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I'm studying general topology and a question has come to my mind.
Given a sequence x (i.e. map from the set N of natural numbers) in a topological space and given a point p of the latter, we say that *p is a special point for x* if the x-preimage of every neighbourhood of p is cofinal in the canonical order of N (th... |
In this
>answer to [Is there any valid complex or just real solution to $\sin(x)^{\cos(x)} = 2$?](https://math.stackexchange.com/a/4693242/905886),
one must calculate $$\frac{d^{n-1}}{dw^{n-1}}\left.\frac{4^{-\frac{n}{\sqrt w}}}{\sqrt w}\right|_1=\sum_{m=0}^\infty\frac{\Gamma\left(\frac12-\frac m2\right)(-\ln(4)... |
Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$, and constants $b_i$, $i=1,\dots,n$. Consider the following mixed integer program (MIP) with decision variable $c_i\in\{0,1\}$, and $X=[x_1,\dots,x_n]\in\mathbb{R}^{l\times n}$ with $x_i\in\mathbb{R}^{l}$ for $i=1,\dots,n$.
Ob... |
I have this series
$$\sum_{n=-\infty}^\infty \frac{x+n}{|x+n|^3}$$
Which seemed quite similar to that of (csc(x))^2:
$$\sum_{n=-\infty}^\infty \frac{1}{(x+n)^2}$$
The only difference is the change of sign throughout the sum, does this series also converge to some function? |
How can I get this estimation? |
Pauli matrices satisfy
$$
\sigma_i \sigma^{\dagger}_j = \delta_{ij} \sigma_0 + \epsilon_{ijk}\sigma_k\,.
$$
Can one construct a set of three complex-valued $2\times 2$ matrices $a_i$, $i=1,2,3$, such that
$$
a_i a^{\dagger}_j = \delta_{ij} \sigma_0 - \epsilon_{ijk}\sigma_k\,?
$$
If not, is it possible for matri... |
>Solve $y+3=3\sqrt{(y+7)^2}$
$\Rightarrow y+3=3(y+7)$
$\Rightarrow y+3=3y+21$
$\Rightarrow 2y=-18$
$y=-9$
But $-9+3\ne3\sqrt{(-9+7)^2} \Rightarrow-6 \ne6$
How is this humanely possible? What's going on here??? |
Pauli matrices satisfy
$$
\sigma_i \sigma^{\dagger}_j = \delta_{ij} \sigma_0 + I \epsilon_{ijk}\sigma_k\,,
$$
with $I$ being the imaginary unit.
Can one construct a set of three complex-valued $2\times 2$ matrices $a_i$, $i=1,2,3$, such that
$$
a_i a^{\dagger}_j = \delta_{ij} \sigma_0 - I \epsilon_{ijk}\sigma_k\... |
> The variable $(X,Y)$ is uniformly distributed over $$D=\{(x,y)\in\mathbb R^2:|x|+|y|\le1\}.$$ Let $$A=X-Y,B=X+Y.$$
Are $A$ and $B$ independent?
I tried to prove $F(AB)=F(A)F(B)$, I tried to find the joint function of $AB$ through changing variables. $X=(A-B)/2$ and $Y=(A+B)/2$ and got that the joint function is ... |
Let the (finite dimensional) Hilbert space $\mathcal{H}$ be the direct sum of $\mathcal{H}_A$ and $\mathcal{H}_B$.
Let $A$ be a linear operator on $\mathcal{H}_A$ and $B$ be a linear operator on $\mathcal{H}_B$.
Let $A = \sum_j \lambda_j^A |\psi_j^A\rangle \langle \psi_j^A|$ and $B = \sum_j \lambda_j^B |\psi_j^B ... |
**Question 1:** Does anyone know a name, or have a reference, for the following lemma?
> $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\opera... |
Natural transformations of Hom-sets “transport” natural transformations from one pair of functors to another? (Reference) |
Let the (finite dimensional) Hilbert space $\mathcal{H}$ be the direct sum of $\mathcal{H}_A$ and $\mathcal{H}_B$.
Let $A$ be a linear operator on $\mathcal{H}_A$ and $B$ be a linear operator on $\mathcal{H}_B$.
