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What is the Second mean value theorem for integral? |
After reading the [post](https://math.stackexchange.com/q/2076791/732917) with the result
$$\int_0^{\frac\pi2} {\sin^2{x} \ln{(\tan x)} \,dx}=\frac{\pi}{4}$$
I was attracted by its decency and wanted to generalise it as
$$
I_n=\int_0^{\frac{\pi}{2}} \sin ^{2 n} x \ln (\tan x) d x \stackrel{x\mapsto\frac \pi 2- x}{... |
A construction is as follows:
>Given a fixed line $L$ and two fixed points $S,S'$.\
>For any point $P$ on the plane, let the line $PS'$ intersect $L$ at $Z$.\
>Draw the line through $S'$ parallel to $SP$ and this line intersect the line $SZ$ at $P'$.\
>The map $P\mapsto P'$ maps lines through $S$ to parallel lines ... |
I was wondering, if you have $f:A \rightarrow B$ a mapping onto sets, is there appropriate terminology for when, for a subset $C \subseteq A$, you have $g:C \rightarrow D$ that verifies $\forall e \in C$, $g(e) \subseteq f(e)$. If so, is there a proper notation for this? |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. The latter example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already ... |
I needed to prove that the interior of a convex set is convex in a Normed Linear space.
I tried a proof using the geometrical and intuitive ideas... I need someone to verify and list out the mistakes, if there are any...
#### Theorem
Interior of a convex set S, is convex
#### Proof:-
If the interior of S c... |
What is the Second mean value theorem for integrals? |
I encountered an interesting problem in “Practical Methods of Optimization” by R. Fletcher, which utilizes the penalty method to solve a constrained optimization problem:\
We have the following problem
\begin{align}
{\mathop{\text{minimize} }} \quad & \left \| Ay \right \|^2 \\
\text{s}\text{.t}\text{.} \quad... |
I came across this summation while solving harmonic chain.
\begin{align}
\sum_{l=1}^N \sin^2{(\frac{m\pi}{N+1}\cdot l)}
\end{align}
for an integer m that can range from 1 to N.
So the question is if the value of the summation changes with m? I checked for until N=5. And it remains the same for all integer... |
Are these 2 equations equivalent? If so how can I prove it, if not how can I disprove it?
$CI(t)=\prod_{t_i<t}(d_i/n_i)$
$CI(t)=1-\prod_{t_i<t}(1-(d_i/n_i))$ |
Is $\prod_{i\in I}x_i$ equal to $1-\prod_{i\in I}(1-x_i)$? |
how to simplify this expression so that I get an expression only?
when $t = 1, 2,.... $
$$y = \sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$
I know that for t = 1, $\lceil \frac{t}{2}\rceil$ = 1
for t = 2, $\lceil \frac{2}{2}\rceil$ = 1
for t = 3, $\lceil \... |
Let $(X,d)$ be a metric space and define for each $x\in X$ and for $\varepsilon >0$ the ball $B_\varepsilon^X (x) := \{ y: y\in X \land d(x,y) < \varepsilon \}$. Now, consider a subset $A\subseteq X$ and define the quotient space $X/_\sim$ as $X/A := \{[x]: x\in X\}$ and with equivalence relation $x\sim y \Longleftrigh... |
I'm interested in sampling from a Gaussian with zero-mean and covariance given by:
$$
\Sigma = \begin{bmatrix}
\Sigma_{11} & \Sigma_{12} & \cdots &\Sigma_{1,100}\\
\Sigma_{21} & \Sigma_{22} & \cdots &\Sigma_{2,100}\\
\vdots & \vdots & \cdots &\vdots \\
\Sigma_{100,1} & \Sigma_{100,2} & \cdots &\Sigma_{100,100}
\... |
There is an engineering question that occurred to me several years ago, but I could not solve it
