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Let $B$ be a real valued function on $[0, 1]$ such that $\|B'\|_{L_2}\le 1$ and $B(0) = 0$. Denote
\begin{gather}\label{small var condition}
\varepsilon = \int_0^1 B^2(s)\, ds - \left(\int_0^1 B(s)\, ds\right)^2.
\end{gather}
**Question.** How large can the following quantities be in terms of $\varepsilon$:
$... |
I'm learning about planes in linear algebra, and my professor said that to define a plane two lines must intersect. However, I was thinking, what if the two lines are parallel? In that case, they would never intersect.
Can't two parallel lines still define a plane, you can think of the plane as a piece of paper and ... |
Let $B$ be a real valued function on $[0, 1]$ such that $\|B'\|_{L_2}\le 1$ and $B(0) = 0$. Denote
\begin{gather}\label{small var condition}
\varepsilon = \int_0^1 B^2(s)\, ds - \left(\int_0^1 B(s)\, ds\right)^2.
\end{gather}
**Question.** How large can the following quantities be in terms of $\varepsilon$:
$... |
If $x, y, z $ are independent random variables, such as all following a uniform distribution between $0$ and $1$, are $x-y, y-z, ,x-z$ also independent of each other? |
If $x, y, z$ are independent random variables, are $ x-y, y-z, x-z$ also independent of each other? |
I am working on an exercise from a book, that asks to show that a Rationalising factor of $x^{1/3}+y^{1/3}+z^{1/3}$ is $$(x^{2/3}+y^{2/3}+z^{2/3}-y^{1/3}z^{1/3}-z^{1/3}x^{1/3}-x^{1/3}y^{1/3})\\ \times (x^{2/3}+y^{2/3}+z^{2/3}+2y^{1/3}z^{1/3}-z^{1/3}x^{1/3}-x^{1/3}y^{1/3}) \\ \times(x^{2/3}+y^{2/3}+z^{2/3}-y^{1/3}z^{1/3... |
We have $C = 2^\mathbb{N}$ and we identify the Cantor set with the image of $C$ via:
$
t\colon C \rightarrow [0,1], \left(x_n\right)_n \mapsto \sum_{n=1}^{\infty}\frac{2x_n}{3^n}
$\
Now if $x,y\in t(C)$ and $\lvert x - y \rvert < 3^{-N}$ for some $N\in \mathbb{N}$ then the first $N$ digits of $x$ and $y$ coincide... |
One can show that the expression
$$ \frac{e^{-i\omega s} - 1}{e^{-i\omega} - 1} $$
approaches $s$ as $\omega\rightarrow 0$ by L'Hopital's rule.
However, it seems there is no convergence when considering $\omega\rightarrow 2\pi n$ and $n$ is a nonzero integer (unless we plug special values of $s$).
My question ... |
Good morning to everybody. I'm involved in the study of reflective categories (from Borceux' book) and I have a silly doubt.
Recall that if $F: \mathcal{A} \to \mathcal{B}$ is a functor and $B$ is an object in $\mathcal{B}$, a reflection of $B$ along $F$ is a pair $(R_B,\eta_B)$ where $R_B$ is an object of $\mathcal... |
Let $X=(X_1,\ldots,X_n)^T$ be a random vector with mean $m$ and covariance $C$. Assume $C$ is positive definite, so it has inverse $C^{-1}$. Let $C=A A^T$.
I saw there are mainly two approaches to define a Gaussian random vector. One uses $X=AY+m$ (where $Y=(Y_1,\ldots,Y_n)^T$ is a random vector with $Y_i$'s indepen... |
(1)Let's start from definition of regular function - $f: \mathbb{R^n}\ni U \rightarrow \mathbb{R^m} $, where $U$ is an open subset in $\mathbb{R^n}, n \geq m$ and $f$ is function of class $C^1$; we call $f$ a regular function if $\forall a \in U \: rank f'(a)=m$, where $f'(a)$ is a linear mapping.