Let $A = \sum_j \lambda_j^A |\psi_j^A\rangle \langle \psi_j^A|$ and $B = \sum_j \lambda_j^B |\psi_j^B ... |
In the lecture on Gagliardo-Niremberg inquality, there was mentioned the fact that for functions:
$$f_{\lambda}(x) = (\lambda + \vert x \vert^q)^\frac{p-n}{p}, \; \text{ where } \frac{1}{q}+\frac{1}{p}=1$$
equality holds.
On checking this, I got stuck on calculating the gradient $\nabla f$. Knowing that $f$ is rad... |
Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$, and constants $b_i$, $i=1,\dots,n$. Consider the following mixed integer program (MIP) with decision variables $c_i\in\{0,1\}$ and $X=[x_1,\dots,x_n]\in\mathbb{R}^{l\times n}$ with $x_i\in\mathbb{R}^{l}$ for $i=1,\dots,n$.
Ob... |
Can ZFC define every n-ary function and relation on the natural numbers that Peano Arithmetic can define, plus more? If so, what is the proof, and also, can someone give an example of a number theoretic function or relation that ZFC can define but PA can't. |
Can ZFC define every function and relation that PA can define, plus more? |
Let the (finite dimensional) Hilbert space $\mathcal{H}$ be the direct sum of $\mathcal{H}_A$ and $\mathcal{H}_B$.
Let $A$ be a linear operator on $\mathcal{H}_A$ and $B$ be a linear operator on $\mathcal{H}_B$.
Let $A = \sum_j \lambda_j^A |\psi_j^A\rangle \langle \psi_j^A|$ and $B = \sum_j \lambda_j^B |\psi_j^B ... |
Suppose that $\int_0^1 x^nf(x)=0$ for all nonnegative integers $n$, where $f$ is a Lebesgue measurable function that is bounded. How do you prove that $f(x)=0$ a.e. on $[0,1]$.
I've seen this problem before, except with the condition that $f$ is continuous. In this case, we can use the Weierstrauss Approximation The... |
Let the (finite dimensional) Hilbert space $\mathcal{H}$ be the direct sum of $\mathcal{H}_A$ and $\mathcal{H}_B$.
Let $A$ be a linear operator on $\mathcal{H}_A$ and $B$ be a linear operator on $\mathcal{H}_B$.
Let $A = \sum_j \lambda_j^A |\psi_j^A\rangle \langle \psi_j^A|$ and $B = \sum_j \lambda_j^B |\psi_j^B ... |
Disclaimer : Not a direct answer but a methodology.
As your question can be understood as "how can I attack such an issue ?", I would like here to propose two natural tools for such questions involving angle bissectors in a triangle : [**trilinear coordinates**](https://en.wikipedia.org/wiki/Trilinear_coordinates) $... |
> $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\operatorname{Ob}}$**Lemma:** Given functors $G_1,G_2: \C \to \G$ such that there exists a natur... |
Given some implicit equations or a definition of a subset of a vector space, what is the requirement for it to define an affine subspace. I'm looking for something analogous to the test for vector subspaces (closed under adittion and scalar multiplication):
Let $V$ be vector space over a field $\mathcal{K}$ and $S\... |
Does $A/B \cong C/D$ and $B \cong D$ imply $A \cong C$? |
I am a high school student and there is something I want to ask about the application of digital sums.
Let's say there is a fraction "**520/7",** let **520/7=a**, so **520= a × 7**, so if we now calculate the digital sums, it would be like **7= a × 7** , it means the digital sum of a should be 1 and nothing else so ... |
> $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\operatorname{Ob}}$**Lemma:** Given functors $G_1,G_2: \C \to \G$ such that there exists a natur... |
**Theorem** (Birkhoff) Let $(E, \mathcal{E}, \mu)$ be a $\sigma$-finite measure space and let $T : E \to E$ be a measure-preserving transformation. Suppose that $f \in L^1(\mu)$. Then, the approximations:
$$F_n : =\frac{\sum_{k=0}^{n-1} f \circ T^k}{n}$$
converge almost everywhere to a $T$-invariant integrable ... |
> $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\operatorname{Ob}}$**Lemma:** Given functors $G_1,G_2: \C \to \G$ such that there exists a natur... |
> Let $\omega(n)$ denote the number of prime factors of a positive integer $n$. Prove that \begin{equation}\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}\end{equation}
My attempt: note that $2^{\omega(n)}$ is multiplicative, since $\omega(n)$ is clearly additive. Therefore the Dirichlet se... |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
Give an interpretation where
$$∃x(\neg P(x) ∨ Q(x)) \to (∃xP(x) ∧ ∀x\neg Q(x))$$
is false.
