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In the theory of Brownian motion, Spitzer's Law states that:
$$\lim_{t\rightarrow\infty}\mathbb{P}\bigg\{\frac{2\theta(t)}{\log(t)}\leq x\bigg\} = \int_{-\infty}^x\frac{1}{\pi (1+y^2)}dy$$
where $\theta(t)$ denotes the angular component of the skew-product representation of a planar Brownian motion:
$$B(t) = \exp\{W... |
If $X$ is an $\mathbb{R}$-valued r.v., with density $g(x)$, then $\mathbb{E}(X) := \int xg(x)dx$. But what if say $X \in \mathbb{R}^{3}?$ It can still have a density $g(x)$ such that the integral of $g$ over the whole 3-dimensional space is 1. I know for a high dimensional Gaussian, as long as we have its pdf, we know ... |
How is the mean of a high dimensional random variable defined? |
Here's the full question
> In Poisson Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in
between. The times at which the bus arrives at Stop A are a Poisson point process with one bus
arriving every five minutes on average, day and night, at which point it immediately picks up a... |
I know well that the Jacobi-Anger expansion is
$$e^{iz\sin (x)}=\sum_{n=-\infty}^\infty J_n(z)e^{inx}$$
or
$$e^{iz\cos (x)}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{inx}$$
My question is how to expand with Jacobi-Anger expansion if the subscript of the exponential is a Fourier series? For example what is the s... |
While teaching a course on ODE, I needed to introduce the notion of matrix logarithms. I intend to define it as follows.
**Definition (Matrix Logarithm)**
Let $A\in GL_n(\mathbb{C})$. We define
**A) Unipotent case:** When $A$ is unipotent, i.e. $A=I+N$, where $N\in M_n(\mathbb{C})$ is nilpotent, we defi... |
Consider the problem of finding the maximum possible error of the 3rd-degree Taylor Series (centered at $x = 1$) of $y = \ln{x}$ on the domain $[0.3, 1.7]$.
My idea was to use the Lagrange Error Bound, which states that the maximum possible error is
\begin{equation*}
\dfrac{M}{(n + 1)!}|x - a|^{n + 1}
\end{equa... |
For the Airy equation, which is a third-order linear partial differential equation given by:
$$ \partial_{t}u+\partial_{xxx}u=0.$$
Suppose furthermore that we consider solutios of Airy equation which are spatially periodic $$u(x+L,t)=u(x,t)$$
for a given period $L$ and for $(t,x)$ in $[0, \infty) \times \mathbb{R}... |
Consider the problem of finding the maximum possible error of the 3rd-degree Taylor Series (centered at $x = 1$) of $y = \ln{x}$ on the domain $[0.3, 1.7]$.
My idea was to use the Lagrange Error Bound, which states that the maximum possible error is
\begin{equation*}
\dfrac{M}{(n + 1)!}|x - a|^{n + 1}
\end{equa... |
Is it possible for a group generated by three rational points to have three generators? |
Here's the full question
> In Poisson Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in
between. The times at which the bus arrives at Stop A are a Poisson point process with one bus
arriving every five minutes on average, day and night, at which point it immediately picks up a... |
I have two loss functions $\mathcal{L}_1$ and $\mathcal{L}_2$ to train my model. The model is predominantly a classification model. Both $\mathcal{L}_1$ and $\mathcal{L}_2$ takes two variants of the focal loss [[reference][1]]. $\mathcal{L}_1$ and $\mathcal{L}_2$ takes as input the same class probabilities and a hyper-... |
How to comment on goodness of loss functions? |
While teaching a course on ODE, I needed to introduce the notion of matrix logarithms. I intend to define it as follows.
**Definition (Matrix Logarithm)**
Let $A\in GL_n(\mathbb{C})$. We define
**A) Unipotent case:** When $A$ is unipotent, i.e. $A=I+N$, where $N\in M_n(\mathbb{C})$ is nilpotent, we defi... |
The two types of triangular octahedra can be characterized by the degree sequences of their vertices. The regular octahedron has six vertices with $4$ incident edges, whereas the other one has degree sequence $(3,3,4,4,5,5)$. A vertex $v$ having degree $d$ means that $v$ can “see” $d$ other vertices; that is, if you pr... |
It is well known that if $f$ defined on a compact interval $[0,T]$ is a continuous function with finite variation, then $f$ induces a signed measure $\mu$ on $[0,T]$. Let $|\mu|$ be the total variation measure associated to $f$, that is, $|\mu|=\mu^++\mu^-$ where $\mu=\mu^+-\mu^-$ is the Jordan decomposition of a signe... |
I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you!
