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I started reading spivak, just today, and on page 6 of the prologue. He states that it is not possible to prove that $1\ne0$ for the real numbers, with the basic properties of the ring.
But I got the following argument:
Assuming that $1=0$, it follows that $a(1)=a(0)$ for all $a \in{R}$, where $a(1)=a$ and $a(0)=... |
Is this proof of $1\ne0$ in real numbers (set/field) valid? |
Let $G=\langle a,b\rangle$ be a finite $p$-group such that $a^p=b^p=1$. Is there any result about the size of the set of $p$-elements $\Omega(G):=\{g\in G\mid g^p=1\}$? In particular, I'm interested in the fraction $|\Omega(G)|/|G|$. |
Let $(M,d_{M})$ and $(N,d_{N})$ be metric spaces and let $f:M\longrightarrow N$ be a function;
**Metric space definition of continuity:** $f$ is continuous at a point $x\in M$ if for every $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that for all $y\in M$
\begin{align*}
d_{M}(x,y)<\delta(\epsilon)\Longr... |
Why is the shell method not
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}{n}\right) \cdot \left(\frac{1}{n}\right)^2.$$?
This means the shells are concentric circles going outward, the ... |
Hoping I can get some help in understanding the raw math behind this. I understand basic probabilities, but sometimes these things are either a bit more complicated than I can work out - or I tend to overthink it and miss the obvious O.o
I'm playing a game, in which we roll a certain # of dice (all 6 sided - stand... |
I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines".
The rules of a normal 9x9 sudoku puzzle are: Each row and column and 3x3 box contain the digits 1-9 once each.
The rules for entropic lines are: ... |
In general, tensor products don't commute with direct products. As such, I understand that the natural map $(\prod\mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Q} \to \prod (\mathbb{Z}\otimes_{\mathbb{Z}} \mathbb{Q}) \cong \prod \mathbb{Q}$ is not an isomorphism since the denominators of $\prod \mathbb{Q}$ are unbounded. Ho... |
Is $\mathbb{Z}^{\mathbb{N}} \otimes_\mathbb{Z} \mathbb{Q} \cong \mathbb{Q}^{\mathbb{N}}$, non-canonically? |
The Jackson Kernel is
$$
K_n(x) = C_n\frac{\sin^4(nx/2)}{\sin^4(x/2)}
$$
Where $C_n = \frac{3}{2\pi n(2n^2+1)}$ is chosen to make sure that $\int_{-\pi}^\pi K_n(x) dx =1.$
How could I evaluate this integral? I have not found a reference for this. |
Computing the normalization of Jackson Kernel? |
I have questions about the proof of Theorem 3.4.12 in Emily Riehl's *Category Theory in Context*.
The theorem states that the colimit of a small diagram $F\colon \mathsf J \to\mathsf C$ can be expressed as a coequalizer between two arrows $c,d\colon\coprod_{f\in\operatorname{mor} J}F(\operatorname{dom} f) \rightarrow ... |
I am reading an article in which they state the following fact:
If $a,b,x>0$, then $x\leq a\sqrt{x}+b$ implies that $x\leq a^2+b$.
I think the proof of this should be pretty trivial, but I cannot figure it out. I have tried squaring both sides of the first inequality and then using the quadratic formula, but it d... |
I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines".
The rules of a normal 9x9 sudoku puzzle are: Each row and column and 3x3 box contain the digits 1-9 once each.
The rules for entropic lines are: ... |
***Background:***
-----------------
*I recently got an idea. My idea was analyzing primes whose digit sum is prime.*
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
I will call a special prime like this $P$.
For primes in the range $1<p<100$, these are all $P$:
$$... |
Let $f,g:X\to \mathbb{R}$ be integrable function on measure space $(X,\mathcal{F}, \mu)$. Define $v=(f,g)$ and $\int vd\mu=(\int fd\mu, \int gd\mu)$. Define $2$-norm $\|(f,g)\|_2=\sqrt{f^2+g^2}$. Show that
$$
\|\int vd\mu\|_2\le\int\|v\|_2d\mu
$$
---
My idea is to define a orthogonal transformation that $$F(v)... |
I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines".
