Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof of an integral property $$\int_0^1 f(x)g'(x)dx=\pi$$ If $f(1)g(1)=f(0)g(0)$ then $$\int_0^1 f'(x)g(x)dx= -\pi$$
So I have to prove this and I have absolutely no idea how to do it. I am guessing I will have to use the fundamental theorem of calculus and it show that the rate of change is $1$ because it didn't cha... | Hint:
$$\dfrac{d(f(x)\cdot g(x))}{dx}=?$$
Integrate both sides with respect to $x$ between $[0,1]$
| {
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"url": "https://math.stackexchange.com/questions/3120855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Discretization matrix for 3D Poisson equation It is known that the 2D Poisson equation defined on a domain $\Omega$ (let's say $\Omega := (0,1)^2$) with Dirichlet boundary conditions $u(x,y)_{|\partial \Omega}=g(x,y)$,
$$u_{xx} + u_{yy}=f$$
can be discretized, using finite differences, to obtain a system of linear equa... | I like a lot the Kronecker product representation. If $A_1$ is the one-dimensional second-derivative approximation, i.e.
$$
A_1 = \begin{pmatrix} 2 & -1 & \\ -1 & 2 & \ddots \\ & \ddots & \ddots \end{pmatrix},
$$
then you can write the $3d$ Laplacian as
$$
A_3 = I \otimes I \otimes A_1 + I \otimes A_1 \otimes I + A_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If a set $A$ is finite then $A\cap B$ is a finite set. Background:
Theorem - If $A\subseteq \mathbb{N}_n$ then $A$ is a finite set and $|A|\leq n$.
Question:
Let $A$ be a finite set and $B$ be some set. If $A$ is a finite set, then $A\cap B$ is a finite set.
Attempted proof:
Let $A$ be a finite set and let $C = A\ca... | Suppose $A\cap B$ is infinite. Then we have $x_1,x_2,x_3,....$ all in $A\cap B$. But then each of them is also in $A$. Then $A$ has ininite cardinality. A contradicition.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 3
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References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ I have read (Theorem 29.3, line 7, pg 144) the following statement:
$\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$.
and
Therefore $\pi_i(St_k(U(n))=0$ for $i \le 2(n-k)$.
... | I think they have made an off-by-$1$ error, they might have been thinking about the map $BU(n-k) \to BU(n)$. (The notes you're reading are unpublished and there is a disclaimer at the start about not being proofread, so an off-by-$1$ error isn't so bad.)
A great reference for a lot of results like this is Mimura and To... | {
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Is this subspace isomorphic to $\ell^1$? Let $X$ be a Banach space.
Suppose there exists a sequence $(x_n)$ in $X$ such that for all finite $A\subseteq\mathbb{N}$ we have that $\|\sum_{n\in A}x_n\|$ equals the number of elements in $A$.
Does this imply that the subspace spanned by $\{x_n\}$ is isomorphic to $\ell^1$?... | In order that the subspace spanned by $(x_n)$ is isomorphic to $l^1$ it has to be closed. Hence, we have to talk about the closure of the span of $(x_n)$.
If $X$ is reflexive, then this subspace is also reflexive, and cannot be isomorphic to $l^1$.
If $X$ is a Hilbert space (or strictly convex space) then no such sequ... | {
"language": "en",
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Prove convexity of set I've had a hard time proving this statement. The objective is to prove that the set $M$ is convex where $f(y)$ can be any function. The task is to prove it using triangle inequality. I've looked at threads like Proving Convexity of an Open Disk but i still can't wrap my head around it. If anyone ... | Another hint: $M$ is the intersection of a certain collection of closed balls.
| {
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Sample Space of a Fair Coin A coin is tossed until, for the first time, the same result appears twice in succession. Define a sample space for this experiment.
The solution in the back of the book is:
$[x_1x_2...x_n: n \ge 1, x_i \in [H, T]; x_i \ne x_{i+1}, 1 \le i \le n-2; x_{n-1}= x_n]$
I don't have a clue how this... | Its a list of the first $n$ coin flips given as $x_i$, with the restriction that the only consecutive tosses with the same outcome are $x_n$ and $x_{n-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3121503",
"timestamp": "2023-03-29T00:00:00",
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Can we prove that infinite many primes begin with any given digitstring? With Dirichlet's theorem, we can easily prove that infinite many primes end with a given digitstring with final digit $1,3,7$ or $9$.
Can we also prove that infinite many primes begin with a given digit-string with first digit non-zero ?
Intuiv... | Yes. All you need is a prime gap of the form $p_{n+1}-p_n\lt (p_n)^\theta$ for $\theta\lt 1$; this is well-known (Wikipedia suggests that Hoheisel was the first to prove a bound of this form). Once you've got that, it becomes a matter of simple math; let $K$ be the initial digit string, of length $k=\lceil\log_{10}(K)... | {
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Determining the basis for strings with cyclic and permutation symmetry I'd like to determine a basis set of strings for strings of length m composed of n letters that are compatible with cyclic symmetry of the string and permutation symmetry of the letters.
It's relatively straightforward to do this by hand in individu... | The problem of counting the basis strings is an instance of Power
Group Enumeration as defined by Harary and Palmer in the text
Graphical Enumeration. We have the cyclic group acting on the slots
with cycle index
$$Z(C_m) = \frac{1}{m} \sum_{d|m} \varphi(d) a_d^{m/d}.$$
The group acting on the colors is ... | {
"language": "en",
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Integral $\int_0^{2π} e^{e^{ix}} dx$
Work out the integral $$\int_0^{2π} e^{\large e^{ix}} \, dx.$$
I am now stuck with this for $2$ days, so please help!
Here is my try:
$$I=\int_0^{2π} e^{\large e^{ix}} dx=\int_0^{2\pi} e^{\large{\cos x+i\sin x}}dx$$
$$=\int_0^{2\pi}e^{cos x}\left(\cos(\sin x) +i\sin(\sin x)\... | Use $z= e^{it}$ and integrate in the circumference of radius 1. Sou your integral becomes
$$ \int_{C} \frac{e^z}{iz}dz$$ which has a singularity in $z=0$ which is $\frac{1}{i}$ and using the resiude theorem then the integral is $2\pi i \; Res_0(f) = 2\pi i \frac{1}{i} = 2\pi$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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"answer_id": 2
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Is there a name for a square matrix with constant diagonal and off-diagonal elements? I am interested in real symmetric matrices of the form:
$$\mathbf{M} = \begin{bmatrix}
a & t & t & \cdots & t \\
t & a & t & \cdots & t \\
t & t & a & \cdots & t \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
t & t & t & \cdots & a ... | In old statistics literature, these are sometimes called completely symmetric matrices. I have made a light effort to revive the term in my writing or teaching (which is usually a combinatorial context).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3121971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why $P(X=Y)=0$ if $X$ and $Y$ are i.i.d continous r.v.s? I read this conclusion in text book:
If $X$ and $Y$ are i.i.d continuous r.v.s, then $P(X=Y)=0$, but it doesn't give any proof. I am a bit confused here, what is the rational for this conclusion? Is it because for a continuous r.v., the probability of it equaling... | For a legalistic argument: $P(X=Y)=E( \mathbb 1_{X=Y}) = E( E( \mathbb 1_{X=Y}|Y)) = E( 0) = 0$ since $E(\mathbb 1_{X=Y}|Y) = 0 \text{ a.s.}$. That is, the OP's observation that continuous random variables take on particular values with probability zero, when used with Fubini's theorem (a.k.a. the law of total expecta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3122192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A noncomplete graph $H$ of order $5$ or more with the property that $deg (u) ≠ deg (v)$ for every pair $u, v$ of nonadjacent vertices of $H$. I'm working in the following graph theory excercise:
Give an example or explain why no such example exists of a noncomplete graph $H$ of order $5$ or more with the property that... | Does
$$
H = ([5], \{(1,2),(2,3),(2,4),(3,4),(3,5),(4,5)\}).
$$
satisfy your requirements?
I found it by considering the complement, which has the property that every pair of adjacent vertices has distinct degrees.
| {
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Is the percentage symbol a constant?
*
*Isn't the percentage symbol actually just a constant with the value $0.01$? As in
$$
15\% = 15 \times \% = 15 \times 0.01 = 0.15.
$$
*Isn't every unit actually just a constant? But why do we treat them in such a special way then?
| Well, it really depends. In Chinese schools, students are told that $100\%=1,40\%=2/5$, so % is a constant. In the UK examination system, it appears that % is treated as a unit. Students are NOT expected to write the above two expressions.
