instruction stringlengths 12 30k |
|---|
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous and nowhere differentiable, and $g: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere and non constant. Can $f \circ g$ ever be differentiable?
My intuition is no, but I'm not sure. Perhaps $g$ is differentiable but not continuously so and its de... |
I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.
I am wondering how is this true, can someone explain this to me please?Is there a proof for this fact? |
I'm having a bit of trouble with this integral and I'm not sure if I'm correct.
Let $f_n(x) = \cos(x) (1 - \frac{\lvert x \rvert}{n} )^n $ when $ \lvert x \rvert < n $ and $0$ when $ \lvert x \rvert \ge n$.
I'm asked to find the limit of $$\lim_{n\to \infty}\int_{-\infty}^{\infty}f_n(x)\,dx$$
Here is my attem... |
>Also, is there a better way to show that the right angle at the center of the square is preserved by the perspective projection and $xw$-rotation when $\text{distance}=\sqrt{2}$?
The center of the square $P_2P_3P_{10}P_{11}$ is $(0,1,-1,0)$.
In the post the "view space" is $w=\text{distance}-1$. We change this t... |
I tried to find a *small* natural deduction proof for Meredith's single axiom
| Infix notation | Polish notation |
| -------------- | --------------- |
[((((ψ→φ)→(¬χ→¬ξ))→χ)→τ)→((τ→ψ)→(ξ→ψ))](https://www.umsu.de/trees/#((((a%E2%86%92b)%E2%86%92(%C2%ACc%E2%86%92%C2%ACd))%E2%86%92c)%E2%86%92e)%E2%86%92((e%E2%86%92a)... |
>Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$.\
$AF,BF$ are arbitrary perpendicular lines through $F$ on the plane $y=\sqrt{2}$.\
These lines project to the lines $DF,EF$ under the perspective projection centered at $C$ onto $xy$-plane.\
Then the lines $DF,EF$ perpendicular?
This questi... |
Consider a complex, separable Hilbert space $H$ and a (densely defined) self-adjoint operator $A: \mathcal D(A)\to H$. Assume that $A$ admits an orthonormal basis of eigenvectors $(\varphi_n)_{n\in \mathbb N}$ with $A\varphi_n=\lambda_n \varphi_n$.
Next, define for $\psi\in \mathcal D(A^2)$ with $\|\psi\|=1$ the foll... |
I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc.
I have two proofs that I'm now sure how to handle.
**Problem 1**.
Prove that an integrable stochastic process $\{X(t),\mathcal{F}_t, t\in \mathbb{T}\}$ is a martingale if and only if for a... |
Let $f:X\to Y$ be a map between matric spaces. I'm interested in the following property:
> For every $x\in X$, there is a sequence $x_n\to x$, with $x_n\ne x$ for every $n$, such that $f(x_n)\to f(x)$.
Question: What's known about functions with this property? Does some well studied class of functions have it?
... |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) 1946 book [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA17&dq=%22directly+based+on+the+golden+section+and+on+the+formula%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj... |
There is one step the necessity of which I don’t understand in the proof of Inverse Function Theorem in Jerry Shurman’s $Calculus$ $and$ $Analysis$ $in$ $Euclidean$ $Space$ (available here https://www.professores.uff.br/diomarcesarlobao/wp-content/uploads/sites/85/2017/09/Multi_calculus_BOOK.pdf).
After having esta... |
I am working on exercise II.4.8 in Hartshorne, and am stuck on part (e). The problem asks to show that a property of morphisms of schemes ($\mathscr{P}$) such that
1. a closed immersion has $\mathscr{P}$
2. a composition of two morphisms having $\mathscr{P}$ has $\mathscr{P}$
3. $\mathscr{P}$ is stable under b... |
>Also, is there a better way to show that the right angle at the center of the square is preserved by the perspective projection and $xw$-rotation when $\text{distance}=\sqrt{2}$?
The center of the square $P_2P_3P_{10}P_{11}$ is $(0,1,-1,0)$.
In the post the "view space" is $w=\text{distance}-1$. We change this t... |
It can be reduced to an implicit solution including special functions. Multiply by $\theta'$ and integrate to get
\begin{align}
\theta'\theta''+b\sin(\theta)\theta'-a^2\cot(\theta)\csc(\theta)^2\theta'=0,\\\\
\frac12 (\theta')^2-b\cos(\theta)-\frac12 a^2\csc(\theta)^2=C,\\\\
\int\frac{\mathrm d\theta}{\sqrt{C+2b\co... |
Let $f:X\to Y$ be a map between matric spaces. I'm interested in the following property:
> For every $x\in X$, there is a sequence $x_n\to x$, with $x_n\ne x$ for every $n$, such that $f(x_n)\to f(x)$.