Assuming $H$ is the point of intersection of the heights, $I$ is the point of intersection of the bisectors, and $G$ is the point of intersection of the averages in triangle $∆ABC$, and the points $H$, $I$, and $G$ are ... |
`Determine the number of 6-combinations of multiset s = {3·a, 4·b, 5·c}`
I can solve this problem by inclusion-exclusion principle, but is there any more clever way to solve it? (e.g. By solving the 12-permutations first...anyway)
Learning "Introductory Combinatorics". |
The map $\mathbb{Z}\hookrightarrow \mathbb{Z}[i]$ induces an isomorphism of fields
$$\mathbb{Z}/173 \simeq\mathbb{Z}[i]/(13+2i)$$
So we could work with representatives in $\mathbb{Z}$, which is sometime easier. By the above $ {80}\mapsto i$, and $107\mapsto 1-3i$. Note that $107$ is a primitive root $\mod 173$. ... |
Let $(X,d)$ be a metric space and define for each $x\in X$ and for $\varepsilon >0$ the ball $B_\varepsilon^X (x) := \{ y: y\in X \land d(x,y) < \varepsilon \}$. Now, consider a subset $A\subseteq X$ and define the quotient space $X/_\sim$ as $X/A := \{[x]: x\in X\}$ and with equivalence relation $x\sim y \Longleftrigh... |
The simplest formulation of the Lebesgue Differentiation Theorem (LDT) (that I am aware of) is the following:
**LDT (Simplest formulation).** Given a function $f \in L^1_{\text{loc}}(\mathbb R^n)$, we have that
$$ \lim_{r \to 0} \frac{1}{|B(x,r)|} \int_{B(x,r)} f(y) \, dy = f(x), $$
for almost every $x \in \m... |
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
>Show that $P(ab>c)=\frac12$.
The result is strongly suggested by simulations, and by my attempt shown below.
The simplicity of the result suggests that there may be an intuitive explan... |
>If the exhaustive set of $x\in(0,2\pi)$ for which $\forall n$ the inequality $$\sin x+\sin2x+\sin3x+\cdots+\sin nx\le\frac{\sqrt3}{2}$$ is valid is $l_1\le x\le l_2$, find $l_1$ and $l_2$.
Let $\displaystyle\sum_{i=1}^n \sin(ix)$=$S$ then I have managed to show that $$S=\frac{\sin(nx+x)\cdot\sin(nx)}{\sin\left(\fra... |
I am working in $L^2$ and I want to change from some measure, $\rho$, to the usual measure. This measure is continuous w.r.t. the Lebesgue measure, and so, I can write it as $\rho(dt) = \rho(t)dt$. I am trying to prove the following thing:
$W_{\rho} = W_{dt}\sqrt{\rho}$, where W is some function in L².
Indeed, I ... |
The question can be expressed by following:
> How many n-dimensional vector sets containing $n$ vectors with elements of $-1$ and $1$, orthogonal to each other pairwise, are there?
when $n=4$, a sample vector set can be this:
$\{ (1,1,1,1),(-1,-1,1,1),(1,-1,1,-1),(1,-1,-1,1) \}$
<hr>
Assume that we have... |
How to calculate the number of n-dimensional Hadamard Matrix? |
I am working in $L^2$ and I want to change from some measure, $\rho$, to the usual measure. This measure is continuous w.r.t. the Lebesgue measure, and so, I can write it as $\rho(dt) = \rho(t)dt$. I am trying to prove the following thing:
$W_{\rho} = W_{dt}\sqrt{\rho}$, where W is some function in L².
Indeed, I ... |
Given two stochastic processes, $X$ on a filtered probability space $(\Omega^X,(\mathcal{F}^X_t)_t), \mathcal{F}^X, P^X)$ and $Y$ on a filtered probability space $(\Omega^Y,(\mathcal{F}^Y_t)_t), \mathcal{F}^Y, P^Y)$, is it possible to construct a filtered probability space $(\Omega,(\mathcal{F}_t)_t,\mathcal{F},P)$ suc... |
Can we construct a probability space encompassing two stochastic processes on different probability spaces? |
Note :This question was in our final exam and I didn't know how to approach it.