However, it is agreed around the world that you should not write something like ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
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The set of even numbers less than 6 The set of even numbers less than 6 is ?
E = {$0, 2 ,4$}
Or
E= {...., $-2 , 0 , 2 ,4$ }
Which answer is correct?
The definition of the even number is any number can be written as $x = 2n$.
Is $$n \in \mathbb Z$$
or
$$n\in \mathbb N ? $$
| The even integers less than $6$ are $\ldots, -2, 0, 2, 4$.
The even whole numbers less than $6$ are $0,2,4$.
The even natural numbers less than $6$ are either $0,2,4$ or perhaps only $2,4$.
"Even number" is even more ambiguous than "even natural number".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3122635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Approximate monotonicity of $\epsilon$-covering number This is from Exercise 4.2.10 in Roman Vershynin's book, High-Dimensional Probability: An Introduction with Applications in Data Science.
Let $(T;d)$ be a metric space and $N(A,d,\epsilon)$ be the $\epsilon$-covering number of $A\subset T$ (the minimal cardinality o... | Yes, we can use the exterior covering numbers :
\begin{equation}
N(A,d, \epsilon) \le N^{ext}(A,d,\epsilon/2) \le N^{ext}(B,d,\epsilon/2) \le N(B,d,\epsilon/2)
\end{equation}
The first and third inequalities are from exercise 4.2.9 and the second one follows from the inclusion and the definition of exterior covering nu... | {
"language": "en",
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Circle measurement of Archimedes Let $f_n$ or $F_n$ be areas of the regular $n-$ polygon described to the unit circle or circumscribed. Show
$f_{2n}=\sqrt{f_nF_n}$ and $F_{2n}=\frac{2f_{2n}F_{n}}{f_{2n}+F_n}$
In the solutions there is only the fact that this follows from elementary geometric considerations.
I tried to... | Let $l_n$ and $L_n$ be the lengths of the sides of the regular $n$-polygon inscribed in, or circumscribed to, the unit circle. The inscribed polygon with $n$ sides is formed by $n$ equal triangles, with base $l_n$ and altitude $\sqrt{1-l_n^2/4}$. We have then:
$$
f_n={n\over2}l_n\sqrt{1-l_n^2/4},\quad F_n={n\over2}L_n.... | {
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Finding a smooth atlas of $M=\{ (x,y,z) \in \mathbb{R}^3 \; | \; x^2+y^2-z^2=1 \}$ We know that $M=\{ (x,y,z) \in \mathbb{R}^3 \; | \; x^2+y^2-z^2=1 \}$ is a submanifold of $\mathbb{R}^3$, with dimension $2$. The point is that I have to find an atlas which define the structure of $C^{\infty}$ manifold for $M$. I just d... | The space is topologically equivalent to a punctured plane, which suggests that we can cover $M$ with a single smooth chart. Since the defining equation has the form $F(x^2 + y^2, z) = C$, $M$ is invariant under rotations about the $z$-axis $Z$, which suggests that we look for a map $M \to \Bbb R^2 - \{ 0 \}$ that is s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Derivative of sigmoid function that contains vectors Could someone show me how to take the derivative of this function with respect to $w_i$?
$f(w) = \frac{1}{1+e^{-w^Tx}}$
$w$ and $x$ are both vectors $\in \mathbb{R}^D$
How would this be different from taking the derivative with respect to $w$ itself?
| With
$x = (x_1, x_2, \ldots, x_n)^T \tag 1$
and
$w = (w_1, w_2, \ldots, w_n)^T, \tag 2$
we have
$w^Tx = \displaystyle \sum_1^n w_i x_i; \tag 3$
we observe that
$\dfrac{\partial (w^Tx)}{\partial w_j} = x_j, \; 1 \le j \le n; \tag 4$
we may write
$f(w) = \dfrac{1}{1 + e^{-w^Tx}} = (1 + e^{-w^Tx})^{-1}, \tag 5$
and deploy... | {
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"timestamp": "2023-03-29T00:00:00",
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Klein bottle with constant curvature is flat A torus (equipped with a Riemannian or Lorentzian metric) which has constant curvature must be flat because of Gauss-Bonnet theorem.
Is it true that a Klein bottle (equipped with a Riemannian or Lorentzian metric) which has constant curvature must also be flat?
My first idea... | The Klein bottle, like the torus, has $\Bbb R^2$ as universal covering and the group of deck transformations is made of translations.
If we endow the Klein bottle with a metric of constant curvature and pull it back to $\Bbb R^2$ we get a metric of constant curvature which admits a group of translations acting as isome... | {
"language": "en",
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Show that all normals to $\gamma(t)=(\cos(t)+t\sin(t),\sin(t)-t\cos(t))$ are the same distance from the origin. Show that all normals to $\gamma(t)=(\cos(t)+t\sin(t),\sin(t)-t\cos(t))$ are the same distance from the origin.
My attempt:
Let $\vec{p}=(\cos(t_0)+t_0\sin(t_0),\sin(t_0)-t_0\cos(t_0))$ be any arbitrary poi... | As a variant: The point $\gamma(t)$ is obtained by taking the point on the circle $r(t)=(\cos(t), \sin(t))$ and following circle's tangent line at $r(t)$, which we denote by $L$, and which is in the direction $Rot_{90} r(t)=(\sin(t), -\cos(t))$, for time $t$. Thus after we compute the tangent to $\gamma(t)$ to have dir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3123301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 5
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Prove that there exists a $k$ such that the sequence obtained by listing each element of $S$ a total $k$ times is a degree sequence of some graph. I'm currently working in the following graph theory excercise:
Let $S = \{2, 6, 7\}$. Prove that there exists a positive integer $k$ such that the sequence obtained by list... | This is a partial answer which addresses only the first part of the question: proving the existence of $k$, without trying to determine the minimal value.
Given any set $S=\{s_1,s_2,\dots,s_n\}$ of $n$ positive integers, let $k$ be the least common multiple of the numbers $s_1+1,s_2+1,\dots,s_n+1$. Then the graph $G=\f... | {
"language": "en",
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How many ways are there to place $5$ balls into $8$ boxes? $8$ boxes are arranged in a row. In how many ways can $5$ distinct balls be put into the boxes if each box can hold at most one ball and no two boxes without balls are adjacent?
| HINTS:
Start by satisfying the claim that "no two empty boxes are adjacent".
For this you will need $4$ balls and I let you figure out in how many ways these can be arranged so that there are no empty boxes next to each other.
Finally you have $1$ ball left. How many empty boxes are left?
Hope this helped
If you need ... | {
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"timestamp": "2023-03-29T00:00:00",
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Interior solution of a Variational Inequality in Hilbert Space I'm trying to understand the proof of the following claim:
Let $K \subset \mathbb{R}^n$ be compact and convex and let $$F:K \rightarrow (\mathbb{R}^n)'$$ be continuous. Then there exists an $x\in K$ s.t. the dual pairing $$\langle F(x), y-x \rangle \geq 0 ... |
*
*I don't see any translation to the origin or anything, so this confused me, are they assuming $x$ is the origin or something?
They chose $x$ as the origin of a new coordinate system by the change of variables $y\mapsto y-x$ i.e. they simply translate the starting coordinate system.
*How does $F(x)=0$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Confused about $\csc(x)$ The title says it all, let's take this simple example:
I want to find $c$.
Using $\sin$ I get:
*
*$\sin(30)$ = $4 / c$
*$c = 4 / \sin(30) = 8$
Now, using $\csc$:
*
*$\csc(30) = c / 4$
*$c = \csc(30) * 4 = 8$
But also:
*
*$\csc(30) = c / 4$
*$\tan(30) = 4 / a$
*$4 = \tan(30) * a$
... | $$\sec30^{\circ}=\frac{1}{\cos30^{\circ}}=\frac{2}{\sqrt3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3123998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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"Expectation exists" vs "Expectation is finite" Are the statements
*
*"Expectation exists"
*"Expectation is finite"
equivalent? If not, could someone please provide a counterexample.