The rules of a normal 9x9 sudoku puzzle are: Each row and column and 3x3 box contain the digits 1-9 once each.
The rules for entropic lines are: ... |
Is there a Sudoku solution with a entropic line covering all cells? |
Ok so I came upon this question, and I have absolutely no idea how do we proceed.
I had done a similar kind of question which was :-
>Find the number of ways to express 12 as a product of 2 factors
In this case it was much simpler, in which I was only required to find out the total number of factors and then div... |
***Background:***
-----------------
*I recently got an idea. My idea was analyzing primes whose digit sum is prime.*
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
I will call a special prime like this $P$.
For primes in the range $1<p<100$, these are all $P$:
$$... |
Given a function $f: A \rightarrow B$ images of subsets of $A$ preserve only inclusion and union of sets whereas pre-images of subsets of $B$ are better behaved, so to speak, and preserve (in addition to the above) intersections and differences of sets as well.
I have both the intuition and the proofs for the this sta... |
Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, then it is standard that $R$ is also a normal domain. This leads me to ask the following question:
Let $X, Y$ be Noe... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
Let $H$ be a Hilbert space. Define the full fock space $F(H)=\bigoplus\limits_{n=0}^\infty H^{\otimes n}$ where $H^{\otimes 0}=\Bbb{C}\Omega$, $\Omega$ is an element of $H$ of unit norm. Let us define the map $\phi:F(H)\otimes F(H)\to F(H)$ by
$$\phi((a\Omega\oplus x_1\oplus x_2\oplus\cdots)\otimes(b\Omega\oplus y_1\o... |
Is true that $F(H)\otimes F(H)\cong F(H)$? where $F(H)$ is the full fock space of Hilbert space $H$ |
The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, no rigorous method I have found that attempts to derive all the well-developed characteristics for the LTI systems in the $s$ domain such as absolute stability, relative stabilit... |
Let's take the following random variable
$$ L = \sum_{i=1}^IB_i $$
with
$$B_{i}=1_{\{X_{i}\leq c_{i}\}}~~\hbox{ where }~~ X_{i} := \sqrt{a_i}Y + \sqrt{1-a_i}\varepsilon_{i}$$
where $Y,\varepsilon_{1},...,\varepsilon_{I}$ are i.i.d. standard normal random variables; $c_i$ is a real number and $a_i \in [0,1)$. Addi... |
Given a function $f: A \rightarrow B$ images of subsets of $A$ preserve only inclusion and union of sets whereas pre-images of subsets of $B$ are better behaved, so to speak, and preserve (in addition to the above) intersections and differences of sets as well.
I have both the intuition and the proofs for the this sta... |
The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, no rigorous method I have found that attempts to derive all the well-developed characteristics for the LTI systems in the $s$ domain such as absolute stability, relative stabilit... |
Let's take the following random variable
$$ L = \sum_{i=1}^IB_i $$
with
$$B_{i}=1_{\{X_{i}\leq c_{i}\}}~~\hbox{ where }~~ X_{i} := \sqrt{a_i}Y + \sqrt{1-a_i}\varepsilon_{i}$$
where $\varepsilon_{1},...,\varepsilon_{I}$ are interdepedent standard normal random variables but independent from $Y$; $c_i$ is a real nu... |
There are 25 red marbles and 25 blue marbles divided between two buckets. You
1) select a bucket uniformly at random
2) select a marble from that bucket uniformly at random
Find initial arrangements of marbles into the buckets in order to maximize $\mathbb P(\text{marble picked is red})$.
[Source][1] (this ... |
On Wikipedia page of **continuum hypothesis (CH)**. It is given that, following is an equivalent statement of continuum hypothesis.