The question:
In the University there are $6$ six courses in Math, $6$ courses in CS, and $3$ in Physics (each course is different). In how many ways we can choose $6$ courses from the database s.t one course from each major will be c... |
In how many we ways can choose at least three courses from each major? |
Consider a linear representation of a group $G$ on a ($n$-dimensional) Hilbert space $H$ (complex vector space with sesquilinear inner product $\langle\cdot,\cdot\rangle$ ):
$$ g\in G \mapsto \rho(g)\in B(H),\quad |\psi\rangle\in H \mapsto |\phi\rangle = \rho(g)|\psi\rangle\in H $$
where $B(H)$ is the space of op... |
Does this group representation have a name? What are its properties? |
Note :This question was in our final exam and I didn't know how to approach it.
The question:
In the University there are $6$ six courses in Math, $6$ courses in CS, and $3$ in Physics (each course is different). In how many ways we can choose $6$ courses from the database s.t one course from each major will be c... |
In how many we ways can choose at least three courses (one from each major)? |
I came across the following Theorem in Folland's book, Theorem 8.33
1. Where do we use the fact that $f$ is continuous in this proof?
2. Why can't we state a similar theorem for $\mathbb{T}^n$ without using Sobolev space?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/RMgmw.p... |
Consider a function $f$ that satisfies the following differential equation:
\begin{equation}
\Delta f - \lambda^2 ((\Delta f)^2) = 0,
\end{equation}
where $\lambda$ is some real constant. Expressing $\Delta$ in spherical coordinates, does the differential equation admit separable solutions of the type $f(r,\the... |
I am trying to solve the following equations:
\begin{equation*}
\begin{aligned}
-\frac{\partial u}{\partial t}&=\Delta u+Cu, (x,t)\in\Omega_T\\
\frac{\partial u}{\partial x}&=0, x\in \partial \Omega_T \\
u(T,x)&=0, \forall x\in\Omega.
\end{aligned}
\end{equation*}
where I employ the BTCS(backwar... |
I encountered the following problem but I have no idea how to proceed.. Basically, I have a class of periodic function satisfying certain properties, and since I do not have explicit form of them I use their Fourier expansions, and hope to arrive the minimum value of $g(x)+g(\alpha-x)$ over interval $x\in(0,\alpha]$, w... |
There are two ways that a function can fail to achieve a maximum:
1. It can increase to $\infty$
2. It can approach the maximum, without quite reaching it, as $1-\frac1x$ increases toward $1$ without getting there
Since the function you are looking for is bounded, (1) is ruled out. It must be (2).
Similarly, ... |
Consider a FIFO queue with an upper bound of queue size $U$. $N$ people already in the queue when this person A arrives at time $t_0$. Now there are $N+1$ people in the queue ($N+1\leq U$). Suppose A is served and leave the queue at time $t_1$. Also in time window $(t_0, t_1)$ a total of $M<U$ other person joins the qu... |
I saw this problem in Sheldon Axler's Measure,Integration and Real Analysis Ex. $10B$, problem $16$ and $17$.
Let's just restrict to the self adjoint case. What I am trying to show is that $||T^{n}||=||T||^{n}$ for all natural numbers. However, I am stuck with this problem.
Firstly, I can directly show that $||T... |
I'm having trouble with the following question:
Let $a_n$ be the number of compositions of $n$ into an odd number of parts, each of which is at least $3$. Prove that
$a_n = [x^n]\frac{x^3-x^4}{1-2x-x^2-x^6}.$
This is what i've done so far:
Let $S=N_{\text{odd}\geq 3}^k$ where $N_{\text{odd}\geq 3}= \{3,5,7,..... |
It is easy to show using the analytic continuation of $\sinh^2(x)$ and $\cosh^2(x)$ that the identity$$\cosh^2(x)-\sinh^2(x)=1$$holds. However, what I want to know is, is there any way to prove this geometrically?
