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I'm faced with the following problem:
"Consider a cubic in $P^2(\mathbb{C})$ with a cusp. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds to the unique singular point, and the sum of the parameters of three collinear points is equal to 0."
I think this problem is related to a question that I... |
You have switched from using subscripts to index the sequence of vectors (in $x_{n+1} = D^{-1} (b - (L + U) x_n)$, expressing the $(n+1)$st vector $x_{n+1}$ in terms of the $n$th vector $x_n$) to using superscripts to index the sequence of vectors and subscripts to denote the entries of vectors (in the equations that g... |
As you note on MO, algebras, as well as groups. One way to view Simon's claim is that we are taking a $*$-representation of the $*$-algebra (i.e., algebra with complex-anti-linear involution) $\mathcal A = \mathbb C[A, A^*]/(A^2 = 0, A^* A + A A^* = 1)$—where $A$ and $A^*$ are treated as formal symbols, subject only t... |
I sort of intuitively see why we care about similar matrices, i.e., when $A=S^{-1}BS$ for some invertible matrix $S$. But I want to make this intuition more precise and abstract.
**Matrices:** First of all, as mentioned [here][1],
> Because matrices are similar if and only if they represent the same linear opera... |
In what sense are similar matrices "the same," and how can this be generalized? |
I'm faced with the following problem:
>Consider a cubic in $P^2(\mathbb{C})$ with a cusp. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds to the unique singular point, and the sum of the parameters of three collinear points is equal to 0.
I think this problem is related to a question that I ... |
I'm studying general topology and a question has come to my mind.
I am referring to the class of theorems that in algebra go by the name of "homomorphism theorems". In my topology course, we have seen some results alike, but it lacks the general result on when, given two continuous maps with the same domain, there e... |
If $f(x)$ and $g(x)$ are two convex functions defined in $\mathbb{R}^n \to \mathbb{R}$. Suppose $f(x)$ has a global minimizer $x_1$ but not necessarily the unique global minimizer.
Similarly, $g(x)$ has a global minimizer $x_2$ but not necessarily the unique global minimizer.
Does there exist a global minimizer for t... |
Does the sum of two convex functions with global minimizers have a global minimizer? |
Just a only doubt. Supposing that I have these three series:
$$\sum_{n=1}^\infty\left[\arcsin(p(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arccos(q(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arctan\left(g(x)\right)\right]^n$$
Supposing that the three goniometric inverse function ar... |
For a local ring $R$, we have a bijection $\mathbf P_R^n(R) \cong (R-\left\lbrace 0\right\rbrace)/R^\times$. This allows us to refer to points $\text{Spec} R \to \mathbf P_R^n$ using homogeneous coordinates. I know the same is not true for general rings $R$.
However, I am wondering about the case where we have an in... |
Do homogeneous coordinates on $\mathbf P_R^n(R)$ extend to $\mathbf P_R^n(S)$? |
Let $f: \Omega \to \mathbb{R}$, where $\Omega$ is a convex compact subset of $\mathbb{R}^n$ with a non-empty interior, denoted by $Int(\Omega)$. Assume that on $Int(\Omega)$, $f$ is the restriction of a linear function $L: \mathbb{R}^n \to \mathbb{R}$. Is $f$ necessarily the restriction of $L$ to $\Omega$ ?
Note th... |
If $f(x)$ and $g(x)$ are two convex functions defined in $\mathbb{R}^n \to \mathbb{R}$. Suppose $f(x)$ has a global minimizer $x_1 \in \mathbb{R}^n$ but not necessarily the unique global minimizer.
Similarly, $g(x)$ has a global minimizer $x_2 \in \mathbb{R}^n$ but not necessarily the unique global minimizer.
Does th... |
Define $\tau_k = 1^{*k}$ where $*$ is the Dirichlet convolution. Then it's known that
$$\sum_{n \le x} \tau_k(n) = x P_k(\log x) + O(x^{1-1/k} \log^{k-2}(x))$$ where $P_k$ has leading coefficient $\frac{1}{(k-1)!} x^{k-1}$.
