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While studying Functional Analysis, Sobolev Spaces and Partial Differential Equations (Brezis), I came across this theorem ($E$ is a Banach space):
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/A4ene.png
In class we studied the problem
$$
\left\{ \begin{array}{lcc} (p(x)y'(x))'+q(x... |
I want to prove the equivalence of strong operator convergence and weak operator convergence for a sequence of orthoprojectors {$P_n$}. These orthoprojectors act from one Hilbert space H into subspaces $H_n$ of the space H, which, as I understand it, do not have to coincide for different n.
Of course, if the dimens... |
Suppose $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$, $\text{rank}(Df(x))=k.$
Now Define a function $H: \mathbb{R}^m\times \mathbb{R}^n \rightarrow \mathbb{R}^k$, $H(x,x')=f(x)$.
Does $\text{rank}(DH(x,x^{\prime}))= \text{rank}(Df(x))$? Can I conclude that $\text{nullity}DH(x,x') = \text{nullity}(Df(x))=n-k$? |
let's say I have an event A happening at a rate X per second. Every event A will trigger event B that will be repeated at a rate Y per second. How can I calculate the total number of A and B events in a period of time Z seconds? |
A 3-Dimensional ellipsoid is given by the equation:
````
x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
````
If a=20, b=15 and c=10. Then as per my understanding, a is the major axes and c is the minor axes. What axes is b called? I am confused. Please provide some references too.
|
There's something I can't quite wrap my head around, and I am not satisfied with my professor's "that's just defined that way" answer. So suppose we have:
$$f_1(x) = x - 5$$
$$f_2(x) = \frac{1}{x - 5}$$
$$f_3(x) = 1$$
By simple algebra, we can see that
$f4(x) = f_1(x) \cdot f_2(x) = f_3(x)$
What I mean... |
Let $k \in K$. As $K$ is a field then
$$
k-\frac{1}{n} \in K
$$
By density there exists $q_n \in \mathbb{Q}$ such that $k-\frac{1}{n} < q_n < k-\frac{1}{n+1}$
We have
$$
\lim_{n \to \infty} q_n= k
$$
Indeed, for any $\epsilon>0$ there exists $N= \lceil \frac{1}{\epsilon} \rceil$ such that, for each $n>N$ ... |
Hint:
1) $D\ge 0,$ and $y \ne 0$ (Why?)
2) $y^2+4y \ge 0$. or $(y+2)^2 \ge 4;$
Can you finish?
|
A 3-Dimensional ellipsoid is given by the equation:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$
Let $a=20, b=15$ and $c=10$. Then as per my understanding, $a$ is the major axes and $c$ is the minor axes. What axes is $b$ called? I am confused. Please provide some references too.
|
I am looking into repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
Fo... |
I am looking into repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
Fo... |
Suppose $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$, $\text{rank}(Df(x))=k.$
Now Define a function $H: \mathbb{R}^m\times \mathbb{R}^n \rightarrow \mathbb{R}^k$, $H(x,x')=f(x)$.
Does $\text{rank}(DH(x,x^{\prime}))= \text{rank}(Df(x))$? Can I conclude that $\text{nullity}DH(x,x') = m+n-\text{rank}(Df(x))=m+n-k$? |
I am examining the problem of determining unknown angles in an antenna array system, where the phase shift equation is given as:
$$
\Delta \Phi = \pi \left[ \sin(\theta_e) \sin(\theta) \cos(\phi_e - \phi) + \cos(\theta_e) \cos(\theta) \right]
$$
In this scenario, $ \Delta \Phi $ represents the phase shift, and ... |
Solving $ \Delta \Phi = \pi \left[ \sin(\theta_e) \sin(\theta) \cos(\phi_e - \phi) + \cos(\theta_e) \cos(\theta) \right]$ for unknown angles? |
I was reading Prof Gaurav Bhatnagnar, "How to prove Ramanujan q-continued fractions", on the first page he mentions:
$$\text{the q-generalisation of } \\
1+1+1 \cdots + 1 = n \\
\text{is } 1+q+q^2+ \cdots + q^{n-1} = \frac{1-q^n}{1-q}
$$
Which turns out to be the geometric series when $q < |1|$. What does a q-gen... |
What is meant by "q-generalisation"? |
Suppose that $f$ is integrable on $[0,\infty)$ and that $f(x)\rightarrow 0$ as $x\rightarrow \infty$.