How does someone even begin with questions like this? I have interpreted it in my head and I kind of get it in a sense. But seems like the only thing I know is that since it is an implication, the only way it will be false i... |
> $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\operatorname{Ob}}$**Lemma:** Given functors $G_1,G_2: \C \to \G$ such that there exists a natur... |
I was reading J. Renault's paper "The Fourier Algebra of a Measured Groupoid" and I am confused about his approach to the predual of a von Neumann Algebra.
Let $M:= VN(\mathcal{G})$ be the von Neumann algebra of a measured groupoid; I think the definition is irrelevant, we just need to know that $M$ is a von Neumann... |
I was reading Marcinkiewicz-Zygmund(MZ) law of large numbers for random fields and came across necessary and sufficient condition $E(|X|\log^+|X|)< \infty$ for MZ-SSLN to hold true. I have a question about this function $\log^+|X|$. Why don't they just need condition without this max, that is, $E(|X|\log|X|)< \infty$ ?... |
Disclaimer : Not a direct answer but a methodology.
As your question can be understood as "how can I attack such an issue ?", I would like here to propose two natural tools for such questions involving angle bisectors in a triangle : [**trilinear coordinates**](https://en.wikipedia.org/wiki/Trilinear_coordinates) $(... |
I can’t post my ask on new thread.
**Problem 1.** Let be $a$ is positive integer $(a\le 5)$. Define $\|1,1,1,a;n\|$ are Number of non-negative integer solutions of equation $\quad x_1+x_2+x_3+ax_4=n$
Proof
$\|1,1,1,a;n\|=\left\lfloor\dfrac{(n+2)(n+a+2)(2n+a+1)}{12a}\right\rfloor$
**Problem2.** Applies resul... |
"Every open subset of R is a union of closed intervals." I just don't get it. What could be an examples? |
Why is this statement correct? |
It's called ... the domain.
*Every* element of the domain is mapped and you *always* use every single element. You are not allowed to *ever* have an element unused.
Worth noting if you have a function such as $\frac 1{x-5}$ the domain is *NOT* $\mathbb R$. The domain is $(-\infty, 5)\cup (5, \infty)$ (which ... |
Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\rho(A) < 1$ and $x_0 = 0$. Assume $r_k\in\mathbb{R}^{n}$ is a bounded reference for the output sequence. Define the t... |
I'm really confused and I can't get the concept. How is $14$ in base $8 = 16$? And how does $8$ in base $8 = 10$?
In case of $14$ isn't $1\times8 + 4\times1 = 12$?
Shouldn't any number in base $8$ be lower than a number in base $10$? |
For instance, can a number like $0.1111111\cdots$ in base $3$ be represented as $0.23515613\cdots$ (non-repeating) in base $8$? I imagine the answer would be a resounding NO but it would be interesting to see a proof of why. |
If a martingale $M$ has $\sup_{t\geq 0} \mathbb{E}(M_t^2)<\infty$, then how do we prove that we also have $\mathbb{E}(\sup_{t\geq 0}M_t^2)<\infty$?
The hint in the book is to use Doob's $L^p$ inequality, to obtain $\mathbb{E}(\sup_{N\geq t\geq 0}M_t^2)<4\mathbb{E}(M_N^2)$, and then take the limits (w.r.t N) and use ... |
How to prove $\sup_{t\geq 0} \mathbb{E}(M_t^2)<\infty$ implies $\mathbb{E}(\sup_{t\geq 0}M_t^2)<\infty$? |
I was reading Marcinkiewicz-Zygmund (MZ) law of large numbers for random fields and came across necessary and sufficient condition $E(|X|\log^+|X|)< \infty$ for MZ-SSLN to hold true. I have a question about this function $\log^+|X|$. Why don’t they just need condition without this max, that is, $E(|X|\log|X|)< \infty$?... |
Is there a term for a natural number $N$ which cannot be expressed in the form $a^b$ where $a$ and $b$ are both naturals not equal to $N$? |
> Consider the following equations $$\begin{cases} 2(x^2+y^2)-z^2=0\\
x+y+z-2=0\end{cases}$$Prove that the above system of equations defines
> a unique function $\phi: z\mapsto (x(z),y(z))$, from a neighborhood of
> $z=2$ to a neighbor hood $V$ of $(1,-1)$ and $\phi\in C^1$ on $U$.
