instruction stringlengths 12 30k |
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Your Fibonacci identity
> $$ F_{i-j}F_{i+j} - F_{i-k}F_{i+k} = F_{k-j}F_{j+k} (-1)^{(i + k)} $$
is derived from the identity
$$ 0 = (a-b)(a+b) -(a-c)(a+c) +(b-c)(b+c) $$
with label $\texttt{id3_3_1_2b}$ in my collection of
[Special Algebraic Identities](https://grail.eecs.csuohio.edu/~somos/ident04.gp).
... |
The problem stated in the Evan's book is
>Integrate by parts to prove:$$\int_U|Du|^p\;dx\le C \left(\int_U|u|^p\;dx\right)^\frac12\left(\int_U|D^2u|^p\;dx\right)^\frac12$$
for $2\le p< \infty$ and all $u\in W^{2,p}(U)\cap W^{1,p}_0(U)$.
I have managed to get the result for all $v\in C^\infty_c(U)$ and would like ... |
We have $C = 2^\mathbb{N}$ and we identify the Cantor set with the image of $C$ via:
$
t\colon C \rightarrow [0,1], \left(x_n\right)_n \mapsto \sum_{n=0}^{\infty}\frac{2x_n}{3^n}
$\
Now if $x,y\in t(C)$ and $\lvert x - y \rvert < 3^{-N}$ for some $N\in \mathbb{N}$ then the first $N$ digits of $x$ and $y$ coincide... |
It's known that [Fibonacci numbers][1] satisfy the following relation:
>$$F_mF_{n+1}-F_{m+1}F_n=(-1)^nF_{m-n}$$
Which is called [d'Ocagne's identity][2].
This identity with the following identities are well-known:
>$$F_{n-1}F_{n+1}-F_{n}^2=(-1)^n\tag{Cassini's identity}$$
>$$F_{n}^2-F_{n-r}F_{n+r}=(-1)^{n-... |
Why do Wolframalpha and Symbolab say
$$\int_0^8\frac1{\sqrt[3]x}dx=6$$
when the function is not continuous in the given interval?
[![image][1]][1]
[1]: https://i.stack.imgur.com/TsorS.png |
Let $K\to L$ be an arbitrary field extension. $K$ has characteristic zero. Let $X$ be a variety over $K$. Let $Y=X\times_K spec L$.
1)Is it true that $Y$ is reduced?
2)Let $Y=\cup Y_i$. We have the natural maps $Y_i\to X$. Is it true that these maps are dominant? |
Let $K\to L$ be an arbitrary field extension. $K$ has characteristic zero. Let $X$ be a variety over $K$. Let $Y=X\times_K spec L$.
1)Is it true that $Y$ is reduced?
2)Let $Y=\cup Y_i$ be decomposition into irreducible components. We have the natural maps $Y_i\to X$. Is it true that these maps are dominant?
I... |
Too long for a comment.
The brief search showed several related papers and even questions at Mathematics StackExchange. I list them below as the references. Unfortunately, now I have no time to study them.
It seems that all of these concern natural solutions, so we show the following proposition.
**Proposition... |
I am unable to find any clear definition of the concept of a *closed submodule* of a module. I'd really appreciate it if someone might elaborate on it, or point me in the right direction.
As far as I can tell, it seems to be an abstraction of the concept of a closed subspace of a vector space. This makes me think th... |
Too long for a comment.
The brief search showed several related papers and even questions at Mathematics StackExchange. I list them below. Unfortunately, now I have no time to study them.
[Eck] M.W. Ecker, *[When Does a Sum of Positive Integers Equal Their Product?](https://www.researchgate.net/publication/27018... |
I'm trying to follow the algebra leading from the gradient function to the Hessian in Logistic Regression, but I can't quite understand where I have gone wrong.
I have the gradient function as:
$$
\sum_i^m \left[x_i\cdot \frac {\exp\{-\theta^T x_i\}} {1+\exp\{-\theta^T x_i\}} - (y_i-1) x_i \right]
$$
After d... |
I want to prove the inequality with analysis methods instead of method of
PDE
Actually we can consider the following PDE:
$$\left\{\begin{array}{l}
u''(x)=u''(x)\\
u(0)=u(1)=0
\end{array}\right.$$
and we can get $u(y)$ by Green's function directly: G(x; y), that is $u(y)=\int_{0}^{1}u''(x)G(x;y)dx$
est... |
Too long for a comment.
