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I recently learned about the formula for finding the surface area of a function rotated about the x-axis:
$$S = \int 2\pi y \sqrt{1+\bigg[ \frac{dy}{dx} \bigg]^2}dx$$
I tried to apply this formula to a problem involving the function $y = x^2$ rotated around the x-axis. The standard approach seems to involve ta... |
Are you sure about your statement?
I've tried to draw the situation and in my case, the point $F$ is located somewhere between points $A$ and $B$, so it can't be tangent to a circle, containing those points:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/jBajw.png |
Consider a non-zero positive element $a$ of a simple C*-algebra $A$. Suppose it has a sub-projection which is Murray-vN equivalent to an infinite projection, i.e. there exists $h$ s.t. $h\precsim a$. Then is $a$ properly infinite?
Here are the steps and I don't see why exactly each one is true:
1. It is true that t... |
I have tried many ways, including denote a tangent point $F'$ and prove $F'\equiv F$ or denote the points at which the incircle touches the sides but I can't go further. Can someone help me in this problem?
The incircle (its center is point $I$) of $\Delta ABC$ touches the side $AB$ at point $D$. Point $E$ is the re... |
One odd thing about the question is the apparent belief that the decimal notation of a number is somehow fundamental to its meaning, and so an unending decimal expansion is a sign that the number itself is somehow unending or imprecise.
First, a simple factual point: the question claims:
> We know that an irratio... |
Why is a rigid rotation on a torus ergodic? |
Let $G$ be a finite group and $H<G$ a subgroup. Let $V$ be a representation of $H$ with character $\chi$. The Frobenius formula states that the character of the induced representation $\text{Ind}_H^G(V) := \mathbb{k}G\otimes_{\mathbb{k}H}V$ is given by:
$$\chi_{\text{Ind}_H^GV}(g) = \frac{1}{|H|} \sum_{\substack{x\i... |
I've been trying to fit some data with polynomials using a pseudo-inversion library (numpy.linalg). There are two real functions $f$ and $g$ such that $f=-\alpha g$, $\alpha > 0$, and
$$
f=f(t_1,\ldots,t_5),
$$
$$
g = g(t_1,\ldots,t_5),
$$
where $t_1,\ldots,t_5$ are real parameters. The parameter $\alpha$ ha... |
I am unable to understand the following recurrence relation from the paper: https://people.inf.ethz.ch/emo/PublFiles/SubexLinProg_ALG16_96.pdf. Why is the probability for $t_k(n-1)$ not being accounted for and how is bound on probability for the second term used in the recurrence can someone help?
$t_k(n) \leq t_k(... |
I am trying to solve the following equations:
\begin{equation*}
\begin{aligned}
-\frac{\partial u}{\partial t}&=\Delta u+Cu+Z, (x,t)\in\Omega_T\\
\frac{\partial u}{\partial x}&=0, x\in \partial \Omega_T \\
u(T,x)&=0, \forall x\in\Omega.
\end{aligned}
\end{equation*}
where I employ the BTCS(backw... |
I have a kind of complicated second order diff eqn that is in the form of
$$y''(x) = 0$$
I have two initial conditions given as:
$$y(x) = \mathcal{C}x, ~~ y'(x)=x$$
where $\mathcal{C}$ is a constant number. Is there a general (known) numeric method to find the $\mathcal{C}$, which satisfies the above equati... |
Assume that $\lim_{x \to 0} f(x) = L.$ Prove that $\lim_{x \to 0} f(x^3) = L.$ Is this proof sufficient? |
I am new to StackExchange so apologies if my question is poorly asked or does not abide by the standards.
Referring to the 2016 edition of Tomkowicz and Wagons' book on the Banach-Tarski Paradox, looking specifically at the second part of corollary 5.12, it states:
Suppose $G$ acts on $X$ and a proper system of ... |
I'm currently having a confusion dealing with the conditional expectation. Let's recall the definition first:
Let $X$ be an integrable function (or a random variable) defined on a probability space $(\Omega,\mathfrak{M},\mu)$ and let $\mathfrak{A}$ be a $\sigma$-subalgebra of $\mathfrak{M}$. The **conditional expect... |
I am new to StackExchange so apologies if my question is poorly asked or does not abide by the standards.