There is a common/standard treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, a... |
I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you!
There is a common/standard treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, a... |
The two types of triangular octahedra can be characterized by the degree sequences of their vertices. The regular octahedron has six vertices with $4$ incident edges, whereas the other one has degree sequence $(3,3,4,4,5,5)$. A vertex $v$ having degree $d$ means that $v$ can “see” $d$ other vertices; that is, if you pr... |
**Question.**
I am having difficulty with both intuitive understanding and necessity of, a technicality in a proof of the Vapnik-Chervonenkis inequality in the notes of [Robert Nowak (2009)](https://nowak.ece.wisc.edu/SLT09/lecture19.pdf#page=2). The proof strategy is taken from the yellow book by [Luc Devroye, Lász... |
I was doing some question on Fibonacci Sequence and came across this formula
$$\large F_n = \frac{\left(\frac{1+\sqrt5}{2}\right)^n - \left(\frac{1-\sqrt5}{2}\right)^n}{\sqrt5}$$
I started thinking “Is there a better way to write this formula? Because this seems ugly, look at the denominator, it has surd and the nume... |
The two types of triangular octahedra can be characterized by the degree sequences of their vertices. The regular octahedron has six vertices with $4$ incident edges, whereas the other one has degree sequence $(3,3,4,4,5,5)$. A vertex $v$ having degree $d$ means that $v$ can “see” $d$ other vertices; that is, if you pr... |
Answers:
1. No. Or more precisely, the "drudgery" is not worth it. By drudgery I mean the computational effort. There are simpler procedures.
2. Maybe yes.
About the computational effort.
IMO, in contrast to the function you suggest the continued fraction Euler shows in §24..25 of [Eneström No. 606][1] is m... |
A filtered bialgebra $(A,m,u,\delta,\epsilon)$ with $A_j,j\in\mathbb Z$ filtration is defined as a bialgebra so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$. The assotiated graded space $\text {gr}A=\bigoplus_{j}A_{j}/A_{j-1}$ has a multiplication and a comultiplication indu... |
(Now asked on MO: https://mathoverflow.net/questions/467368 )
I would like to determine whether the polynomial $x^n+5x+3$ is irreducible over $\mathbb{Q}$ for $n\ge 2$. I thought of considering $\bmod{2}$ so that we obtain a polynomial $x^n+x+1$. However the polynomial $x^n+x+1$ is not always irreducible over $\math... |
> Let $n=p^{\alpha}, p$ be an odd prime, $\alpha\in\mathbb{N}^*$. Assume
> that $\{x_1,...,x_k\}$ is a reduced residue system modulo $n$. Prove that
> $$x_1x_2...x_k\equiv -1\pmod{n}.$$
My attempt:\
I have already shown that $x_1^2x_2^2...x_k^2\equiv 1\pmod{n}.$ So we have 2 cases here:
$$x_1x_2...x_k\equiv -1\p... |
Let $\mathscr{P}\left(A\right)$ denote the set of all subsets of a set $A$.
Show that the relation R defined on $\mathscr{P}\left(A\right)$ by aRb $\forall x(x\in A\Leftrightarrow x\in B)$ is an equivalence relation on$\mathscr{P}\left(A\right)$
My idea is Let a$\epsilon $A,b$\epsilon $BaRa，aRb $\Leftrightarrow$bRa... |
So going off of Kurt G, for $\alpha = 1$ on the boundaries, I should get:
$$\oint_C 1\nabla \phi_i \cdot \hat{\textbf{n}} ds = \int\int_D \nabla \cdot \nabla\phi_i dA$$
And computing the double integral, I should get:
$$-2(n\pi)^2\int_0^1 \int_0^1 \sin(n\pi x)\sin(n \pi y)dxdy = \\ 2n\pi \int_0^1 \Big(\cos(n\p... |
Let $A,B \subset \Bbb{R}$ be Borel measurable with $m(A),m(B) \in (0,\infty)$, then prove $\chi_A \ast \chi_B$ is continuous.