For $x\geq 0$, define $$g(x)=\int_0^\infty f(x+r) dr.$$ Using the Fundamental Theorem of Calculus, if $f'$ exists and is integrable on $[0,\infty)$, then I can justify:
$$g'(u)=\int_0^\infty f'(u+r)dr=\int_0^\infty... |
**The Question**
$\newcommand{\img}{\operatorname{img}}$
$\newcommand{\bm}[1]{\boldsymbol{#1}}$
I am self-studying solutions of matrix ODEs. Below I have written the cases I know how to deal with, but I have some trouble with formalising the higher order ODE case when the matrix $\bm{B}_{m+1}$ is singular. Similar... |
Context: We were taught collinearity of 3 points (in vectors) and a method for checking 4 points for coplanirity simultaneously.. Searching around i found a better method for checking for coplanirity in 4 points i.e the 4x4 determinant.
In a 2d plane there is a 3x3 determinant to check for collinearity of 3 points ... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases. It seems to me that more can be said under the assumption of prime $a$.
**If $a>2$ is prime and $b^{2a}-4a^b$ is a perfect square, then either $b \equiv 0 \mod{a^2}$ or $b \equiv a \mod{a^2}$... |
**The Question**
$\newcommand{\img}{\operatorname{img}}$
$\newcommand{\bm}[1]{\boldsymbol{#1}}$
I am self-studying solutions of matrix ODEs. Below I have written the cases I know how to deal with, but I have some trouble with formalising the higher order ODE case when the matrix $\bm{B}_{m+1}$ is singular. Similar... |
I have a rather peculiar contrained quadratic programming problem where I try to fit a left-stochastic matrix and I am not sure how to properly solve it. Consider matrices $S\in\mathbb R^{D\times N}$, $C\in \mathbb R^{D\times K'}$, $G \in \mathbb [0, 1]^{K\times N}$. I am interested in solving the following QP:
$$\ope... |
According to the relevant <a href="https://en.wikipedia.org/wiki/Product_integral">wiki article</a>, there are three types of product integrals. Type II, the *geometric integral*, is named as the continuous analogue of the discrete product operator.
The geometric integral is defined as follows:
$$\prod_{a}^{b} ... |
I came across this fun little math / programming problem.
[![enter image description here][1]][1]
*The programming language is Java*
Now, to shortcut a few things...
* The `n1 = n0 + (n0 = n1))` line affects a Fibonacci sequence.
* The loop is flawed, because `i` never increments
* `int` in Java is a... |
I'm asked to find the integral of the function $ xy e^{x^2 - y^2} $, on $$ D = {\{(x,y) \in R^2: 1 \leq y, \sqrt2 \le x \le 9, 1 \le x^2 - y^2 \le 9 \} } $$
I think I don't understand the necessary steps to get the bounds of integration.
I know that $ x^2 - 9 \le y^2 \le x^2 - 1 $. I know that $y^2 \ge 1$ so we can... |
I came across this fun little math / programming problem.
[![enter image description here][1]][1]
*The programming language is Java*
Now, to shortcut a few things...
* The `n1 = n0 + (n0 = n1))` line affects a Fibonacci sequence.
* The loop is flawed, because `i` never increments
* `int` in Java is a... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
**If $a>2$ and $b^{2a}-4a^b$ is a perfect square, then $a$ cannot be a prime.**
*Proof.*
Assume that $p := a > 2$ is a prime number and let $b^{2p} - 4p^b = t^2$ for some positive in... |
While trying to get an expression for [the number of deals from a generalised bridge deck with nobody being void in any suit](https://math.stackexchange.com/q/3443423/357390) I encountered the following subproblem.
From a generalised bridge deck with $r$ ranks instead of just $13$ deal four hands of $r$ cards each. ... |
How do I solve $x^x - 5x + 6 = 0$ using the Lambert W function?
**EDIT:**
I solved equations $2^x - 5x + 6 = 0$ and $3^x - 4x - 15 = 0$ using Lambert W function, but not able to solve $x^x - 5x + 6 = 0$ equation. How and which method can be used to solve this equation?