(2)Definition of o... |
Good morning to everybody. I'm involved in the study of reflective categories (from Borceux' book) and I have a silly doubt.
Recall that if $F: \mathcal{A} \to \mathcal{B}$ is a functor and $B$ is an object in $\mathcal{B}$, a reflection of $B$ along $F$ is a pair $(R_B,\eta_B)$ where $R_B$ is an object of $\mathcal... |
The question comes from Liu's book. Let $f_1 : X_1 \mapsto Y , f_2 : X_2 \mapsto Y$ be morphisms of locally Noetherian
schemes of finite type. Let us suppose that for every $y \in Y$ , there exists
i = 1 or 2 such that $f_i$ is unramified (resp. étale; resp. smooth) at all
points of $f_{i}^{−1} (y)$.
**I want to sho... |
If you want a specific matrix $g$, you can take
$$
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.
$$
The unique fixed-point... |
Compactness and connectedness are preserved by homeomorphism, in the sense that if two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are homeomorphic and $(X,d_X)$ is compact then it follows that $(Y,d_Y)$ is also compact (and similarly for connectedness). However, if $(X,d_X)$ is complete it does not necessarily follow that $... |
What properties of metric spaces are not preserved by uniformly continuous isomorphism? |
We want to know if a smooth surface under a $C^{\infty}$ diffeomorphism (infinitely differentiable with a infinitely differentiable inverse) map to a smooth surface too?
Further, it would be nice to understand what happens if the diffeomorphism was $C^k$ for some $k\geq 1$.
So far I have looked at these books:
... |
> Consider the nonlinear wave problem $u_t + (x+u)u_x = 0$; $u(x,0) = x$
>
> a) Write down Monge's equations in the for $du/d\tau$...,etc, and solve them.
>
> b) Plot the characteristics in the tx plane
So, I honestly don't know how to do this. I know the process when we have an equation of the form $u_t + c(u)u... |
Recall the following version of the Riemann-Lebesgue theorem:
`Theorem:` Let $1 \leq p \leq \infty$, $\Omega = \Pi_1^n(a_i,b_i)$ and $u \in L^p(\Omega)$. Let $u$ be extended by periodicity from $\Omega$ to $\mathbb{R}^n$ and define $$u_\nu(x) = u(\nu x), \ \ \text{ and } \ \ \bar{u} = \frac{1}{meas \Omega}\int_\Omeg... |
What's failing on the assumptions? |
I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.
I need some help with question 39 in Section 1.5 problems.
The expression is this:
$$\frac{\epsilon + \delta}{\sqrt{\epsilon^2 + \delta^2}}$$
It also gives the following hint:
> (Hint: Assume $... |
$\epsilon$, $\delta$ are positive infinitesimal, is the following expression infinitesimal, finite, or infite? |
**Background**
The Euler-Maclaurin (E-M) <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">formula</a> is a formula for the difference between the sum and the integral of a real or complex continuous function on the interval $[m,n]$. It expresses this difference in terms of the higher derivatie... |
Is there an Euler-Maclaurin-like formula for products? |
When I was trying to prove the following theorem:
> Theorem. Let $E$ be an extension field of the field $F$ and let $a \in E$
> .If $a$ is algebraic over $F$, then $F(a) \cong F[x]/ \langle p(x)\rangle$
> , where $p(x)$ is a polynomial in $F[x]$ of minimum degree
> such that $p(a)=0$. Moreover, $p(x)$ is irreduci... |
Recall the following version of the Riemann-Lebesgue theorem:
`Theorem:` Let $1 \leq p \leq \infty$, $\Omega = \Pi_1^n(a_i,b_i)$ and $u \in L^p(\Omega)$. Let $u$ be extended by periodicity from $\Omega$ to $\mathbb{R}^n$ and define $$u_\nu(x) = u(\nu x), \ \ \text{ and } \ \ \bar{u} = \frac{1}{meas \Omega}\int_\Omeg... |
If I consider $z_0 \in \mathbb{C}$ and denote $g(z)=\bar{z}$, then I can verify that:
- For $\lambda \in \mathbb{R^*}$:
$$lim_{\lambda \to 0} \frac{g(z_0+\lambda)-g(z_0)}{\lambda}=\frac{\lambda}{\lambda}=1$$
- In other hand, if we consider $w \in i\mathbb{R^*}$:
$$lim_{\lambda \to 0} \frac{g(z_0+w)-g(z_0)... |
Let $a$ be in a point in $\mathbb{R}^m$.