**Attempt:**
Note
$$f(x)=\chi_A \ast \chi_B (x)=\int \chi_A(x-y)\chi_B(y)dy.$$
As $C_c(\Bbb{R}) \subset L^1(\Bbb{R})$ is dense, there exists $g_n \in C_c(\Bbb{R}$) such that
$$\... |
Prove that the function $\sqrt{x}$ is continuous on its domain $[0,\infty)$.
Proof.
Since $\sqrt{0} = 0, $ we consider the function $\sqrt{x}$ on $[a,\infty)$ where $a$ is real number and $a \neq 0.$ Let $\delta=2\sqrt{a}\varepsilon.$ Then, $\forall x \in \mathit{dom},$ and $\left | x-x_0\right | < \delta \Righta... |
I don't have much background in MDPs but I was reading a text in which I came across the following which I am finding difficult to understand. Consider a Markov decision process with state space $S$ where the only action possible for state $s=0$ is $a=0$ and the reward on completing the actin is $r_a$ and the task time... |
Let $\mathscr{P} \left( A \right)$ denote the set of all subsets of a set $A$.
Show that the relation $R$ defined on $\mathscr{P} \left( A \right)$ by $a R b, \forall x (x \in A \Leftrightarrow x \in B)$ is an equivalence relation on $\mathscr{P} \left( A \right)$.
My idea is to let $a \in \mathscr{P} (A), b \in ... |
**The importance of $\lambda(n)$ dividing $n$**: The unit group $(\mathbb{Z}/n)^{\times}$ is a $\mathbb{Z}$-module and $(\lambda(n))$ is by definition, the annihilator of the module. In general, there is no canonical way to turn to raise elements of the unit group to an exponent in $(\mathbb{Z}/n)$. Of course, you can ... |
Consider three points on the plane: $(1,0), (0,1) $and $(\frac{p}{q},\frac{m}{n})$, where $p,q,m,n\in\mathbb Z, q,n\neq 0,\gcd(p,q)=\gcd(m,n)=1.$
The group generated by $(1,0), (0,1) $and $(\frac{p}{q},\frac{m}{n})$ with respect to addition can be denoted as
$$\Gamma=\{k_1(1,0)+k_2(0,1)+k_3(\frac{p}{q},\frac{m}{n})... |
A group generated by three rational points has actually two generators? |
There is a similar post regarding the integral $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ on Stack. The reason I have some trouble with this post is because it gives an analytic continuation of the function $\sin^3(x)/x^3=\left(\frac{e^{iz}-e^{-iz}}{2iz}\right)^3$, namely $h(z)=-\frac{e^{3iz}-3e^{iz}}{4z^3}-\frac{1}{2z^3... |
Is it possible to evaluate $\int_{-\infty}^\infty \sin(x^3)/x^3$ without finding an analytic continuation? |
My progress so far:
We claim that the unique solution is $$f(x) \equiv x^3.$$ Plugging this in, this works because $$(x + y)^3 = x^3 + y^3 + 3(x + y)xy$$ is true after expanding.
We wish to show that this is the only valid solution. Plug in $(x, x)$ to get $$f(2x) = 2f(x) + 6x\sqrt[3]{f(x)^2}.$$ So, $$\frac{f(2x) -... |
*For exponential family distributions, is the log-partition function continuous and differentiable in the whole natural parameter space?*
This maybe a trivial question but I couldn't find any reference or show it rigorously. Any help is highly appreciated.