My idea is to use The [implici... |
We consider two $n \times n$ projections, $A$ and $B$. In particular, this means that $A^2 = A$ and $B^2 = B$.
Given this, I was curious on if products of projection matrices would have the same rank? For instance, if $\text{rk}(PQ) = \text{rk}(QP)$ or $\text{rk}(PQP) = \text{rk}(QPQ)$?
We have that for projectio... |
**Background**
This is Problem 5-17 of John Lee's *Introduction to Topological Manifolds*.
Suppose $\sigma=[v_0,\ldots,v_k]$ is a simplex in $\mathbb{R} ^n$ and $w\in \mathbb{R} ^n$. If $\{w,v_0,\ldots,v_k\}$ is an affinely independent set, we say that **$w$ is affinely independent of $\sigma$**. In this case, th... |
Let $n\in\mathbb{N}$ a fixed natural number and $I_\varphi\subset\mathbb{R}$ a fixed compact interval. Consider the space of all $\cal{C}^1$ functions $\varphi:I_\varphi\to \mathbb{R}^n$. Define an equivalence relation as $\varphi \cal{R} \phi$ iff there exists a function $\lambda:I_\varphi\to \phi$ that is bijective, ... |
No, you must check that the $2\times 2$ Jacobian matrix $\dfrac{\partial (F,G)}{\partial (x,y)}$ is invertible at the given point $(1,-1,2)$. With regard to your final question, read the statement of the Implicit Function Theorem very carefully. What specific question do you have? |
This message has been encoded by a monoalphabetic function $f(p)=p+b~ (mod 26)$:
$$ APHUO~~ EGEHP~~ PEXOV ~~FKEUH ~~CKVUE~~ CHKVE~~ APHUO,$$
where we digitize the alphabet by letting A = 00, B = 01, . . . , Z = 25.
I want to find the original message. For this purpose, I tried to identify $b$ by finding the mo... |
How to decode the message? |
Consider the following statement:
$$ A \subseteq B $$
It's negation is the following
$$ A \not \subseteq B$$
Negating statements can be required in proofs by contraposition, for example.
---
Now consider the case that $A$ is empty, $A=\emptyset$.
The first statement $A \subseteq B$ is fine, since t... |
This message has been encoded by a monoalphabetic function $f(p)=p+b~ \pmod{26}$:
$$ APHUO~~ EGEHP~~ PEXOV ~~FKEUH ~~CKVUE~~ CHKVE~~ APHUO,$$
where we digitize the alphabet by letting A = 00, B = 01, . . . , Z = 25.