$f$ is a partial function from $\mathbb{R}^m$ to $\mathbb{R}^n$, where $m\geq n$, that is differentiable in a neighbourhood $V$ of $a$ and continuously differentiable at $a$.
**I'm trying to prove that if $J_{f_a}$ has full rank then $f(V)$ is a neighbourhood of $f(a)$.... |
(1)Let's start from definition of regular function - $f: \mathbb{R^n}\ni U \rightarrow \mathbb{R^m} $, where $U$ is an open subset in $\mathbb{R^n}, n \geq m$ and $f$ is function of class $C^1$; we call $f$ a regular function if $\forall a \in U \: rank f'(a)=m$, where $f'(a)$ is a linear mapping.
(2)Definition of o... |
Why does a summation vanish for a constant $f(x)$ in the Deutsch–Jozsa algirthm? |
I would like to know if there a functor from Simplicial Sets $\rightarrow$ Simplicial Sets that resemble the plus construction in Top?
More precisely, does there exist a functor, (abuse of notation) $-^{+}$: Simplicial sets $\rightarrow$ Simplicial sets, such that
$(BG)^{+} \cong |(B_{n} G)^{+}|$ for any group $G... |
Plus construction on Simplicial Sets? |
I would like to know if there a functor from Simplicial Sets $\rightarrow$ Simplicial Sets that resemble the plus construction in Top?
More precisely, does there exist a functor, (abuse of notation) $-^{+}$: Simplicial sets $\rightarrow$ Simplicial sets, such that
$(BG)^{+} \cong |(B_{n} G)^{+}|$ for any group $G... |
I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the Poincare upper half plane. So, with the parametrization
$$\sigma(t): x=r\cos(t), y=r\sin(t),\; \alpha\leq t\leq \beta$$
the distance is just
$$\mathrm{dist}((x_{1},y_{1}),(x_{2},y_{2}))=\int_{\alpha}^{\be... |
> [Big Question] Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider $p:\mathbb{P}_X(\mathscr{E})\to X$ be a projective bundle in the sense of Grothendieck.
Is there some useful method to find the positivity of $T_{\mathbb{P}(\mathscr{E})}$ or $\Omega^1_{\mathbb{P}(\mathscr{E})}$?
In this case we ha... |
**Background**
The Euler-Maclaurin (E-M) <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">formula</a> is a formula for the difference between the sum and the integral of a real or complex continuous function on the interval $[m,n]$. It expresses this difference in terms of the higher derivatie... |
I did check for O(n^3) by $$ T(k) <= ck^3-k^2 $$for all k<n. It is O(n^3) however checking for O(n^2), can I pick T(k) as $$ T(k) \leq ck^2 - k^{1.5} $$
or something like that. Then we can show that is O(n^2). It is also Ω(n) by T(k) <= ck for all k < n. How do I further improve this using substituion method? |
I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.
I'm doing an even numbered one and there are only answers for odd numbered problems. Q. 42, section 1.5 problems.