The brief search showed several related papers and even questions at Mathematics StackExchange. I list them below as the references. Unfortunately, now I have no time to study them.
But it seems that all of these references concern natural solutions, so we show the following proposition.
... |
In his book *Number Systems and the Foundations of Analysis*, Elliot Mendelson first defines a Peano system (page 53). He then goes on to prove that any two Peano systems are isomorphic — they are essentially the same (page 80). He then makes a basic existence assumption — there exists a Peano system (page 83).
Etha... |
In the axiomatic treatment of natural numbers, can we define what a natural number is? |
If $f$ is a polynomial of degree $n$ such that
$\int_0^1 f(x)x^kdx=0$ for all $k=1,2,\dots,n$, then $\int_0^1 f^2(x)dx=(n+1)^2\left(\int_0^1 f(x)dx\right)^2.$
This implies that $\begin{vmatrix}
1-\frac{1}{(n+1)^2}& \frac{1}{2}& \frac{1}{3} & \cdots & \frac{1}{n+1} \\
\frac{1}{2}& \frac{1}{3} & \frac{1}{4}& \cdot... |
**I have this problem**: *Two circles shown in the figure, of centers $O$ and $O'$,
both have radius equal to $r$, are externally tangent
to each other and are further tangent to the line
$t$. The square $ABCD$ has side $AB$ on line $t$ and the other
two vertices, $C$ and $D$, which belong respectively
to the circ... |
**The statement of the problem** : Find all monotone functions $f : \mathbb R \rightarrow \mathbb R $ that verify the following $f(x) + f(y) = f ( xy - 2024x - 2024y + 2024*2025 ) ,\forall x,y \in (2024,\infty)$
**My ideas** (some of them are wrong ) : First of all, by making $x = y = 2025$ we easily get that f(2025... |
The book I'm reading showed a the promise without proving it, after the bolanzo Weirestrass theorem. The theorem is :
Let $\left\{{a_n}\right\}^\infty _{n=1}$ , $\left\{{b_n}\right\}^\infty _{n=1}$ be real number sequences
Prove:
$$\scriptsize\ \lim_{\overline{n\to\infty}}a_n+\lim_{\overline{n\to\infty}}b_n\le\lim_{... |
Most (all?) textbooks on calculus state unconditionally that a function *f* is differentiable at _x_ **if and only if** both of the one-sided derivatives $f_+'(x)$ and $f_-'(x)$ exist and are equal. On the other hand, answers to [this question][1] implies that that is not the case. So, are the textbooks or answers to t... |
Suppose you want to find the average of an $n\times n$ **diagonal** matrix $A$ over all possible rotations,
$$
\langle A\rangle = \int\limits_\text{SO($n$)} Q^T A Q \; dQ.
$$
It's easy enough to do this for $n=2$ because you have only one degree of freedom. I suspect that $n=3$ would be "simple", but tedious.
... |
How would one integrate over SO(n)? |
Let $X$ be an infinite set. Then there exists a sequence $(x_n)_n \subset X$ with $x_n \neq x_m$ for $n\neq m$. To see this we can argue inductively: choose $x_1$ arbitrarily. If $x_1, \dots, x_n$ Are already chosen there must exist a $x_{n+1} \in X\setminus \{x_1,\dots, x_n\}$ since otherwise $X$ is finite.
But why... |
Axiom of choice for choosing a sequence? |
The book I'm reading showed a the promise without proving it, after the bolanzo Weirestrass theorem. The theorem is :
Let $\left\{{a_n}\right\}^\infty _{n=1}$ , $\left\{{b_n}\right\}^\infty _{n=1}$ be real number sequences
Prove:
$$\scriptsize\ \lim_{\overline{n\to\infty}}a_n+\lim_{\overline{n\to\infty}}b_n\le\lim_{... |
Consider this argument:
I am given some constants $a$,$b$,$c$, and I want to find the value of variable $d$ for which some relation $X$ based on the given. I somehow know that this value must be unique.<br>
I start solving this question. I find that in this case, $d = f(a,d) + f(b,d) + f(c,d) - g(a,b,c,d)$ for some... |
The answer [here][1] makes sense to me, in the sense that the stochastic integral does not preserve the local martingale property if the integrand is not predictable, but I am confused as to why. I have been taught the following definition of the stochastic integral:
We first define the quadratic variation for a RC... |
Consider this argument:
I am given some constants $a$,$b$,$c$, and I want to find the value of variable $d$ for which some relation $X$ holds based on the given. I somehow know that this value must be unique.<br>
I start solving this question. I find that in this case, $d = f(a,d) + f(b,d) + f(c,d) - g(a,b,c,d)$ fo... |
Is there a specific notation for eigenvalues? specifically, I'd like to write:
$$m\equiv \text{smallest eigenvalue of }H$$
I've seen some sources write this as: $H\succeq mI$, where "$\succeq0$" means the matrix is positive semi-definite, but it seems a bit convoluted.