Apparently this can be proven through applying the Dirichlet Hyperbola Method with a thresh... |
## Recent Open Access Books:
(2020) [Making up Numbers, A History of Invention in Mathematics][1]
### Standalone, $2000-2020$ Open Access E-Books:
[Mathematics and Its History][2]
[History of Mathematics by Burton][3]
[A History of Mathematics by Florian Cajori][4]
[A Short Account of the History of Mat... |
This is an old question, but I would like to provide another angle which touches directly the statement used in the Wikipedia article and whose gist has been hinted at it in the other answers:
> -finiteness can be compared in this respect to separability of topological spaces
In a separable topological space a c... |
How can we find the all positive integers $n$ for which there exists positive integers $a_1,a_2,\cdots,a_n$ such that
$\sum\limits_{k=1}^{n}\frac{2^{k-1}}{a_k^2}=1$?
My attempt:
For any odd $n=2k+1$, I guess that the solution will in this form:
$a_1=a_2=2,a_{2k+1}=4^k,$ and $
a_{2j+1}=a_{2j+2}=4a_{2j−1},1<j<k$... |
All positive integers $n$ such that $\sum\limits_{k=1}^{n}\frac{2^{k-1}}{a_k^2}=1$? |
I'm faced with the following problem:
>Consider a cubic in $P^2(\mathbb{C})$ with a cusp. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds to the unique singular point, and the sum of the parameters of three collinear points is equal to 0.
I think this problem is related to a question that I ... |
How can we find the all positive integers $n$ for which there exists positive integers $a_1,a_2,\cdots,a_n$ such that
$\sum\limits_{k=1}^{n}\frac{2^{k-1}}{a_k^2}=1$?
My attempt:
For any odd $n=2k+1$, I guess that the solution will in this form:
$a_1=a_2=2,a_{2k+1}=4^k,$ and $
a_{2j+1}=a_{2j+2}=4a_{2j−1},1<j<... |
Assumptions for reaching the [logistic map equation][1]
---
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
... |
Assume that the non-compact $n$-dimensional Riemannian manifold $M$ satisfies a curvature-dimension $\mathrm{CD}(K,N)$, $K \le 0$, with respect to a measure $\mu = e^{-V} d\mathrm{Vol}$; i.e.
\begin{equation*}
\mathrm{Ric} + \mathrm{Hess} V - \frac{\nabla V \otimes \nabla V}{N - n} \ge Kg.
\end{equation*}
I wonder ... |
We show that if $\sigma(n)$ is exponential then the lacunary (note spelling) serolies tends towards $1/2$ but retains an oscillation about this value and does not completely converge.
Let $\sigma(n)=3^n$ and sum from $n=0$ to $n=\infty$, thus
$f(x)=\lim_{x\to1^-}\sum\limits_{n=0}^\infty (-1)^n x^{3^n}.$
The fo... |
I'm faced with the following problem:
>Consider a cubic in $P^2(\mathbb{C})$ with a cusp. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds to the unique singular point, and the sum of the parameters of three collinear points is equal to 0.
I think this problem is related to a question that I ... |
Just a only doubt. Supposing that I have these three series:
$$\sum_{n=1}^\infty\left[\arcsin(p(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arccos(q(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arctan\left(g(x)\right)\right]^n$$
Supposing that the three goniometric inverse function ar... |
## Question: Find the number of solutions of equation $f(x)=2^x-x^2+x+\cos x=0$ ##
----------
Given answer is $1$
*My Attempt*
I have checked the derivative of it. $$f'(x) = 2^x\ln2-2x+1-\sin x\\f''(x)=2^x(ln2)^2-2-\cos x$$ but $f'(x)$ is both +ve and -ve so I could not make any conclusion. I had projected... |
Proof of the first inequality :
Let us give a geometric interpretation allowing to conclude that expression :
$$S=\sin(x_1+x_2) + \sin(x_2+x_3) + \sin(x_3+x_4) + \sin(x_4+x_5)+\sin(x_5+x_1)$$
is positive.