> any subset of the real numbers is finite, is countably infinite, or has the same cardinality as the real numbers
Now, **If** CH is False! Then, there must be some subset $S\subset... |
Is such a weird subset of irrationals exists? (Question related to CH) |
The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, no rigorous method I have found that attempts to derive all the well-developed characteristics for the LTI systems in the $s$ domain such as absolute stability, relative stabilit... |
I am trying to solve $$\sum_{j=0}^{n-1}j2^j$$ but I don't know how to proceed with the $j$ in front of $2^j$. What is making it difficult for me is it is an indexing variable, so I don't think I can factor it out. How would I proceed?
Thank you |
I am looking for a good survey of techniques of summation of series. Trying to search this, I can only find undergraduate course notes covering very elementary series (geometric, telescoping, etc.). It would be very nice to have a reference for seasoned mathematicians, that goes into more depth and discusses more techn... |
I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines".
The rules of a normal 9x9 sudoku puzzle are: Each row and column and 3x3 box contain the digits 1-9 once each.
The rules for entropic lines are: ... |
Let's take the following random variable
$$ L = \sum_{i=1}^IB_i $$
with
$$B_{i}=1_{\{X_{i}\leq c_{i}\}}~~\hbox{ where }~~ X_{i} := \sqrt{a_i}Y + \sqrt{1-a_i}\varepsilon_{i}$$
where $\varepsilon_{1},...,\varepsilon_{I}$ are interdepedent standard normal random variables but independent from $Y$; $c_i$ is a real nu... |
On Wikipedia page of **continuum hypothesis (CH)**. It is given that, following is an equivalent statement of continuum hypothesis.
> any subset of the real numbers is finite, is countably infinite, or has the same cardinality as the real numbers
Now, **If** CH is False! Then, there must be some subset $S\subset... |
My textbook has a question that I'm unsure of my answer on, specifically part (a) and (d):
>Suppose that a point $(X,Y)$ is chosen at random from
the circle $S$ defined as follows:
$$S=\{(x,y):x^2+y^2\le 1\}$$
>a) Determine the joint p.d.f. of $X$ and $Y$
**My Approach:** My guess is that we're basically looki... |
**Fact 1**. The function $1/|\xi|^{s}$ is integrable if and only if $s<n$.
**Fact 2**. If $f\in L^1(\mathbb{R}^n)$, then $\widehat{f}\in\mathcal{C}^{\infty}(\mathbb{R}^n)$ (bounded).
Let $H^{2}:=\left\{u\in L^2(\mathbb{R}^n): \mathcal{F}^{-1}\left((1+|\xi|^2)^2\widehat{f}(\xi)\right)\in L^2(\mathbb{R}^n)\right\... |
Let's take the following random variable
$$ L = \sum_{i=1}^IB_i $$
with
$$B_{i}=1_{\{X_{i}\leq c_{i}\}}~~\hbox{ where }~~ X_{i} := \sqrt{a_i}Y + \sqrt{1-a_i}\varepsilon_{i}$$
where $\varepsilon_{1},...,\varepsilon_{I}$ are interdepedent standard normal random variables but independent from $Y$; $c_i$ is a real nu... |
I am looking for a good survey of techniques of summation of series.
Trying to search for this, I can only find undergraduate course notes covering very elementary series (geometric, telescoping, etc.). It would be very nice to have a reference for seasoned mathematicians, that goes into more depth and discusses... |
I'm building a three legged sculpture stand. I would like to construct a hexagon in which:
- Every other side is of equal specific lengths (Lengths of sides: *a b a b a b*).
- Opposing sides are parallel.
I hope the above summarises in words what I want to do. It should look approximately like this:
[ is the operation $\mathcal{A}$ such that for any *Suslin scheme* $\{A_s : s \in \omega^{<\omega}\}$ of subsets of $X$, we have:
$$
\mathcal{A}\{A_s : s \in \omega^{<\omega}\} := \bigcup_{a \in \omega^\omega} ... |
I am stuck in understanding how I can deduce if the solution I found is a max or a min. The exercise reads:
$$\max \quad f(x, y) = 2x - y \qquad \text{subject to} \qquad \begin{cases} x^2+y = 2 \\ x+y \geq 0 \\ x \geq 0 \end{cases}$$
I wrote the problem in standard form, that is:
$$\max \quad f(x, y) = 2x - y... |
How do I know when a point is a max or a min (multivariable optimisation problem)? |
I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines".