Question: What's known about functions with this property? Does some well studied class of functions have it?
... |
Instead of looking at the collection $X_1,...,X_n$ we can look at the vector $Y = (X_1,...,X_n)$ for which $\sigma(Y) = \sigma(X_1,...,X_n)$, so it is enough to consider case of single random variable $Y$ (but taking values in some product space $\mathbb R^n$ or any different).
Thus, let us assume that $(\Omega,\mat... |
I am referring to the ISS definition by Sontag of
${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$
I understand that 0-GAS is a necessary condition for ISS. But is GAS for all constant u a sufficient condition? |
Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? |
We are given two quadratic equations
$$ 0= a x^2 + b x + c + d y $$
and
$$ 0= A y^2 + B y + C + D x $$
with real constants $a,b,c,d$ and $A, B, C, D$.
Under which conditions on the constants do (one or several) real solutions $(x,y)$ exist?
For one variable ($d=D=0$) we obtain two uncoupled equations that ar... |
I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.
I am wondering how is this true, can someone explain this to me please?Is there a proof for this fact?
**Edit:**
It seems like my question was not clear, here is what exactly I am a... |
Every source I've found so far says that first order logic is sufficient for defining ZFC set theory.
My lecture notes write the Axiom of Comprehension as follows:
> For any formula $\phi(x)$ with parameters and any set $y$, there exists a set $z$ whose elements are precisely those elements $x$ of $y$ that satisf... |
Does ZFC's Axiom of Comprehension require second-order logic? |
Let $H=L^2(\mathbb{R}^3, \mathbb{R}^3)$ be a Hilbert space and $V=\{u \in H \mid \langle x, u(x) \rangle = 0 \text{ for a.e. } x\in \mathbb{R}^3\}$ a subspace.
Is $V$ closed? What is the orthogonal projection of $H$ onto $V$?
I think V consists of "rotations"? At least that's how it would work in the $\mathbb{R}^... |
I am currently reading up on cohomology with compact supports, and am looking for a reference as to whether the De Rham isomorphism $H_\textrm{DR}^*(M)\simeq H^*(M;\mathbb R)$ exists for the compactly supported cohomologies $H_\textrm{DR,c}^*(M)$ and $H_c^*(M;\mathbb R)$ as well. I think it probably does, because I sus... |
I have a smooth function $f : P \rightarrow R$, where $P$ is the positive orthant of $R^3$. I know that $\partial f / \partial x,\; \partial f / \partial y$ are both $> 0$ everywhere on $P$ and that $\partial f / \partial z < 0$, also everywhere on $P$. Finally, I know that all three partials are functions only of $z$... |
Let $(X,\vec X,+)$ a real affine space, $a\in X, b\in X, \vec A $ and $\vec B$ two subspaces of $\vec X$.
$A=a+\vec A$ and $B= b+\vec B$ two affine subspaces.
I want to understand what the "affine subspace generated by $A$ and $B$" is, because Jean Frenkel speaks of it in *Géométrie pour l'élève professeur*(Geom... |
I'm seeing two different definitions for analytic functions.
1. A function $\Phi: \mathcal{D}\rightarrow \mathbb C$ is analytic if, for every $z_0 \in \mathcal{D}\subseteq \mathbb C$, there exists a power series converging to $\Phi(z)$ on some neighborhood of $z_0$.
2. A function $\Phi: \mathcal{D}\rightarrow \mathbb... |
It is well know the following resul holds.
>**Theorem**
>
>For any $(M,\bot,e)$ 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
$$
x\curlywedge_\bot 0=e
$$
> 2. the equality holds
$... |
A complex Hadamard matrix of order $N$ is a square matrix $A = [a_{ij}]$ of size $N$ with entries $|a_{ij}| = 1$ for all $i,j$; and further that $A^\dagger A= N I$ (or $A/\sqrt{N} $ is a unitary matrix).
Further, a complex Hadamard matrix of Butson-type is such that $a_{ij}$ is the $q$-th root of unity for some posi... |
You're lucky I like investigating strange statements given without context such as "this answer is incorrect." I eventually found that particular problem on the website you mentioned and played with it long enough to find the issue.