Indeed, angles $x_k$, having their sum equal to $2 \pi$, can be interpreted as "deviation angles" $(\... |
As you note on MO, [algebras](https://en.wikipedia.org/wiki/Algebra_representation), as well as [groups](https://en.wikipedia.org/wiki/Group_representation), have representations. One way to view Simon's claim is that we are taking a [$*$-representation](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Se... |
This specific question has arisen out of my attempts to do a wider exercise ([link][1]).
---
**Context**
We say two functions, $f$ and $g$, are equal if all three of the following conditions are met:
- (i) they have the same **domain**
- (ii) they have the same **codomain**
- (iii) they map the same el... |
How can I show that the quotient function to the real projective space is closed? |
As part of a proof of the admissibility of the complex Morlet wavelet, I am trying to show that the following integral is positive and finite
$$
0<\int_0^\infty{\frac{(e^{\sigma \omega}-1)^2e^{-\omega^2}}{\omega}}d\omega<\infty
$$
Where $\sigma > 0$. Is anyone willing to give any guidance, or provide an explicit ... |
Let $\mathcal{R}$ an o-minimal expansion of $\overline{\mathbb{R}}$ and $g$ a unary definable function such that $g'\sim 1/x$ (by the monotonicity theorem $g$ is ultimately differentiable)
By applying the mean value theorem on $g$ (by the monotonicity theorem $g$ is ultimately differentiable) there exists a function... |
When I learned this $\sigma$-algebra, which is the smallest $\sigma$-algebra that contains all the open intervals of $\mathbb{R}$ (let's denote it by $\mathcal{B}$), I've got the feeling that no subset of $\mathbb{R}$ is not in it. Because:
1. Of course $\emptyset \in \mathcal{B}$ and $\mathbb{R} \in \mathcal{B}$
... |
The inequality in the OP holds for $$n\ge3.$$
I used Wolfram numerical integration to show:
for $n=1$, it [becomes][1]: 0.761578
for $n=2$, it [becomes][2]: 0.980789
for $n=3$, it [becomes][3]: 1.09101.
In the links, I used
$$\Phi(x)=\frac{1}{2}+\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}} \right).... |
You suggest a theorem of the form:
> If $X$, $Y$, and $Z$ are structures, $f\colon X\to Y$ and $g\colon X\to Z$ are morphisms (respecting the structure), then provided that `blah` holds, there exists a morphism $h\colon Y\to Z$ such that $g=h\circ f$.
In groups, this is not a standard theorem (in the way that, sa... |
Proof By Induction - How to apply Induction Hypothesis? |
>Let $f : A \rightarrow B$ and $g : A \rightarrow C$ be homomorphisms of Noetherian rings.Suppose \
1)$ B \otimes_A C$ is Noetherian,\
2) $f$ is flat and\
3) $g$ is non-degenerate.\
Then $1_B \otimes g : B \rightarrow B \otimes C$ is also non-degenerate.
This is a proposition in Mat80, but is left as... |
In orthogonal cartesian coordinates c*c = square of length of vector c
In oblique coordinates square of length of c = c * g * c where g is the metric tensor and g*c is a covector
If ex and ey are the unit vectors of vector c then how does one calculate the new unit vectors ex' and ey' of g*c?
Am I correct in a... |
To be honest I dont know how to do this problem, I only know that we should get a second order linear differention equation for the position x. And we have to solve the problem using theta.
I know that X(t)= rsin(theta), where r would be the length of the rod but I dont know where to go from there.
Were also given ... |
I came across an issue in code whereby I noticed the standard deviation, $\sigma$, of a set of constant values of e.g. 0.33333333 = 1/3 was not equal to zero. Rather, $\sigma$ was some small near-zero value.
Below is an example Excel spreadsheet I learned could easily replicate the issue after converting 1/3 to de... |
Double precision: When is near-zero considered zero? |
To be honest I dont know how to do this problem, I only know that we should get a second order linear differention equation for the position x. And we have to solve the problem using theta.
I know that X(t)= rsin(theta), where r would be the length of the rod but I dont know where to go from there.