In case it's relevant, I don't know measure theory, but am confortable with probability theory & statistics at the undergraduate le... | A counterexample is the Cauchy distribution. It is symmetric around $0$ but both tails of the distribution are 'too' heavy:
$$E[X]=\int_{-\infty}^{\infty} x\cdot\frac{1}{\pi(1+x^2)}\,\mathrm{d}x = \int_{-\infty}^{0} x\cdot\frac{1}{\pi(1+x^2)}\,\mathrm{d} x + \int_{0}^{\infty} x\cdot\frac{1}{\pi(1+x^2)}\,\mathrm{d} x =\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all the group morphisms from $C_3$ to $C_4$ I know that since the gcd of 3 and 4 is 1, there can only be one group morphism but I'm unsure of how to find it. Any help would be greatly appreciated!
| Hint: If $\phi:C_{3}\to C_{4}$ is a group homomorphism, what will be the kernel and image of $\phi$? Especially, what is an order of $\ker(\phi)$? And also an order of $\mathrm{img}(\phi)$? Think about the Lagrange's theorem and the 1st isomorphism theorem.
| {
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"url": "https://math.stackexchange.com/questions/3124236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\operatorname{Tr}(AB)=\operatorname{Tr}(A)\operatorname{Tr}(B)$
Problem. Show that $\operatorname{Tr}(AB)=\operatorname{Tr}(A)\operatorname{Tr}(B)$, if $A^2+3AB+B^2=BA$ and $\det(A)=0$, where all matrices involved is real valued $2\times2$ matrices.
First I rewrote the $A^2+3AB+B^2=BA$ as
$$ A(A+3B)=B(A-B) $$
and I... | Here is a bit messy solution:
Lemma. Let $F$ be a field with characteristic $\neq 2$ and $A$ and $B$ be $2\times 2$ be matrices over $F$.
*
*If $A$ and $B$ commute, then either $A = pB + qI$ or $B = pA + qI$ for some $p, q \in F$.
*If $A^2 = k(AB - BA)$, then $A^2 = 0$. In particular, if $k \neq 0$, then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3124375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
If $S$ is compact then the separating classes are same as convergence determining classes Suppose $(S,\rho)$ is a metric space with Borel $\sigma-$algebra $\mathcal S$.
A class of subsets $\mathcal A\subset\mathcal S$ is called a separating class if whenever two Borel probability measures $P$ and $Q$ satisfy $P(A)=Q(A... | In the case of a compact metric space the space of all Borel probability measures is relatively compact (by Prohorov's Theorem). Let $A$ be a $P-$ continuity set in $\mathcal S$. It is enough to show that every subsequence of $\{P_n(A)\}$ has a further subsequence converging to $P(A)$. But every subsequence of $\{P_n\}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3124517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limiting value of a sequence when n tends to infinity Q) Let, $a_{n} \;=\; \left ( 1-\frac{1}{\sqrt{2}} \right ) ... \left ( 1- \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$
(A) equals $1$
(B) does not exist
(C) equals $\frac{1}{\sqrt{\pi }}$
(D) equals $0$
My Approach :- I am n... | As we multiply positive factors below $1$, the sequence is positive and strictly decreasing, hence convergent (ruling out B). As already $a_1<1$, we also rule out A.
If we multiply $a_n$ by $b_n:=\left(1+\frac1{\sqrt 2}\right)\cdots \left(1+\frac1{\sqrt {n+1}}\right)$, note that the product of corresponding factors is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3124615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability of getting a sequence of at least $k$ heads in $n$ coin flips. Let's say we are flipping a coin $n$ times. What is the probability of getting a sequence of at least $k$ heads?
So far I've come up with this:
If we assume $k=2$, and the number of events satisfying the conditions is $m$ for $N$ throws, then fo... | This is not a complete answer to your question but only works under the extra condition that $n\leq2k$.
Flip a fair coin again and again (so do not stop).
Let $A_{k,n}$ denote the event that among the first $n$ tosses there is a consecutive sequence of at least $k$ heads.
Let $T$ denote the index of the first occurence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3124684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Isomorphic field extensions have the same degree
Let $k_1 \subseteq k'_1$ and $k_2 \subseteq k_2'$ be field extensions. Suppose there is a field isomorphism $\phi: k'_1 \to k'_2$ where $\phi(k_1)=k_2$. Show that $[k_1':k_1]=[k'_2:k_2]$.
Now my first instinct was to try and show $k_1'$ is isomorphic to $k'_2$ as vecto... | Let $1_{k_1},a_1,\dots, a_n$ a linear basis of $k_1'$ over $k_1$.
Show that $\phi(1_{k_1}),\phi(a_1),\dots, \phi(a_n)$ is a linear basis of $k_2'$ over $k_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3124767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Is $a=b$ the only possibility?
In the following: $$2a-2b+(b^2-a^2)\operatorname{sinh} (by)\cos(ax)=0, \forall x,y \in \mathbb{R},$$ does it result automatically that $a=b$ in order for this to hold?
Is there any other possibility? Why?
| Say $a\ne b$ then we have $$2-(b+a)\sinh(by)\cos(ax)=0$$ for each $x,y$. So for $y=0$ we get $2=0$.
A contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3125035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that embeddings, diffeomorphisms, etc. are stable classes of maps This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds.
Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ for some smooth $F:N\times S\to M.$ A class $\mathca... | I found solution on the page 13 of
http://math.berkeley.edu/~kozai/m141f14/m141_notes.pdf
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove that $\det(A^{2019} +B^{2019} )$ and $\det(A^{2019} -B^{2019} )$ are divisible by $4$
Let $A, B \in M_2(\mathbb{Z})$ so that $$\det A=\det B=\frac{1}{4} \det(A^2+B^2)=1$$ Prove that $\det(A^{2019} +B^{2019} )$ and $\det(A^{2019} -B^{2019} )$ are divisible by $4$.
The only observation I have made is that $\... | The other answer demonstrates that the result is false unless $AB=BA$; here's a proof in that case. Note that $A^2+B^2=\dfrac12\left((A+B)^2+(A-B)^2\right)$. Therefore,
$$4=\det(A^2+B^2)=\frac14\det\left((A+B)^2+(A-B)^2\right),$$
i.e.
$$\det\left((A+B)^2+(A-B)^2\right)=16.$$
But because $\det(X+Y)=2\det X+2\det Y-\det(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3125319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Wave Equation with Initial Conditions on Characteristic Curves I am trying to solve the initial value problem:
$$\begin{cases}
u_{tt}-u_{xx} =0\\
u|_{t = x^2/2} = x^3, \quad |x| \leq 1 \\
u_{t}|_{t = x^2 / 2} = 2x, \quad |x| \leq 1 \\
\end{cases}
$$
I'm unsure if we can use D'Alambert's formula in this problem, so what... | The general solution is $u(x,t)=f(x-t)+g(x+t)$
$u|_{t=\frac{x^2}{2}}=x^3$ :
$f\left(x-\dfrac{x^2}{2}\right)+g\left(x+\dfrac{x^2}{2}\right)=x^3......(1)$
$u_t(x,t)=f_t(x-t)+g_t(x+t)=g_x(x+t)-f_x(x-t)$
$u_t|_{t=\frac{x^2}{2}}=2x$ :
$g_x\left(x+\dfrac{x^2}{2}\right)-f_x\left(x-\dfrac{x^2}{2}\right)=2x$
$g\left(x+\dfrac{x^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Image of $\gamma(t)=(\sin(t+k)\cos(t), \sin(t+k)\sin(t))$ is a circle. I would like to show the image of $\gamma(t)=(\sin(t+k)\cos(t), \sin(t+k)\sin(t))$ is a circle.
I was hinted that I should appeal to the Isoperimetric Inequality. Hence I calculate the area that the simply closed positively oriented curve $\gamma(t... | The problem is in setting $T = 2\pi$, because your function is actually $\pi$ periodic. Hence the area enclosed is
$$A = \frac 1 2 \int_0^{\pi} x\dot y - y \dot x = \frac 1 2 \int_0^{\pi} \sin^2(t + k) \, dt = \frac{\pi}{4}$$
while the corresponding length is
$$\ell(\gamma(t)) = \int_0^{\pi} \, dt = \pi.$$
Now the equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3125586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Preduals and $c_0$ I know that $c_0$ does not have a predual, but if we put an equivalent norm on $c_0$, can this space have a predual?
| No. In a separable dual space, every closed convex set is the closed convex hull of its extreme points. (Krein-Milman property). But in $c_0$ with an equivalent norm, the original unit ball is still a closed convex set with no extreme points.