The rules of a normal 9x9 sudoku puzzle are: Each row and column and 3x3 box contain the digits 1-9 once each.
The rules for entropic lines are: ... |
I have the following channel with $\mathcal X = \mathcal Y$:
$
p(y|x) =
\begin{bmatrix}
1/2 & 1/2 & 0 \\
0 & 1/2 & 1/2 \\
1/2 & 0 & 1/2\end{bmatrix}
$
Is it possible to decrease the capacity by adding a row to the channel matrix?
I think not, this is what I have so far,
I added the following last ro... |
Is it possible to decrease the channel capacity by adding a row to the coding channel matrix? |
Let $A \subset \mathsf{SL}(2; \mathbb{C})$ be a finite set of matrices. Consider the set $$S = \overline{\langle A \rangle},$$ where we take the closure in $\mathsf{SL}(2; \mathbb{C})$, and where $\langle A \rangle$ denotes the free group of $A$.
Clearly $S$ is a group, and by construction $S$ is closed in $\mathsf{... |
Im trying to understand the linked paper (cf [paper][1]), please correct me if my understanding is wrong !
Im looking to get an interpolation function from the result of a Dormand Prince (RK45) integration step, first introduced in the [paper][1]. In order to get the polynomial coefficients, it is said that only 3 ... |
I was wondering about this problem: say I have to take a test made of $31$ questions chosen among a database of $140$ questions total.
Those questions are open questions (that is, not multiple choice questions).
Say that I know all the database of questions, because it's public, but I have studied poorly and I on... |
I am stuck in understanding how I can deduce if the solution I found is a max or a min. The exercise reads:
$$\max \quad f(x, y) = 2x - y \qquad \text{subject to} \qquad \begin{cases} x^2+y = 2 \\ x+y \geq 0 \\ x \geq 0 \end{cases}$$
I wrote the problem in standard form, that is:
$$\max \quad f(x, y) = 2x - y... |
Consider the set of all N-dimensional copulas denoted by $\mathcal{C}^N$. Notice that each copula $C$ is a joint distribution on the state space $[0,1]^N$ with uniform marginal distributions. Now consider an arbitrary joint distribution P on $[0,1]^N$, whose marginals are not necessarily uniform.
Question: What is ... |
The triangular number $T(b)$ is $(b^2+b)/2$, so a number is twice a triangular number iff it has the form $b^2+b$.
The [OEIS entry](https://oeis.org/A230312) you link to provides the answer, in the form of a conjecture credited to Alonso del Arte in 2013: that a number $n^2$ is in A230312 iff $4n^2+1$ is prime.
L... |
>Assume $K/F$ is an inseparable extension. Is it necessary that $K$ contains an element $u \notin F$ that is purely inseparable over $F$?
I also posted in MO. |
I was wondering about this problem: say I have to take a test made of $31$ questions chosen among a database of $140$ questions total.
Those questions are open questions (that is, not multiple choice questions).
Say that I know all the database of questions, because it's public, but I have studied poorly and I on... |
> Is $\text{BRANCH}(n)$ finite for $n > 2$? Define $\text{BRANCH}(n)$ as
> the maximum length of a string that is composed of at most $n$ unique
> characters AND meets the following condition:
>
> Define a substring as all letters from positions $i$ to $2i$
> (inclusive) (where $i= 1, 2, 3...$). Then, a certain s... |
I stumbled upon a problem which seems easy but is actually hard to answer. It involves the sudoku game and a commonly used custom constraint rule called "entropic lines".
The rules of a normal 9x9 sudoku puzzle are: Each row and column and 3x3 box contain the digits 1-9 once each.
The rules for entropic lines are: ... |
I want to test the primality of some numbers.