Is there a simpler, more accepted way to wri... |
Suppose that $(A, ∥ · ∥_A)$ and $(B, ∥ · ∥_B)$ are two Banach algebras such that $B$
is an ideal in $A$ and $∥·∥_A ≤ ∥·∥_B$. We know that each Banach $A$-bimodule $X$
is also a Banach $B$-bimodule. But the converse is not true in general!
Do you have an example of a $B$-bimodule $X$ that is not an $A$-bimodule? |
Posts like [this one](https://math.stackexchange.com/questions/1021316/mapping-a-region-between-two-circles-on-a-half-plane/4219901#4219901) have always baffled me. I have taken a few complex variables courses in my time and it seems that every time I take a course like this, either the textbook isn't great at explaini... |
I am still new to Lyapunov stability and I have a question:
The system is:
$\dot{x}_1 = x_2(1-x_1^2)$ and $\dot{x}_2=-(x_1+x_2)(1-x_1^2)$
I used $V(x)=\frac 1 2(x_1^2+x_2^2)$
Then, I get $\dot{V}(x)= -(1-x_1^2)x_2^2.
Can I define $X=\{x_1^2+x_2^2<1\}$ where $\dot{V}(x)<0$ $\forall x\in X$ and conclude it i... |
I am still new to Lyapunov stability and I have a question:
The system is:
$\dot{x}_1 = x_2(1-x_1^2)$ and $\dot{x}_2=-(x_1+x_2)(1-x_1^2)$
I used $V(x)=\frac 1 2(x_1^2+x_2^2)$
Then, I get $\dot{V}(x)= -(1-x_1^2)x_2^2$.
Can I define $X=\{x_1^2+x_2^2<1\}$ where $\dot{V}(x)<0$ $\forall x\in X$ and conclude it ... |
I am still new to Lyapunov stability and I have a question:
The system is:
$\dot{x}_1 = x_2(1-x_1^2)$ and $\dot{x}_2=-(x_1+x_2)(1-x_1^2)$
I used $V(x)=\frac 1 2(x_1^2+x_2^2)$
Then, I get $\dot{V}(x)= -(1-x_1^2)x_2^2$.
Can I define $X=\{x_1^2+x_2^2<1\}$ where $\dot{V}(x)<0$ $\forall x\in X$ and conclude it ... |
I am still new to Lyapunov stability and I have a question:
The system is:
$\dot{x}_1 = x_2(1-x_1^2)$ and $\dot{x}_2=-(x_1+x_2)(1-x_1^2)$
I used $V(x)=\frac 1 2(x_1^2+x_2^2)$
Then, I get $\dot{V}(x)= -(1-x_1^2)x_2^2$.
Can I define $X=\{x\in \mathbb {R}^2 | x_1^2+x_2^2<1\}$ where $\dot{V}(x)<0$ $\forall x\i... |
I'm trying to solve the following differential equation
$$
y'' + \dfrac{1}{2}(y')^2 = A \delta(x) + B \delta(x-a) + C
$$
I tried two times, the first one using Laplace transforms, but I don't really know how to deal with the $(y')^2$ term. I found some papers discussing it, but I couldn't really use them since ... |
Here is the question:
> we want to hold a math contest for 8 groups, giving each group 6 questions with the following condition : no group has more than one question from each year and each possible pair of the years that the questions are chosen from is not repeated in any other group. what is the minimum number of y... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
There is a dog at (x,y) = (37,10) with units say feet. The dog wants to go home located at origin (0,0). The dog speed is 6 feet/seconds on the x,y axes and 6.5 anywhere else. What's the fastest path to the origin?
