \in G$.
Attempted solution:
Let $f:G\mapsto\hat{G}$ be a group homomorphism such that $\mbox{Ker}f \subseteq H$ we will try to show that $[G:H]=[f(G):f(H)]$.
Define a map $\phi: xH\... | Hint: construct directly a set theoretic bijection (not a homomorphism) between the cosets of $H$ and those of $gHg^{-1}$.Fix a $g \in G$ and let $\mathcal{H}=\{xH: x \in G\}$ the set of left cosets of $H$ and let $\mathcal{H}'=\{xH^g: x \in G\}$ the set of left cosets of $H^g:=gHg^{-1}$. Define $\phi:\mathcal{H} \ri... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Unit circle - how to prevent backward rotation Let's assume we have a unit circle (0, 2$\pi$).
Basically I have a point on this circle who is supposed to move only forward. This point is controlled by the user mouse and constantly calculate 25 times per seconds.
For the moment I calculate the new angle ( based on the u... | It is not trivial. As long as you update frequently enough that the user cannot move more than $180$ degrees between updates you can just find whether to add or subtract $360$ to get the change as close to $0$ as possible, so
If (new_angle - old_angle > 180) do not update because real rotation is negative
If (-360 < ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/605716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $f : X → Y$ be a continuous closed surjection such that $f^{–1}(y)$ is compact for all $y ∈ Y .$ Let $f : X → Y$ be a continuous closed surjection such that $f^{–1}(y)$ is compact for all $y ∈ Y .$ Suppose that $X$ is Hausdorff. Prove that $Y$ is Hausdorff.
I have that $f$ is a qoutient map, but I can not think of ... | Just take $y, y' ∈ Y$. Their preimages are compact disjoint subsets of $X$ and so can be separated by disjoint open sets. Complements of images of these sets are open disjoint and separating $y, y'$ in $Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/605784",
"timestamp": "2023-03-29T00:00:00",
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Borel function which is not continuous (in every point) Give example function $f: \mathbb{R} \rightarrow \mathbb{R}$ which $\forall x \in \mathbb{R}$ is not continuous function but is Borel function.
I think that I can take $$f(x) = \begin{cases} 1 & x \in \mathbb{R} \setminus \mathbb{Q} \ \\ -1 & x \in \mathbb{Q} \en... | Yes, you're correct. To show that it's a Borel function, begin by showing that if $\mathcal{O} \subseteq \mathbb{R}$ is an open set, then $f^{-1}(\mathcal{O})$ is one of exactly four possible sets: $\emptyset, \mathbb{Q}$, $\mathbb{R} \setminus \mathbb{Q}$, and $\mathbb{R}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to verify $y_c(x)+y_p(x)$ is a solution to the differential equation? I am given a nonhomogeneous differential equation:
$$y''+4y'+3y=g(x)$$
where $g(x)=3 \sin 2x$.
After working through the problem, I have
$$y_c(x)=C_1e^{-3x}+C_2e^{-x}$$
(I was to find a general solution for which $g(x)=0$)
$$y_p(x)=-(24/65) ... | Well, your solution $y_{c}(x)$ satisfies the problem $y'' + 4y' + 3y = 0$ and $y_{p}(x)$ satisfies the problem $y'' + 4y'+ 3y = g(x)$. So, $(y_{c}+y_{p})'' + 4(y_{c}+y_{p})' + 3(y_{c}+y_{p}) = [y_{c}'' + 4y_{c}' + 3y_{c}] + [y_{p}'' + 4y_{p}' + 3y_{p}] = 0 + g(x) = g(x)$. Hence, $y(x)$ satisfies the ODE. Note that deri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/605968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Double harmonic sum $\sum_{n\geq 1}\frac{H^{(p)}_nH_n}{n^q}$ Are there any general formula for the following series
$$\tag{1}\sum_{n\geq 1}\frac{H^{(p)}_nH_n}{n^q}$$
Where we define
$$H^{(p)}_n= \sum_{k=1}^n \frac{1}{k^p}\,\,\,\,\,H^{(1)}_n\equiv H_n =\sum_{k=1}^n\frac{1}{k} $$
For the special case $p=q=2$ in (1) I f... | In here we provide a generating function of the quantities in question. Let us define:
\begin{equation}
{\bf H}^{(p,r)}_q(t) := \sum\limits_{m=1}^\infty H_m^{(p)} H_m^{(r)} \frac{t^m}{m^q}
\end{equation}
In here we take $q\ge1$. We have:
\begin{eqnarray}
&&{\bf H}^{(p,1)}_q(t) = Li_p(1) \cdot \frac{1}{2} [\log(1-t)]^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Epsilon delta proof min http://www.milefoot.com/math/calculus/limits/DeltaEpsilonProofs03.htm
I've been studying these épsilon delta proofs. In the non-linear case, he gets:
$$\delta=\min\left\{5-\sqrt{25-\dfrac{\epsilon}{3}},-5+\sqrt{25+\dfrac{\epsilon}{3}}\right\}$$
Well, I know that these $\delta$ are not equal the ... | It is because $\delta$ has to be acceptable in the worst case. Say we are proving $\lim_{x \to 0} f(x)=L$ and for (the given) $\epsilon$ we are within $\epsilon$ over the interval $\delta \in (-1,0.1)$ The definition of limit is symmetric: it says whenever $x$ is within $\delta$ of $0$, then $|f(x)-L|\lt \epsilon$ s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $f(x)=\chi_{(0,\infty)}\exp(-1/x)$, show that $f\in C^{\infty}$. Define the function $f:\mathbb{R}\to\mathbb{R}$ as follow:
$f(x)=\chi_{(0,\infty)}\exp(-1/x)$
In other words: $f(x)=0$ if $x\le 0$, and $f(x)=\exp(-1/x)$ if $x>0$.