Reference
Diestel, J.; Uhl, J. J. jun., Vector measures, Mathematical Su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How is this property called for mod? We have a name for the property of integers to be $0$ or $1$ $\mathrm{mod}\ 2$ - parity.
Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use parity in a broader sense, just to name the remainder $\mathrm{mod}\ n$?
| Either AlessioDV's good answer, or you could say: "of the form $nq+r$". as a representation of $\equiv r \pmod n$ for example a number $N$ with $$N\equiv 1\pmod 3$$ has property $N=3q+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3125933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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Interpolation between "arithmetic mean" and "geometric mean" Define a function $f(X; p)$, which measures a statistic on a set of points $X = {x_1, ..., x_n}$ (and $x_i \in (0, 1)$), with respect to a parameter $p$. I am looking for ways to define the function $f(.)$, such that on one extreme it becomes arithmetic mean,... | $$f(X;p) = M_{1-p}(x_1, \ldots, x_n)=\left( \frac1n \sum_{i=1}^n x_i^{1-p}\right)^\frac1{1-p}$$
When $p=0$, we clearly have the AM.
Also, $M_p$ also satisfies the condition that $\lim\limits_{p \to 1}M_{1-p}(x_1, \ldots, x_n)=\lim\limits_{p \to 0}M_p(x_1, \ldots, x_n)$ to be equal to the geometric mean.
This is known... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Stuck on a Geometry Problem
$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of the segment $FG$.
How can I approach this problem, preferably without trigonometry?
| Ok after some calculus, I figured out to solve this problem. Let's draw the vertical V that goes through E, and call $\alpha$ and $\frac{\pi}{4}-\alpha$ the angles we get on the left and right of the 45° angle cut by V and call c the length of the side of the square. We can then do some trigonometry to get :
*
*$c \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3126169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 12,
"answer_id": 9
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Negating statements help I need to negate (move the negation inside) or write the contrapositive statements of the following statements:
*
*Write the contrapositive of: If everyone is here, then someone will leave.
*Write the negation of: If Alice and Bob go, then Carol or Dave will come.
*Write the negation of Ev... |
*
*Write the contrapositive of: If everyone is here, then someone will leave.
*
*If someone didn’t leave, then not everyone was here.
"Did not" is not the complement for "will". "Was not" is not the complement for "is". Don't change the tenses.
*Write the negation of: If Alice and Bob go, then Carol or ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3126230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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If Two Independent Geometric Random Variables are equal Let $W_1$ and $W_2$ be independent geometric random variables with parameters $p_2$ and $p_2$ , respectively.
Find $P (W_1 = W_ 2 )$.
To me this question seems unsolvable. I think we can assume $W_1=p_1$ and $W_2=P_2$. Also, I think we can assume that $W_1+W_2=1$.... | A geometric random variable with parameter $p$ has probability mass function
$$
p(k) = (1-p)^{k-1}p,\quad k=1,2,\ldots.
$$
So if $W_1\sim\mathrm{Geo}(p_1)$ and $W_2\sim\mathrm{Geo}(p_2)$ are independent, we have
\begin{align}
\mathbb P(W_1=W_2) &= \mathbb P\left(\bigcup_{k=1}^\infty\{W_1=W_2, W_1=k\} \right)\\
&= \sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3126349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Determinant of $E^T H E$ where $E$ is semi-orthogonal and $H$ is positive definite Is there a way to simplify/obtain alternative forms of $\text{det} \left(E^T H E \right)$ where $E \in \mathbb{R}^{m \times n}$, $m > n$ is semi-orthogonal (meaning that its columns are orthonormal) and $H \in \mathbb{R}^{m \times m}$ is... | The Cauchy interlacing theorem states that if $\lambda_1,\dots,\lambda_m$ are the eigenvalues of $H$ in decreasing order, and $\mu_1,\dots,\mu_n$ are the eigenvalues of $E^THE$, then we have the inequalities
$$
\lambda_j \geq \mu_j \geq \lambda_{j+(m-n)}
$$
for $j = 1,\dots,n$. Conversely, for any tuple of real number... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3126467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus. Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus. This inequality is equivalent to
$$
2018^{1/2018}>2019^{1/2019}
$$
One of my 'High school' stu... | It is the same as $2018>(1+\frac{1}{2018})^{2018}$, so we only need to show that $$a_n=\left(1+\frac{1}{n}\right)^n$$
is bounded. There are many proofs of this. Here is one possibility.
Let $b_n=(1+\frac{1}{n})^{n+1}$. We observe that $b_n$ is decreasing since
$$\begin{aligned}b_{n-1}=\left(\frac n{n-1}\right)^n&=\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3126568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Shift in Time Domain by altering Fourier-Coefficients I want to apply a circular shift to a time-series, by changing the phase of the fourier-coefficients. But I run into trouble. In the current state the true sequence starts in the middle of the time-series. So thought that, I could do a phase shift in the frequency d... | A delay of $\tau$ seconds corresponds to a multiplication with $e^{-\tau s} = e^{- i \tau \omega}$ in the Laplace/Fourier domain.
You need to add a multiplication with a radial frequency vector in the exponent exp(-i*pi).
The Matlab code becomes
[x,fs]= audioread("sound.wav");
x= x(:,1)';
N = length(x);
w = 2*pi*(0:N-1... | {
"language": "en",
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Q: on Lemma I.5.11 proof from Tourlakis 2003 I'm unsure of the exact form of the I.H. in this case (the $\exists_x B \in Wff(M)$ rule).
I was thinking it could be $(B[x \leftarrow t])^\mathcal{J} = (B[x \leftarrow \bar i])^\mathcal{J}$, but it is possible that $w$ occurs free in $B$, so interpretation may not be define... | First line of the proof : the formula $\mathscr A$ is written in full.
Second step, the substitution $[x \leftarrow t]$ acts only on the formula $\mathscr B$ in the scope of the quantifier.
Third line : the semantic clause for the existential quantifier is applied (Def.1.5.6, page 55). Here we have an object $j \in M$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Geodesically complete implies exponential map defined on all of $T_pM$ This should be completely straightforward as every book I have read on the Hopf-Rinow theorem states this is "obvious". But I can find the life of me to justify this.
Geodesically complete means every geodesic $\gamma$ extends to all of time $\math... | Let $(M,g)$ a Riemannian manifold. Recall that for all $p\in M$ and $v \in T_pM$ there exists some $\varepsilon > 0$ and a $C^\infty$curve $\gamma(t)\colon (-\varepsilon, \varepsilon)\to M$ such that $\gamma$ is the unique geodesic passing through $p$ at time zero with velocity $\dot{\gamma}(0) = v$. The fact that the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3127076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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When is maximising a definite integral the same as maximising the integrand? When is maximising
$$\int_a^b f(x) \text{d}x$$
the same as maximising $f(x)$?
Context: I was trying to find the most probable location of an electron in the ground state hydrogen atom, where the wavefunction is $\text{exp}(-r/a_0)$, and hence ... | Think of the definite integral as giving you the area under the curve of $f(x)$ above the interval $[a,b]$. Suppose at all values on the interval, you had another function $g$ which is always more than $f$ (i.e. $g(x)\geq f(x)$ for all $x\in [a,b]$). The area under $g$ is obviously greater. Therefore, when the integran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3127189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that $\chi(G)=\omega(G)$ if the complement of G is bipartite. ***Please note I am aware that this question was already asked here: Proving that $ \chi(G) = \omega(G) $ if $ \bar{G} $ is bipartite.
however, I have a reason to believe the answer given in that topic is flawed.
What I so far tried: I divided the co... | Using the notation $G^c$ for complement, and $\alpha(G)$ for the stability number of $G$
The proposed proof went just a bit fast on the definition of $U$ and $V$. With its definition, you could have that $V$ is not the biggest independent set in $G^c$.
The key point is that you need $\vert V \vert \geq \vert U \vert$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3127355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does a generalization of the Borel-Cantelli lemma mean intuitively? I know the first two BC lemmas.
However, I was looking at this generalization of it and I don't understand the intuition behind it.
In the 2nd BC lemma, we require independence of events. The generalization is used to replace the condition of pa... | This is known as the "pairwise negative correlation" property. This is found for example in the paper Balanced Matroids by Feder, Mihail, available here.