The exponential family (in canonical form) is
$$f_{X}(x|... |
There is a similar post regarding the integral $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ on Stack. The reason I have some trouble with this post is because it gives an analytic continuation of the function $\sin^3(x)/x^3=\left(\frac{e^{iz}-e^{-iz}}{2iz}\right)^3$, namely $h(z)=-\frac{e^{3iz}-3e^{iz}}{4z^3}-\frac{1}{2z^3... |
Name the first two axioms as follows:
$$K: a → b → a,\quad S: (a → b → c) → (a → b) → a → c,$$
where we adopt the prevailing convention of of associating to the right, i.e. $a → b → c = a → (b → c)$. Name the third axiom
$$Z: (¬a → ¬b) → b → a,$$
and for modus ponens, applied to $f: a → b$ and $g: a$, write $fg: b$... |
I'm currently working on a physics project where I am basically using logger pro's video analysis to find speeds for an object being accelerated by an object. Furthermore, I am conducting multiple trials in order to get averaged values. I have propagated the uncertainty assuming that the major source of uncertainty is ... |
When should I use a weighted mean vs a regular mean? |
Name the first two axioms as follows:
$$K: a → b → a,\quad S: (a → b → c) → (a → b) → a → c,$$
where we adopt the prevailing convention of of associating to the right, i.e. $a → b → c = a → (b → c)$. Name the third axiom
$$Z: (¬a → ¬b) → b → a,$$
and for modus ponens, applied to $f: a → b$ and $g: a$, write $fg: b$... |
I am reading some context from computer graphics about using eikonal equation to computer geodesic on triangular mesh, I find some reference e.g [text](https://www.sciencedirect.com/science/article/abs/pii/S0019357718304294)
The Aurhors state that if $\| grad \: \textit{f}\|_g=1$ then we get $$\nabla_{grad \: \texti... |
Is it possible to evaluate $\int_{-\infty}^\infty \sin(x^3)/x^3 dx$ without finding an analytic continuation? |
Find all functions from positive integers to positive integers such that $$f(x + y) = f(x) + f(y) + 3(x + y)\sqrt[3]{f(x)f(y)}$$ for all positive integers $x$ and $y$.
My progress so far:
We claim that the unique solution is $$f(x) \equiv x^3.$$ Plugging this in, this works because $$(x + y)^3 = x^3 + y^3 + 3(x + y... |
Is the Series : $\sum_{n=1}^{\infty} \left( \frac 1 {(n+1)^2} +\ldots+\frac 1 {(n+n)^2} \right) \sin^2 n\theta $ convergent? |
Is this proof vigorous enough? (Feel free to provide any suggestions for improvement as well).
(A centre of a connected graph $G$ is a vertex $v$ with the property that the maximum of the distances between $v$ and all other vertices of $G$ is as small as possible.)
Note: I will not prove the tree properties liste... |
First, let's just make things simpler by making everything integer: So instead of $(1,0)$, $(0,1)$, and $(p/q,m/n)$, we can use $(qn,0)$, $(0,qn)$, and $(pn,mq)$ by multiplying all the vectors by a constant factor of $qn$. This group is isomorphic to our original group, since we're just scaling everything with a consta... |
First, let's just make things simpler by making everything integer: So instead of $(1,0)$, $(0,1)$, and $(p/q,m/n)$, we can use $(qn,0)$, $(0,qn)$, and $(pn,mq)$ by multiplying all the vectors by a constant factor of $qn$. This group is isomorphic to our original group, since we're just scaling everything with a consta... |
*(It may be noted that the author was not aware of Endomorphisms & Automorphisms at the time of writing this question)*
I'm trying to better understand the concept of a linear transformation acting as an isomorphism on a specific subspace of the vector space, even though it may not be an isomorphism on the entire ve... |
For the first few parts of a question, I have used the definition of partial derivatives to show that $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ both exist at the point $(0,0)$ and are equal to $0$. I have also found the partial derivatives for $f$ at a point away from $(0,0)$ using normal... |
Is the function $(x^2+y^2)sin(\frac{1}{(x^2+y^2)}$ differentiable at the point (0,0)? |
Is it possible to evaluate $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ without finding an analytic continuation? |
Is the function $(x^2+y^2)sin(\frac{1}{x^2+y^2}$ differentiable at the point (0,0)? |
Is the function $(x^2+y^2)\sin{\frac{1}{x^2+y^2}}$ differentiable at the point (0,0)? |
Is the function $(x^2+y^2)\sin{(\frac{1}{x^2+y^2})}$ differentiable at the point (0,0)? |
I'm given a table that shows the percent of Canadian households that used e-mail from year A to year B. Then what is the unit of the average rate of change of the percent of households using e-mails from year A to year B? Is it *percent per year*? I'm fine with dollars per year, miles per hour and so forth, but percent... |
what is the unit of this rate of change? |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
I have a blue curve, a simple circle $x^2+y^2=1$, onto which I am projecting, like a shadow, an exact function $F$. Let's say it's a simple Gaussian $F(x)=e^{-x^2}$ visualized in red below.