I want to find the original message. For this purpose, I tried to identify $b$ by finding the m... |
Let $n\in\mathbb{N}$ a fixed natural number and $I_\varphi\subset\mathbb{R}$ a fixed compact interval. Consider the space of all $\cal{C}^1$ functions $\varphi:I_\varphi\to \mathbb{R}^n$. Define an equivalence relation as $\varphi \cal{R} \phi$ iff there exists a function $\lambda:I_\varphi\to \phi$ that is bijective, ... |
Why is the difference of consecutive primes from Fibonacci sequence devisible by 4? |
Are there any theorems about the existence of feasible solutions to the Mixed-Integer Nonlinear Programming (MINLP) problem? |
Recently, i've been experimenting with the map $N \rightarrow N^N \% (2N+1)$. There's a lot of cycles with this map, but i found a particularly massive one:
$13612, 15106, 27724, 27553, 29074, 53239, 76162, 135319, 103369, 201064, 323761, 202351, 24889, 15556$
This cycle have period of $14$. It is very long and i... |
Recently, i've been experimenting with the map $N \rightarrow N^N \% (2N+1)$. What i would do is repeated apply this map to some numbers. There's a lot of cycles with this map, but i found a particularly massive one:
$13612, 15106, 27724, 27553, 29074, 53239, 76162, 135319, 103369, 201064, 323761, 202351, 24889, 155... |
Recently, i've been experimenting with the map $N \rightarrow N^N \bmod (2N+1)$. What i would do is repeated apply this map to some numbers. There's a lot of cycles with this map, but i found a particularly massive one:
$13612, 15106, 27724, 27553, 29074, 53239, 76162, 135319, 103369, 201064, 323761, 202351, 24889, ... |
$$a(n)=a(\lceil \mathop{\rm abs}(a(n-1)) \rceil\bmod n)) + a(\lceil \mathop{\rm abs}(a(n-2)) \rceil\bmod n))$$
For starting values, $a(0)=a(1)=1$, the sequence has a cycle starting with $n=441329$ having a period of $63584$ (source: https://oeis.org/A330615) For starting value $a(0)=i$ and $a(1)=1$, the cycle starts a... |
I am trying to show:
$$\sum_{k=0}^{n} \sum_{r=k+1}^{n+1} \binom{n}{k} \binom{n+1}{r} = 2^{2n}$$
Could I have a hint?
I have tried rewriting $ \sum_{k=0}^{n} \sum_{r=k+1}^{n+1} \binom{n}{k} \binom{n+1}{r} = \sum_{k=0}^{n} \sum_{r=0}^{n-k} \binom{n}{k} \binom{n+1}{r} $
but I'm not sure what to do next. I can... |
Given a multivariate function, how to derive the equations for its contour plot or the level sets? |
Why is the difference of consecutive primes from Fibonacci sequence divisible by $4$? |
The inequality $0 < x+y \leq 60$ and the equation $17x+29y=1222$. How do you find x and y other than bashing? |
From what I've seen, most of the proofs of convergence for gradient descent on convex functions assume that there exists at least one minimizer, i.e. for a convex $f: \mathbb{R} \rightarrow \mathbb{R}^d$ with $\inf_{x \in \mathbb{R}^d} f(x) > -\infty,$ there exists an $x_* \in \mathbb{R}^d$ such that $f(x_*) = \inf_{x ... |
**Exercise 3.F.29(a)**
Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V,W)$.
(a) Prove that if $\varphi \in W'$ and $\text{null} \ T' = \text{span} \ (\varphi)$, then $\text{range} \ T = \text{null} \ \varphi$.
----------
**Source.**
Linear Algebra Done Right, Sheldon Axler, 4th e... |
The inequality $0 < x+y \leq 60$ and the equation $17x+29y=1222$. How do you find x and y other than bashing?
Edit: I forgot to mention x and y are integers |
Determine the largest number $R$ such that the Laurent series of
$$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$
about $z=-2$ converges for $0<|z+2|<R$?
I know the maclaurin series for sine and cosine which are valid for all complex numbers.
For $\frac{1}{z^2-4} = -\frac{0.25}{z+2} + \frac{0.25}{... |
Suppose I have two monomial ideals $I$ and $J$ of $\mathbb{C}[\overline{x}]$. Can I claim that if the Hilbert series of $\mathbb{C}[\overline{x}] / I$ is equal to the Hilbert series of $\mathbb{C}[\overline{x}] / J$ then $I = J$?. I feel this is supposed to be a standard fact of monomial ideals / Groebner bases but I d... |
In PDE basic theory by the author Taylor I can't understand two facts:
Prop 1. If $u\in \mathcal{S}'(\mathbb{R}^n)$ is supported by $\left\{0\right\}$, then there exists $k$ and complex numbers $a_\alpha$ such that
\begin{align}
u=\sum_{|\alpha|\leq k} a_\alpha D^\alpha \delta
\end{align}
Prop 2. Suppose $u\i... |
This message has been encoded by a monoalphabetic function $f(p)=p+b~ \pmod{26}: APHUO~ EGEHP~ PEXOV~ FKEUH~ CKVUE~ CHKVE~ APHUO,$ where we digitize the alphabet by letting $A = 00, B = 01, . . . , Z = 25.$
I want to find the original message. For this purpose, I tried to identify $b$ by finding the most occurring ... |
Consider $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$ and $T_f$ is the regular distribution associated to $f$.