To do this I have tried PARI/GP and from the command line:
isprime(1797985968535216664944496004358374343509215681442061471412191560717902567100922783048272325884204171542595289833292525444657649015706240748603258905587228070218364321463272748372578381297752753873677... |
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... |
Let $E$ be normed space and $I: E \to \mathbb{R}$ be an operator given by
$$I(u) = \int_{\Omega} \dfrac{|\nabla u|^{p(x)}}{p(x)} \, \mathrm{d}x,$$
where $p \in C^{0,1}(\overline{\Omega})$. Knowing that its Fréchet derivative $I'(u)$ in $u \in E$ is a linear and continuous application such that
$$\lim_{||v|| \to 0} \... |
Context: We were taught collinearity of 3 points (in vectors) and a method for checking 4 points for coplanarity simultaneously. Searching around I found a better method for checking for coplanarity in 4 points i.e. the 4x4 determinant.
In a 2d plane there is a 3x3 determinant to check for collinearity of 3 points ... |
Let $X$ be a Polish space. Recall that the [Suslin operation](https://en.wikipedia.org/wiki/Suslin_operation) is the operation $\mathcal{A}$ such that for any *Suslin scheme* $\{A_s : s \in \omega^{<\omega}\}$ of subsets of $X$, we have:
$$
\mathcal{A}\{A_s : s \in \omega^{<\omega}\} := \bigcup_{a \in \omega^\omega} ... |
First of all, the list of closed connected subgroups of
$SL(2, {\mathbb C})$ is not very long, see my answer [here](https://math.stackexchange.com/questions/646183/list-of-connected-lie-subgroups-of-mathrmsl2-mathbbc/650285#650285). From that, you conclude that a subgroup $\Gamma< SL(2, {\mathbb C})$ is either *virtu... |
This is the pairing algorithm I have written and it seems to work.
numlist =(1,2,3,4)
letter_combinations=("pa","pb")
i=0
for num1 in numlist:
for num2 in numlist:
if num1 != num2:
used_combinations = set()
for comb1 in letter_combina... |
is there a closed solution to counting pairwise combinations of the elements of two sets without resampling? |
There are two notions of homotopy limits, one for triangulated categories and one for model categories and I wonder whether these two coincide. More concretely, let $\mathcal{T}$ be a triangulated category having countable direct sums and $X_0 \to X_1 \to \cdots $ a sequence of morphisms in $\mathcal{T}$, we define the... |
Is it possible to make a pseudo-euclidean space $ \mathbb{R}^{p, q} $ into a differentiable manifold?
My intuition tells me that yes since in special relativity we deal with the flat Minkowski space $ \mathbb{R}^{1, 3} $ however i tried to approach the problem of proving that for any given pseudo-euclidean space we... |
>Give an example of extension $K/F$ that is neither separable nor purely inseparable.
Take $F = \mathbb{F}_{2}(x)$ and $K = F(\sqrt[6]{x})$. We can write $K = F(\sqrt{x},\sqrt[3]{x})$. Thus $(\sqrt{x})^{2^{1}} = x \in F$ so, $\sqrt{x}$ is purely inseparable over $F$. Also, $(\sqrt[3]{x})^{2^{n}} = x^{\frac{2^{n}}{3}... |
> (The Five Lemma). Let $\require{AMScd}$
\begin{CD}
A_1 @>{f_1}>> A_2 @>{f_2}>> A_3 @>{f_3}>> A_4 @>{f_4}>> A_5
\\
@V{\alpha_1}VV @V{\alpha_2}VV @V{\alpha_3}VV @V{\alpha_4}VV @V{\alpha_5}VV
\\
B_1 @>{g_1}>> B_2 @>{g_2}>> B_3 @>{g_3}>> B_4 @>{g_4}>> B_5
\end{CD}
>
> be a commutative diagram of R-modules and R-... |
Let $H$ be a Hilbert space. Define the full fock space $F(H)=\bigoplus\limits_{n=0}^\infty H^{\otimes n}$ where $H^{\otimes 0}=\Bbb{C}\Omega$, $\Omega$ is an element of $H$ of unit norm. Let us define the map $\phi:F(H)\otimes F(H)\to F(H)$ by
$$\phi((a\Omega\oplus x_1\oplus x_2\oplus\cdots)\otimes(b\Omega\oplus y_1\o... |
I am wondering if the following is true.