Referring to the 2016 edition of Tomkowicz and Wagons' book on the Banach-Tarski Paradox, looking specifically at the second part of corollary 5.12, it states:
Suppose $G$ acts on $X$ and a proper system of ... |
What formula could calculate the speed of music playback in a variable length of time? |
I have a kind of complicated second order diff eqn that is in the form of
$$y''(x) = 0$$
I have two initial conditions given as:
$$y(x) = \mathcal{C}x, ~~ y'(x)=\mathcal{C}$$
where $\mathcal{C}$ is a constant number. Is there a general (known) numeric method to find the $\mathcal{C}$, which satisfies the ab... |
I am reading [Super-Golden-Gates for $PU(2)$](https://arxiv.org/abs/1704.02106) by Ori Parzanchevski and Peter Sarnak. If I understand correctly then in section 4.1.3 they seem to be saying that the matrices
$$
F:= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1\\ i & -i \end{bmatrix}, S:=\zeta_8^7 \begin{bmatrix} 1 & 0\\ ... |
I have a second order diff eqn that is in the form of
\begin{equation}
y^{\prime\prime} + \Big(\frac{3}{x} - \frac{3x^{-4}}{2[x^{-3} + B]}\Big)y^{\prime} - \frac{3}{2}\frac{1}{x^5[x^{-3} + B]}y = \mathcal{F}\,.
\end{equation}
where $B$ and $\mathcal{F}$ are **known** constant. I have two initial conditions ... |
I will follow the definition of the volume of a Riemmanian manifold from page 422 of [Introduction to Smooth Manifolds][1]
Let $M$ be an open and convex subset of $\mathbb{R}^n$ with non-empty interior and compact closure (e.g. $(0,1)\times(0,1)\subset\mathbb{R}^n$). Such $M$ clearly defines a smooth manifold, and w... |
Let $(M,g)$ be a Riemannian manifold. I know that the gradient of a smooth function $f: M \rightarrow \mathbb R$ can be defined implicitly using the metric:
$$
g(\nabla f, X) = df(X), \quad \forall X \in \Gamma(TM)
$$
I have seen the higher order gradient $\nabla^k f$, $k \geq 1$ in the same context, without an exp... |
Higher order gradients defined on a Riemmanian manifold? |
Some problematic points in your proof:
> Proof: let $\{a_1,a_2,...,a_i,v_1,v_2...v_n\}$ be basis for $W_1$, then $\dim(W_1)= i+n$ and $\{b_1,b_2,...,b_j,v_1,v_2...v_n\}$ be basis for $W_2$, then $\dim(W_2)= j+n$ .
You are missing some fundamental information. What are $i,j,n$? Why do both bases contain the sam... |
Suppose $T$ is a compact operator from a Hilbert Space. Then for $a\neq 0$, we have that $\text{range}(T- aI)^{n-1}=\text{range}(T-aI)^{n}$ for some $n$ and $\ker(T-aI)^{m-1}=\ker(T-aI)^{m}$ for some $m$.
Firstly, if $a$ is a not an eigen value, then by Fredholm alternative we know that $T-aI$ is surjective.
... |
Essentially the subject line: Suppose there is a matrix of shape $[M,M]$ where $M \ge N$. We perform eigen decomposition and observe that the leading $N$ eigenvalues have the same, real value. Is this hypothetical scenario possible?
In my anecdotal experience, the leading eigenvector (largest eigenvalue) dominates a... |
Can first N eigenvalues have the same real value? |
Here is a question from my calc test:
Consider the series $\sum_{n=1}^\infty (-1)^n a_n $ where $a_n$ is positive for all n, Which of the following guarantees that the series converges?
Option 1:
$$\int_{0}^{\infty} f(x)dx$$ converges, where $f(n)=a_n$ for all $n$.