- Assume that there exists $m>0$ such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right)$, prove then $T_f \in$ $\mathscr{S}^{\prime}\left(\mathbb{R}^n\right)$(all the linear functionals on Schwartz spa... |
I need to calculate $\int_\Gamma ze^{z^2} dz$ where $\Gamma = \sqrt2 t + (1-t)i$ where $0 \leq t \leq 1$.
I write $$\int_\Gamma ze^{z^2} dz = [\frac{1}{2}e^{z^2}]_i^\sqrt2 = \frac{1}{2}e^2 - \frac{1}{2}e^{i^2}$$
Is this correct? |
> Proposition: If $x,y\in\mathbb{N}$ then for any $\varepsilon>0,$ there
> are infinitely many pairs $(n,m)$ such that $$\frac{\left\lvert
y^m-x^n \right\rvert}{y^m} < \varepsilon,$$
>
> i.e. $\displaystyle\large{\frac{x^n}{y^m}} \to 1\ $ as these pairs
> $(n,m) \to \infty.$
I think this is true, and I want t... |
I am currently studying the LSV map, a variant of the class of Pomeau-Manneville maps, which has the form:
$$T(x) =\begin{cases}
x+2^\alpha x^{1+\alpha} & 0 \leq x < \frac{1}{2} \\
2x-1 & \frac{1}{2} \leq x < 1 \\
\end{cases}
$$
In the original [paper][1] in the first paragraph on the second page, the... |
I am currently studying the LSV map, a variant of the class of Pomeau-Manneville maps, which has the form:
$$T(x) =\begin{cases}
x+2^\alpha x^{1+\alpha} & 0 \leq x < \frac{1}{2} \\
2x-1 & \frac{1}{2} \leq x < 1 \\
\end{cases}
$$
In the original [paper][1] proposing the LSV map in the first paragraph o... |
> Proposition: If $x,y\in\mathbb{N}$ then for any $\varepsilon>0,$ there
> are infinitely many pairs $(n,m)$ such that $$\frac{\left\lvert
y^m-x^n \right\rvert}{y^m} < \varepsilon,$$
>
> i.e. $\displaystyle\large{\frac{x^n}{y^m}} \to 1\ $ as these pairs
> $(n,m) \to \infty.$
I think this is true, and I want t... |
If $x,y\in\mathbb{N},\varepsilon>0$ then are there infinitely many integer pairs $(n,m)$ s.t. $\vert\frac{x^n}{y^m}- 1\vert < \varepsilon?$ |
The paper uses power $j$ instead of $2$ in $(y^2-1)$.
But with this definition, doesn't it just hold that $P_{i,j}(x,y) = P_i(x) P_j(y)$?
If I'm not missing something and it's true, you can just do
$$
\sum\limits_{i=0}^\infty \sum\limits_{j=0}^\infty P_{i,j}(x,y) a^i b^j = \left(\sum\limits_{i=0}^\infty P_i(x) ... |
For every positive integer $m$ let $f(m)=n$ be the $m$-th highly composite number. A positive integer is called highly-composite if it has more divisors then every smaller positive integer.
Suppose , some positive integer $k$ is given.
> Can we efficiently determine
> - whether there is some positive integer $... |
Can we detect a highly composite number and determine its number in the list? |
I am currently studying the LSV map, a variant of the class of Pomeau-Manneville maps, which has the form:
$$T(x) =\begin{cases}
x+2^\alpha x^{1+\alpha} & 0 \leq x < \frac{1}{2} \\
2x-1 & \frac{1}{2} \leq x < 1 \\
\end{cases}
$$
In the original [paper][1] proposing the LSV map in the first paragraph o... |
I was reading J. Renault's paper "The Fourier Algebra of a Measured Groupoid" and I am confused about his approach to the predual of a von Neumann Algebra.
Let $M:= VN(\mathcal{G})$ be the von Neumann algebra of a measured groupoid; I think the definition is irrelevant, we just need to know that $M$ is a von Neumann... |
I am starting a course in Manifolds, and would like to see how a solution to this typical problem might appear.