Were also given ... |
Just a only doubt. Supposing that I have these three series:
$$\sum_{n=1}^\infty\left[\arcsin(p(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arccos(q(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arctan\left(g(x)\right)\right]^n$$
Supposing that the three goniometric inverse functions a... |
To be honest I dont know how to do this problem, I only know that we should get a second order linear differention equation for the position x. And we have to solve the problem using theta.
I know that X(t)= rsin(theta), where r would be the length of the rod but I dont know where to go from there.
Were also given ... |
The rules of a lottery are as follows: You select 10 numbers between 1 and 50. On lottery night, the celebrity mathematician Richard Thomas chooses at random 6 'correct' numbers. If your 10 numbers include all 6 correct ones, you win.
How many ways are there to win the lottery?
From my understanding, the requested ... |
In the paper [Perona-Malik equation and its numerical properties](https://arxiv.org/pdf/1412.6291.pdf), the following PDE is considered:
[![enter image description here][1]][1]
The $u_0$ I'm (and so is the author) interested in is given by an image and hence decomposes into pixels. The author is proposing the fol... |
Numerical solution of Perona-Malik equation: How to handle the boundary properly? |
I'm reading Serre's A course in Arithmetic and I have the following question about Proposition 14 in Chapter VI. It uses this version of Dirichlet's theorem: Let $m\ge 1$, $(a,m) = 1$. Let $P_a$ be the set of prime numbers such that $p\equiv a \mod m$. The set $P_a$ has density $1/\phi(m)$.
**Proposition 14** Let $... |
Is there a version of the Poincaré inequality which gives the estimate $$\int_\Omega |u-(u)_\Omega|^2~dx\le C\left(\int_\Omega |Du|~dx\right)^2\quad (u\in H^1(\Omega))$$ for a bounded, open domain $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, where $(u)_\Omega= \frac{1}{\mathcal{L}^n(\Omega)}\int_\Omega u ~dx$ ... |
I attempted to prove the above but I am not sure whether this is correct. It goes as follows:
Consider:
For $(t, y), (t_0, y) \in [a,b] \times [c,d]$
\begin{align*}
|F(t, y) - F(t_0, y_0)| & = |\int_{a}^{t} f(x,y) - \int_{a}^{t_0} f(x, y_0)\ dx| \\
& = |\int_{a}^{t_0} f(x,y) dx - \int_{a}^{t_0} f(x,y) dx + \... |
Consider $n$ equally sized circles $c_i$ in the plane with their centers $x_i$ in the vertices $v_i$ of a regular $n$-gon and all meeting in the center of that $n$-gon.
Examples:
- circle pattern for $n=5$:
[![enter image description here][2]][2]
- circle pattern for $n=15$:
[), I found the following approximation for the [Softplus function](https://en.wikipedia.org/wiki/Rectifier_(neural_networks)#Softplus) $f(x)=\ln(1+e^x)$:
$$\ln(1+e^x) \approx \begin{... |
> Let $f : [a,b] \times [c,d] \to \mathbb{R} $ be a continuous function. Let $F(t,y) = \int_{a}^{t} f(x,y) dx$ be a function. Show that F is continuous.
I attempted to prove the above but I am not sure whether this is correct. It goes as follows:
Consider:
For $(t, y), (t_0, y) \in [a,b] \times [c,d]$
\begi... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
Let $b$ be a base, that is, an integer greater than or equal to $2$, and let $n$ be a positive integer that has $d$ digits in base $b$. What is the formal definition of the function that takes a positive integer $m$ less than or equal to $d$, and outputs the $m$-th digit of $n$, starting from the left? I know what it i... |
I am currently trying to work through the proof of Stokes' Theorem given in Lee's Introduction to Smooth Manifolds (2. Edition). I've understood everything for the special cases of $\mathbb{H}^n$ (the upper half space) and $\mathbb{R}^n$, but I'm now stuck at the following step where we assume the (compact) support of ... |
This is problem 7 from last year's IMC (2023), as can be found <a href="https://www.imc-math.org.uk/?year=2023§ion=problems&item=day2">here</a>.