Option 2:
$$\lim_{n\to\infty}\frac{a_{n+1}}{a... |
Can you use the integral test on the positive part of an alternating series, and then conclude that the alternating series converges? |
I have the following exercise:
$$(2x-y)e^{\frac{x}{y}}dx + (y+xe^{\frac{x}{y}})dy=0$$
After some substitution and putting $v=\frac{x}{y}$ i have the following equation:
$$\frac{1-2v}{2v^2+e^{-v}}dv=\frac1ydy$$
but i don't know how continue. Is it correct? Thanks. |
We shall consider functions $h=h(t, x):[0, \infty) \times \mathbb{R} \rightarrow(0, \infty)$ which are $2 \pi$-periodic with respect to $x$, belong to $C^{\infty}([0, \infty) \times \mathbb{R})$ and satisfy the $1 \mathrm{D}$ heat equation
$$
h_t-h_{xx}=0, \quad \text { in }(0, \infty) \times \mathbb{R} .
$$
and t... |
A topology T is a collection of subsets of some set where T is closed under arbitrary union and finite intersection. Do we get a meaningful generalization if instead of union and intersection, we choose other kind of constructions in the category of sets (maybe some limits, some colimits or hom-sets) such that T is clo... |
Is there a meaningful generalization of topology? |
Solve $\begin{equation}u''+u=\delta(x)\end{equation}.$
Using the Fourier integrals we obtain:
$\mathscr{F}\{u''\}= -(\omega)^2\hat{u}$,
$\mathscr{F}\{u\}= \hat{u}$,
$\mathscr{F}\{\delta\}=\frac{1}{\sqrt{2\pi}}$.
We then have $\begin{equation}-(\omega)^2\hat{u}+\hat{u}=\frac{1}{\sqrt{2\pi}}.\end{equa... |
I got this problem in my home exercise at the University. I think I solved it, but I am not sure, that my solution is right. **Can somebody check it?**
Let $A$ and $B$ be two events on a probability space $(\Omega, \mathbb{P})$, with $\mathbb{P}(A)=\frac{1}{3}$ and $\mathbb{P}(B)=\frac{1}{4}.$
Determine the v... |
I got this problem in my home exercise at the University. I think I solved it, but I am not sure, that my solution is right. **Can somebody check it?**
> Let $A$ and $B$ be two events on a probability space $(\Omega, \mathbb{P})$, with $\mathbb{P}(A)=\frac{1}{3}$ and $\mathbb{P}(B)=\frac{1}{4}.$
> Determine t... |
Let
$$S=\sum_{0\le i<j\le n} {n \choose i}~~~~~~~~~~(*)$$
Consider the summation:
$$\sum_{j=0}^{n}\sum_{i=0}^{n} A_i B_j=\sum_{i=0}^n A_i B_i+\sum_{0\le i<j\le n} A_i B_j+ \sum_{n\ge i>j\ge 0} A_i B_j$$
If $B_j=1$, then the second and the third sums in RHS above are equal, then
$$\sum_{0\le i<j\le n}A_i=\frac{1}{... |
Let
$$S=\sum \sum_{0\le i<j\le n} {n \choose i}~~~~~~~~~~(*)$$
Consider the summation:
$$\sum_{j=0}^{n}\sum_{i=0}^{n} A_i B_j=\sum_{i=0}^n A_i B_i+\sum_{0\le i<j\le n} A_i B_j+ \sum_{n\ge i>j\ge 0} A_i B_j$$
If $B_j=1$, then the second and the third sums in RHS above are equal, then
$$\sum_{0\le i<j\le n}A_i=\fra... |
I've been trying to fit some data with polynomials using a pseudo-inversion library (numpy.linalg). There are two real functions $f$ and $g$ such that $f=-\alpha g$, $\alpha > 0$, and
$$
f=f(t_1,\ldots,t_5),
$$
$$
g = g(t_1,\ldots,t_5),
$$
where $t_1,\ldots,t_5$ are real parameters. The parameter $\alpha$ ha... |
Let
$$S=\sum_{0\le i<j\le n} {n \choose i}~~~~~~~~~~(*)$$
Consider the summation:
$$\sum_{j=0}^{n}\sum_{i=0}^{n} A_i B_j=\sum_{i=0}^n A_i B_i+\sum_{0\le i<j\le n} A_i B_j+ \sum_{n\ge i>j\ge 0} A_i B_j$$
If $B_j=1$, then the second and the third sums in RHS above are equal, then
$$\sum_{0\le i<j\le n}A_i=\frac{1}{... |
As title says I have to show that the equilateral triangle has the minimum perimeter between all the triangles with fixed area.