My answer says a direct path but that seems to easy (this is supposed to be a trick question). Am i m... |
I am reading Richard Sutton's textbook Reinforcement Learning, chapter $11.4$, and I am confused by his definition of the *Projected Bellman Error*.
He defines a norm on value functions $v: S \mapsto \mathbb{R}$ where $S$ is the set of states, as
$$||v||_{\mu} = \sum_{s \in S} \mu(s) v(s)^2$$
Then assuming lin... |
I am reading Richard Sutton's textbook Reinforcement Learning, chapter $11.4$, and I am confused by his definition of the *Projected Bellman Error*.
He defines a norm on value functions $v: S \mapsto \mathbb{R}$ where $S$ is the set of states, as
$$||v||_{\mu} = \sum_{s \in S} \mu(s) v(s)^2$$
Then assuming lin... |
I am solving the following problem from Sheldon Ross:
Lobsters are caught in morning and afternoon sessions. The number of lobsters that can be caught in each session are 0, 1, 2, 3, 4, or 5, each with probability 1/6. Find the probability of at least one lobster being caught in the afternoon given that more lobsters ... |
$1 + \dim(U \cap W) = \dim(U+W) \geq \dim(U) \geq \dim(U \cap W)$, and we see that at least one of these inequalities must be an equality. So, either $\dim(U + W) = \dim(U)$, or $\dim(U) = \dim(U \cap W)$. Can you take it from here? |
Let $C_i\sim C$ and $R_i \sim R$ denote the durations (in ticks) of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$... |
Let $\mathcal{C}$ be an $\infty$-category, and let $G$ be a group. Denote by $\mathcal{C}^{\text{BG}}$ the functor $\infty$-category $\text{Fun}(\text{BG}\longrightarrow \mathcal{C})$. The homotopy orbit functor is given by
$$-_{\text{hG}}: \mathcal{C}^{\text{BG}}\longrightarrow \mathcal{C}: (F: \text{BG}\longrighta... |
The hypperreal numbers are an extension of the reals that allow for a rigorous treatment of infinitely small and infinitely large values. Specifically it includes the number $\varepsilon$ where
$$
0 < \varepsilon < x \tag{1}
$$
for all $x \in \Bbb R$. As such, the hypperreal field $^*\Bbb R$ is both non-Archimede... |
What is the motivation for building the hyperreals using an ultrapower construction? |
If I ask a man if he can say "no" and he says "no". Is it a paradox?
If he answers "no" it means he can't say "no", but he just said it |
Let G be a locally compact group with Haar measure $\mu$. The left regular representation of G on $L^2(G, \mu)$ is given by $L(g)(f): x \mapsto f(g^{-1}x)$. It's unitary and in the case where G is compact, the 3rd part of the Peter-Weyl theorem asserts that L decomposes $L^2(G)$ into the direct sum of all the unitary ... |
If I ask a person if they can say "no" and they say "no". Is it a paradox?
If they answer "no" it means they can't say "no", but they just said it |
Let G be a locally compact group with Haar measure $\mu$. The left regular representation of G on $L^2(G, \mu)$ is given by $L(g)(f): x \mapsto f(g^{-1}x)$.
If G is abelian, the Plancherel theorem says that the Fourier transform can be extended to an unitary equivalence $L^2(G) \cong L^2(\widehat{G})$, so this imp... |
First, I'll define the following unit vectors:
$u_0 = \dfrac{b - a}{\| b - a\|}$
$u_1 = \dfrac{a + b}{\| a + b \| }$
$u_2 = \dfrac{a \times b }{ \| a \times b \| }$
then the axis of rotation that takes $a$ to $b$ is given by
$ U = \cos \alpha \ u_1 + \sin \alpha \ u_2 $
unit vectors $v_1$ and $v_2$ ar... |
Dedekind gave a construction and explanation of integers and rational in 1858. I'm not sure if this was the standard view of these objects after this point.
If there can be no conclusive "when", are there any textbooks that include general constructions or something like this?
When did integers start being viewe... |
When were negative numbers fully accepted into mathematics? |
Dedekind gave a construction and explanation of integers and rational in 1858. This was as ordered pairs of natural numbers. I'm not sure if this was the standard view of these objects after this point.