I know, because I composed this problem :]
For these kinds of problems, you often want to use Rolle's Theorem, which states that if $a < b$ and $f$ is continuous on ... |
Let's use the classical definition
$$
\vartheta _1\left( z,q \right) =-i\sum_{n\in \mathbb{Z}}{\left( -1 \right) ^nq^{\left( n+\frac{1}{2} \right) ^2}e^{i\left( 2n+1 \right) z}}\,\,\,\,\,\
q=e^{i\pi \tau}
$$
Now consider function
\begin{equation}
g\left( z \right) =\frac{\vartheta _1\left( z-a,q^{1/3} \right)}{... |
This is problem 7 from last year's IMC (2023), as can be found <a href="https://www.imc-math.org.uk/?year=2023§ion=problems&item=day2">here</a>.
I know, because I composed this problem :]
For these kinds of problems, you often want to use Rolle's Theorem, which states that if $a < b$ and $f$ is continuous on ... |
In orthogonal cartesian coordinates c*c = square of length of vector c
In oblique coordinates the square of the length of c = c * g * c where g is the metric tensor and g*c is a covector
If $e_x$ and $e_y$ are the basis vectors of vector c then how does one calculate the new basis vectors $e_x'$ and $e_y'$ of g*c... |
We show that if $\sigma(n)$ is exponential then the series tends towards $1/2$ but retains an oscillation about this value and does not completely converge.
Let $\sigma(n)=3^n$ and sum from $n=0$ to $n=\infty$, thus
$f(x)=\lim_{x\to1^-}\sum\limits_{n=0}^\infty (-1)^n x^{3^n}.$
The following graph shows a plot ... |
>**Question.**
Let $R$ be a Noetherian ring. Let $M\subset N$ be finitely generated $R$-modules such that $\dim(M/N)>0$. Take a prime filtration $0=K_0\subsetneq K_1\subsetneq ...\subsetneq K_t = M/N$, and let $P_i\in\text{Spec}(R)$ such that $K_i/K_{i-1}\cong R/P_i$ for $1\leq i\leq t$.
I am trying to show that ... |
Let $\chi$ be a $C_c^\infty$ function which equals $1$ on $|x|<1.$ Define
$$
I_{\Phi, \epsilon} = \int e^{i\Phi(x,\theta)}a(x, \theta) \chi(\epsilon \theta) d \theta
$$
for a phase function $\Phi$ with suitable assumptions so that $\lim_{\epsilon} I_{\Phi, \epsilon}$ converges to a distribution in $\mathcal D'(\ma... |
Characterisation of what can be written as an oscillatory integral? |
*Added a few years later: In this answer I found some improvements to traditional bounds (aren't discoveries tho, there are some references, but with less detail), but finally I realized that a time-limited continuous function could be of unbounded rate of change, as counter-example:* $$f(t) = \begin{cases} t\ln(|t|)\e... |
The [Wikipedia stated that][1]
>Let C be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes).
>For all objects A and B in C we define two functors to the category of sets as follows:
>Hom$(-,B):C\rightarrow$ Set
>This is a contravariant functor given by... |
What does $Hom(h, B)$ mean in the contravariant functor? |
I'm currently studying a proof regarding the minimal polynomial's degree, which states that it's less than or equal to \( n^2 \), where \( n \) is the dimension of the vector space on which the operator acts. In the proof, it's mentioned:
*"The dimension of the transformations acting on \( V \) is \( n^2 \)"*
I'm... |
Let X be a random variable that distibuted exp(L) when L is random variable that distibuted exp($\lambda$)
I need to find $f_x(x)$
My first thought was maybe to use the rule of multiplication but it makes it even harder, next I thought about total probability but I can't figure out how to calculate this. Any hint... |
Given a $d$-dimensional box, using only orthogonal cuts and not weird cuts to try to be 'creative', it takes $\sum\limits_{i=1}^{d}\lceil \log_2(s_i)\rceil$ cuts given that the sidelengths of the box are $(s_1,s_2,...,s_d)$. For example, the minimum number of cuts to turn a cube with side lengths of $4$ into $64$ unit ... |
Betweenness is an algorithmic problem in order theory about ordering a collection of items subject to constraints that some items must be placed between others. It has applications in bioinformatics and was shown to be NP-complete by Opatrný (1979).