> Let $x$, $y$ be positive hyperreal numbers. Can $\frac{x}{y}+\frac{y}{x}$ be infinit... |
Let $x$, $y$ be positive hyperreal numbers. Can $\frac{x}{y}+\frac{y}{x}$ be infinite? finite? infinitesimal? |
I was reading Fischer's complex analytic geometry book. In page 64, we proved the proper mapping theorem. Which says
> Let $\varphi: X \rightarrow Y$ be a proper holomorphic map and denote by $\mathcal{K} \subset \mathcal{O}_Y$ the coherent sheaf of ideals which is the kernel of the canonical homomorphism
$$
\til... |
I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.
I need some help with question 39 in Section 1.5 problems.
$\epsilon$, $\delta$ are positive infinitesimal, is the following expression infinitesimal, finite, or infite?
The expression is this:
... |
Suppose $\mathcal{Z} = \{z_1, \dots, z_n\}$ is the set of points in $d$-dimensional Euclidean space. The aim is to partition the dataset into $(K\leq n)$ distinct clusters $R_1,\dots, R_K$ where $R_i\cap R_j = \emptyset$, for $i\neq j$, and $\cup_{j=1}^{K} R_j = \mathcal{Z}$ such that some empirical loss function is mi... |
> [Big Question] Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider $p:\mathbb{P}_X(\mathscr{E})\to X$ be a projective bundle in the sense of Grothendieck.
Is there some useful method to find the positivity of $T_{\mathbb{P}(\mathscr{E})}$ or $\Omega^1_{\mathbb{P}(\mathscr{E})}$?
In this case we ha... |
I did check for $O(n^3)$ by $$ T(k) \leq ck^3-k^2 $$for all k<n. It is O(n^3) however checking for O(n^2), can I pick T(k) as $$ T(k) \leq ck^2 - k^{1.5} $$
or something like that. Then we can show that is $O(n^2)$. It is also $Ω(n)$ by $T(k) \le ck$ for all $k < n$.
How do I further improve this using substituion... |
$T:R^2\rightarrow R^3$
$T(x)=
\begin{bmatrix}1&0\\0&1\\0 &0\end{bmatrix}x, \forall x\in R^2$
I know that a linear map is invertible if and only if the map is bijective. By looking at the matrix which is injective the map is not invertible. But I thought could it be an invertible map if the codomain $R^3$ is chan... |
Is this an invertible linear map? |
There are two issues with your proof. Firstly, apriori to $\lim\sup$ and $\lim\inf$ of a real sequence existing and being equal, the term "$\lim$" does not make sense. So you should say $\{x:\lim\sup I_{F_{n}}\neq 0\}$. Also note that since this is a sequence of $0$s and $1$'s, the limit exists and is $0$ if and only i... |
I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.
I need some help with question 39 in Section 1.5 problems.
$\epsilon$, $\delta$ are positive infinitesimal, is the following expression infinitesimal, finite, or infite?
The expression is this:
... |
Here is the idea for the set. Let $\alpha$ be a real number. Then $x_\alpha$ be a set of digits with the property that a digit is in the set if it appears infinitely many times in $\alpha$.
For many numbers $\alpha$ it is easy to determine $x_\alpha$. For some it is challenging like $\alpha = \pi$. However, what if ... |
Is this a finite set for which it is impossible to list its elements? |
I've found the following statement (but it is rather an example) in a book that states if $X$ is a local martingale and its quadratic variation has the form
$$\left[X\right]=t+cX$$
where $c$ is a constant, then $X$ is a compensated Poisson process. The example wanted to emphasize the similarity between this for... |
I don't understand a question and I need clarification as to what is being asked. The question is:
Assume that $\forall n, \frac{n}{n+1}$ is a partial limit of the sequence $\left\{{a_n}\right\}^\infty _{n=1}$, and that there is no partial limit bigger than 1, prove that there exists a subsequence $\left\{{a_{n_k}... |
How to compute $\displaystyle\int\dfrac{1}{x}\left(\displaystyle\sum_{n\geq 0}\frac{x^{2n}}{2^{2n}(n!)^2}\right)^{-2}\text dx$? |
I'm using the following definitions:
**Definition 1 (Linear Span)** Let $(V,+,\cdot)$ be a vector space over a field $\mathbb{K}$. The linear span of a subset $X\subseteq V$ is defined as
\begin{align*}
\mathrm{span} (X):=\left\{\sum\limits_{i=1}^{n} \lambda_{i}\mathbf{v}_{i}\; \Bigg | \; n\in\mathbb{N},\;\lambda... |
The following is presented as an example to motivate my question. It's my paraphrase of the principle of recursion theorem and proof. From [Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra][1]
>*Let $c$ be a number and let $F$ be a function of two arguments defined
in $\ma... |
Is this proof of the principle of recursion theorem proving anything? |
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... |
While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia.