Show that $f\in C^{\infty}$.
So I think I want to show the nth derivative $f^{(n)}$ is ... | Just note that taking derivatives (by the definition, not by the algorithm) always leaves you with terms of the form $1/P(x) \cdot e^{1/x}$. Then use that the exponential grows faster than any polynomial at infinity, noting that as $x\to 0$, we have $\frac1{x}\to \infty$. More explicitly, for $f'(x)$,
$$\lim_{x\to 0}\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606219",
"timestamp": "2023-03-29T00:00:00",
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What is the particular solution for $ y''+2y'=2x+5-e^{-2x}$? How is the particular solution for $y''+2y'=2x+5-e^{-2x}$ be the following?
$$y_p = Ax^2 + Bx + Cxe^{-2x}$$
Shouldn't it be $y_p = Ax + B + Cxe^{-2x}$?
Anything of degree one should be in the form $Ax + B$, and $2x+5$ is in degree one and not squared... I ju... | Hint:
The particular solution is of the form:
$$y_p = a x + b x^2 + c x e^{-2x}$$
We have to take $a + b x$ and multiply by $x$ and multiply $e^{-2x}$ by $x$ because we already have a constant in homogeneous and also have $e^{-2x}$ in homogeneous.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Reasoning behind the cross products used to find area Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross product is used instead of the dot product.
I was hoping math.se could... | I think the signed part of area is the most difficult to assign some intuitive meaning.
Consider two vectors in $\mathbb{R^2}$, and let $A : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ be the signed area. Then $A$ should be linear in each variable separately, since we should have $A(\lambda x,y) = \lambda A(x,y)$,... | {
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Properties of the relation $R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$
$A= \Bbb R \\
R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$
Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, (g)equivalence relation.
This is what I did but I'm not sure:
It is reflexi... | Most of your answers are correct, but the justifications given are a little confusing. In general, you should offer a genuine proof. For example:
It is reflexive.
Proof. Let $x \in \mathbb{R}$ be fixed but arbitrary. Then $x-x=0$. Thus $x-x \in \mathbb{Z}.$ So $xRx.$
Anyway, your answers for "reflexive", "symmetric" an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Real polynomial in two variables I have problems proving the following result:
Each $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $\forall a,b \in \mathbb{R} \ : \ f_a(y) := f(a,y), \ f_b(x) := f(x,b) $ are polynomials is a polynomial with two variables.
If I consider $f$ as a function of $x$, then its derivativ... | Set $f(\cdot, y):x\mapsto f(x,y)$. Since this $f(\cdot, y)$ is $x$-continuous, when $y_n\to y$ the polynomials $f(\cdot, y_n)$ converge pointwise everywhere. Hence, for fixed $N$, the sets
$\{y| \deg f(\cdot, y)\leq N \}$ are closed. By Baire there exist $N$ and $Y\subset \Bbb R$ open, such that $\deg f(\cdot, y)\leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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} |
Exercise 3.3.25 of Karatzas and Shreve This is the Exercise 3.25 of Karatzas and Shreve on page 163
Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted process satisfying
$$E\int_0^T|X_t|^{2m}dt<\infty$$
for some real numbers $T>0$ and $m\geq1$,... | First of all, your calculation is not correct. Itô's formula gives
$$\begin{align*} M_T^{2m} &= 2m \cdot \int_0^T M_t^{2m-1} \, dM_t + m \cdot (2m-1) \cdot \int_0^t M_t^{2m-2} \, d \langle M \rangle_t \\ \Rightarrow \mathbb{E}(M_T^{2m}) &= m \cdot (2m-1) \cdot \mathbb{E} \left( \int_0^T M_t^{2m-2} \cdot X_t^2 \, dt \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Matrices such that $A^2=A$ and $B^2=B$ Let $A,B$ be two matrices of $M(n,\mathbb{R})$ such that $$A^2=A\quad\text{and}\quad B^2=B$$
Then $A$ and $B$ are similar if and only if $\operatorname{rk}A = \operatorname{rk}B$.