It's an homework that should revolve around the Lagrange multipliers. I kinda found a way but it looks like a lot of work:
Fix $A$ the area. Heron's formula states that $A^2=s(s-a)(s-b)(s-... |
> Assume that $f(z)$ is holomorphic on $\{|z|<R\}$ and has Taylor
> expansion $f(z)=\sum\limits_{n=0}^{\infty}c_nz^n$. Prove that
> $$\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}|f(re^{i\varphi})|^2d\varphi=\sum\limits_{n=0}^{\infty}|c_n|^2r^{2n},\quad (0<r<R).$$
We have $c_n=\frac{1}{2\pi i}\int\limits_{|z|=r}\frac{f(z)... |
Let
$$S=\sum_{0\le i<j\le n} {n \choose i}~~~~~~~~~~(*)$$
Consider the summation:
$$\sum_{j=0}^{n}\sum_{i=0}^{n} A_i B_j=\sum_{i=0}^n A_i B_i+\sum_{0\le i<j\le n} A_i B_j+ \sum_{n\ge i>j\ge 0} A_i B_j$$
If $B_j=1$, then the second and the third sums in RHS above are equal, then
$$\sum_{0\le i<j\le n}A_i=\frac{1}{... |
Let's say we have an expanding system of slots(you can be keep increasing the number of slots). Now I have a very large number of coins in a bag, all equi-similar, labeled(1,2,..n). Now the experiment is, you take a random coin from the bag and put it in a slot. Now you keep repeating this until you have at-least on... |
Consider the three categories $\mathbf{Top}$ (of topological spaces with continuous maps), $\mathbf{Graph}$ (of simple graphs with maps that preserve or contract edges) and $\mathbf{Pros}$ (of pre-ordered sets with monotone maps). These three categories, albeit rather different in feel, share a common property: They co... |
Let's say we have an expanding system of slots(you can be keep increasing the number of slots). Now I have a very large number of coins in a bag, all equi-similar, labeled(1,2,..n). Now the experiment is, you take a random coin from the bag and put it in a slot. Now you keep repeating this until you have at-least on... |
Conjecture :
----------
$$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$
Where $a_n$ is an integer such that :
$$0\leq a_n\leq n$$
>>Some arguments :
I think the best we can say is until we have the (says 20 because of floating point ) firs... |
**Theorem:1** Let the function
$$f (z) = u(r, θ) + iv(r, θ)$$be defined throughout some neighborhood of a nonzero point $z_0 = r_0 \exp(iθ_0)$ and suppose that
>**(a)** The first-order partial derivatives of the functions u and v with respect to r and θ
exist everywhere in the neighborhood;
>
>**(b)** Those parti... |
How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix}
0&I_6&0\\I_6&0&0\\0&0& -I_{16}\end{bmatrix} \tag{2}$$
I know that a symmetric matrix has $28 \times 29/2=406$ free parameters and I know that the final a... |
Is the integral average of a Sobolev function on a sphere continuous? |
We have two continuous functions $f,g:\mathbb{R}\to\mathbb{R}$, and we are interested in whether or not $$\frac{f(x)}{f(y)}=\frac{g(x)}{g(y)}$$ for all $x$ and $y$. It is obvious that a sufficient condition is that $f(x)=a g(y)$ for some $a\in\mathbb{R}$. But is this condition also necessary? It would not be necessary... |
$\frac{f(x)}{f(y)}=\frac{g(x)}{g(y)}$ iff $f(x)=a g(x)$? |
Let $M$ be a smooth manifold, and $L$ be a smooth real (non-trivial) line bundles over $M$, is it then true that $L\oplus L$ admits a non vanishing section?
Intuitively, I feel like the answer should be yes, since I think I should be able to take a global section $s$ of $L$, and then obtain a new section $t$ by "pus... |
Does the direct sum of two real line bundles admit a non-vanishing section? |
> Assume that $f(z)$ is holomorphic on $\{|z|<R\}$ and has Taylor
> expansion $f(z)=\sum\limits_{n=0}^{\infty}c_nz^n$.