Problem statement
The input to a betweenness problem is a collecti... |
Reading some material on rational maps between projective varieties, I've seen that some people define them differently. For example, some people define a rational map $f : \mathbb P^n \to \mathbb P^m$ by $f(x) = [f_0(x), \dots, f_m(x)]$ for homogeneous functions $f_0,\dots,f_m$ of degree $d$. However, I have also seen... |
Given a $d$-dimensional box, using only orthogonal cuts and not weird cuts to try to be 'creative', it takes $\sum\limits_{i=1}^{d}\lceil \log_2(s_i)\rceil$ cuts given that the sidelengths of the box are $(s_1,s_2,...,s_d)$. For example, the minimum number of cuts to turn a cube with side lengths of $4$ into $64$ unit ... |
I'm assuming by quotient function, you mean the one from $S^n \rightarrow \mathbb{R}P^n$? Regardless, it makes no difference to the outcome. The way that the quotient function/topology is defined immediately gives you the result.
If you are trying to show that $\mathbb{R}P^n$ is Hausdorff, you can construct a CW co... |
I'm currently studying a proof regarding the minimal polynomial's degree, which states that it's less than or equal to $( n^2)$, where $( n )$ is the dimension of the vector space on which the operator acts. In the proof, it's mentioned:
*"The dimension of the transformations acting on $( V )$ is $( n^2 )$"*
I'm ... |
Is the order of sequences preserved in the limit in general Polish space? |
> Consider
$$
\begin{cases}
-\Delta u &= f\quad \text{in} \ \ \Omega\\
\frac{\partial u}{\partial n} &= g \ \ \ \text{on} \ \ \ \ \partial \Omega,
\end{cases}
$$
where $\Omega \subset \mathbb R^n$ is a bounded domain with boundary $\partial \Omega$ , $\Delta$ is the laplace operator, $f$ and $g$ are given smoot... |
Betweenness is an algorithmic problem in order theory about ordering a collection of items subject to constraints that some items must be placed between others. It has applications in bioinformatics and was shown to be NP-complete by Opatrný (1979).
Problem statement
The input to a betweenness problem is a collecti... |
I'm assuming by quotient function, you mean the one from $S^n \rightarrow \mathbb{R}P^n$? Regardless, it makes no difference to the outcome. The way that the quotient function/topology is defined gives you the result.
**Edit: As David pointed out, quotient maps are not always closed. However, due to the specifics o... |
I'm reading Serre's A course in Arithmetic and I have the following question about Proposition 14 in Chapter VI. It uses this version of Dirichlet's theorem: Let $m\ge 1$, $(a,m) = 1$. Let $P_a$ be the set of prime numbers such that $p\equiv a \mod m$. The set $P_a$ has density $1/\phi(m)$.
**Proposition 14** Let $... |
Are $144, 233, 377,$ and $17711$ the only Fibonacci Numbers that contains only two different digits(in base ten)?
I noticed that $144, 233, 377,$ and $17711$ are all Fibonacci Numbers and they contain only two different digits.
But are there any Fibonacci Numbers not listed above that contains only two different ... |
Are $144, 233, 377,$ and $17711$ the only Fibonacci Numbers that contains only two different digits(in base ten)? |
I need your help with the following exercise:
The sphere $x^{2}+y^{2}+z^{2}=4$ is made of certain material whose density is given by $\rho(x,y,z)=y^{2}+xy+2$. I need to find the extreme values of $\rho(x,y,z)$ using the Lagrange multiplier method.
If $g(x,y,z) = x^{2}+y^{2}+z^{2}-4=0$, $\rho_{x}(x,y,z) = y$, $\rh... |
I am currently reading *Measure, Integration, and Real Analysis* by Sheldon Axler and I am in the middle of the proof for the following lemma:
> Suppose $B ⊆ \mathbb{R}$ is a Borel set. Then for every $ε > 0$, there exists a closed set
$F ⊆ B$ such that $|B\setminus F| < ε$.