What this means intuitively is the appearance of one element in the pairwise condition means that the appearance of the other element is less likely.
| {
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"timestamp": "2023-03-29T00:00:00",
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Simplify $\frac{4^{-2}x^3y^{-3}}{2x^0}$ to $\frac{x^3}{32y^3}$ I am to simplify $\frac{4^{-2}x^3y^{-3}}{2x^0}$ and I know that the solution is $\frac{x^3}{32y^3}$
I understand how to apply rules of exponents to individual components of this expression but not as a whole.
For example, I know that $4^{-2}$ = $\frac{1}{4^... | Do it step by step :
First, as you said $4^{-2} = \frac{1}{16}$, replace it in the given expression :
$$ \dfrac{4^{-2}x^3y^{-3}}{2x^0} = \dfrac{x^3y^{-3}}{16\cdot 2 x^0}$$
Then, simplify the denominator, $16\cdot 2 = 32$ and $x^0 = 1$ so that :
$$\dfrac{4^{-2}x^3y^{-3}}{2x^0} = \dfrac{x^3\cdot y^{-3}}{32} $$
Then, be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3127720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Determinant of A transpose time B equals determinant of A times B transpose I am reading Franz Hohn's Elementary Matrix Algebra (1973) and having trouble solving the following exercise:
Prove that, if $A$ and $B$ are both of order $n$,
(a) $\det A^{T}B = \det A B^T = \det A^T B^T = \det AB$
(b) $\det A^*B^* = \overlin... | We can use that $\det(C^T)=\det(C)$ and $\det(AB)=\det(BA)$ as follows:
$$
\det(A^TB)=\det(A^TB)^T=\det(B^TA)=\det(AB^T)
$$
At some point we need to prove the "classical" properties, like $\det(AB)=\det(BA)$ or even better, $\det(AB)=\det(A)\det(B)$. And then we do not care too much whether the text has it at page $xy$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Can a holomorphic function be globally represented by a power series on an open connected set? I have a theorem in my book (Stein) which says:
Suppose $f$ is holomorphic in an open set $\Omega$. If $D$ is a disc centered at $z_0$ and whose closure is contained in $\Omega$, then $f$ has a power series expansion at $z_0... | The set in which a power series converges is always an open disk together with some subset of the boundary of the disk. So, if a function $f$ can be represented by a single power series centered at $0$ on an open set $\Omega$, then $\Omega$ must be contained in an open disk $D$ (possibly of infinite radius) around $0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3128090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Discrepancy between calculus methodologies - Is it significant? Two of the ways of doing calculus with algebra are non-standard analysis NSA and smooth infinitesimal analysis SIA. NSA has a technique called 'taking the standard part' which neglects incremental (or infinitesimal) terms at the end of derivations, whereas... | Taking the standard part is not really a technique for neglecting infinitesimal terms at the end of a derivation. Instead, non-standard analysis has a bona fide standard part function $st: \mathbb{R}^* \rightarrow \mathbb{R}$, a map that assigns an (extended) real number to each hyper-real number.
With this in mind, y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3128254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Conjugacy of CSAs: Proof in Humphries' Intro to Lie Algebras and Rep Theory I am reading Chapter 16 of the title mentioned. On the middle of page 82 the statement is made "it is clear that if $\phi: L \rightarrow L'$ is a surjective homomorphism (of Lie algebras) then $\phi(L_a(ad(y)))={L'}_a(ad(\phi(y)))$". Here $L_... | I think what we have here is this linear algebra set-up. There are finite-dimensional
vector spaces $V$ and $V'$ ($L$ and $L'$ here) and a surjective linear map
$\phi:V\to V'$. I'll let $K$ denote its kernel. One also has
an endomorphism $A$ of $V$ (here, $\text{ad y}$) with $A(K)\subseteq K$
so that $A$ induces an end... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving diffusion equation with variable dissapation I've been asked to solve the diffusion equation with variable dissipation:
$ \frac{∂u}{∂t} - D \frac{∂^2u}{∂x^2} + e^{-pt}u = 0 , x∈(-∞,∞), t∈[0,∞] $
subject to
$u(x,0) = φ(x),$
where $D>0$ and $p>0$ are given constants, and φ(x) is a given function.
I think I need ... | You're on the right track. Let's try to figure out what substitution will work. Let $u(x,t) = h(t)v(x,t)$ then
$$ h_tv + hv_t - Dhv_{xx} + e^{-pt}hv = 0 $$
$$ \implies v_t - Dv_{xx} + \left(\frac{h_t}{h}+e^{-pt}\right)v = 0 $$
$v$ will satisfy the heat equation if the third term on the LHS is zero, i.e.
$$ \frac{h_t}{h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3128597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a proof for the Determinant of the Log-Euclidean-Tensor-Interpolation yielding the geometric mean of the Determinants of its sampling points? Given a set of $i \in \mathbb{N}$ symmetric-positive-definite-tensors $\boldsymbol{A}_i$ and the corresponding weights $w_i$ with $w_i>0$ and $\sum_i w_i = 1$.
The Log-E... | Using some well-known properties connecting exponents, determinants, traces, and logarithms, as well as multiplicativity of determinant and linearity of trace, this boils down to some boring algebra:
$$\det \bar A = \det \exp\sum\limits_i w_i \log(A_i)=\exp\operatorname{tr}\sum\limits_i w_i \log(A_i) = \\
= \exp \sum\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3128712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Graphical examination for existence of 2nd order ODE given a solution curve Given any solution curve $y(t)$, what are some graphical criterion to determine whether there exists some 2nd order linear, homogeneous, with continuous but possibly non-constant coefficients, to which the curve is a solution?
Each of the plot... | Not a complete answer, but too long for a comment.
Upon closer inspection, I'll concede that (b) is in fact not a solution of the original question.
Note that the solution curves in (b) all differ by a constant. If there exists a second-order ODE that admits all of these curves as a solution, then it must also admit a ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Method to find solution for $a^x \equiv \mod n$ The congruence $5^x \equiv 1 \mod 36$ has a solution because $5$ and $36$ are relatively prime, i.e. $5$ and $2^23^2$ have no common factors.
Is there a method to find $x$?
All I can see is that $5^3 \equiv 5 \mod 6$.
| You could always use the Chinese Remainder Theorem for congruences like this.
For
$$
5^x\equiv 1 \ \bmod 36
$$
you should solve
$$
\begin{cases}
5^x\equiv 1 \ \bmod 4\\
5^x\equiv 1 \ \bmod 9
\end{cases}
$$
This way you have to deal with smaller moduli.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3128930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to restrict coefficients of polynomial, so the function is strictly monotonic? I need to restrict the degree of freedom of the coefficients of a polynomial, so the function is always strictly monotonic in the domain $x\in\left[0, 1\right]$ and $y\in\left[0, 1\right]$. The polynomial also has to go through the point... | The $n=3$ case is tractable: Your polynomial is increasing if the derivative is always positive, so it's enough to show that it is never zero. (Never zero means it has the same sign everywhere, $f(0) < f(1)$ ensures that that sign is positive.) Since the derivative is a quadratic, having no zeros is equivalent to its d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show homeomorphism between two quotient topologies Consider two subspaces of $\Bbb R^2$ in the usual topology:
Square $I^2$ := $\{(x, y) ∈ \Bbb R^2: 0 \leq x, y \leq 1\}$
Annulus $A$:= $\{(x, y) \in \Bbb R^2: 1 \leq x^2 + y^2 \leq 4\}$,
Define the following equivalence relations $∼$ and $≈$ on $I^2$ and $A$ respectivel... | Identifying $\mathbb{R}^2$ with $\mathbb{C}$, we have $A = \{ z \in \mathbb{C} \mid 1 \le \lvert z \rvert \le 2 \}$. Define
$$q : I^2 \to A, q(x,y) = 2(x+1)e^{2\pi iy} .$$
This is a well-defined continuous map because $\lvert q(x,y) \rvert = 2(x+1) \in [1,2]$. It is a closed map because $I^2$ is compact and $A$ is Haus... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Two basic questions in Wasserstein spaces We denote by $P (\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$
and for $p\geqslant 1$ the Wasserstein space by
\begin{equation*}
P^p (\mathbb{R}^{d}) = \{ \mu \in P(\mathbb{R}^{d}) \,|\, \int |x|^p d\mu(x) <\infty \}
\end{equation*}
Also, we say that $... | If $p > q$, then a probability distribution $X$ with a $p$th moment has a $q$th moment.