The first implication is pretty easy because the rank is an invariant under matrix similarity. But th... | Your way of thinking is very good.
Hint: If $L:V\to V$ is an idempotent linear transformation ($L^2=L$) then $$V=\ker L\oplus{\rm im\,}L\,.$$
Use the decomposition $v=(v-Lv)+Lv$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/606678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Median and Mean of Sum of Two Exponentials I have a cumulative distribution function:
$$G(x) = -ae^{-xb} - ce^{-xd}+h$$
The associated probability density function is:
$$g(x) = abe^{-xb} + cde^{-xd}$$
My problem concerns $x\ge 0, X \in R$.
I know that the mean (expected value) of $x$ can be computed by:
\begin{align}
E... | Your intuition is correct, the ratio of mean to median of a random variable $X$ with density of shape $abe^{-ax}+cde^{-cx}$ is not always the same as the ratio of mean to median of an exponentially distributed random variable. (The latter ratio, as your post pointed out, is $\frac{1}{\ln 2}$.)
To show this, it is enoug... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many times are the hands of a clock at $90$ degrees.
How many times are the hands of a clock at right angle in a day?
Initially, I worked this out to be $2$ times every hour. The answer came to $48$.
However, in the cases of $3$ o'clock and $9$ o'clock, right angles happen only once.
So the answer came out to be... | Yes, but a more “mathematical” approach might be this: In a 12 hour period, the minute hand makes 12 revolutions while the hour hand makes one. If you switch to a rotating coordinate system in which the hour hand stands still, then the minute hand makes only 11 revolutions, and so it is at right angles with the hour ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 9,
"answer_id": 2
} |
Try to solve the following differential equation: $y''-4y=2\tan2x$ I am trying to solve this equation:
$y''-4y=2\tan2x$
the Homogeneous part is:
$$y_h=c_1e^{2x}+c_2e^{-2x}$$
and I get according the formula:
$$C_1'e^{2x}+C_2'e^{-2x}=0$$
$$2C_1'e^{2x}-2C_2'e^{-2x}=2\tan2x$$
my questions is:
*
*if $y_h$ is right?
*how... | What you have done so far is correct. You should proceed as follows:
Write the last two equations as a system
$$\left(\begin{array}{cc}
e^{2x} & e^{-2x}
\\
e^{2x} & -e^{-2x}
\end{array}\right)
\cdot \left(\begin{array}{c} C_{1}^{\prime}
\\
C_{2}^{\prime}
\end{array}\right)=\left(\begin{array}{c}
0
\\
\tan(2x) \end{arra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Negative exponential distance Let $X := \left\{(a_k)_{k \in \mathbb N}, a_k \in \mathbb C\right\}$. Let $d\left( (a_k)_{k \in \mathbb N}, (b_k)_{k \in \mathbb N} \right) := e^{-u}$ with $u$ the smallest integer $k$ such that $a_k \ne b_k$ be a distance on $X$. Is $(X, d)$ compact/complete/connected?
Here's my not very ... | You are correct that $X$ is not compact, by exactly the example you mention.
On the other hand, $X$ is in fact complete. Observe that if $d((a_k), (b_k)) < e^{-n}$, then $a_i = b_i$ for all $i \leq n$. It follows that if $(\mathbf{a}_n)_{n\in \mathbb{N}} = ((a_{n,k})_{k\in\mathbb{N}})_{n\in \mathbb{N}}$ is a Cauchy s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/606953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that in an impartial Game, the P-Positions all have Sprague-Grundy Value =0 I'm looking at some work with Combinatorial Game Theory and I have currently got:
(P-Position is previous player win, N-Position is next player win)
Every Terminal Position is a P-Position,
For every P-Position, any move will result in a ... | If you can win a given game, you must use the following strategy: always move to a 0-position, so your opponent is forced to move to a non-zero position, then reply to a 0-position again. Eventually you reach a terminal position, because games end in a finite number of turns by definition. Since a terminal position has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/607044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$
I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine.