In short, your answer is correct...
... but it's not simplified enough for that ... |
I'm looking at the Galois group for $F = \mathbb{Q}(\sqrt[3]{2}, \omega)$ where $\omega$ is the first primitive third root of unity. I know this is the splitting field for $x^3 - 2$ and, I'm told, that $G = Gal(F/\mathbb{Q}) \cong S_3$ (which, for now, I believe). I can also directly see how most of the subgroups hold ... |
How can I directly see that this subgroup has this subfield as tis fixed field? |
When $p\equiv7\pmod8$, then $y^2=2$ has solutions $\pmod p$,
which I will write as $y=\pm\sqrt2\pmod p$.
The solutions of the congruence $x^2-34x+1\equiv0\pmod p$
are seen to be $x=17\pm12\sqrt2\pmod p,$ using the quadratic formula.
Can you show that $x=(3\pm2\sqrt2)^2$? |
You're lucky I like investigating strange statements given without context such as "this answer is incorrect." I eventually found that particular problem on the website you mentioned and played with it long enough to find the issue.
In short, your answer is correct...
... but it's not simplified enough for that ... |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) 1946 book [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA17&dq=%22directly+based+on+the+golden+section+and+on+the+formula%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj... |
I have proved that $\phi \in \operatorname{Aut}(\mathbb{D})$ has a fixed point in $\mathbb{D}$ iff its correspondent from $\operatorname{Aut}(\mathbb{H})$ has a fixed point in $\mathbb{H}$, where $\mathbb{D}$ is open unit disk, and $\mathbb{H}$ is upper half-plane.
But I don't know how to show that $\phi \in \opera... |
How can I directly see that this subgroup has this subfield as its fixed field? |
I'm reading *Stochastic Differential Equations* (sixth edition) by Bernt Oksendal and I'm trying to understand the author's definition of the Ito integral. For a given probability space $(\Omega, \mathcal{F},P)$, the author defines the Ito integral as follows:
> **Definition 3.1.6 (The Ito integral)** *Let $f \in \m... |
Let $Spin(p,q)$ be the real spin group to the quadratic space $\mathbb{R}^{p,q}$ defined via Clifford algebras as $Pin(\mathbb{R}^{p,q}) \cap Cl^0(\mathbb{R}^{p,q})$, meaning as the subset of the Clifford algebra which is generated by an even number of elements $v \in \mathbb{R}^{p,q}$ with $Q(v) = \pm 1$, where $Q$ de... |
How do I prove that 5 is the limit of the given function using a delta epsilon proof? |
I am aware that related topics have been discussed in other threads. However, not being an expert, I would like to know the answers to the following questions:\
Let $E$ be a set having n elements ; $ |E|=n.$\
$A_p$ is a subset of $E$ with $p$ elements $(1 \le p \le n)$, chosen uniformly at random from the set of $\b... |
You're lucky I like investigating strange statements given without context such as "this answer is incorrect." I eventually found that particular problem on the website you mentioned and played with it long enough to find the issue.
In short, your answer is correct...
... but it's not simplified enough for that ... |
I am studying Cauchy integrals and their value in points on the integration curve. The principal Cauchy value is defined on these points, however in part of the proof it is used that
$\int_{γ} \frac{1}{|z-z_{0}|^\alpha} dz$
----------------------------------------
exists, where $γ$ is a smooth open or closed cu... |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) 1946 book [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA17&dq=%22directly+based+on+the+golden+section+and+on+the+formula%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj... |
I am studying Cauchy integrals and their value in points on the integration curve. The principal Cauchy value is defined on these points, however in part of the proof it is used that
$\int_{γ} \frac{1}{|z-z_{0}|^\alpha} dz$
----------------------------------------
exists, where $γ$ is a smooth open or closed cu... |
I'm looking at the Galois group for $F = \mathbb{Q}(\sqrt[3]{2}, \omega)$ where $\omega$ is the first primitive third root of unity. I know this is the splitting field for $x^3 - 2$ and, I'm told, that $G = Gal(F/\mathbb{Q}) \cong S_3$ (which, for now, I believe). I can also directly see how most of the subgroups hold ... |
I'm looking for an analog of the adjoint relationship: $\langle\lambda,g(x)\rangle:=\langle\lambda,Ax\rangle=\langle A^\top\lambda,x\rangle$ for the case where $g(x)$ is not linear.