This follows by applying Jensen's inequality to the function $\phi(x) = x^{p/q}$.
Note that $\phi''(x) = (p/q)(p/q - 1) x^{p/q - 1}$, which is convex on $[0,\infty)$ exactly when $p/q > 1$.
So we can compute:
$E [ ( |X|^q)^{p/q} ] ... | {
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Why are there two different recurrences for Gegenbauer polynomials? As I mentioned previously, I've been reading up on Gegenbauer polynomials in preparation for a blog post on the kissing number problem—specifically, the Delsarte method.
To make a long story short, the method involves expressing a particular function a... | I'll use $G$ for the second definition and $\widetilde G$ for the first definition. $G$ has the generating function $(1 - 2 x t + t^2)^{-n}$:
$$(1 - 2 x t + t^2)^{-n} = \sum_{k \geq 0} G_k^{(n)}(x) \,t^k.$$
$\widetilde G$ is constructed by first taking $C$ with the gf $(1 - 2 x t + t^2)^{1 -n/2}$ and then normalizing b... | {
"language": "en",
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"source": "stackexchange",
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If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$ for $R$ a sub-ring of $S$.
Let $R$ be a sub-ring of a ring $S$. Let $f,g$ be non-zero polynomials in $R[x]$ and assume that the leading coefficient of $f$ is a unit in $R$. If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$.
Thoughts:
If... | Well, as I mentioned in the comments and Bill Dubuque has explained it well in his answer, you can use the Euclidean division algorithm. Note that for the Euclidean division algorithm to work, all that you need is to know that the leading coefficient of the divisor is a unit. To see this, try to divide a polynomial $g(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\to S$ satisfying $\sum \limits_{x\in T} x^{f(x)}\equiv a \pmod p$ This question is the third round of Iranian exam questions, which has not been answered for several years now. I think there are many people here, which may be able to solve this problem.
F... | first if you don't have the restriction $ord(y)<p-1$ you can choose $p-2$ elements arbitary and choose $f(g)$(g is an primary root) such that the equality work.the best think after $g$ is $g^{\pm 2}$ so choose f of other elements arbitary and choose f of this two elements such that the equality works this gave the prob... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why would a facility use room numbers $3233$, $53$, $61$, $\infty$, $10^{100}$, $1729$, $4$, $1.61803$, $3.14159$, $1.33333$, $\sqrt{-1}$, $0$, $1$? I was part of TCS Ignite training after my BSc graduation from July 2007 to October 2007. The training space had a numbering system for its rooms such as
$$3233,\; 53,\; ... | *
*3233 is semiprime and a product of other two prime numbers in the list 53 & 61. 3233 is the biggest room in the facility and can accommodate 200 people. Rooms 53 & 61 can accommodate 100 people each. All other rooms are labs accommodating 24 people with computers. In RSA algorithm these three numbers 3233, 53, 61 c... | {
"language": "en",
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Noncentral Wishart Expected Value - Solving a Matrix Integral Let $\mathbf{V} \sim ncWish\left(\nu_1, \, \mathbf{\Sigma}, \, \frac{\nu_1}{\nu_2-p-1} \mathbf{\Theta}\right)$ follow a noncentral Wishart distribution according to Theorem 3.5.1. in Gupta, Nagar - Matrix Variate Distributions.
What is
$$
\mathbf{E}\left[ \... | I suppose that all matrices in question are $p \times p$. It is clear that no closed-form result for the integral can be obtained if $p=1$, so it is doubtful that such an expression exists for general $p$, especially since there is no such expression for the generalized hypergeometric function ${}_0F_1$ of matrix argu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Whitehead product $[i_2, i_2]_W$ Let $i_2$ be the generator of $\pi_2(S^2)$ and $\eta$ be the Hopf fibration from $S^3$ to $S^2$. How would one go about showing that $[i_2,i_2]_W=2\eta$ (up to a sign)?
This is one of the exercises in a course of topology I am currently self-studying. I have seen some solutions using t... | The Hopf fibration $S^1\to S^3\to S^2$ gives the long exact sequence in homotopy groups. Since higher homotopy groups of $S^1$ are trivial, we get $\pi_3(S^3)\cong\pi_3(S^2)$, with the generator given by the Hopf map. We can define the inverse isomorphism $H:\pi_3(S^2)\to \mathbb{Z}$ called the Hopf invariant as follow... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve for $x$ : $\sqrt{x-6} \, + \, \sqrt{x-1} \, + \, \sqrt{x+6} = 9$? I want to solve the following equation for $x$ :
$$\sqrt{x-6} \, + \, \sqrt{x-1} \, + \, \sqrt{x+6} = 9$$
My approach:
Let the given eq.:
$$\sqrt{x-6} \, + \, \sqrt{x-1} \, + \, \sqrt{x+6} = 9 \tag {i}$$
On rearranging, we get:
$$\sqrt{x-6} \, +... | I'm not entirely sure if this approach will find all possible $x$ in any such question, nevertheless here it is giving the correct answer and is relatively short.
Since the RHS $=9$ is an integer, all three terms in the RHS under the radicals must be perfect squares. $$\therefore, x-6=p^2$$
$$x-1=q^2$$ $$x+6=r^2$$
wher... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Finding the position of two lines for each value of k I have the two lines r and s
$$ r: \begin{cases} x+y=1 \\x+z=1 \end{cases} $$ $$ s: \begin{cases} x-ky=k \\z-x=k \end{cases}$$
First, I create one single system with all equations
$$ \begin{cases} x+y=1 \\x+z=1\\x-ky=k\\x-z=k \end{cases} $$
which I rewrite as
$$\be... | You made a sign error when you combined the two systems of equations: the second equation for $s$ is $z-x=k$, but the last equation in your combined system is $x-z=k$, which is not the same thing. If you reorder the variables, it should be either $-x+z=k$ or $x-z=-k$. This error cascades through the further calculation... | {
"language": "en",
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Show that matrix of $T$ has at least dim range $T$ nonzero entries. Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Show that with respect to each choice of bases of $V$ and $W$, the matrix of $T$ has at least dim range $T$ nonzero entries.
I have no idea on how to solve this.
| This is an exercise from Linear Algebra Done Right, so here's an answer in line with Axler's style: Choose bases $v_1,v_2,...,v_n$ of $V$ and $w_1,w_2,...,w_m$ of $W$ and let $\mathcal{M}(T)$ be the matrix of $T$ with respect to these bases. Column $j$ of $\mathcal{M}(T)$ will consist entirely of zeros iff $Tv_j=0$; eq... | {
"language": "en",
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Why this sequence of R.V converge in distribution, but doesnt in probability Why this sequence of R.V converge in distribution, but doesnt in probability?
Probability space $([0,1],B,m)$. ($B$ consists of all Borel sets of $[0,1]$, $m$ is the Lebesgue measure.)
Let $X_{2n}(ω)=ω$, $X_{2n−1}(ω)=1−ω$
My intuiton says that... | You're right about the convergence in distribution part. To show they have a uniform distribution, calculate the CDF. For instance, for $x\in (0,1),$ $P(X_0\le x) = P(\omega < x) = m(\{\omega :\omega <x\})=m((0,x)) = x,$ and you can compute similarly for $1-\omega.$
To show that it doesn't converge in probability, obse... | {
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A Version of Fubini-Tonelli Theorem for Hilbert Space Valued Functions I'm currently working on a project in which we define a new type of integral. And I'm trying to intechange the integral with expectation, something like $\mathbb{E} \left[ (\mathcal{N})\int f dW \right]=(\mathcal{N})\int \left[ \mathbb{E} f dW \righ... | It seems that the Theorem you need is Theorem 2 in Chapter X in the book The Bochner Integral isbn 9780124958500
| {
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Leaving recurrence summation in terms of $k$, $\sum_{i=0}^{k-1}\frac{3^i\sqrt{\frac n{3^i}}}{\log\frac n{3^i}}$ I have an exercise where I need to use the substitution method to solve the following recurrence and determine their corresponding complexity.
$$t(n)=3t(n/3) + \frac{\sqrt n}{\log n}$$
After some iterations, ... | Hint 1. As regards the asymptotic analysis (complexity?), you may use the Master Theorem. Then $c_{crit}=\log_2(3)\approx 1.58$ and $f(n)=\frac{n^{1/2}}{\log(n)}\leq n^{1/2}$. What about $T(n)$?