In the page [Poincaré Half Plane Model][1] it is explicitly stated that the distance of $z,w \in \mathbb{H}$ is:
$$d_{... |
How to compute $\int\frac1x\Bigl(\sum\limits_{n\ge0}\frac{x^{2n}}{2^{2n}(n!)^2}\Bigr)^{-2}\mathrm dx$? |
A while ago I was interested in categorical image of composition of morphisms in abelian categories, see this [post][1]. I learnt that $\text{Im}(gf)=\text{Im}(g i)$ where $i$ is the canonical monomorphism gotten by factorizing $f$ in the abelian category. Recently, I had to deal with the same thing but with kernels.
... |
I am not a mathematics major, and I encountered this problem while self-studying group theory (because it is an important tool in my research field). Here is the problem:
Let $A$ be a subgroup of a group $G$ with finite index. Prove that there exist elements $g_{1}$, $g_{2}$, $\cdots$ in $G$ that can serve as repres... |
This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of sequences by taking general theorems about nets and replacing the term "net" with the term "sequence". For example: A set ... |
In my PhD thesis, I'm interpreting some of my code in Category Theory. I ended up modeling some stuff as a slice category $\mathbf{Set}/C$, and ended up with the question whether I could define a functor $F$ from $\mathbf{Set}$ to $\mathbf{Set}/C$. To my surprise, it seems that this would not be possible... Am I corre... |
**Background**
For a certain project, I collected a considerable number of definite integrals whose evaluations amounted to numbers of the form $P :=a+b \pi^{2} $, where $a, b \in \mathbb{Q}$. I won't list them all here (the document comprises 34 pages, including references), but state the ones that stood out to me ... |
In Bredons Topology book, in the chapter 3. The Cohomologiy cross product, Theorem 3.2 he claims that the map:
$$ \gamma: Hom(H_{\star}(X),\Delta)\otimes Hom(H_{\star}(Y),\Delta) \to Hom(H_{\star}(X) \otimes H_{\star}(Y),\Delta)$$
given by (I suppose?) $\gamma(f, g) = f \otimes g$ (is this right?) is an isomorphism ... |
I understand that structures are means by which we know how to interpret certain languages. So for example, in the language of number theory, we might have $\displaystyle L=\left\langle 0,S,+,\cdot,E,< \right\rangle$. As I understand it, without specifying our strucutre, we do not know how to inerpret the symbols of ou... |
What exactly is a structure of a language? |
$(Z_n)_{n \in \mathbb{N}}$ are random variables such that $\mathbb{E} \left( \sum_{i=1}^\infty |Z_i| \right) < \infty$. Show that $\mathbb{E} \left( \sum_{i=1}^\infty Z_i \right) = \sum_{i=1}^\infty \mathbb{E}(Z_i)$.