The proof begins thus:
> Let $L$ b... |
In the category of sets, $\operatorname{Mor}(\emptyset,A)$ has one element for any $A$, and $\operatorname{Mor}(A,\emptyset)$ has zero elements for any nonempty $A$.
This is because the definition of a function allows for an empty set of ordered pairs.
In set theory, a function is defined as a certain subset of t... |
A sequence of real random variables $X_1,...,X_n$ is said to have a **turning point** at index-value $i$ ($1\lt i\lt n$) iff $X_{i-1}\lt X_{i}\gt X_{i+1}$ or $X_{i-1}\gt X_{i}\lt X_{i+1}$. The number of turning points in the sequence is denoted $T_n$, and we define the **turning-point fraction** as $$R_n={\text{number... |
The "turning-point fraction" of a random sample from a discrete distribution must have expectation less than 2/3? |
You are correct in calling out the constraint $-y_1+y_2+y_3+4y_4\geq 1$, as it shouldn't be $-y_1+y_2+y_3+4y_4\geq 3$. Thus, it appears to be a typo on OP's part.
For the future, an easier way to intuitively convert primal-dual problems is via the [S.O.B. method][1]
\begin{matrix}
&\text{Primal Problem} && \tex... |
Consider two psd and symmetric matrices, namely $\bf{A}$ and $\bf{B}$. Denote $\Vert\cdot\Vert$ as the operator norm. If we have another psd matrix $\bf{B}^*$ that satisfies $\Vert\bf{B}^*\Vert\geqslant\Vert\bf{B}\Vert$, can we get
$$\Vert\bf{A}+\bf{B}\Vert\leqslant\Vert\bf{A}+\bf{B}^*\Vert$$? I have worked out that f... |
Considering the ODE Method for Stochastic Gradient Descent: Suppose $Q$ represents a probability measure on $R^d$, denoting the data distribution, and $L: \mathbb{R}^p \times \mathbb{R}^d \rightarrow \mathbb{R}$ is the loss function, with $\mathbb{R}^p$ as the parameter space and $\mathbb{R}^d$ as the data space. Stoch... |
You are correct in calling out the constraint $-y_1+y_2+y_3+4y_4\geq 1$, as it shouldn't be $-y_1+y_2+y_3+4y_4\geq 3$. Thus, it appears to be a typo on OP's part.
For the future, an easier way to intuitively convert primal-dual problems is via the [S.O.B. method][1]. See [this as well][2].
\begin{matrix}
&\text{... |
I want to compute
$$\lim_{z \to 2e^{i\pi /3}} \frac{z^3+8}{z^4+4z^2+16}$$
Note that if we replace the value
$$(2e^{i\pi /3})^3+8=-8+8=0$$
and
$$(2e^{i\pi /3})^4+4(2e^{i\pi /3})^2+16=16(e^{i 4\pi /3}+e^{i 2\pi /3}+1)=16(-1+1)=0$$
So it is a case of $\frac{0}{0}$ of course my goal is not to use L'Hopital yet, ... |
I want to to implement the Blahut-Arimoto algorithm to evaluate the clustering membership probability, $p(\hat{x}|\mathbf{x})$, with fixed number of clusters, $N_c$, and compression-distortion tradeoff parameter, $\beta$. However, I need help on how to calculate the distortion measure $d(x,\hat{x})$.
I have a data ... |
> Let $f: D(0,1)\to\mathbb{C}$ be holomorphic s.t $\Re
> (f'(z))>0\quad\forall z\in D(0,1)$. Prove that $f$ is injective.
My attempt:\
Since $f'(z)=\dfrac{\partial u}{\partial x}+i\dfrac{\partial v}{\partial x}=\dfrac{\partial v}{\partial y}-i\dfrac{\partial u}{\partial y}$ (Cauchy Riemann), we have $\dfrac{\part... |
I want to to implement the Blahut-Arimoto algorithm to evaluate the clustering membership probability, $p(\hat{x}|\mathbf{x})$, with fixed number of clusters, $N_c$, and compression-distortion tradeoff parameter, $\beta$. However, I need help on how to calculate the distortion measure $d(x,\hat{x})$.
I have a data ... |
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