How to solve this equation without graphi... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
**If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* Assume that $a > 1$ is an odd number and let $b^{2a} - 4a^b = t^2$ for some positive integer $t$. From diff... |
I am stuck at simplifying the following integral:
$$
I =
\int_{-\pi/2}^{+\pi/2}
d\phi
\left[
1-M(\phi) \theta^*(\phi)
\right]
$$
where
$
M=
p \cos \phi + q \sin \phi
$.
Here $p,q$ are fixed.
The $\theta^\star$ is defined through
$\cos \theta^*=
\frac{M}{\sqrt{1+M^2}}
$
Can anybody help me to c... |
How to integrate $$\int_{2}^{\infty} \frac{\pi(x) \ln(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx \quad?$$
Wolfram gives the numerical value
$$\int_{2}^{\infty} \frac{\pi x (1 + x^2) \log(x^{\sqrt{x}})}{(-1 + x^2)^3} \, dx = 1.46036
$$
**My idea**
Let
$$I = \int_{2}^{\infty} \frac{\pi(x) \ln(x^{\sqrt{... |
I would like to understand precisely how the usual topology on $\mathbb{R}$ can be extended to $\overline{\mathbb{R}}:=\mathbb{R} \cup \{-\infty,\infty\}$. The book I'm reading defines therefore the following neighbourhood bases: Additionally to the bases $\{(x-\epsilon,x +\epsilon) : \epsilon >0\}$ for $x\in \mathbb{R... |
I have learnt the Cheeger inequality for a graph $G$:
$\lambda_2/2\le h(G)\le\sqrt{2\lambda_2}$.
But does the equality hold?For a non-connected graph, it is obvious.But for a connected graph,I have tried some examples and find the first equality may hold, but fail at the second one.Are there any conditions for eq... |
What's the conditions for equality of Cheeger inequality? |
*Remark:* My attemps is WRONG as I fasly assumed that the $Y_n$ were independant. I ve corrected this in my own answer to this post.
## Question:
Let $a > 0$. For $n \in \mathbb{N}$, let the random variable $X_n$ on $\{0,n\}$ be defined by
$$
\begin{cases}
P(X_n=0)=1- \frac{1}{n^{a}}\\
P(X_n=n)=\frac{1}{n^{a... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
**If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* Assume that $a > 1$ is an odd number and let $b^{2a} - 4a^b = t^2$ for some positive integer $t$. From diff... |
I'm struggling to grasp the justification behind integrating both sides of an equation. While I understand that operations can be applied to both sides, maintaining equality, it appears that this principle doesn't apply here.
given the equality $x=y$ integrating both sides by dx would give $\frac{x^2}{2} = xy$
... |
does Integrating both sides of an equation in dx will Invalidates the equality? |
I will try to write my proof of the Riemann lebesgues Lemma, correct me if something is wrong.
If $f\in L_{1}(R)$ then its fourier transform $f^{*}$ maps to $C_{0}(R)$, a space of continuous functions that converge to zero in infinity, that is $\lim_{|\zeta|\to\infty} f^{*}(\zeta)=0.$
Proof: Since compactly supp... |
I will try to write my proof of the Riemann lebesgues Lemma, correct me if something is wrong.
If $f\in L_{1}(R)$ then its fourier transform $f^{*}$ maps to $C_{0}(R)$, a space of continuous functions that converge to zero in infinity, that is $\lim_{|\zeta|\to\infty} |f^{*}(\zeta)|=0.$
Proof: Since compactly su... |
Suppose $(M, J)$ is a compact complex manifold of complex dimension $n$ and there exists a Kaehler metric $\Omega$ on $M$. By the global $\partial \bar\partial$ lemma, any Kaehler metric in the same cohomology class as $\Omega$ is of the form $\Omega' := \Omega + \partial\bar\partial f$ for $f \in \mathcal{C}^\infty(M)... |
I'm trying to find the minimum to the functional $J[y]$ given above where the values for $y$ at the boundary values are not specified.