It seems like the result should follow from linearity of expectation, but I assume the key is checki... |
I am given this piecewise function $f: \mathbb{R} \rightarrow \mathbb{R}$,
$f(x)= \left\{\begin{array}{ll}x^2 & x>0 \\ 0 & x ≤ 0 \\ \end{array} \right. $
I have to determine if the function is differentiable over $\mathbb{R}$ or not. The way I think about differentiability is like this:
"Can a function be ... |
How to determine if function is differentiable? |
I have the $D=$ set {$|z-2+5i|<7,\text{Im}(z)>0$}. I sketched it in Desmos, the overlapping region represents the set. I need to check the properties below of this set.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/9U2o0.png
Boundary points - Just the union of these sets: {$|z-2+5i... |
Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that resembles the plus construction in $\mathsf{Top}$?
More precisely, if we write $BG$ for the classifying topological space o... |
Let $ X $ be a locally compact Hausdorff space, and $ C_c(X) $ be the space of all complex-valued continuous functions with compact support on $ X $.
As far as I know, there are two formulations of Riesz–Markov–Kakutani representation theorem.
**Formulation 1 (e.g., D. L. Cohn, *Measure Theory*, 2nd edition, Theo... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
a_{0} = b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{x}{1 - x} - A(\frac{... |
First, we must check the continuity, because continuity is a necessary condition for differentiability. The only point we must check is the junction $x=0$
$$\lim_{x\to 0^+}f(x)=\lim_{x\to 0^+} x^2=0$$
$$\lim_{x\to 0^-}f(x)=\lim_{x\to 0^-} 0=0$$
so $f$ is continuous over $\mathbb{R}.$ Now we want to study if it's dif... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
a_{0} = b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{x}{1 - x} - A(\frac{... |
Enderton, Elements of Set Theory, page 189 just above Theorem 7J mentions that if $<$ is an ordering on $A$ and $C$ is a proper subset of $A$ then $<$ is technically not an ordering on $C$ as $<$ is not on $C$. Similarly, the way that partially ordered sets (or any other types of ordered sets) are defined, if $\langle ... |
I have a question regarding the measurability of a function in the following paper:
https://statistics.berkeley.edu/sites/default/files/tech-reports/638.pdf
Let $\phi$ be a surrogate loss function and $(X,Y)$ a joint random variable where $X\in \mathcal{X}$ and $Y\in\{-1,+1\}$. Define the risk for a measurable f... |
I am working on understanding the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. As part of their definition of equations for a 3D surface, they use a function $F(ξ, \bar{ξ})$ where $ξ$ is "the local complex coordinate on the unit 2-sphere in $^3$ obtained by stere... |
How skew symmetric matrix can be used to express rotations about a given axis?
I came across this concept while dealing with rotation matrices used in robotics.
Can someone detail on this concept? and explain how a vector is converted to a skew symmetric matrix?
Also I would like to go in depth in this. |
Quoting from Rogers and Williams, p. 116, Theorem 21.2 (A necessary and sufficient condition for $\mathscr{L}^1$ convergence):
Let $(X_n)$ be a sequence in $\mathscr{L}^1$,
and let $X\in\mathscr{L}^1$.
Then $X_n\to X$ in $\mathscr{L}^1$,
or, equivalently $\mathbb{E}(|X_n - X|)\to 0$,
if and only if the following... |
Convergence in $\mathscr{L}^1$ implies uniform integrability? |
There is a challenging question assigned regarding Counting. here is the question:
>we call every path from point A to B a **good** path, if in every step it has moved up, down, or right and not crossed the same **horizontal line** more than once. we write down the number of squares beneath every possible *good* pat... |
I am not a mathematics major, and I encountered this problem while self-studying group theory (because it is an important tool in my research field). Here is the problem:
>Let $A$ be a subgroup of a group $G$ with finite index. Prove that there exist elements $g_{1}$, $g_{2}$, $\cdots$ in $G$ that can serve as repre... |
Quoting from Rogers and Williams, p. 116, Theorem 21.2 (A necessary and sufficient condition for $\mathscr{L}^1$ convergence):
Let $(X_n)$ be a sequence in $\mathscr{L}^1$,
and let $X\in\mathscr{L}^1$.