I have been trying to prove it geometrically for around 3 or 4 hours, but I’m stuck on how I should sta... |
Geometric proof of $\cosh^2(x)-\sinh^2(x)=1$? |
A construction is as follows:
>Given a fixed line $L$ and two fixed points $S,S'$.\
>For any point $P$ on the plane, let the line $PS'$ intersect $L$ at $Z$.\
>Draw the line through $S'$ parallel to $SP$, intersecting the line $SZ$ at $P'$.\
>The map $P\mapsto P'$ maps lines through $S$ to parallel lines through $S... |
For a given $n\geq 2$, let $S=(s_1,s_2,\ldots)$ be an infinite sequence of elements in $\{3,\ldots,n+2\}$.
For $i\geq 1$, let $S_i:=S[i:2i]$ be the subsequence $S_i=(s_i,s_{i+1},\ldots,s_{2i})$. We build a rooted tree $T_i$ on $i+1$ vertices as follow. Start with $T_i$ being a single vertex, the root of $T_i$. For $... |
In the definition of **convergence in measure** many authors say:
Let $f_n, f: E \to \overline {\mathbb R} = \mathbb R \cup \{\pm \infty\}$ be measurable and *almost everywhere finite* functions on $E$. If for any $\varepsilon > 0$
$$
\mu(\left\{x \in E: |f_n(x) - f(x)| \geq \varepsilon \right\}) \xrightarrow[n \... |
I just make a quick and dirty code that uses an already found list of primes up to 1 bilion.
The file can be found [here](https://sourceforge.net/projects/billion-primes/)
The code I have used is:
def sum_digit(n):
sum = 0
while (n != 0):
sum = sum + (n % 10)
... |
A question I found and that I could not answer so far. Assuming that $A$ is a $n \times n$ matrix with non-negative entries, that fulfills the equation $A*A = A^2= 0$, where $0$ is the zero matrix. What is the maximal number of positive entries for $A$?
So far I have tried going through the specific entries, namely ... |
If a stochastic process follows $dX(t) = a(t)X(t)dt + \sigma_1(t)X(t)dW_1(t)+ \sigma_2(t)X(t)dW_2(t)$, does the solution take the form
$$ X(t) = X(0) exp^{\int_0^t (a(s) - (1/2) \sigma(s)^2)ds + \int_0^t\sigma_1(s)dW_1(s) + \int_0^t\sigma_2(s)dW_2(s)}
$$
?
$\linebreak $
where $W_1$ and $W_2$ are 2 Brownian motion ... |
The basic idea is this:
> If you know what a linear transformation does on a spanning set, then you know what it does on any vector.
Why? Because if $a_1,\ldots,a_n$ span $V$, and you know that $f\colon V\to V'$ is a linear transformation, *and* you know what $f(a_1),\ldots,f(a_n)$ are, then there is only one th... |
Consider the incidence graph of the matrix obtained by replacing every non-zero entry by 1. A path of length 2 in this graph corresponds to the product of two non-zero coefficients, say $a_{i,j}a_{j,k}$. In your example, the graph would have 5 vertices and edges $(2,1), (3,1), (4,1), (2,5), (3,5), (4,5)$.
Thus the pr... |
I just make a quick and dirty code that uses an already found list of primes up to 1 bilion.
The file can be found [here](https://sourceforge.net/projects/billion-primes/)
The code I have used is:
def sum_digit(n):
sum = 0
while (n != 0):
sum = sum + (n % 10)
... |
Given a *non-normal* square matrix $A\in \mathbb{R}^{N\times N}$, we know its Eigenvalues $\lambda_i$ and its singular values $\sigma_i, i=1,\dots, N$.
Say, I approximate $A$ by $\sigma_i\rightarrow 0\ \forall i\ge k$ for some cutoff index $k$, how can I prove that $\lambda_i=0\ \forall i\ge k$?
