instruction stringlengths 12 30k |
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I have the following function that is for the stereographic projection:
$$f(x, y) = \dfrac{1}{\kappa}(2x, 2y, x^2 + y^2 - 1)\hspace{0.3cm} \text{where} \hspace{0.3cm} \kappa = x^2 + y^2 + 1.$$
So let now I calculate the partial derivatives of the function, giving me this
$$Dfe_{1} = \dfrac{\partial f}{\partial x} = ... |
What is the probability of rolling $n$ consecutive numbers when rolling $k$ dice? |
Consider the problem: $y' = xy(y+2) $<br/>
If we consider the point $I(x,y)$ satisfying $y(y + 2) \neq 0$, we have a normal solution. <br/> But in the opposite case, why do we deduce it only has solutions $y = 0$ and $y = -2$. <br/>If according to my understanding, by Wiki : https://en.wikipedia.org/wiki/Ordinary_dif... |
>Assume that $Q(x)$ is a polynomial with integer coefficients. Is there a prime number $p$ such that the equation $Q(x)=0$ has all its roots in the finite field $\mathbb{Z}/p\mathbb{Z}$?
I asked this question in RG, too.
https://www.researchgate.net/post/Can_a_given_polynomial_with_integer_coefficients_be_spl... |
Consider a second-order parabolic pde
$$
u_t = \Delta u + F(t,x,u,\nabla u), \quad (t,x) \in [0,+\infty) \times M
$$
where $M$ is a compact, smooth manifold with boundary. We assume that $F(x,t,u,p)$ is smooth everywhere except at one interior point $x_0 \in M$ and we set a smooth (parabolic) boundary condition... |
Thomas Kojar has provided an answer and some references already, but here is an intuitive explanation, in order to stay in the same context/spirit as your question.
**1) Why $\mathrm{d}x^2 \sim 0$ in the standard case ?**
It is to be recalled that, in the standard case of a function depending on a non-stochastic ... |
In a [comment](https://math.stackexchange.com/posts/comments/10429851) elsewhere, @Justauser pointed out that, for separably closed fields, your claim follows from the general machinery that algebraic representations carry semisimple (respectively, nilpotent) elements to semisimple (respectively, nilpotent elements). ... |
Consider the following question (I only need help with the last part)
>The dynamics of a directly transmitted microparasite can be modelled by the system
\begin{align*}
\frac{d X}{d t} & =b N-\beta X Y-b X, \\
\frac{d Y}{d t} & =\beta X Y-(b+r) Y, \\
\frac{d Z}{d t} & =r Y-b Z,
\end{align*}
where $b, \beta$ an... |
Consider the following question (I only need help with the last part)
>The dynamics of a directly transmitted microparasite can be modelled by the system
\begin{align*}
\frac{d X}{d t} & =b N-\beta X Y-b X, \\
\frac{d Y}{d t} & =\beta X Y-(b+r) Y, \\
\frac{d Z}{d t} & =r Y-b Z,
\end{align*}
where $b, \beta$ an... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
What is the probability of rolling at least $n$ consecutive numbers when rolling $k$ dice? |
I have several log(y) ~ a + b * log(x) models that I want to express as exponential relationships. I know this involves an expoential transformation, but how do I solve it?
Example:
Step 1: log(y) ~ 2 + 3 * log(x)
Step 2: exp(log(y) ~ exp(2 + 3 * log(x))
Step 3: exp(log(y) ~ exp(2) + exp(3 * log(x))
what... |
Let $f \in \mathbb{Z}[x]$ be non constant. How to prove that there exist infinitely many primes $p$ such $f$ have root in $\mathbb{Z}/p\mathbb Z$? |
I'm trying to understand better optimisation problems and in particular the KKT conditions. To this end, consider the minimisation problem
$\min(x^2+y^2)$ subject to $x\ge1$ and $y\ge -2$.
It's clear that the solution is $x=1, y=0$. However, how do I get this by applying the KKT conditions? And is this a kind of pr... |
How to solve $\min(x^2+y^2)$, $x\ge1, y\ge-2$, using the KKT conditions? |
This is a linear programming problem I was given in my semester examinations. The question is attached as an image. Given its size , I couldn't type it out.
[LPP problem][1]
[1]: https://i.stack.imgur.com/dxnGf.jpg
So basically we have to come up with all the inequalities for this problem given the conditi... |
If $A$ is an unital associative $\mathbb C-$algebra, and it admits a decomposition
$$
A = U V \; ,
$$
where $U,V$ are subalgebras of $A$ and $U \cap V= \mathbb C \textbf 1$, then can we show that
$$
A \cong U \otimes V \; ?
$$
If we consider the homomorphism
$$
\begin{matrix}
\varphi: & UV & \longrightar... |
How to solve the problem $\min c^T x$ subject to $Ax=0$ and $\|x\|_2^2\le1$ using KKT? |
The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation.
\begin{equation}
d X_t = \mu X_t \cdot dt + \sigma X_t^\beta \cdot d B_t \tag{1}
\end{equation}
where $\mu,\sigma,\beta$ are positive real parameters and $B_t$ is t... |
Let's drop the informal terminology of "reaching in", "mutating", "changing arrow directions" and so on and treat this exercise somewhat more formally; this will hopefully clear up the confusion. Namely, let's construct an isomorphism (that is, an invertible functor) $F\colon (c / \mathcal{C}^\mathrm{op})^\mathrm{op} \... |
### Background
In physics, there is the concept of "[mean free path][1]," which is the expected value for the distance a molecule (for example) can travel before it hits another one.