As the function is projected onto the shape, it ***does not wrap around it, but so-to-speak casts a shadow*** as if it were a c... |
EDIT: The question specifies that f(0,0)=0 in a piecewise.
For the first few parts of a question, I have used the definition of partial derivatives to show that $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ both exist at the point $(0,0)$ and are equal to $0$. I have also found the partial ... |
Name the first two axioms as follows:
$$K: a → b → a,\quad S: (a → b → c) → (a → b) → a → c,$$
where we adopt the prevailing convention of of associating to the right, i.e. $a → b → c = a → (b → c)$. Name the third axiom
$$Z: (¬a → ¬b) → b → a,$$
and for modus ponens, applied to $f: a → b$ and $g: a$, write $fg: b$... |
EDIT: The question specifies that $f(0,0)=0$ in a piecewise.
For the first few parts of a question, I have used the definition of partial derivatives to show that $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ both exist at the point $(0,0)$ and are equal to $0$. I have also found the partia... |
I have the following figure:
[![enter image description here][1]][1]
Let's consider two right triangles, with right angles at points $L$ and $M$, respectively. Since the lines passing through points $A$ and $C$ and through points $L$ and $M$ are parallel, intersected by a transversal passing through points $A$ an... |
*Too long for comments.*
Using exact arithmetic, it is "simple" to confirm the results (at least for $m=n$) since if
$$f_n(a) = \prod_{i=0}^n ( \sin(i) +a)$$
$$f_n(a) = i\,\frac {(-1)^n}{2^{n+1} }\,e^{-\frac{i}{2} n (n+\pi +1)}\left(i \left(a-\sqrt{a^2-1}\right);e^i\right){}_{n+1} \left(i
\left(a+\sqrt{a^2-1}... |
In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of observables, representable as a subalgebra of $\mathcal{B}(H)$ for some Hilbert space $H$ via the GNS construction) that are posi... |
Consider the matrix ode $$\frac{\mathrm d x}{\mathrm d t}(t)=x(t)A(t),x(0)=c$$where $x(t)\in\mathbb R^n, A(t) \in M_{n\times n}$ is continuous. It seems that we can use perturbative method (which I'm not sure if this is reasonable) to solve the matrix equaiton $Y_{\epsilon}(t)\in GL_n$
$$Y'_{\epsilon}(t)=\epsilon Y_{\... |
**It is neither right nor wrong , though it is useful to improve basic intuition.**
Euclidean Geometry ( with the list of Axioms & Postulates including the Parallel Postulate ) gives the Pythagorean theorem.
Euclidean Geometry [ with Similarity Property & Pythagorean theorem ] gives rise to trigonometry.
... |
Introduction:
Find a such family of polynomial then we can always use Newton's method provided we have a procedure to start.
We introduce a rational function:
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots x}}}}\right)-C=0... |
Consider the matrix ode $$\frac{\mathrm d x}{\mathrm d t}(t)=x(t)A(t),x(0)=c$$where $x(t)\in\mathbb R^n, A(t) \in M_{n\times n}$ is continuous. It seems that we can use perturbative method (which I'm not sure if this is reasonable) to solve the matrix equaiton $Y_{\epsilon}(t)\in GL_n$
$$Y'_{\epsilon}(t)=\epsilon Y_{\... |
In mathematics we do not consider questions like "are there objects that are XYZ?", which is a rather philosophical question. In mathematics we consider questions like "are there objects that are XYZ under ABC axioms?". Everything in maths is relative to some axiomatic theory.
In particular the answer to this questi... |
It probably doesn't help to think in terms of hard problems versus easy problems. That probably means problems that require some conceptual thought versus relatively straightforward calculations. But being good at calculation is a skill in itself, and as your experience shows, it can't be taken for granted. Indeed, a d... |
**It is neither right nor wrong , though it is useful to improve basic intuition.**
Euclidean Geometry ( with the list of Axioms & Postulates including the Parallel Postulate ) gives the Pythagorean theorem.
Euclidean Geometry [ with Similarity Property & Pythagorean theorem ] gives rise to trigonometry.
... |
Let $(E,N)$ be a normed space, and $M$ be another norm. Can you find a necessary and sufficient condition on $M$ so that $\sqrt{NM}$ is also a norm on $E$?
I hink the condition is that $M$ is proportional to $N$.