I did some numerica... |
Let $(X,d)$ be a metric space and define for each $x\in X$ and for $\varepsilon >0$ the ball $B_\varepsilon^X (x) := \{ y:y\in X \land d(x,y) < \varepsilon\}$. Now, consider a subset $A\subseteq X$ and define the quotient space $X/_\sim$ as $X/A := \{[x]: x\in X\}$ and with equivalence relation $x\sim y \Longleftrighta... |
> Let $f,g$ be holomorphic on a bounded domain $D$ and continuous on
> $\overline{D}$. Assume that $\Re(f)=\Re(g)\forall z\in\partial D$.
> Prove that $\Re (f-g)=0$.
My attempt\
Consider $h=f-g$. Clearly $h$ is holomorphic on $D$, continuous on $\overline{D}$.\
If $h$ is constant, then we finish.\
If $h$ is non... |
Given a *non-normal* diagonalizable square matrix $A\in \mathbb{R}^{N\times N}$, we know its Eigenvalues $\lambda_i$ and its singular values $\sigma_i, i=1,\dots, N$.
Say, I approximate $A$ by $\sigma_i\rightarrow 0\ \forall i\ge k$ for some cutoff index $k$, how can I prove that $\lambda_i=0\ \forall i\ge k$?
I di... |
There is [numerical evidence][1] that $$I=\int_{\arccos(1/4)}^{\pi/2}\arccos\left(\cos x\left(2\sin^2x+\sqrt{1+4\sin^4x}\right)\right)\mathrm dx=\frac{\pi^2}{40}$$
How can this be proved?
I was trying to [answer another question][2], and I got it down to this integral.
Wolfram [does not evaluate][3] the indefi... |
Zero is a singular value of $A$ iff there exists $u\neq 0\in\mathbb{R}^N$ such that $ A^TAu = 0$. However,
$$
A^TAu=0\implies u^TA^TAu = \|Au\|_2^2=0\implies Au=0,
$$
so $u$ is an eigenvector of $A$ corresponding to $\lambda=0$. This trick can be used in the other direction to show that $\lambda=0 \iff \sigma=0$.
... |
I have began to study Riemannian Geometry and there I encountered a statement that I am unable to feel/understand/visualize.
It is the following statement that is still bothering me:
>Any geodesic for a linear connection on a smooth manifold is an integral curve of some vector field in $TM$.
The instructor ... |
I was wondering if it would be possible (and feasible) to analyse a card game called "declare" which I recently discovered. Suppose there are 9 players playing with 4 decks of 52 cards and starting off with 9 cards in hand. If a person deems the sum of their cards small enough, they can show their cards (they win if it... |
>What are the possible dimensions of an irreducible representation of a group of order 21?
I'm not sure what theorems might be useful for solving this problem. I know that every element of the group has order either 1, 3, or 7, and that there is at least one element of order 3 and one of order 7 (because not every e... |
what are the possible dimensions of an irreducible representation of a group of order 21? |
A Poulet-number is a Fermat pseudoprime to base $2$ that is it is a composite number $N$ such that $$2^{N-1}\equiv 1\mod N$$ holds.
Let $m$ be a positive integer and consider $f(m)=(2m^2+1)(8m^2+1)$
> If $f(m)$ is a Poulet-number , must at least one of $2m^2+1$ and $8m^2+1$ be prime ?
Numerical analysis :
-... |
I have questions about the proof of Theorem 3.4.12 in Emily Riehl's *Category Theory in Context*.
The theorem states that the colimit of a small diagram $F\colon \mathsf J \to\mathsf C$ can be expressed as a coequalizer between two arrows $c,d\colon\coprod_{f\in\operatorname{mor} J}F(\operatorname{dom} f) \rightarrow ... |
A question I found and that I could not answer so far. Assuming that $A$ is a $n \times n$ matrix with non-negative entries, that fulfills the equation $A^2= 0$, where $0$ is the zero matrix. What is the maximal number of positive entries for $A$?