>
> a) Let
> $M(r)=\sup\limits_{|z|=r}|f(z)|$. Prove that
> $$|c_n|\leq\dfrac{M(r)}{r^n}\quad (0<r<R).$$ b) Prove that if one of
> the equalities in $a)$ becomes equality, i.e $|c_k|=\dfrac{M(r)}... |
Given that
$$
\forall x,y\in \mathbb R. \frac{f(x)}{f(y)}=\frac{g(x)}{g(y)}.
$$
It follows that $f$ and $g$ are never zero, so we can rearrange the equation:
$$
\forall x,y\in \mathbb R. \frac{f(x)}{g(x)}=\frac{f(y)}{g(y)}.
$$
In particular, if we fix $y = 0$,
$$
\forall x\in \mathbb R. \frac{f(x)}{g(x)}=... |
Find a conformal mapping $f$ that maps $\{z \in \mathbb{C}: Re(z) > 0\}$ onto itself that maps $f(1)=2$.
So we need to ensure $f(1)=2$ and that the right half plane maps to the right half plane. Well, we can choose a Mobius mapping. Notice that the mapping
$$f(z) = \frac{z+1}{\frac{1}{2}z+\frac{1}{2}}$$
will... |
In Serge Lang's Differential and Riemannian manifold, he defined derivative between two topological vector spaces with the concept tangent to 0:
>Let $\mathbf{E}, \mathbf{F}$ be two topological vector spaces, and $\varphi$ a mapping of a neighborhood of 0 in $\mathbf{E}$ into $\mathbf{F}$. We say that $\varphi$ is tan... |
We have two continuous functions $f,g:\mathbb{R}_{++}\to\mathbb{R}$, and we are interested in whether or not $$\frac{f(x)}{f(y)}=\frac{g(x)}{g(y)}$$ for all $x$ and $y$. It is obvious that a sufficient condition is that $f(x)=a g(y)$ for some $a\in\mathbb{R}$. But is this condition also necessary? It would not be nece... |
We have two continuous, nonzero functions $f,g:\mathbb{R}\to\mathbb{R}$, and we are interested in whether or not $$\frac{f(x)}{f(y)}=\frac{g(x)}{g(y)}$$ for all $x$ and $y$. It is obvious that a sufficient condition is that $f(x)=a g(x)$ for some $a\in\mathbb{R}$. But is this condition also necessary? It would not be ... |
Have you tried working through an algebraic derivation of the primitive Pythagorean triple parametric formula in $\mathbf Z$ to get an analogue in $\mathbf Z[i]$? Because $-1$ is a square, the terms in $\alpha^2 + \beta^2 = \gamma^2$ are essentially symmetric in $\alpha$, $\beta$, and $\gamma$, so we can assume withou... |
A teacher leaves his home everyday at 8:00A.M to come to school. If he comes 95 meters per minute. He will arrive one minute late for his class. If he walks 105 meters per minute he will arrive one minute early for his class. At what speed should he walk to arrive perfectly on time?
Is this the correct way to approa... |
When solving one specific PDE to find $f(x,y)$, I came across the following equation through which I would find the eigenvalues $\lambda_l$:
$$a\tanh^{-1}(\frac{\lambda}{b}) - \lambda = 0,$$
in which $a$ and $b$ are real constants.
I found out graphically that when $a < b$, the values of $\lambda_l$ are real, but w... |
I know from [this](https://math.stackexchange.com/questions/4081172/is-it-true-that-range-of-i-t-closed-whenever-t-is-compact) question that this is true for Hilbert spaces. But is the result true for Banach spaces as well?
So suppose $x_{n}$ is a bounded sequence and $(I-T)(x_{n})$ converges. Then upto a subsequenc... |
I found on page 25 of Arthur Besse’s “Einstein Manifolds” that the differential Bianchi identity
$$d^\nabla R^\nabla=0$$
follows from the alternate definition of the Riemann curvature tensor as the second covariant exterior derivative
$$R_{X,Y}^\nabla s = -d^\nabla(d^\nabla s) (X,Y)$$
and the fact that $(d... |
I am reading [Super-Golden-Gates for $PU(2)$](https://arxiv.org/abs/1704.02106) by Ori Parzanchevski and Peter Sarnak. If I understand correctly then in section 4.1.3 they seem to be saying that the matrices
$$
F:= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1\\ i & -i \end{bmatrix}, S:=\zeta_8^7 \begin{bmatrix} 1 & 0\\ ... |
Suppose I have two real functions $f,g$ such that their averages over an interval are equal, i.e.$$\langle f\rangle = \langle g\rangle. $$ For simplicity, we can think that the average is over an angle and the average is over a full period $[0,2\pi)$. It is also the case that both functions depend on more variables asi... |
When two averages of (real) functions are given, when one function is equal to the average of the other? |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
I am trying to use the SQP solver with a nonlinear constraint. The solver requires a linear constraint so I am trying to approximate the constraint with the first-order Taylor approximation. Is this to correct way how to use SQP solvers or is this either a completely wrong or naive approach?