If there can be no conclusive "when", are there any textbooks that include general constructions or something lik... |
I am trying to familiarise myself with isomorphisms of quotient rings. More specifically, I am trying to find out which of the following three relations hold:
1) $\mathbb{Q}/(X^3-1) \cong \mathbb{Q}[i\sqrt{3}]$
2) $\mathbb{Q}/(X^4-4) \cong \mathbb{Q}[i, \sqrt{2}]$
3) $\mathbb{Q}/(X^4-6X^2+9) \cong \mathbb{Q}[i, \s... |
According to the [Wikipedia article on complex Lie algebras](https://en.wikipedia.org/wiki/Complex_Lie_algebra) , any complex Lie algebra $\mathfrak{g}$ that is isomorphic to its conjugate $\overline{\mathfrak{g}}$ admits a real form.
So as I understand it, this isomorphism exists if and only if there is an antiline... |
> If I ask a person if they can say "no" and they say "no", is this a paradox?
If they answer "no" it means they can't say "no", but they just said it |
If I ask a person if they can say "no" and they say "no", is this a paradox? |
I am trying to familiarise myself with isomorphisms of quotient rings. More specifically, I am trying to find out which of the following three relations hold:
1) $\mathbb{Q}[X]/(X^3-1) \cong \mathbb{Q}[i\sqrt{3}]$
2) $\mathbb{Q}[X]/(X^4-4) \cong \mathbb{Q}[i, \sqrt{2}]$
3) $\mathbb{Q}[X]/(X^4-6X^2+9) \cong \mathbb... |
If you take a function:
$$
f( x) =\begin{cases}
x^{2} & ,x\leqslant 0\\
x & ,x >0
\end{cases}
$$
Then at $x=0$ we can't agree on what the derivative is because there are 2 choices. Which one would you choose - the left or the right derivative? That's why the (two-sided) derivative doesn't exist at that point.
... |
Let G be a locally compact group with Haar measure $\mu$. The left regular representation of G on $L^2(G, \mu)$ is given by $L(g)(f): x \mapsto f(g^{-1}x)$.
If G is abelian, the Plancherel theorem says that the Fourier transform can be extended to an unitary equivalence $L^2(G) \cong L^2(\widehat{G})$, so this imp... |
I am solving the following problem from Sheldon Ross:
Lobsters are caught in morning and afternoon sessions. The number of lobsters that can be caught in each session are 0, 1, 2, 3, 4, or 5, each with probability 1/6. Find the probability of at least one lobster being caught in the afternoon given that more lobsters ... |
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). Consider $\bigcap_{n>0} I^n = \underset{\longrightarrow}{\textrm{lim}} \, I^n$ as a colimit. I would like to apply the $\t... |
Let $a$ and $b$ be two relatively prime positive integers. Prove that the integer $ab - a - b$ cannot be written as a positive linear
combination of $a$ and $b$ but that every integer greater than $ab - a - b$ is a positive linear combination of $a$ and $b$.
Let $N_0 = ab-a-b$. I want to show that for $N > N_0$, th... |
In $\mathbb{R}^{n}$, assume $P_{1}$ and $P_{2}$ are both positive definite. Let $A = P_{1}(P_{1} + P_{2})^{-1}$.
Can we conclude the following,
Given every vector $x$, we have
\begin{equation*}
|(Ax)_{i}| \leq |x_{i}|, \quad i = 1,\ldots,n
\end{equation*}
where $(Ax)_{i}$ denote the $i$th component of $Ax$.
... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume there are two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of being 0 or 1 (the information bits) and an additional check bit $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$). You now transf... |
Nobody was touching this question, so i will say some words on the problem, there was a bounty, so there is a special interest in having a parallel opinion and feed-back. (This too long for a comment, became an answer...)
----------
One may use the same plan, but structured to be "more convincing". Since i have... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume there are two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of being 0 or 1 (the information bits) and an additional check bit $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$). You now transf... |
Let $P = \mathbb{R}^2 \setminus \{(0,0)\}$. We define a relation on $P$ as follows: $(x,y) \sim (z,w)$ if and only if there exist $\lambda \in \mathbb{R}\setminus \{0\}$ such that $x = \lambda z$ and $y = \lambda w$.