Hint 2. Note that
$$t(2^n)=3^nt(1)+\sum_{k=0}^{n-1}\frac{3^k\sqrt{2^{n-k}}}{\log (2^{n-k})}
=3^nt(1)+3^n\sum_{k=1}^{n}\frac{(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving $3\mid p^3 \implies 3\mid p$ I want to prove $3\mid p^3 \implies 3\mid p$ (Does it?)
The contrapositive would be $3 \nmid p \implies 3 \nmid p^3$ I believe.
$3\nmid p \implies p = 3q + r$ ($0<r<3$), so $p^3 = 27q^3+27q^2r+9qr^2+r^3$
Dividing by $3$ we get $3(9q^3 + 9q^2r + 3qr^2) + r^3$
Is this correct so far? ... | $$p^3\bmod3=(p\bmod3)^3\bmod 3,$$ so that
$$p\bmod3=0\to p^3\bmod3=0$$
$$p\bmod3=1\to p^3\bmod3=1$$
$$p\bmod3=2\to p^3\bmod3=2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Paper folding and continued fractions This question is suggested by a prior question, which has not received a complete answer. One corner of rectangular sheet of paper $P_1$ is folded down to make a trapezoid, and then a right triangle is cut off to make another rectangle $P_2.$ The process is repeated with $P_2$ to... | There is another possible limit $\color{blue}{3-2\sqrt{2}=(1+\sqrt{2})^{-2}}$. How?
Suppose rectangle $n$ has length $a$ and width $b$ with $a>2b$. Then to get to rectangle $n+2$ you shorten the $a$ dimension twice leaving a rectangle with dimensions $a-2b$ and $b$. If
$\dfrac{a-2b}{b}=\dfrac{b}{a}$
then the rectang... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Strong Induction for Fibonacci I'm a little lost of how to use strong induction to prove the following for the Fibonacci sequence:
$F_n < 2^n$ for all natural numbers
Any help would be very much appreciated!!
| Hint: $F_0=0<2^0$, $F_1=1<2^1$. If the statement holds for every $m<n$, where $n\ge2$, then
$$
F_n=F_{n-1}+F_{n-2}
$$
and $n-1<n$, $n-2<n$, so…
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A Continuous function $f: \overline{B_1(0)} \subset \ell^2\to \mathbb{R}$ which does not reach the maximum? If necessary, recall that
$$
\ell^2 = \{x=\{x_n\}_n\subset \mathbb{R} : \|x\|^2:=\sum_n |x|^2<\infty\}
$$
and $ \overline{B_1(0)} $ is the closed unit ball with respect to that norm.
Can we find an explicit ex... | Try $$f(x) = \sum_{n=1}^\infty (1-1/n) x_n^2$$ Note that $f(x) < 1$ for all $x \in \overline{B_1(0)}$, and you can get arbitrarily close to $1$...
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the following sequence converges. Please Critique my proof. The problem is as follows:
Let $\{a_n\}$ be a sequence of nonnegative numbers such that
$$
a_{n+1}\leq a_n+\frac{(-1)^n}{n}.
$$
Show that $a_n$ converges.
My (wrong) proof:
Notice that
$$
|a_{n+1}-a_n|\leq \left|\frac{(-1)^n}{n}\right|\leq\fr... | Consider $b_n = a_n + \sum_{k=1}^{n-1} \frac{(-1)^{k-1}}{k}$. Then
$$ b_{n+1}
= a_{n+1} + \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k}
\leq a_n + \frac{(-1)^n}{n} + \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k}
= b_n, $$
which shows that $(b_n)$ is non-increasing. Moreover, since $\sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k}$ converges by al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3131816",
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How fast is the height of the water in a cylindrical tank increasing? A cylindrical tank with radius $5m$ is being filled with water at a rate of $3m^{3}/\min$. How fast is the height of the water increasing.
The radius $r=5m$
The rate of water is $\dfrac{dV}{dt}=3m^{3}/\min$
The height of the water in the cylinder is... | Each quantity is a function of time except for the radius $r$ because it does not change with time (well, technically speaking, you still could think of it as a function of time, but it would be a constant function then: $r(t)=5\ m$). All those quantities are related by this expression:
$$V(t)=\pi r^2 h(t)$$
This equal... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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direct image functor $f_*$ left exact I would have to ask for apology for following question by everybody who is familar with algebraic geometry since this might be a quite trivial problem that cause some irritations for me: we consider a morphism $f: Z \to Y$ between schemes. then the induced direct image functor $f_*... | Left exact on all open sets implies left exact on stalks. This follows from exactness of direct limits for categories of modules, cf. Why do direct limits preserve exactness?.
| {
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Why does wolfram answer as such in this example for surface area and volume of revolution on an area crossing the axis? Compute the volume and surface area of the solid $S$ obtained by revolving the region $R$ (pictured below) enclosed by the graphs $y=q(x)=-(2x^2-7x+3),x=1,x=2$ and $y=-1$, around the $x$-axis, which c... | It appears that in the interpretation of Wolfram Alpha, two regions on opposite sides of the axis cancel each other out rather than simply reaffirming that the points swept by both are part of the solid.
Hence as we see in the figures on the two pages you linked to,
the figure that is bounded at the $x$-axis produces a... | {
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"answer_id": 0
} |
Show that $\left<\mathcal{M}\right>=V$ iff $v_i$ is a linear combination of the elements of $\mathcal{M}$. I'm struggling with the following problem:
Let $V$ be a finitely generated vector space over the field $F$. Let
$\left( v_1, \ldots, v_n \right)$ be a basis of $V$. Let
$\mathcal{M}$ be a family of vectors ... | Do note that it is not necessary for $M$ to have $n$ elements. It is only a generating set, so all you know is that it has at least $n$ elements.
Suppose $\langle M\rangle=V$. So, every element of $V$ can be written as a linear combination of elements of $M$. Now, $(v_i)_1^n$ is a basis of $V$, so each $v_i$ must be an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Area between trigo function and a line How can I compute the area of region enclosed by $\cos^2x$ and $y=1-\frac{2}{\pi}x$?
I am just stuck on the first step, finding their intersection point, I know I can plot the graph using online resources but is there any numerical method to obtain the intersection?
| Define $f(x)=\cos^2x$ and $g(x)=1-2/\pi \cdot x$.
Two intersection points can simply be found by plugging in $x=0$ and $x=\pi/2$. For the third intersection point, you may use numerical methods such Newton-Raphson, Secant and the alike. But if this is a problem on a test which it seems to be because of the "nice" $x$-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Closed form for these polynomials? I have a recurrence relation for these polynomials $p_i(x) $:
\begin{align}
p_0(x)&=0 \\
p_1(x)&=1 \\
p_{2i}(x)&=p_{2i-1}(x)-p_{2i-2}(x) \\
p_{2i+1}(x)&=xp_{2i}(x)-p_{2i-1}(x) \\
\end{align}
I've been unable to find a closed form, but the recurrence is simple so it seems there might b... | Writing $v_n=\begin{pmatrix} p_{2n+1} \\ p_{2n}\end{pmatrix}$, the recurrence can be written as $v_{n+1}=Av_n$ where $$A=\begin{pmatrix} x-1 & -x \\ 1 & -1\end{pmatrix}.$$ So, we have $v_n=A^nv_0$ for all $n$, and the question is just about computing powers of the matrix $A$. This can be done by diagonalizing: the ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
How can the correct form of the partial fractions decomposition be found for arbitrary rational functions? What is the reasoning or intuition that leads to the assumption that
$$r(x) =\frac{x^2 + 2}{ (x+2)(x-1)^2}$$
can be expressed as
$$r(x) = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+2}$$
(For the sake of con... | Start with the rational function $\frac{N(x)}{D(x)},$ where $N(x)$ and $D(x)$ are polynomials over whichever field we happen to be working in (such as the real numbers or complex numbers) and the degree of $N(x)$ is less than the degree of $D(x).$
Consequences of Bezout's Identity
Suppose we can factor $D(x) = P_1(x)P_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Where $a$ is introduced in $a + bi$ when solving polynomials using complex numbers Trying to wrap my head around complex numbers and almost there. I am looking for problems that show me how to introduce $i$ into an equation. What I'm finding a lot of is "Simplify 2i + 3i = (2 + 3)i = 5i", where the $i$ has already been... | A complex number is always of the form $a+bi$, where $a,b\in \mathbb R$ but $a$ and/or $b$ can be zero. Also, when you were solving the equation $(x+3)^2=-10$ you forgot the second solution $x=-3-\sqrt{10} i$. Remember that $-10=(\pm \sqrt{10}i)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Integral inequality of length of curve Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function. Prove that for any $a.b\in \mathbb{R}$
$$\left (\int_a^b\sqrt{1+(f'(x))^2}\,dx\right)^2\ge (a-b)^2+(f(b)-f(a))^2$$.
source: This is from TIFR GS stage 2.