Of course, the only thing to investigate is the triangle inequality: we want to prove that if for a... |
Given $a,b\in \mathbb{R}$ with $a<b$ and $t_1,...,t_n\in [a,b]$ and $a_1,...,a_n\in \mathbb{C}$ how to define a complex-valued continuous function $f$ on $[a,b]$ such that $f(t_i)=a_i$ for each $1\leq i\leq n$ and $\max_{t\in [a,b]}|f(t)|\leq \max_{1\leq i\leq n}|a_i|$ ?
It is easy to define such a real-valued conti... |
from [The influence of conjugacy class sizes on the
structure of finite groups](https://www.bing.com/ck/a?!&&p=c3f89a7137a3353aJmltdHM9MTcxMTA2NTYwMCZpZ3VpZD0xNTYyNGMzZC0xNWI3LTY5OWEtM2Y1ZS01Zjk0MTQ2NDY4NWMmaW5zaWQ9NTE5Mw&ptn=3&ver=2&hsh=3&fclid=15624c3d-15b7-699a-3f5e-5f941464685c&psq=influencbof+conjugacy+classes&u=... |
In a paper it is claimed that the matrix algebra generated by a matrix $J=\oplus_i J_{\lambda_i,m_i}$ in Jordan normal form "is the algebra of block diagonal matrices with the same block partition, where each block contains an arbitrary upper triangular Toeplitz matrix". I take issue with the arbitrary Toeplitz blocks... |
What is the exact form of the matrix algebra generated by a Jordan matrix? |
from [The influence of conjugacy class sizes on the
structure of finite groups](https://www.bing.com/ck/a?!&&p=c3f89a7137a3353aJmltdHM9MTcxMTA2NTYwMCZpZ3VpZD0xNTYyNGMzZC0xNWI3LTY5OWEtM2Y1ZS01Zjk0MTQ2NDY4NWMmaW5zaWQ9NTE5Mw&ptn=3&ver=2&hsh=3&fclid=15624c3d-15b7-699a-3f5e-5f941464685c&psq=influencbof+conjugacy+classes&u=... |
From [The influence of conjugacy class sizes on the
structure of finite groups](https://www.bing.com/ck/a?!&&p=c3f89a7137a3353aJmltdHM9MTcxMTA2NTYwMCZpZ3VpZD0xNTYyNGMzZC0xNWI3LTY5OWEtM2Y1ZS01Zjk0MTQ2NDY4NWMmaW5zaWQ9NTE5Mw&ptn=3&ver=2&hsh=3&fclid=15624c3d-15b7-699a-3f5e-5f941464685c&psq=influencbof+conjugacy+classes&u=... |
Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? |
I'm trying to understand why the fundamental theorem of arithmetic is phrased like this:
> Every positive integer, except 1, is a product of primes.
A prime number is an integer but it is not a product of primes. It seems to me that "every positive integer" generalization is not justified. Something like, "every ... |
Let $X$ and $Y$ be independent random variables on $\mathbb{R}$. If they have an absolutely continuous distribution (w.r.t. the Lebesgue measure), we know that $P(X=Y)=0$, due to the fact that the event $X=Y$ is supported on a set with zero (Lebesgue) measure on $\mathbb{R}^2$. Now assume the variables have a discrete ... |
There is a similar post regarding the integral $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ on Stack. The reason I have some trouble with this post is because it gives an analytic continuation of the function $\sin^3(x)/x^3=\left(\frac{e^{iz}-e^{-iz}}{2iz}\right)^3$, namely $h(z)=-\frac{e^{3iz}-3e^{iz}}{4z^3}-\frac{1}{2z^3... |
In a paper it is claimed that the [matrix algebra][1] generated by a matrix $$J=\oplus_i J_{\lambda_i,m_i}$$ in Jordan normal form "is the algebra of block [diagonal matrices][2] with the same block partition, where each block contains an arbitrary upper triangular [Toeplitz matrix][3]".
I take issue with the arbit... |
In a paper it is claimed that the [matrix algebra][1] generated by a matrix $$J=\oplus_i J_{\lambda_i,m_i}$$ in Jordan normal form "is the algebra of block [diagonal matrices][2] with the same block partition, where each block contains an arbitrary upper triangular [Toeplitz matrix][3]".
I take issue with the arbit... |
Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.
Given a commutative ring with unity $R$, $R\operatorname{-Mod}$ denotes the category of $R$-modules.