My first idea was to find the partial derivatives of $L(x,y,y')=\frac{1}{2}y'^2+yy'+y'+y'$ which are
\begin{align}
&\frac{\partial L}{\partial y}=y'+1, \\
&\frac{\partial L}{\p... |
A non-trivial finite group $G$ of order $n$ is said to be sequenceable if its elements can be arranged in a sequence ($g_{1}, g_{2}, \dots ,g_{n}$) in such a way that the partial products ($a_{1}, a_{2}, \dots , a_{n}$), where $a_{i}=g_{1}g_{2} \dots g_{i}$ are distinct. The sequence ($g_{1}, g_{2}, \dots g_{n}$) is ca... |
Are sequenceable groups self-reducible? |
A non-trivial finite group $G$ of order $n$ is said to be sequenceable if its elements can be arranged in a sequence ($g_{1}, g_{2}, \dots ,g_{n}$) in such a way that the partial products ($a_{1}, a_{2}, \dots , a_{n}$), where $a_{i}=g_{1}g_{2} \dots g_{i}$ are distinct. The sequence ($g_{1}, g_{2}, \dots g_{n}$) is ca... |
I'm trying to find the minimum to the functional $$J[y]=\int_0^1 \left(\frac{1}{2}y'^2+yy'+y'+y'\right)\, \mathrm dx$$ where the values for $y$ at the boundary values are not specified.
My first idea was to find the partial derivatives of $L(x,y,y')=\frac{1}{2}y'^2+yy'+y'+y'$ which are
\begin{align}
\frac{\parti... |
How to solve this double series problem:
$$\sum_{m=0}^{\infty}\sum_{n=0}^{m}\left[\mathrm{C}_{n}^{m}(m+n+2)\left(\frac{1}{2}\right)^{m+n}\right]$$
where $\mathrm{C}_{n}^{m}\equiv\frac{m!}{(m-n)!n!}$ is "combination without repetition".
The answer is 24.
Note. This double summation is midway in solving a "to... |
I'm trying to find the minimum to the functional $$J[y]=\int_0^1 \left(\frac{1}{2}y'^2+yy'+y'+y\right)\, \mathrm dx$$ where the values for $y$ at the boundary values are not specified.
My first idea was to find the partial derivatives of $L(x,y,y')=\frac{1}{2}y'^2+yy'+y'+y'$ which are
\begin{align}
\frac{\partia... |
How to integrate $$\int_{2}^{\infty} \frac{\pi(x) \ln^2(x^{\sqrt{x}}) \cdot (x^2 + 1)}{(x^2 - 1)^3} \,dx \quad?$$
Wolfram gives the numerical value
$$\int_{2}^{\infty} \frac{\pi x (1 + x^2) \log^2(x^{\sqrt{x}})}{(-1 + x^2)^3} \, dx = 1.46036
$$
**My idea**
Let
$$I = \int_{2}^{\infty} \frac{\pi(x) \ln^2(x^{... |
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
>I need asymptotic formula of $I_n$ as $n\to\infty$
Firstly we see that since $x(1-x)\leq \frac{1}{4}$, so the defini... |
I try to numerically solve (by finite difference method) the beam deflection equation
$$EJ\cdot\frac{\partial^4 v}{\partial x^4} = -g\cdot F\cdot\rho$$ with following boundary conditions
$$v(0)=0;\frac{\partial v(0)}{\partial x} = 0$$
$$\frac{\partial^2 v(L)}{\partial x^2} = 0;\frac{\partial^3 v(L)}{\partial x^3} = ... |
The high-school classes from a well-known high-school teacher of my country justifies the limit of $x \ln(x)$ at $0$ by composing (i.e. doing a change of variable $X= \ln(x)$) on the right by the exponential function. The limit of $x \exp(x)$ at $-\infty$ being known, we can conclude that $\lim_{x\to 0} x \ln(x) = 0$.
... |
Finding $\lim_{x\to 0} x \ln(x)$ by $\lim_{x\to -\infty} x \exp(x)$ - is the composition to the right licit? |
Recall that
$$\int_{|z|=R} |dz| = 2 \pi R$$
Now, by the triangle inequality:
$$\color{blue}{|z|^2-|a|^2} \leq |z^2-a^2| \leq \color{red}{|z-a||z+a|}$$
Hence,
$$\int_{|z|=R} \frac{|dz|}{\color{red}{|z-a||z+a|}} \leq \int_{|z|=R} \frac{|dz|}{\color{blue}{|z|^2-|a|^2}} = \frac{2\pi R}{|R^2-|a|^2|}$$
Sti... |
**The Question**
$\newcommand{\img}{\operatorname{img}}$
$\newcommand{\bm}[1]{\boldsymbol{#1}}$
I am self-studying solutions of matrix ODEs. Below I have written the cases I know how to deal with, but I have some trouble with formalising the higher order ODE case when the matrix $\bm{B}_{m+1}$ is singular. Similar... |
I am trying to learn some basic number theory now, and I apologize for this trivial question, but I just can't see how to get $p_n\sim n\log(n)$ from $\pi(x)\sim\frac x{\log x}$?