Then $X_n\to X$ in $\mathscr{L}^1$,
or, equivalently $\mathbb{E}(|X_n - X|)\to 0$,
if and only if the following... |
Eight boys and seven girls went to the movies and sat in the same row of 15 seats. Assuming
that all the 15! permutations of their seating arrangements are equally likely, compute the
expected number of pairs of neighbors of different genders.
I believe the answer to the above question is 7.67. Could someone confi... |
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 understand that structures are means by which we know how to interpret certain languages. So for example, in the language of number theory, we might have $\displaystyle L=\left\langle 0,S,+,\cdot,E,< \right\rangle$. As I understand it, without specifying our strucutre, we do not know how to inerpret the symbols of ou... |
I have the $D=$ set {$|z-2+5i|<7,\text{Im}(z)>0$}. I sketched it in Desmos, the overlapping region represents the set. I need to check the properties below of this set.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/9U2o0.png
Boundary points - Just the union of these sets:
$$\unde... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given with a definition that is equivalent to solvable groups.
>Equivalent definition: $G$ is obtained from the group $\{1\}$ by a finite number of extensions with abelian kernels.
In the same section, the following claim is g... |
Both the $5x+1$ and $7x+1$ variant of the Collatz sequence are conjectured to have large number of divergent trajectory. Here, i combined the two. As always, when you encounter even $x$, you apply $x\rightarrow x/2$, but if you encounter odd $x$, you have the option of applying either $x \rightarrow 5x+1$ or $x \righta... |
Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that resembles the plus construction in $\mathsf{Top}$?
More precisely, let $G$ be a group and $|BG|$ the classifying space of $... |
There is a challenging question assigned regarding Counting. here is the question:
>Consider a square lattice $(\mathbb{Z}_{\ge 0}^2)$ with point $A$ on its bottom left (the origin) and point $B$ on its top right. We call a path from $A$ to $B$ a **good** path if it does not ever move left and it does not visit any ... |
Let $R$ an equivalence relation over a set $A$. For $a \in A$ we define $a / R = \{ b \in A: (a, b) \in R\}$ and call this set the equivalence class of $A$. Furthermore, we define $A / R = \{a / R : a \in A\}$ the set of all equivalence classes for the equivalence relation $R$ over $A$.
As a last definition, if $f : ... |
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given with a definition that is equivalent to solvable groups.
>Equivalent definition: $G$ is obtained from the group $\{1\}$ by a finite number of extensions with abelian kernels.
In the same section, the following claim is g... |
This is from my uni's algebra textbook.
Given vector $u=(1_5,3_5,4_5)$ and basis $B=\left \{ (2_5,3_5,4_5), (1_5,2_5,3_5), (3_5,3_5, 2_5) \right \}$, find the coordinates of the vector $u$ in the basis $B$.
I get that from the definition, the coordinates of $u$ are $(\alpha,\beta, \gamma )$ where $\alpha,\beta, \... |
Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that resembles the plus construction in $\mathsf{Top}$?
More precisely, let $G$ be a group and $|BG|$ the classifying space of $... |
$\DeclareMathOperator{\R}{\mathbb R} T:\R^2\rightarrow \R^3$
$T(x)=
\begin{bmatrix}1&0\\0&1\\0 &0\end{bmatrix}x, \forall x\in \R^2$
I know that a linear map is invertible if and only if the map is bijective. By looking at the matrix which is injective the map is not invertible. But I thought could it be an inver... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
a_{0} = 1, b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{1}{1 - x} - A(\fr... |
I tried something like that:
For $R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}$ when tried to take the limit for $c=0$ I get;
$lim_{n\rightarrow\infty}\frac{(2n+1)(x^{2n})(x^{n+1})}{(2n+1!)}$ (Should i look it in two cases when z=c and z=x ? since for $x>c$ wlog $z\in[c,x]$ ) After that the limit gets the form;
$l... |
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