Problem formulation
$$\... |
How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix}
0&I_6&0\\I_6&0&0\\0&0& -I_{16}\end{bmatrix} \tag{2}$$
I know that a symmetric matrix has $28 \times 29/2=406$ free parameters and I know that the final a... |
A function similar to Minkowski's question-mark [function][1] pairs periodic 2-adic numbers with quadratic irrationals, and pairs these with rectangle-shapes.
It is likely that periodic 5-adic numbers can be paired with quintics and with 5-hyperrectangles.
Are the 5-hyperrectangles known to break some complexity ... |
Set of infinite well founded trees if complete coanalytic? |
I started experimenting with bootstrapping and noticed that using a bigger sample size gives a tighter confidence interval, especially at very low sample size.
I made a test and created a bootstrap function that upscale the sample size (during the resampling step). The value it gives is very close than without upsca... |
Suppose $T$ is a compact operator from a Hilbert Space. Then for $a\neq 0$, we have that $\text{range}(T- aI)^{n-1}=\text{range}(T-aI)^{n}$ for some $n$ and $\ker(T-aI)^{m-1}=\ker(T-aI)^{m}$ for some $m$.
Firstly, if $a$ is a not an eigen value, then by Fredholm alternative we know that $T-aI$ is surjective.
... |
The goal is to find $m$ swaps $s_1, s_2, \cdots, s_m$ such that any $n-$permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \cdots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2), s_2=(2,3), s_3... |
The goal is to find $m$ swaps $s_1, s_2, \cdots, s_m$ such that any $n-$permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \cdots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2), s_2=(2,3), s_3... |
**Background**
The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the limit is taken over all partitions of the interval
$[a,b]$ whose norms approach zero.
We can do something r... |
The goal is to find $m$ swaps $s_1, s_2, \cdots, s_m$ such that any $n-$permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \cdots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2), s_2=(2,3), s_3... |
The goal is to find $m$ swaps $s_1, s_2, \cdots, s_m$ such that any $n-$permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \cdots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2), s_2=(2,3), s_3... |
It is well know the following resul holds.
>**Theorem**
>
>For any $(M,\ast,e)$ monoid there exists a unique esternal operation $\wedge_\ast$ from $X\times\omega$ into $X$ such that for any $x$ in $M$ the following results hod:
>
> 1. the equality
$$
x\wedge_\ast 0=e
$$
> 2. the equality holds
$$
x... |
Is possibile to define an exponentiation into the ordinal class? |
The goal is to find $m$ swaps $s_1, s_2, \cdots, s_m$ such that any $n-$permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \cdots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2), s_2=(2,3), s_3... |
I know there are $n$ zeros in the open ball $B_1(1)$ from Rouche's Theorem. However, how do I show that the order of these zeros are all one? I don't know where the zeros are, and this should be true for any arbitrary $n$. One thing I want to try is to show that $f'(z_0)\neq 0$ for any zero $z_0, so we have
$$f'(z)=... |
I came across a proof of Euclid's Lemma from https://www.sci.brooklyn.cuny.edu/~mate/misc/euclids_lemma.pdf that used another lemma to assist in the proof:
> Let a and c be positive integers and let t be the smallest positive integer such that c | at. Let b be a positive integer such that c | ab. Then t | b. In part... |
It is well know the following resul holds.
>**Theorem**
>
>For any $(M,\ast,e)$ monoid there exists a unique esternal operation $\wedge_\ast$ from $X\times\omega$ into $X$ such that for any $x$ in $M$ the following results hod:
>
> 1. the equality
$$
x\wedge_\ast 0=e
$$
> 2. the equality holds
$$
x... |
Is that possible to have an example on $$\int_{-2}^2f(x)dx=4 \text{, and $f$ is an even function}$$ but $\int_0^2f(x)dx$ does not exist?
|
Find all field homomorphisms $f:\mathbb{Q}(\sqrt{2}) \rightarrow \mathbb{Q}(\sqrt{2})$.