>The prime number theorem is equivalent to the statement that the $n$th prime number $p_n$ satisfies $\pi(x)\sim\frac x{\log x}$?
**I tr... |
How does $p_n\sim n\log(n)$ come from $\pi(x)\sim\frac x{\log x}$? |
I'm trying to get more knowledge on Riemann, Ricci and Einstein tensors and while reading Differential Geometry Curves Surfaces Manifolds Third Edition - W. Kühnel I came across the sectional curvature of the Riemann manifold with respect to the plane $\sigma$. I don't really see what the purpose of this curvature is, ... |
The high-school classes from a well-known high-school teacher of my country justifies the limit of $x \ln(x)$ at $0$ by composing (i.e. doing a change of variable $X= \ln(x)$) on the right by the exponential function. The limit of $x \exp(x)$ at $-\infty$ being known, we can conclude that $\lim_{x\to 0} x \ln(x) = 0$.
... |
Finding $\lim_{x\to 0} x \ln(x)$ via $\lim_{x\to -\infty} x \exp(x)$ - is the composition to the right licit? |
The high-school classes from a well-known high-school teacher of my country justify the limit of $x \ln(x)$ at $0$ by composing (i.e. doing a change of variable $X= \ln(x)$) on the right by the exponential function. The limit of $x \exp(x)$ at $-\infty$ being known, we can conclude that $\lim_{x\to 0} x \ln(x) = 0$.
... |
I'm trying to find the minimum to the functional $$J[y]=\int_0^1 \left(\frac{1}{2}y'^2+yy'+y'+y\right)\, \mathrm dx$$ where the values for $y$ at the boundary values are not specified.
My first idea was to find the partial derivatives of $L(x,y,y')=\frac{1}{2}y'^2+yy'+y'+y$ which are
\begin{align}
\frac{\partial... |
Let $$c_n=n!\sum\limits_{k=0}^n (-1)^k \frac{1}{k!}$$ be the number of derangements of $n$ elements. The following combinatorial identity is coming up in my research:
$$\sum\limits_{j=1}^{n-2}c_{n-j}{n-2\choose j-1}=(n-2)!(n-2)$$
Is there a known reference or proof for this inequality? Simulations do support this... |
I read about Hahn-Banach theorem from the book "Introduction to topology and modern analysis" by Simmons. There it was stated like this:
**Let $M$ be a linear subspace of a normed linear space $N$, and let $f$ be a functional defined on $M$. Then $f$ can be extended to a functional $f_0$ defined on the whole space $... |
I came across this fun little math / programming problem.
[![enter image description here][1]][1]
*The programming language is Java*
Now, to shortcut a few things...
* The `n1 = n0 + (n0 = n1))` line affects a Fibonacci sequence.