So far I have tried going through the specific entries, namely to sol... |
**I would like to know what assumptions and/or properties characterize the Gaussian copula.** I am interested in the case of arbitrary dimensionality, and am not so interested in properties specific to the bivariate case.
# What do I mean by characterizing assumption/property? #
Many univariate distributions have... |
Given the following equation:
$$
\dot{\Theta} = \operatorname{sgn}(z) \sqrt{| z |}
$$
$$
z = \dot{\Theta}^2 \operatorname{sgn}(\dot{\Theta})
$$
$$
\dot{z} = 2 \sqrt{| z |} \ddot{\Theta}
$$
how can I get the vector field $\boldsymbol{g}$?
$$
\dot{\boldsymbol{q}} = \boldsymbol{g} (\boldsymbol{q})
$... |
In this [answer][1], there's the following line:
> If $m_*(\mathcal N^c)<1$, then $\forall\ \varepsilon>0$, $\exists\ U$ is measurable s.t. $\mathcal N^c \subset U \subset I$ and $m_*(U)<1-\varepsilon$.
(Here $m_*$ denotes **outer** measure.)
I don't see immediately why such measurable set $U$ always exists. C... |
Consider the incidence graph of the matrix obtained by replacing every non-zero entry by 1. A path of length 2 in this graph corresponds to the product of two non-zero coefficients, say $a_{i,j}a_{j,k}$. In your example, the graph would have 5 vertices and edges $(2,1), (3,1), (4,1), (2,5), (3,5), (4,5)$.
Thus the pr... |
Assume a static (time-independent) random variable $r$ for which we know its probability distribution $P(r)$. Consider this to be a Gaussian distribution, such that $\langle r \rangle=0$ and $\langle r^2 \rangle=r_0$, etc.
Consider now the following differential equation: $\frac{d}{dt}X(t)=r A X(t)$, where $X=(x_1,x... |
Let $S$ be a scheme, which we will conflate with a site on $S$ -- the etale, fppf or fpqc site. Let $\mathcal G$ be a gerbe over $S$, i.e. a stack on $S$ fibered in groupoids such that (1) there is some $T/S$ such that $\mathcal G(T)$ is nonempty, and (2) for any two objects $x, y \in \mathcal G(T')$ there is a coverin... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. on the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
I have trouble with the implication that in the "shell method" to calculate volumes of revolution, the left end point of the first piece is $0$, the left end point of the second piece is $\dfrac{(b-a)}{n}$, the left end point of the third piece is $2\dfrac{b-a}{n}.$
Why do you use the left end point of the pieces in... |
How to prove every shell is non-overlapping in the Riemann sum of volume of revolution by "shells"? |
Given the following equation:
$$
\dot{\Theta} = \operatorname{sgn}(z) \sqrt{| z |}
$$
$$
z = \dot{\Theta}^2 \operatorname{sgn}(\dot{\Theta})
$$
$$
\dot{z} = 2 \sqrt{| z |} \ddot{\Theta}
$$
how can I get the vector field $\boldsymbol{g}$?
$$
\dot{\boldsymbol{q}} = \boldsymbol{g} (\boldsymbol{q})
$... |
Why is the shell method not
$$\sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}{n}\right) \cdot \left(\frac{1}{n}\right)^2.$$?
I have trouble with the implication that in the "shell method" to calculate volumes of revol... |
Why is the shell method not
$$\sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}{n}\right) \cdot \left(\frac{1}{n}\right)^2.$$?
I have trouble with the implication that in the "shell method" to calculate volumes of revol... |
I have the equation
$$f(x) = \arccos(x) + (C(1 - x^2) - x)\sqrt{1 - x^2}$$ where the parameter $C$ is a real number. When $-\frac{2}{3} ≤ C$, the function has one real zero at $x = 1$. When $C$ drops below $-\frac{2}{3}$, a second zero appears, starting at $x = 1$ and decreasing towards $x = -1$ as $C$ tends towards ... |
Can the zero of $f(x)=\arccos(x)+(C (1 - x^2) - x) \sqrt{1 - x^2}$ be calculated explicitly, or is it transcendental? |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
Why is the shell method not
$$\sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}{n}\right) \cdot \left(\frac{1}{n}\right)^2.$$?