For example, suppose we have $g_i(x)=\max\bigl\{\langle a_i,x\rangle, \langle b_i,x\rangle\bigr\}$ is piecewise linear convex, for $i=1... |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) 1946 book [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA17&dq=%22directly+based+on+the+golden+section+and+on+the+formula%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj... |
Suppose we are working over the real numbers $\mathbb{R}$. Is the cosine function $cos(x)$ a function of the sine function $sin(x)$? I define a function $f$ to be a function of a function $g$ if there is a function $h$ such that $h$ composed with $g$, in that order, equals $f$. |
Is cosine a function of sine? |
I'm working on a problem,
>Show that in an elliptic curve $E/\mathbb{Q}$ the sum of three colinear rational points of it is equal to $O$ exactly when the neutral element of the group $E(\mathbb{Q})$, $O$ is an inflection point of the curve.
I found the following in my notes.
> Let $C$ be a cubic curve that is ... |
I saw this problem in Sheldon Axler's Measure,Integration and Real Analysis Ex. $10B$, problem $17$ and $18$.
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... |
How do I prove that -5 is the limit of the given function using a delta epsilon proof? |
Suppose we are working over the real numbers $\mathbb{R}$. Is the cosine function $\cos(x)$ a function of the sine function $\sin(x)$? I define a function $f$ to be a function of a function $g$ if there is a function $h$ such that $h$ composed with $g$, in that order, equals $f$. |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) 1946 book [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA17&dq=%22directly+based+on+the+golden+section+and+on+the+formula%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj... |
I'm curious about a exercise of Hirsch's book. Let $U$ be a open subset (nonempty) of $\mathbb{R}^2$. Suppose given a $C^r$ vector field on $U$ without zeros, such that each integral curve is closed in $U$. Let $M$ be the identification space obtained by collapsing each curve to a point. Then $M$ is a $C^r$ 1-manifold ... |
I'm looking for an analog of the adjoint relationship: $\langle\lambda,g(x)\rangle:=\langle\lambda,Ax\rangle=\langle A^\top\lambda,x\rangle$ for the case where $g(x)$ is not linear.
For example, suppose we have $g_i(x)=\max\bigl\{\langle a_i,x\rangle, \langle b_i,x\rangle\bigr\}$ is piecewise linear convex, for $i=1... |
*Salvador Dali* applied in his 1947–1949 painting "*Leda Atomica*" a concept that relates to the golden ratio or its conjugate. It is well known that Dali used mathematical formulas to elaborate his works (e.g. golden ratio, catastrophe theory, etc.), this said, we are not talking here about a layman.
[![enter image... |
I'm wondering if reindexing a divergent or conditionally convergent infinite series changes the value of the series. Specifically, if we have a series $\sum_{n=1}^{\infty} a_n$ that is either divergent or conditionally convergent, does rearranging the terms by reindexing the series (e.g. $\sum_{n=2}^{\infty} a_{n-1}$ o... |
Does reindexing an infinite series change its value? |
I encountered the following integral question:
Show that
$$\frac{3}{8} \leq \int_{0}^{\frac{1}{2}} \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx \leq \frac{\sqrt{3}}{4}$$
I know I can just integrate it, but is there a way to obtain the result above without directly integrating? I tried doing rectangles and trapezoids b... |
You're lucky I like investigating strange statements given without context such as "this answer is incorrect." I eventually found that particular problem on the website you mentioned and played with it long enough to find the issue.
In short, your answer is correct...
... but it's not simplified enough for that ... |
How do I find the area in the region inside the cardioid r = 6 + 6sin(theta) and above the line y = 9/2? |
Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$).
Let $T_n\subseteq [0,1]$ be a measurable set for each $n$ such that $h_n$ vanishes almost everywhere on $T_n$.
Suppos... |
I'm looking for an analog of the adjoint relationship: $\langle\lambda,g(x)\rangle:=\langle\lambda,Ax\rangle=\langle A^\top\lambda,x\rangle$ for the case where $g(x)$ is not linear.
For example, suppose we have $g_i(x)=\max\bigl\{\langle a_i,x\rangle, \langle b_i,x\rangle\bigr\}$ is piecewise linear convex, for $i=1... |
The constructions of complex number that you must have seen rely on people's intuitions and experience with arithmetic, in order to avoid certain formalities that are deemed too confusing or too demotivating for most people. However, if one is very careful, one discovers that they do not hold water, just like you did: ... |
There are several different approach to induction of characters, and depending on the approach some things become definitions and some become actual results. My favourite is the following one.