Q: Consider $\mathbb{R}^2$ (with complete standard atlas containing the chart $(\mathbb{R}^2, id_{\mathbb{R}^2})$ - i.e. standard differentiable structure). Find all points $p\in{\mathbb{R}^2}$ in a neigh... |
Given that we are allowed $N>0$ turns total (in our case $N=100$), we want to choose "place" for $(N-n)$ turns, followed by choosing "take" for the remaining $n$ turns. (We should prove that, whenever we choose "take" on $n$ turns, it is best if those $n$ turns are at the end. One idea might be: given any arrangement ... |
**Setup**
Let there be a board looking like a rectangular table. A piece is placed at any square of the board. Two players play a game. They move the piece in turns. The piece can only be moved to an adjacent square (no diagonal moves). The piece can’t be moved to a square that it has already visited (the starting s... |
The Batista-Costa surface is a triply periodic minimal surface. Three photos of part of the same surface are below:
[![This is the Batista-Costa surface with many units][1]][1]
[![This is the unit surface of the Batista-Costa surface.][2]][2]
[![This highlights the unit patches of the Batista-Costa surface.][3]][3... |
Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$. Let $k \in K$ be an extreme point.
**Question 1:** Does there exist a supporting hyperplane of $X$ containing $k$?
I _think_ the answer is “yes” via some Hahn-Banach argument, although I’m a little confused about this at the moment. But... |
Is every extreme point in a compact convex set contained in a defining supporting hyperplane? |
It's called ... the domain.
*Every* element of the domain is mapped and you *always* use every single element. You are not allowed to *ever* have an element unused.
Worth noting if you have a function such as $\frac 1{x-5}$ the domain is *NOT* $\mathbb R$. The domain is $(-\infty, 5)\cup (5, \infty)$ (which ... |
Is there a name for the following partial differential equation?
$$(\nabla V) \cdot (\nabla V) = c$$
Is it known what type of scalar fields $V$ satisfy this? In two dimensions wolfram claims the solution is just $f(x,y) = c_1 x + c_x y.$ Can this be shown for arbitrairly higher dimensions?
Intuitively, for $A= \n... |
I am interested in counting the number of times a given part (some integer) occurs in the weak compositions (i.e. 0 is admitted as a summand) of some restricted size and some restricted maximum summand.
E.G., in the compositions of 10 of length 4 with a maximum of 5 allowed for summands, how many times does some in... |
A disjoint union of open balls is of course disconnected. [Here](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/spaces-which-cannot-be-written-as-a-countable-disjoint-union-of-closed-subsets/D24FAD263DF49239740F1A548826CA1B) it is proved that a locally compact, connected, Hausdorff space... |
In a town N, every person is either a truth-teller, who always tells the truth, or a liar, who always lies. Every person in town N took part in a survey. "Is winter your favorite season?" was answered "yes" by 40% of respondents. A similar question about spring had 30% of affirmative answers, about summer had 50%, and ... |
Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$. Let $k \in K$ be an extreme point.
**Question 1:** Does there exist a supporting hyperplane of $X$ containing $k$?
I _think_ the answer is “yes” via some Hahn-Banach argument, although I’m a little confused about this at the moment. But... |
**TL/DR: Skip to the end for examples!**
Given that we are allowed $N>0$ turns total (in our case $N=100$), we want to choose "place" for $(N-n)$ turns, followed by choosing "take" for the remaining $n$ turns. (We should prove that, whenever we choose "take" on $n$ turns, it is best if those $n$ turns are at the en... |
In the context of proving the [Hoeffding Lemma ][1] I came across a slightly weaker statement in the form of an exercise:
"If $X$ is a real valued random variable and $|X| \leq 1$ a.s. then there exists a random variable $Y$ with values in $\{ -1, +1 \}$ such that
\begin{equation}
E[Y|X] = X \qquad a.s.
\end{e... |
This came up on my homework and I don't understand how to calculate $|u+2w|$. How do I get from $|u+w|$ to $|u+2w|$? I'm guessing that I have to square $|u+w|$ and then add $|u|$ and $|w|$ in a way that which would have a square root of $|u+2w|$ but I don't know how to get there. |
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