I think mean value theorem kills it but can't do i... | An easy way to do this is to note that since distance is invariant under rotations, without loss of generality, we may assume that $f(a)=f(b).$ And now, since $\sqrt{1-f'(x)}\ge 0$ on $[a,b]$, the function in $C^1([a,b])$ that minimizes the integral coincides with the function $f$ that minimizes the integrand, and clea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Absolute convegence of Fourier series implies uniform convergence? I've read many statements in my textbook that include something along the lines of "the Fourier series of $f$ converges absolutely, and hence uniformly to $f$". I'm not sure why this is true.
If the Fourier series converges absolutely, then it converge... | Let $$S_n(\theta)=\sum_{|k|\leq n}c_ke^{2i\pi k\theta}$$ be the partial Fourier series.
Now, suppose that for all $\theta \in [0, 2\pi]$, we have absolute convergence $S_n(\theta)\rightarrow f(\theta)$.
Let $\varepsilon>0$. By definition of the series being absolutely convergent, $$\sum_{k\in\mathbb Z}|c_k|<+\infty$$
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Help with an application of Young's inequality I am reading through a set of notes about concentration of Gaussian measure, and on page 56, they make the following claim that I am failing to see the proof of:
Now we estimate the second summand at the right hand side. Using
Young's inequality with $y = e^{\lambda \ti... | They use a generalized version of Young's inequality (see here, the last example under "Generalization using Fenchel-Legendre transform"). In particular, given two real numbers $x > 0$, $y$, one gets:
$$xy \leq x \ln x - x +e^y.$$
This version of Young's inequality was already used in the document you linked, at the bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Non nesting action on Real Trees Background
Let $G$ be a group acting by homeomorphisms on an $\mathbf{R}$-tree $T$. The element $ g \in G $ is elliptic if the fixed point set $\operatorname{Fix}(g)$ is non-empty. The group action is non-nesting if no $ g \in G $ maps an arc properly into itself.
Claim
Let $ g \in G $ ... | First, the property of being at most a singleton is true without assumption on $g$:
Let $T$ be an $\mathbf{R}$-tree. Then for an arbitrary self-homeomorphism $g$ of $T$ and $x\in T$, the intersection $\mathrm{Fix}(g)\cap [x,gx]$ is at most 1 point.
Indeed, if $y,z$ are fixed points in $[x,gx]$, say with $y\in [x,z]$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Derivative of $tr(A(C \circ X)BB'(C' \circ X')A')$ Can we differentiate this function: $tr(A(C \circ X)BB'(C' \circ X')A')$ w.r.t $X$?
Also, $tr(A(C \circ X)Y)$ w.r.t $X$.
| The Frobenius product is a convenient notation for the trace
$\,A:B = {\rm Tr}(A^TB)$
Define the matrices
$$\eqalign{
Y &= C\circ X &\implies dY &= C\circ dX \cr
W &= AYB &\implies dW &= A\,dY\,B \cr
}$$
Write the function in terms of these new variables. Then calculate its differential and gradient.
$$\eqalign{
\phi &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\lim\limits_{n \to \infty} \int_{1-\epsilon}^1 \ln(1+x+...+x^{n-1})dx$
Evaluate $$\lim_{n \to \infty} \int_{1-\epsilon}^1 \ln(1+x+...+x^{n-1})dx$$
for $0<\epsilon<1$.
The problem is at the point $x=1$, because there the formula $$1+x+...+x^{n-1}=\frac{1-x^n}{1-x}$$ can't be applied. I think of using the m... | You can replace the upper limit with $k$ and apply $\lim_{k\rightarrow 1}$ to the whole integral. Then that formula can be applied.
So the integral becomes
$$\int_{c}^{k} (\log(1-x^n) - \log (1-x))dx$$
Since $x<1$ in the given interval $\log(1-x^n)$ is $0$ as n tends to infinity.
So
\begin{align*}
& =-\int_{c}^{k} \lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the number of ways to express 1050 as sum of consecutive integers I have to solve this task:
Find the number of ways to present $1050$ as sum of consecutive
positive integers.
I was thinking if factorization can help there:
$$1050 = 2 \cdot 3 \cdot 5^2 \cdot 7 $$
but I am not sure how to use that information... | For the sum of next integer we may use formula for the sum of arithmetic sequence
$$\sum_{k=1}^na_k=\frac n2(a_1+a_n)$$
So
\begin{aligned}
1050 &= \frac n2(a_1+a_n) \\
2100 &= n(a_1+a_n) \\
2100&= n(a_1+a_n) \\
2100&= n(a_1+a1+(n-1)) \\
2^2\cdot3\cdot5^2\cdot7&= n(2a_1+n-1)
\end{aligned}
Now:
*
*If $n$ is even, then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
4 Spheres all touching each other?? If there are 4 spheres all touching each other and 3 of them have diameters 4, 6 and 12 what is the diameter of the fourth one?
I imagine it like 3 balls on a flat table touching each other and then we are supposed to put another one on top of them but in my imagination the top sphe... | Not any size. If the fourth sphere is sufficiently small, it can fit in the hole in the middle, also resting on the table, without touching any of the other three spheres.
I’m guessing this problem is asking for the minimum diameter of the fourth sphere which guarantees contact with all of the other three.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Determine whether the given map $\phi$ makes these two binary structures isomorphic $\langle \mathbb{Q}, \mathbb{+} \rangle$ with $\langle \mathbb{Q}, \mathbb{+} \rangle$ where $\phi(x) = x + 1$ for $x \in \mathbb{Q}$
In order to show that two binary structures are isomorphic, I need to check whether $\phi$ is one-to-o... | Your check for a) is fine, though you might want to start with "For all $x,y\in\Bbb{Q}$...".
Your check for b) is incomplete, and has a typo (you write $\Bbb{Z}$ in stead of $\Bbb{Q}$). It would suffice to say
For all $x\in\Bbb{Q}$ we have $x=\phi(x-1)$, so $\phi$ is surjective.
Your check for c) is fine, and show th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
question on the meaning of supremum I know that supremum means least upper bound. If I have a sequence of events, $\{A_n\}_{n=1}^\infty$
then $$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$
I'm having trouble understanding this statement:
"The supremum of a collection of elements ... | Probability events are (the measurable) subsets of the probability space $\Omega$, and the partial order among them is simply the relation 'being subset of'. So, for any countable set of events $B_i$, their supremum is just $\bigcup_i B_i$ (which is still measurable).
By the way, we also have
$$\limsup_n A_n\ =\ \lim_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Having the sequence $(a_{n})_{n\geq1}$, $a_{n}=\int_{0}^{1} x^{n}(1-x)^{n}dx$, find $\lim\limits_{n{\rightarrow}\infty} \frac{a_{n+1}}{a_{n}}$ Having the sequence $(a_{n})_{n\geq1}$, $a_{n}=\int_{0}^{1} x^{n}(1-x)^{n}dx$, find $\lim\limits_{n{\rightarrow}\infty} \frac{a_{n+1}}{a_{n}}$. Is this limit equal to $\lim\limi... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
How to show that "If cancellation laws hold in $\mathbb{Z}_n$, then $n$ must be prime" I was assigned to prove that the following statements are equivalent:
*
*$\mathbb{Z}_n$ has no zero divisors.
*$\mathbb{Z}_n$ is an integral domain.
*The cancellation laws hold in $\mathbb{Z}_n$
*$n$ is prime.
For the proof ... | You are almost there, here's a hint: suppose that $n$ is not prime, so that $n = st$ with $s,t \geq 1$. Look at this equation modulo $n$.
Derive a contradiction from the fact that $st \equiv 0 = s \cdot 0\pmod{n}$ but $0 < s \neq n$ and in particular, $n | s$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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