We know that:
$
\mathbb{Q}(\sqrt{2}) = \text{span}_\mathbb{Q} \{ 1, \sqrt{2} \} = \{ a+\sqrt{2}b \,|\, a,b\in \mathbb{Q} \}
$
And that a field homomorphism fulfills the following criteria:
- $\forall x \in \mathbb{Q}(\s... |
How can I find nicely behaving functions which are easy to compute and which fit well to the upper limits of the (2-logarithm) of the nCr function?
What I am interested in in practice is to be able to model "how many binary digits (bits) will a digital representation of nCr(a,b)" have? A guess of 1 or 2 too many bit... |
Not related to the body of the post, but rather the title.
One can provide an explicit example to show the Fourier transform is not surjective: https://mathoverflow.net/a/319225/112504 proves that $$g(t) := 1_{[0\leq t\leq e]}\cdot \frac te + 1_{[t>e]}\cdot \frac{1}{\ln t},$$ **extended to be an odd continuous funct... |
Is possibile to define an exponentiation with respect an ordinal operation? |
I am trying to use the SQP solver with a nonlinear constraint. The solver requires a linear constraint so I am trying to approximate the constraint with the first-order Taylor approximation. Is this to correct way how to use SQP solvers or is this either a completely wrong or naive approach?
Problem formulation
$$\... |
I'm trying to understand the following theorem:
Theorem: Suppose that $x=x(t)$ is a solution of $\dot{x}=F(x)$ where $F$ is continuous. Suppose that $x(t)$ approaches a finite limit $\textbf{a}$ as $t \to \infty$. Then $\textbf{a}$ must be an equilibrium state for the equation, that is, $F(\textbf{a})=0$.
I found... |
I have two theories and I was pondering on what basis we are choosing one over the other. Let me take specific example and describe the two theories and any discussion in this regard would be helpful to me.
> $\textbf{Question}$ There were two partners $A$ and $B$ who had invested $Rs. 50,000$ and $Rs. 80,000$ resp... |
I'm not sure about the integral associated with the example below. I see different options and I don't know exactly which one is the right one. My found options are the following:
$$ \int_{0}^{100} (1000+60x)*10*4 \,dx $$ or $$ \int_{0}^{400} (1000+60x)*10 \,dx $$
I also doubt whether my integral takes into acco... |
Suppose one has a smooth curve $\gamma:\mathbb{R}\to M$ on a manifold $M$ and a smooth one-parameter family $f_t:M\to M$ of diffeomorphisms (not necessarily a one-parameter subgroup). Then, the curve $\rho(t)=f_t(\gamma(t))$ is smooth.
My question is: is there a way to relate the velocity of $\gamma$ with the one of... |
Consider the matrix defined by block :
$$
A =
\begin{bmatrix}
D_1 & B_{1,2} & \cdots & B_{1,n-1} \\
B_{1,2}^T & & \cdots & B_{2,n-1} \\
\vdots & \vdots & \ddots & \vdots \\
B_{1,n-1}^T & B_{2,n-2}^T & \cdots & D_n
\end{bmatrix}
$$
where $B_{i,j} = \Pi_{k=i}^{j-1} B_{k,k+1}$ (the $B$s are iterated products) a... |
Integral is continuous, whilst $\sum$ is discrete.
You don't want $1.00000000000000000001$ passengers on the lift do you? |
I'm not sure about the integral associated with the example below. I see different options and I don't know exactly which one is the right one. My found options are the following:
$$ \int_{0}^{100} (1000+60x)*10*4 \,dx $$ or $$ \int_{0}^{400} (1000+60x)*10 \,dx $$
I'm not sure about my limits. Maybe it should be... |
Is there a prime number $p$ with $p \mid \sum_{j=1}^{p-1} j!^2$?
I checked the primes upto $600\ 000$ without finding a solution.
Heuristic : If we can assume that the probability that $p$ is a solution is $\frac{1}{p}$ , then there should be infinite many solutions and therefore the desired prime should exist. B... |
It is well know the following resul holds.
>**Theorem**
>
>For any $(M,\ast,e)$ monoid there exists a unique esternal operation $\wedge_\ast$ from $X\times\omega$ into $X$ such that for any $x$ in $M$ the following results hod:
>
> 1. the equality
$$
x\wedge_\ast 0=e
$$
> 2. the equality holds
$$
x... |
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