$\DeclareMathOperator{\CF}{CF}\DeclareMathOperator{\Res}{Res}\DeclareMathOperator{\Ind}{Ind}$
Let $\CF(G)$ denote the space... |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) 1946 book [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA17&dq=%22directly+based+on+the+golden+section+and+on+the+formula%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwj... |
There are several different approaches to induction of characters, and depending on the approach some things become definitions and some become actual results. My favourite is the following one.
$\DeclareMathOperator{\CF}{CF}\DeclareMathOperator{\Res}{Res}\DeclareMathOperator{\Ind}{Ind}$
Let $\CF(G)$ denote the spa... |
I'm trying to solve a brainteaser:
A 2D circle (cake) is cut randomly with straight lines n-times. What's the expected number of pieces?
Could you give me a hint on how to approach this? Is it possible to solve it analytically or by simulation?
Thanks! |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
Recently on MSE's chat, user "Simd" raised the following problem (I have rephrased and introduced some notation):
> For $n \geq 1$ let $S_n \subseteq \mathbb{R}^2$ denote the Cartesian product of $\{0,1,2,\dots,n\}$ with itself, and let $\|\cdot\|$ denote the usual Euclidean distance on $\mathbb{R}^2$. Define $m_n... |
How close do distinct distances to $0$ determined by a square integer lattice in $\mathbb{R}^2$ get? |
x = (3^n - 2^n) / (2^(m+n) - 3^n)
n, m, and x must be positive integers >= 1
x must also be odd
the trivial solution is
n=1, m=1, x=1
are there any other solutions? is there a way to prove there are no other solutions? |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
$x = (3^n - 2^n) / (2^{m+n} - 3^n)$
$n$, $m$, and $x$ must be positive integers $\geq 1$
$x$ must also be odd
The trivial solution is
$n=1, m=1, x=1$
Are there any other solutions? If not, is there a way to prove there are no other solutions? |
I consider a subset $\mathcal{W}$ of $L^2(\Omega, \mathcal{F},\mathbb{P})$.
In a paper I am reading it is said that a random variable $X\in\mathcal{W}$ is optimal if it is of the form
$$
X = \mathbb{E}\left[Y\mid \sigma(Z)\right]
$$
where $Y$ is **an arbitrary element** of $\mathcal{W}$.
My question is... |
From my lectures I know that $a$ is a quadratic residue mod $m$ if there is a $b$ such that $b^2\equiv a \pmod m$.
But in several book I now read that $a$ and $m$ have to be coprime.
That would mean that $4$ is not a quadratic residue $\mod 6$.
But obiously we have $2^2\equiv 4\pmod 6$.
So I wonder why some authors... |
Do we need relative prime in the definition of a quadratic residue? |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
I came upon the following specific property for a normed space $X$, and I am looking for a characterization of the normed spaces where it holds true:
If a sequence $x_n$ in $X$ satisfies $\displaystyle \lim_{n\to\infty}(||x_n+y||-||x_n||)=||y||$ for all $y\in X$, then $\displaystyle \lim_{n\to\infty}x_n=0$.
This... |
Suppose I have a matrix $A \in \mathbb{R}^{m \times n}$ with singular value decomposition (SVD) $A = U S V^T$. I then stack $A$ on top of itself $k$ times:
$A' = [A; A; ...; A] \in \mathbb{R}^{km \times n}$
Is there any way to relate the SVD of $A'$ to the SVD of $A$? I'm equally happy with any result that goes e... |
How does the SVD of a matrix relate to the SVD of multiple stacked duplicates of the matrix? |
From [here](https://math.stackexchange.com/questions/1044249/inverse-of-a-2-times2-matrix-with-noncommuting-entries), the inverse of a $2\times2$ quaternion matrix is given by
$$ \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}^{-1}
=\frac{1}{\mathrm{Nrd}}
\begin{bmatrix}
|d|^2\overline{a}-\overline{c... |
Inverse of a 2x2 quaternion matrix: whence this formula? |
A very important example of this is the incredible expressive power of first-order arithmetic. For instance, in the structure $(\mathbb{N},+,\cdot)$ it is possible to express the fundamental theorem of arithmetic as a first-order sentence. At first glance this seems very unlikely: to state that for all $n\in\mathbb{N... |
I came upon the following specific property for a normed space $X$, and I am looking for a characterization of the normed spaces where it holds true:
If a sequence $x_n$ in $X$ satisfies $\displaystyle \lim_{n\to\infty}(||x_n+y||-||x_n||)=||y||$ for all $y\in X$, then $\displaystyle \lim_{n\to\infty}x_n=0$.