I have trouble with the implication that in the "shell method" to calculate volumes of revol... |
We know that for a good summability kernel ${k_{n}}$: $k_{n}*f$ converges to f in $L^p$ norm for 1 <= p < $\infty$. But I have trouble finding a counter-example for the case p = $\infty$. Can someone please help me with this one ?
Best regards |
Why is the shell method not
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}{n}\right) \cdot \left(\frac{1}{n}\right)^2.$$?
I have trouble with the implication that in the "shell method" t... |
Let's take the following random variables $X,Y,Z_1,...,Z_I$ on the same probability space (or associated with the same experiment), where
$$ Y = \sum_{i=1}^I Z_i $$
Let's say I am interested in computing the conditional variance of $X$ knowing $Y$, similarly as in this [question][1]. Applying the law of total var... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
So we where given the homework of formulating and prooving the fundamental theorem of homomorphisms for G-Sets, which factors equivariance maps. (this is not the homework question though, but a follow up).
So take two G-Sets M,N and an equivariance map ($\varphi:N\rightarrow M$ with $\varphi(gn)=g\varphi(n)$). It's ... |
Which elements have equal image under equivariance maps? |
For context, the question popped up while studying the homotopical version of Cauchy's Integral Theorem, and comparing it with the homological version.
**Definition:** two $γ_0, γ_1 : [0, 1] → U$ be two closed rectifiable curves in
a region $G$ are said to be ***homotopic*** iff there is a continuous function
$Γ :... |
Does being homotopic to $0$ imply being homologous to $0$ or viceversa? |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
I've been looking all over the internet and have been having trouble finding good uses of a covariance matrix to find the correlation coefficient.
I know that, from a simple $2 \times2$ variance-covariance matrix, the correlation is given by $\mathrm{COR}\left(X,Y\right)=\frac{\mathrm{COV} \left(X,Y\right)}{\sqrt{Va... |
I'm starting to self-study differential geometry with Guillemin & Haine. They introduce a Lie derivative and in exercises they want to prove formula for lie bracket, i.e.:
Let $U\in \mathbb{R}^n$ be open and $v_1, v_2$ be vector fields on $U$. Show that there exists a unique vector field $w$ on $U$ with the property... |
How to effectively determine whether a point (N-dim) from a point cloud belongs to its convex hull without computing the hull? |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
How to prove that $\frac{2^{\sqrt{n}}}{n!}\left(\frac{n}{e}\right)^n$ diverges as $n \rightarrow \infty$? |
I started reading spivak, just today, and on page 6 of the prologue. He states that it is not possible to prove that 1≠0 for the real numbers, with the basic properties of the ring.
But I got the following argument:
Assuming that $1=0$, it follows that $a(1)=a(0)$ for all $a \in{R}$, where $a(1)=a$ and $a(0)=0$, ... |
This proof is valid for 1≠0 in real numbers (set/field)? |
I have doubts for what concerns the end of this exercise. I have the function $f(x, y, z) = x$ and I have to find min and max subject to the constraints $x^2+y^2+z^2 = \frac{5}{2}$ and $y + z = 1$.
I calculated the Jacobian matrix of the constraints and I found it has maximum rank of $2$, unless $x = y = z = 0$ thou... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
Find the number of sequences $(a_1,\cdots,a_{100})$ where all $a_i$'s are even numbers which forms $0\leq a_1 \leq \cdots \leq a_{100} \leq 200$.
I know that the number of strictly incremental sequences of length $K$ with elements from $\{1, 2,\cdots,N\}$ is ${N}\choose{K}$. However, here the sequence is not strictl... |
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