Given $R$-modules $A$ and $B$, $A \cong B$ denotes isomorphism of $A$ and $B$ as $R$-modules.
---
Does there exist an example of th... |
I am following Roger's *Supermanifolds: Theory and Applications* and I might be getting something wrong, because I reach a definition that, as I understand it, doesn't imply what the author states.
Letting $\mathbb{R}_{\infty}$ be a real Grassmann algebra generated by anticommuting $\{\xi_j\}_{j\in\mathbb{N}}$, with... |
Given $a,b\in \mathbb{R}$ with $a<b$ and $t_1,...,t_n\in [a,b]$ (where $t_i$'s are distinct) and $a_1,...,a_n\in \mathbb{C}$ how to define a complex-valued continuous function $f$ on $[a,b]$ such that $f(t_i)=a_i$ for each $1\leq i\leq n$ and $\max_{t\in [a,b]}|f(t)|\leq \max_{1\leq i\leq n}|a_i|$ ?
It is easy to de... |
I have managed to solve this ODE
$$y''+4y=cos(2x)$$
using comparison of coefficient, and I get:
$$y(x) = C_1cos(2x)+C_2sin(2x)+\frac{1}{4}xsin(2x)$$
Using parameters variation I get:
$$y(x) = C_1cos(2x)+C_2sin(2x)+\frac{1}{4}cos(2x)cos(4x)+\frac{1}{2}xsin(2x)+\frac{1}{8}sin(4x)sin(2x)$$
However, trying to ge... |
Is the Cartesian product of two bounded sets bounded? What about for compact sets? |
Let X_1, X_2, ..., X_n be a random sample from N(\theta+2, \theta^2). Find the MLE of theta.
I went through some work and I could not solve for theta. Am I doing anything wrong?
[**My Work**][1]
[1]: https://i.stack.imgur.com/WmNGW.png |
Let $X_1, X_2, ..., X_n$ be a random sample from $N(\theta+2, \theta^2)$. Find the MLE of $\theta$.
I went through some work and I could not solve for $\theta$. Am I doing anything wrong?
[**My Work**][1]
[1]: https://i.stack.imgur.com/WmNGW.png |
Consider a dynamical system:
$\dot{x} = f(x,\lambda,\Lambda)$.
For a specific value of $\Lambda=\Lambda_1$, the system admits a fold bifurcation at $(\lambda_1,x_1)$. I want to find out two things, I want to find how the fold bifurcation evolves, i.e I want to:
1. find $(\lambda(\Lambda),x(\Lambda))$ such tha... |
Consider a dynamical system:
$$\dot{x} = f(x,\lambda,\Lambda)$$
For a specific value of $\Lambda=\Lambda_1$, the system admits a fold bifurcation at $(\lambda_1,x_1)$. I want to find out two things, I want to find how the fold bifurcation evolves, i.e I want to:
1. find $(\lambda(\Lambda),x(\Lambda))$ such th... |
Consider a dynamical system:
$\dot{x} = f(x,\lambda,\Lambda)$.
For a specific value of $\Lambda=\Lambda_1$, the system admits a fold bifurcation at $(\lambda_1,x_1)$. I want to find out two things, I want to find how the fold bifurcation evolves, i.e I want to:
1. find $(\lambda(\Lambda),x(\Lambda))$ such tha... |
Let $X_1, X_2, ..., X_n$ be a random sample from $N(\theta+2, \theta^2)$. Find the MLE of $\theta$.
I went through some work and I could not solve for $\theta$. Am I doing anything wrong?
L( $\theta$ ) = $\prod _ { i = 1 } ^ { n } \frac { 1 } { \theta \sqrt { 2 \pi } } \exp \bigg[- \frac{( x _ { i } - \theta - 2 ... |
I have managed to solve this ODE
$$y''+4y=\cos(2x)$$
using comparison of coefficient, and I get:
$$y(x) = C_1\cos(2x)+C_2\sin(2x)+\frac{1}{4}x\sin(2x)$$
Using parameters variation I get:
$$y(x) = C_1\cos(2x)+C_2\sin(2x)+\frac{1}{4}\cos(2x)\cos(4x)+\frac{1}{2}x\sin(2x)+\frac{1}{8}\sin(4x)\sin(2x)$$
However, t... |
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