Define the function $\phi : \mathbb{R} \to {^{P}\big/_{\sim}}$ by $\phi(t) = [(t,1)]$ for $t \in \ma... |
As part of a personal project, I found the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1], which discusses deriving equations for a 3D surface I want to parameterize. However, despite my efforts to decipher the paper, I can't follow everything it is describing. The ... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume there are two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of being 0 or 1 (the information bits) and an additional check bit $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$). You now transf... |
Let $X_1, \ldots, X_n$ be i.i.d r.v. with the p.d.f.
$$
f(x; \theta_1, \theta_2) =
\begin{cases}
\frac{1}{\theta_2} \exp\left(-\frac{x - \theta_1}{\theta_2}\right), & \text{if } x > \theta_1, \\
0, & \text{otherwise},
\end{cases}
$$
where $\theta_1 \in \mathbb{R}$, $\theta_2 > 0$ are unknown parameters.
... |
My question is basically in the title. When we move from real numbers to complex numbers, is the reason we can talk about expressions such as $e^{i\pi}$ the transfer principle? |
I am studying the book 'Automorphic forms and representations' by Daniel Bump. Currently, I'm reading chapter $4$.
There are many good references for the first section, so I understood almost every theorems in this section.
Currently, I'm stuck in section $4.5$, i.e., 'The principal series representations'. I h... |
In MacKay's "Information theory, inference, and learning algorithms" book, in chapter 26 he covers the sum-product algorithm for calculating marginal probabilities, partition functions, etc. The same info can be found in this [wikipedia article][1].
Assume we have an unnormalized probability distribution $P^\ast(x)$... |
Does the standard sum-product message passing algorithm save any computation over just doing this? |
My question is basically in the title. When we move from real numbers to complex numbers, is the reason we can talk about expressions such as $e^{\pi i}$ the transfer principle? |
I would like to evaluate the mean and variance of the edge density for subgraphs obtained by repeatedly subsampling nodes.
Specifically, suppose we have an undirected graph $G$ with $N$ vertices and adjacency matrix $y$. We obtain a subset of nodes $S$ by [Bernoulli sampling](https://en.wikipedia.org/wiki/Bernoulli_... |
Fundamentally, your confusion stems from not properly making a distinction between parameters (more specifically, moments) of a probability distribution, and *estimators* of those parameters. This becomes especially apparent in your use of notation.
When we talk about the mean or variance of a probability distribut... |
Posts like [this one](https://math.stackexchange.com/questions/1021316/mapping-a-region-between-two-circles-on-a-half-plane/4219901#4219901) have always baffled me. I have taken a few complex variables courses in my time and it seems that every time I take a course like this, either the textbook isn't great at explaini... |
My question is basically in the title. When we move from real numbers to complex numbers, is the reason we can talk about expressions such as $e^{\pi i}$ the transfer principle?
EDIT: Since clarification was needed, I was referring to something similar to the transfer principle for hyperreal numbers, where it's used... |
My question is basically in the title. When we move from real numbers to complex numbers, is the reason we can talk about expressions such as $e^{\pi i}$ the transfer principle?
EDIT: Since clarification was needed, I was referring to something similar to the transfer principle for hyperreal numbers, where it's used... |
If you want a specific matrix $g$, you can take
$$
g=\left[ \begin{array}{cc}
1&0\\
\sqrt{2}&1\end{array}\right].
$$
Here is the reason. Let $U_\infty$ denote the subgroup of strictly upper triangular matrices in $G$. Then for $g\in G$,
$$
U_\infty g = g U, \quad U=g^{-1}U_\infty g.
$$
The unique fixed-point... |
I am studying the book 'Automorphic forms and representations' by Daniel Bump. Currently, I'm reading chapter $4$.
There are many good references for the first section, so I understood almost every theorems in this section.
Currently, I'm stuck in section $4.5$, i.e., 'The principal series representations'. I h... |
Short answer: No :)
The [transfer principle](https://en.wikipedia.org/wiki/Transfer_principle) is a powerful principle of mathematical logic, most famously used in nonstandard analysis. In that context, it states that the real numbers $\mathbb{R}$ and hyperreal numbers $\mathbb{R}^*$ satisfy exactly the same set of ... |
How would one integrate over $SO(n)$? |
Let $G_1\subseteq \Bbb R^n$ and $G_2\subseteq \Bbb R^m$ be open sets, then show that $G_1\times G_2\subseteq \Bbb R^n\times \Bbb R^m$ is an open set.