* The loop is flawed, because `i` never increments
* `int` in Java is a... |
Find every real numbers x so that $\log_2 (\frac{x+1}{x^2+2} )= x^2-2x-1$
Of course $x=${$0$,$2$} are solutions and because $x^2-2x-1$ is strictly increasing from $[1,\infty )$ and $\log_2 (\frac{x+1}{x^2+2} )$ is decreasing in that interval ,$2$, is the only solution in $[1,\infty )$. How can i prove that $0$ is t... |
Find every real numbers x so that $\log_2 (\frac{x+1}{x^2+2} )= x^2-2x-1$
Of course $x=${$0$,$2$} are solutions. Because $x^2-2x-1$ is strictly increasing from $[1,\infty )$ and $\log_2 (\frac{x+1}{x^2+2} )$ is decreasing in that interval ,$2$, is the only solution in $[1,\infty )$. How can i prove that $0$ is the ... |
Let $T$ and $S$ be stopping times wrt a filtration $(\mathcal{F}_n)$ and assume that $S \leq T<\infty$ a.s. The question is whether $TS$ is also a stopping time. In continuous time, I believe it isn't, but, in my course, we have only been considering discrete time, in which case I think I have a proof that it is a stop... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. The reciprocal triangle inequality $\frac{1}{a} + \frac{1}{b} \ge \frac{1}{c}$ is not true in general. Experimental data shows that the probability
$$
P\lef... |
According to Wikipedia, for integer $m$ and large real $x$,
\begin{aligned}I_{m }(x)={\frac {1}{{\sqrt {2\pi x}}{\sqrt[{4}]{1+{\frac {m^{2}}{x^{2}}}}}}}\exp \left(-m \operatorname {arsinh} \left({\frac {m}{x}}\right)+x{\sqrt {1+{\frac {m^{2}}{x^{2}}}}}\right)\left(1+{\mathcal {O}}\left({\frac {1}{x{\sqrt {1+{\frac ... |
Fisrt, let $k$ be a **number field**, and let $A/k$ be an abelian variety. I know that by the Mordell–Weil theorem, $A(k)$ is finitely generated, but I saw [here][1] (in the introduction) that $A(k)_{tors}$ is finite, but I don't understand why. I tries to search for a proof and I saw that is also true in some other se... |
How to prove $\lim_{n\to \infty} \left(\frac{S_n}{n^{1-\alpha}}\right) = \frac{1}{1-\alpha}$? |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. The regular triangle inequality states that the sum of any two sides is greater than the third side. But what happens if we take the sum of the reciprocal of any... |
Context: We were taught collinearity of 3 points (in vectors) and a method for checking 4 points for coplanarity simultaneously. Searching around I found a better method for checking for coplanarity in 4 points i.e the 4x4 determinant.
In a 2d plane there is a 3x3 determinant to check for collinearity of 3 points
... |
How to determine the convergence/divergence of an improper integral:
$$
\int_{0}^{\infty} \frac{\arctan(ax) - \arctan(bx)}{x}dx
$$
I know that this integral can be written into the sum of two integrals by the additivity property:
$$
\int_{0}^{1} \frac{\arctan(ax) - \arctan(bx)}{x}dx + \int_{1}^{\infty} \frac{\arc... |
Let $$c_n=n!\sum\limits_{k=0}^n (-1)^k \frac{1}{k!}$$ be the number of derangements of $n$ elements. The following combinatorial identity is coming up in my research:
$$\sum\limits_{j=1}^{n-2}c_{n-j}{n-2\choose j-1}=(n-2)!(n-2)$$
Is there a known reference or proof for this identity? Simulations do support this. |
When $x$ is in $(0,2)$, $\frac{x+1}{x^2+2}$ is in $(1/2,1)$. Indeed, $\frac{x+1}{x^2+2}>1/2$ is equivalent to $2x+2>x^2+2$, or $2>x$, which is true. And $\frac{x+1}{x^2+2}<1$ comes from $x+1<x^2+2$, or $x^2-x+1>0$, which is always true.
Then $\log_2\frac{x+1}{x^2+2}$ is in $(-1,0)$ for $x\in (0,2)$.
Also, for $x\... |
Consider the following definitions taken from the book *Topics in Algebra* by I. N. Herstein.
> An integral domain $D$ is said to be of characteristic $0$ if the
> relation $ma=0$, where $a\neq 0$ is in $D$, and where $m$ is an
> integer, can hold only if $m=0$.
>
> An integral domain $D$ is said to be finite c... |
I don't understanding the estimation of IV thoroughoutly. The following is my puzzle.
In the first stage, we have
$$D_i = \alpha + \beta Z_i + u_i$$
then we get predicotrs $\hat{D_i}$.
In my mind, $\hat{D_i}$ is a mixture of "compliers, always-takers, never-takers and Defiers" given the IV variable, $Z$
... |
I want to show the following:
Let $X$ be a topological vector space (and Hausdorff), $f:X \rightarrow \mathbb{K}$ a linear functional. Then the following are equivalent:
1.$f$ is continuous
2.$f$ is continuous at $0$
3.$ker(f)$ is closed
From 1 $\Rightarrow$ 2 is obviuos.
So let's start with 2 $\Rig... |
Let $R$ be an integral domain, is it true that if $R$ is atomic, then it must contain a prime element?