| $$\large \sin^2\theta + \cos^2\theta
=\sin\theta\sin\theta+\cos\theta\cos\theta
=\cos(\theta-\theta)
=\cos0
=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/607103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 16,
"answer_id": 10
} |
Find $f$ if $f(f(x))=\sqrt{1-x^2}$ Find $f$ if $f(f(x))=\sqrt{1-x^2} \land [-1; 1] \subseteq Dom(f)$
$$$$Please give both real and complex functions. Can it be continuous or not (if f is real)
| I guessed that $f$ is of the form $\sqrt{ax^2+b}$. Then, $f^2$ is $\sqrt{a^2x^2 + \frac{a^2-1}{a-1}b}$. From here on in, it is algebra:
$$
a^2 =-1 \implies a = i ~~~~\text{and}~~~~\frac{a^2-1}{a-1}b = 1 \implies b = \frac{1-i}{2}
$$
So we get $f(x) = \sqrt{ix^2 + \frac{1-i}{2}}$. I checked using Wolfram, and $f^2$ appe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/607234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Probability of an Event defined by two continuous random variables I'm having trouble solving this word problem. I have the answer, but do not know how to get there.
An electronic gadget employs two integrated circuit chips: a signal processing chip and a power condition chip, which fail independently of each other. ... | Hint: Let $Y$ be the random time taken for the power conditioning chip (PC) to fail and $X$ be the random time taken for the signal processing (SP) chip to fail. How do you denote the following event?$$\mathbb{P}[\{\text{Time required for SP chip to fail} > \text{Time required for PC chip to fail}\}]$$ If you figure ou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/607343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What do polynomials look like in the complex plane? I have a hard time visualizing the fundamental theorem of algebra, which says that any polynomial has at least one zero, superficially I know this is true as every polynomial must have either an imaginary zero or real zero, but how do I visualize this in the complex p... | See these:
*
*Visual Complex Functions by Wegert.
*Phase Plots of Complex Functions: A Journey in Illustration by Wegert and Semmler.
*The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
Try an interactive demo at http://www.math.osu.edu/~fowler.291/phase/.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/607436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 6,
"answer_id": 5
} |
Simple examples of $3 \times 3$ rotation matrices I'd like to have some numerically simple examples of $3 \times 3$ rotation matrices that are easy to handle in hand calculations (using only your brain and a pencil). Matrices that contain too many zeros and ones are boring, and ones with square roots are undesirable. A... | Some entries of the rotation matrix, whether $2 \times 2$ or $3 \times 3$, are the trigonometric functions; to ensure that entries of the matrix are simple numbers that are less computationally expensive, pick integer multiples of $\pi$ on which the trigonometric functions are either $1, -1$ or $0$.
Is that what you me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/607540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
Can a function be both upper and lower quasi-continuous? Can you give me a non-trivial example? Below is the definition I am using:
A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty ope... | For a slightly non-trivial example, consider
$$f(x)=\begin{cases}\sin\Bigl(\dfrac1x\Bigr)&x\ne0,\\a&x=0.\end{cases}$$
I think you will find that this function is quasi-continuous (i.e. upper and lower) if $\lvert a\rvert\le1$, more generally upper quasi-continuous iff $a\ge -1$ and lower quasi-continuous iff $a\le1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/607637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $\max \cos(x)$ is $1$ and $\min \cos(x)$ is $ -1$ Prove $\max \cos(x)$ is $1$ and $\min \cos(x)$ is $-1$
How to prove it with only calculus and not multivariable calculus?
Please notice that this is not a homework question, but a pre-exam question. Thanks a lot.
| It is $\cos{(\varphi)}=\Re{({e^{i\varphi}})}$ and $|e^{i\varphi}|=1$.
$|\Re{(z)}|\leq|z|$ together with the evaluation at $0$ and $\pi$ proves your question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/607729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Checking irreducibility of polynomials over number fields Are there general methods for checking irreducibility of polynomials over number fields? For instance, letting $F = \mathbb{Q}(\sqrt{3})$, I want to know whether $x^3 - 10 + 6\sqrt{3}$ is irreducible over $F$ (I know that it is but it's not trivial). There is th... | The "usual" algorithm for factoring polynomials over number fields go back to Kronecker, and uses the idea that it is essentially sufficient to factor the norm. for a bit of a discussion, see my recent preprint (which is really about something else, but gives a description when talking about an algorithm to compute the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/607842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
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