Dem:
Given that $G_1,G_2$ open sets, there exists $B_1(g_1,\epsilon)\subseteq G_1$ and $B_2(g_2,\epsilon')\subseteq G_2$, and by the definition of cartesian produ... |
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a continuously differentiable function and assume its gradient $\nabla f$ is Lipschitz continuous with a constant $L$, i.e., for every $x_1,x_2 \in \mathbb{R}^n$, it holds that
$$
\left\| \nabla f(x_1) - \nabla f(x_2) \right\| \leq L \|x_1 - x_2\|.
$$
I am reading a ... |
I would like to evaluate the mean and variance of the edge density for subgraphs obtained by repeatedly subsampling nodes.
Specifically, suppose we have an undirected graph $G$ with $N$ vertices and adjacency matrix $y$. We obtain a subset of nodes $S$ by [Bernoulli sampling](https://en.wikipedia.org/wiki/Bernoulli_... |
Let $\mathcal{M}$ be a (not necessarily complete) Riemannian manifold. As a result, we cannot claim that closed and bounded subsets are compact. Pick $x \in \mathcal{M}$. Let $r = inj(x)$ be the injectivity radius at $x$. Take the geodesic ball $B:=B(x,r/2)$. Set $B_x:=\exp_x^{-1}(B)$. We know that $\exp_x^{-1}(\bar{B}... |
Are sufficiently small closed geodesic balls compact? |
Let $\mathcal{M}$ be a (not necessarily complete) Riemannian manifold. As a result, we cannot claim that closed and bounded subsets are compact. Pick $x \in \mathcal{M}$. Let $r = \text{inj}_{\mathcal{M}}(x)$ be the injectivity radius at $x$. Then $\exp_x$ is continuous and invertible on $B_{T_x\mathcal{M}}(0,r):=\{v \... |
Let $a,b,c\ge 0:ab+bc+ca=3.$ Find the minimal value $$P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}.$$
I think $2+\sqrt{2}$ is desired result. $P$ attain this value when $a=b=c=1.$
I use $a+b+c\ge \sqrt{3(ab+bc+ca)}=3$ and we need to prove $$a^2b+b^2c+c^2a+abc\ge 4.$$
But it is wrong when $c=0,b=3,a=1$
I got stuck her... |
Motivation:
---
>**Puzzle:** Draw $\odot$ on paper without lifting your pen.
*Solution:* Start by putting your pen in the centre of the paper to create the dot. Then fold the paper such that the edge of it touches your pen tip. Draw a line outward from the dot but on the other side of the paper. With the paper u... |
I'm reading An introduction to homological algebra by Charles A. Weibel and I'm trying to prove the snake lemma 1.3.2.
[Using this diagram][1] (don't have enough points to embbed it in my question) it says that $\partial=c_0^{-1} f_1b_1^{-1}$ makes the sequence $\ker(f_0)\to\ker(f_1)\to\ker(f_2)\overset{\partial}{\to}... |
Is my proof for the following problem correct?
Let $f:X\rightarrow(0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable.\
Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable.
Proof:
"$\implies$" Let f be $\lim \limits_{n \to \infty}s_n(x)=f(x)$ $\forall x\in X$(where $s_n$ is a sequence of ... |
Is my proof for the following problem correct?
Let $f:X\rightarrow[0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable.\
Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable.
Proof:
"$\implies$" Let f be $\lim \limits_{n \to \infty}s_n(x)=f(x)$ $\forall x\in X$(where $s_n$ is a sequence of ... |
Let $G_1\subseteq \Bbb R^n$ and $G_2\subseteq \Bbb R^m$ be open sets, then show that $G_1\times G_2\subseteq \Bbb R^n\times \Bbb R^m$ is an open set.
Dem:
Given that $G_1,G_2$ open sets, there exists $B_1(g_1,\epsilon)\subseteq G_1$ and $B_2(g_2,\epsilon')\subseteq G_2$, and by the definition of cartesian produ... |
Let $a$ and $b$ be two relatively prime positive integers. Prove that the integer $ab - a - b$ cannot be written as a positive linear
combination of $a$ and $b$ but that every integer greater than $ab - a - b$ is a positive linear combination of $a$ and $b$.
Let $N_0 = ab-a-b$. I want to show that for $N > N_0$, th... |
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