Let $G$ be a Lie group, $V$ a vector space, $\rho$ a representation of $G$ on V, and $\pi: E\rightarrow M$ a vector bundle with fibre $V$. Does there exist a unique (up to principal bundle isomorphism) principal $G$-bundle, such that $E\cong P\times_\rho V$, where $P\times_\r... |
Let $R$ be an arbitrary ring. I want to know how many ring homomorphisms $\phi: \mathbb{Z} \to R$ are? For instance, if $\phi(1) = 0$, that's the trivial morphism.
My intuition says, that if $\phi(1) \neq 0$, then $\phi$ is unique.
I already saw other answers but from my personal understanding it is still not cle... |
How many ring homomorphisms $\phi: \mathbb{Z} \to R$ are? |
Considering $k,x\in \mathbb{R}, k^x=1+\sum_{n=1^{\infty\frac{x^n(\ln(k))^n}{n!}$.
Does it have an elementary proof?
I attempt to solve by Maclaurin Series:
$f(x)=\sum_{n=1}^{\infty}\frac{x^n f^{(n)}(0)}{n!}$.
It is well known that the $n$th derivatives of the function $k^x$ are $\frac{d}{dx}k^x=k^x\ln (k)$; $... |
[The problematic triangle in question][1]
My HS math teacher (who also taught at a local university) gave my class this problem in 2019 during my last year as a challenge. After a week without anyone finding an answer, we gave up. Now I'm almost done with my Masters in Data Science and I found it among my notes - I ... |
I am working on a mathematical model describing the crystallization process, represented by an integral equation involving the crystalline fraction $\xi_V(T, t)$, where $T$ is the temperature and $t$ is time. The equation is as follows:
$$
\xi_V(T(t)) = \frac{4 \pi}{3} \int_0^t I_V(T(z))\left[\int_z^t G(T(\tau)) \,... |
>If the exhaustive set of $x\in(0,2\pi)$ for which $\forall n$ the inequality $$\sin x+\sin2x+\sin3x+\cdots+\sin nx\le\frac{\sqrt3}{2}$$ is valid is $l_1\le x\le l_2$, find $l_1$ and $l_2$.
Let $\displaystyle\sum_{i=1}^n \sin(ix)$=$S$ then I have managed to show that $$S=\frac{\sin\left(\frac{nx+x}{2}\right)\cdot\si... |
What is the discriminant of a degree $n$ polynomial? |
How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted? |
I've been trying to solve the following question:
Show that the polynomial ring $\mathbb{Z}[x,y,z]$ is a finitely generated module over its subring generated by the following three elements:$$x+y+z,\ xy+xz+yz,\ xyz.$$
Could you give me a hint? Say $R$ is the ring generated by the above three elements. My idea wa... |
Is such a subset of irrationals exists? (Question related to CH) |
I do not know much about the Gamma function, $\Gamma(n)$, except its series expansion (I am from the physics stackexchange), but I was wondering if there is a way to reduce $\Gamma(1-d/2)$ for $d\in\mathbb{C}$ (namely for $d\geq 0$) to something that looks likes $(1-d/2)\Gamma(-d/2)$? I know there are properties such a... |
let $A = ( a+bi: \in C : a > 4)$ and $B = ( a+bi: \in C : a >= 4)$
and let $d(z,w)$ be defined as
$$ 0 , z\neq w$$
$$ |z| +|w| , z=w$$
my questions are:
1) Is A and b open under (C,de) ? where de is the usual metric under C
2) Is A and B open under (C,d(z,w))?