If not, what is a counterexample? |
y = {1,2,3,4,5,6}
My first try is (1/6)(y/6)^n-1 , but this doesn’t work, should I consider the order? |
Let Y denote the largest number obtained in n independent rolls of a fair die, where n is a positive integer . What is the probability mass function? |
$$\int_{2}^{343}\{x-\lfloor{x}\rfloor\}^2dx = \int_{2}^{3}x^2dx + \int_{3}^{4}x^2dx+\cdots+\int_{342}^{343}x^2dx$$
This is what I did, How can I proceed further with this? Any help is appreciated. |
Suppose I have a function $f$ defined on $[0,\infty)$ and differentiable on $(0,\infty)$. Define a function $g$ on $(-\infty,\infty)$ by $g(x)=f(|x|)$. Then do I get that $g$ is differentiable on $(-\infty,\infty)$, or could $g$ not be differentiable at $0$? |
I read about Hahn-Banach theorem from the book "Introduction to topology and modern analysis" by Simmons. There it was stated like this:
**Let $M$ be a linear subspace of a normed linear space $N$, and let $f$ be a functional defined on $M$. Then $f$ can be extended to a functional $f_0$ defined on the whole space $... |
I shall outline what is basically going on in the way I am familiar with, because, I suppose, once the main idea has been grasped, the rest could be worked out according to the specific details of a particular exposition.
Our intention is to build an increasing sequence of consistent set of sentences in an expanded ... |
y = {1,2,3,4,5,6}
My first try is $(1/6)(y/6)^n-1 , but this doesn’t work, should I consider the order? |
$$\int_0^\frac{\pi}{2} \frac{tan(x)}{(2ln(tan(x)))^2 + \pi^2}dx$$ <br>I have solved it as such: <br>$$\int_0^\frac{\pi}{2}\frac{tan(x)}{(2ln(tan(x)))^2+\pi^2}dx=\int_0^\frac{\pi}{2}\frac{tan(\frac{\pi}{2}-x)}{(2ln(tan(\frac{\pi}{2}-x)))^2+\pi^2}dx=\int_0^\frac{\pi}{2} \frac{cot(x)}{(2ln(tan(x)))^2 + \pi^2}dx$$<br>$$2I=... |
y = {1,2,3,4,5,6}
My first try is $(1/6)(y/6)^n-1$ , but this doesn’t work, should I consider the order? |
y = {1,2,3,4,5,6}
My first try is $(1/6)(y/6)^(n-1)$ , but this doesn’t work, should I consider the order? |
y = {1,2,3,4,5,6}
My first try is $(1/6)(y/6)^n-1$ , but this doesn’t work, should I consider the order? |
Let $R$ be an integral domain, is it true that if $R$ is atomic, then it must contain a prime element?
If not, what is a counterexample?
I know that if an element is prime, then if $I$ is the ideal generated by it, then $R/I$ must be an integral domain. However, $R$ being an atomic integral domain doesn't really ... |
y = {1,2,3,4,5,6}
My first try is (1/6)*(y/6)^(n-1) , but this doesn’t work, should I consider the order? |
y = {1,2,3,4,5,6}
My first try is $(1/6)*(y/6)^{n-1}$ , but this doesn’t work, should I consider the order? |
**Def (Sequential Accumulation Point):** Given a topological space $(X,\tau)$ and a subset $S\subseteq X$, we say that $a\in X$ is a *sequential* accumulation point if there exists $(s_n)_{n\geq1}\subseteq S\setminus\{a\}$ that converges to $a$.
The name comes from the following analogy. We say that $a\in X$ is an a... |
Must an infinite subset of a compact Hausdorff space contain a *sequential* accumulation point? |
I am reading [The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces](https://arxiv.org/abs/1811.02086), and I came across a usage of the words "covariant" and "contravariant" that I did not understand.
Here is the relevant passage:
> Generalizing the construction of $C... |
$$\int_0^\frac{\pi}{2} \frac{\tan(x)}{(2\ln(\tan(x)))^2 + \pi^2}dx$$ <br>I have solved it as such: <br>$$\int_0^\frac{\pi}{2}\frac{\tan(x)}{(2\ln(\tan(x)))^2+\pi^2}dx=\int_0^\frac{\pi}{2}\frac{\tan(\frac{\pi}{2}-x)}{(2\ln(\tan(\frac{\pi}{2}-x)))^2+\pi^2}dx=\int_0^\frac{\pi}{2} \frac{\cot(x)}{(2\ln(\tan(x)))^2 + \pi^2}d... |
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