This... |
As far as I'm concerned to show that something is true, proving that something is true for an example is never enough, you have to be able to prove that it is true for all statements/numbers with the same property. However the answer to a problem that I was trying for a while ends the proof with an example. Maybe it is... |
I have a bit of a problem with the Extended Cauchy Schwarz Inequality, specifically at the line
"with equality if and only if b = cB^(-1)d for some constant c."
[![enter image description here][1]][1]
---
As such, I'm having problems understanding the following lemma, at a similarly worded line:
"with th... |
This may be best answered by looking for an improvement to that formula, and explaining how to find it.
The basic idea for all those approximations for $e$ is to write some large number $N$ in two different ways, using exactly the digits from $2$ to $9$, and then take $(1+\frac{1}{N})^{N}$ as an approximation for $e... |
By Gödel's second incompleteness theorem, no sufficiently powerful formal system can prove its own consistency.
I was wondering what happens if one tries to manually append an axiom stating a formal system's own consistency to an existing formal system.
By the $2$nd incompleteness theorem (assuming the consisten... |
I'm studying algebraic geometry and I tried to understand th concept of Transcendental degree of a polynomial like the following.
> [Wiki][1] \
The field of rational functions in $n$ variables $K(x_1, \ldots, x_n)$ (i.e., the field of fractions of the polynomial ring $K[x_1, \ldots, x_n]$) is a purely transcendenta... |
Here's one way to do it:
Suppose $F(x)=\sum_{n=0}^{\infty}a_nx^n=a_0+a_1x+a_2x^2+...$. Then $f(x)=\sum_{n=1}^{\infty}na_nx^{n-1}-x=a_1+(2a_2-1)x+3a_3x^2+...$
If these functions are to be equal, then we must have the following:
$$a_0=a_1$$
$$a_1=2a_2-1$$
$$a_2=3a_3$$
$$a_3=4a_4$$
$$a_4=5a_5$$
And so on. ... |
>all imaginary numbers are irrational, because they are not rational.
Does ‘insensitive’ refer to every entity that isn't sensitive, or is this adjective *understood* to apply only to humans? This is how I persuade myself that it's not illogical to define the irrationals as $\mathbb{R} {\setminus} \mathbb{Q}.$
Si... |
During my attempt to formulate a Lipschitz bound in a course extra material, I encountered an expression in the form $a\|x-y\|+b$, with $a,b>0$. Is it possible to adjust the expression to a form of the type $C(a,b)\|x-y\|$, so that $C(a,b)$ serves as Lipschitz constant? |
I'm wondering how many ways n different objects can be split up into r groups, where each group is of size q, and where n >= r * q. I know that, when n = r * q, there is a straightforward solution ([here][1] is the formula). When n > r * q, though, does the formula change?
I believe (from simulations) that the formu... |
I have a real signal $f(t) $that is periodic on $[0,L]$ where $L=1$. I sampled it uniformly $64$ times and computed FFT (in python). I was asked to find the central frequency, namely the frequency with the largest magnitude. I found the index $i$ of the largest value in the transformed signal.
How do I find the freq... |
I'm wondering how many ways n different objects can be split up into r groups, where each group is of size q, and where n >= r * q. I know that, when n = r * q, there is a straightforward solution (the formula, as seen [here][1], is (n!) / (((q!) ^ r) * (r!)). When n > r * q, though, does the formula change?
I belie... |
I want to proof the following result:
1) Let $(X,\left \| \cdot \right \|)$ be a Banach Space. Let $(x_n)_{n\in \mathbb{N}}\subseteq X$ be such that $x_n\neq 0$ for all $n\in \mathbb{N}.$
2) There is a constant $K$ such that, for every choice of scalars $(a_i)_{i=1}^{\infty}$, and integers $n<m$, we have
$$... |
I'm wondering how many ways n different objects can be split up into r groups, where each group is of size q, and where n >= r * q. I know that, when n = r * q, there is a straightforward solution (the formula, as seen [here][1], is (n!) / (((q!) ^ r) * (r!)). When n > r * q, though, does the formula change? (The extra... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.