my solution:
1) from my understanding, A is... |
My only problem is this step:
From making this:
∑{n=-∞,+∞}((-1)^n)(z^n)(q^n^2)*∑{n=-∞,+∞}(z^n)(q^n^2)=
=(A^2(q)/A(q^2))*∑{n=-∞,+∞}((-1)^n)(z^2n)(q^2(n)^2)
To this:
∑{n=-∞,+∞}((-1)^n)(q^2(n)^2)=(A^2(q)/A(q^2))
in this proof:
https://math.stackexchange.com/questions/434959/motivation-for-history-of-jac... |
Following this question: https://math.stackexchange.com/questions/242779/limit-of-lp-norm?noredirect=1&lq=1
>Given a finite measure space $(X, \mathcal{M}, \mu)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$
Can we find a counterexample when $\mu$ i... |
I believe that the problem stated in the title is false. I have attempted to prove it by manipulating the assumption that there is some $h_1 \in H_1$ and some $h_2 \in H_2$ such that $a_1h_1 = a_2h_2$. The idea was to come to the conclusion that $H_1 \subset H_2$ and $H_2 \subset H_1$ showing they were equal. After try... |
Consider the following stochastic differential equation for geometric Brownian motion:
$$
{\displaystyle dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}}
$$
I was reading on Wikipedia about geometric Brownian motion (https://en.wikipedia.org/wiki/Geometric_Brownian_motion) and this question popped into my mind. To be cle... |
Given a topological spaces
$(X,\tau)$ and a subset $S\subseteq X$, we say that $X$ is $S$-contractible if
$$X\simeq X/R$$
Where $$R= S^2 \cup ( \Delta_R/ \Delta_S )$$
The idea is that pulling $S$ to a single point doesn't change $X$. I have two questions:
1. Is it possible to classify all spaces $X$ with... |
I'm going through relative homology in Hatcher's *Algebraic Topology*.
Just as there are induced homomorphisms for nonrelative homology, a map $f: (X,A) \to (Y,B)$ — i.e., such that $f(A) \subset B$ — induces homomorphisms $f_\sharp: C_n(X,A) \to C_n(Y,B)$. Of course, for this to work, $f_\sharp: C_n(X) \to C_n(Y)$ ... |
Let Y be the set of all non injective functions from N → N (All n ∈ N,
|f−1[{n}] is different from 1)
I need to prove this cardinality equivalence.
So to do that i need to find an injective function from Y to N → N and another one from N → N to Y.
The first one is obvious beacause Y ⊆ N → N.
Having trouble findi... |
Let Y be the set of all non injective functions from N → N (All n ∈ N,
|f−1[{n}]| is different from 1)
I need to prove this cardinality equivalence.
So to do that i need to find an injective function from Y to N → N and another one from N → N to Y.
The first one is obvious beacause Y ⊆ N → N.
Having trouble find... |
I read in [On the spectral gap for finitely-generated subgroups of SU(2)](https://link.springer.com/article/10.1007/s00222-007-0072-z) that every finitely generated dense subgroup of $ SU(2) $ contains a free subgroup.
Is it true in general that every finitely generated dense subgroup of $ SU(n) $ contains a free s... |
Does every finitely generated dense subgroup of $ SU(n) $ contain a free subgroup? |
[The problematic triangle in question][1]
This problem with finding an unknown angle inside an isosceles triangle puzzled me for 5 years now.
My HS math teacher (who also taught at a local university) gave my class this problem in 2019 during my last year as a challenge. After a week without anyone finding an ans... |
So we were given the homework of formulating and proving the Fundamental Theorem of Homomorphisms for G-Sets, which factors equivariance maps. (this is not the homework question though, but a follow up).
So take two G-Sets $M$,$N$ and an equivariance map ($\varphi:N\rightarrow M$ with $\varphi(gn)=g\varphi(n)$). I... |
**Prove that in each ear decomposition of a 4-regular n-vertex 2-connected graph G, the number of ears (counting the initial cycle) is n+1.**
Definition of ear decomposition: (P_0, P_1,...,P_k) of the edge set of G such that (a) P_0 is a cycle of length at least 3, and (b) for i=1,...,k, P_i is an ear of P_0 ∪ P_1 ∪... |
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