If they're all the same size, and the neighbor molecules are motionless, this [can be calculated][2] as $\frac{1}{\pi d^2 n_v}$ (wh... |
How to determine the direction of rotation for purely imaginary eigenvalues by conditions on eigenvalues and eigenvectors? |
I have been long time puzzled by the correct application of Fubinis theorem. Is it true that the bounds need to be constant if the integrals can be interchanged? Please correct if not.
Let's consider the following Laplace transform pair.
\begin{align}
\mathcal{L}^{-1}\Bigl[\frac{2(\cos(at)-\cos(bt))}{t}\Bigr] = ... |
It is well known that $\int_0^\pi \sin(nx) \sin(m x)dx = \dfrac{\pi}{2}I(m = n)$. By using an online integration calculator, I found that $$\int_0^\pi \sin^2(x) \sin(nx) \sin(mx) = \dfrac{\pi}{4}I(m=n)$$, when $m,n\neq 1$. (I just tried various $n,m$ and observe this pattern).
I have two question. First, how can we ... |
Given convex polyhedron with N vertices, what is the largest number of vertices that can have a polygon which is intersection of polyhedron with some plane? My lower bound formula is M = 2*N - 4 |
For the sake of familiarity, let's suppose that you have a large number $n$ written in base ten, and want to know if it's a prime number. A simple test is to look at the last digit.
* If it's 0, 2, 4, 6, or 8, then $n$ is a multiple of 2 (aka “even”), so not prime.
* If it's 0 or 5, then $n$ is a multiple of 5, ... |
It is well known that $\int_0^\pi \sin(nx) \sin(m x)dx = \dfrac{\pi}{2}I(m = n)$. By using an online integration calculator, I found that $$\int_0^\pi \sin^2(x) \sin(nx) \sin(mx) = \dfrac{\pi}{4}I(m=n)$$, when $m,n\neq 1$. (I just tried various $n,m$ and observed this pattern).
I have two questions. First, how can w... |
I have this equation:
$y = e^{t(A + W)} x_0 $
where A is a diagonal matrix and W is a symmetric matrix.
I need to find $\frac{\partial y}{\partial W}$.
If A and W commute then I could use the fact that
$e^{t(A + W)} = e^{tA} . e^{tW} $ and then use the kronecker product:
$\frac{\partial y}{\part... |
I have several log(y) ~ a + b * log(x) models that I want to express as exponential relationships. I know this involves an expoential transformation, but how do I solve it?
Example:
Step 1: log(y) ~ 2 + 3 * log(x)
Step 2: exp(log(y)) ~ exp(2 + 3 * log(x))
Step 3: exp(log(y)) ~ exp(2) + exp(3 * log(x))
wh... |
I am trying to solve the particle in a disk problem and arrived at the radial part ODE, shown in bellow.
$$r^2\frac{\mathrm{d^2} \phi_r}{\mathrm{d^2} r} + r\frac{\mathrm{d} \phi_r}{\mathrm{d} r} + (\omega^2r^2-\ell^2)\phi_r = 0$$
What I found, after some research, is that this ODE is a special case of the Bessel ... |
The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation.
\begin{equation}
d X_t = \mu X_t \cdot dt + \sigma X_t^\beta \cdot d B_t \tag{1}
\end{equation}
where $\mu,\sigma,\beta$ are positive real parameters and $B_t$ is t... |
For $u,v\in \mathcal{S}(\mathbb R^d)$, $s,s_1,s_2\in\mathbb R$,
$$
\|uv\|_{H^s}\lesssim_{d,s,s
_1}\|u\|_{H^{s_1}}\|v\|_{H^{s_2}}.
$$
where $s_1+s_2=s+d/2>0$.
**My attempts:** We work in a direct estimate in Fourier side except introducing Bony's decomposition. Indeed, $$
|(\hat{u}\ast \hat{v})(\xi)|\le \left... |
### Background
In physics, there is the concept of "[mean free path][1]," which is the expected value for the distance a molecule (for example) can travel before it hits another one.
If they're all the same size, and the neighbor molecules are motionless, this [can be calculated][2] as $\frac{1}{\pi d^2 n_v}$ (wh... |
I'm self-studying from Abbott's Real Analysis. They begin with an attempt to prove that the pointwise limit of continuous functions is continuous, and show that it can't be done.
Assume $(f_n)$ is a sequence of continuous functions on $A\subseteq \mathbb{R}$ converging pointwise to limit $f$. Fix $c\in A$, and let ... |
# Problem
> Let $X=(X_t)_{t\geq0}$ be an (a.s.) continuous $\mathbb{R}$-valued process with $X_0=0$ such that $(e^{i\lambda X_t + \frac{1}{2}\lambda^2 t})_{t\geq 0}$ is a $\mathbb{C}$-valued local martingale for all $\lambda\in\mathbb{R}$. Show that $X$ is a standard $\mathbb{R}$-valued $(\mathcal{F}_t)_{t\geq 0}$-B... |
Is $\exists y\forall x(Pxy\land Qy)$ always false? I ask because of the following:
1. Let the domain of discourse be the set where x and y are natural numbers and y is always greater than x.
2. Let P be the set of Prime Numbers
3. $\exists x\exists y\left(y>x\land y\in P\right)\rightarrow\exists y\forall x\left(y>x\... |
Is $\exists y\forall x(Pxy\land Qy)$ always false? |
Suppose that $\Omega \subseteq \mathbb R^n$ non-empty, open, and $n \ge 1$.
We have an increasing collection $(K_i)_{i \in \mathbb N}$ of compact subsets of $\Omega$ with union $\Omega$.
For $K \subset \Omega$ fixed and compact, is it necessarily the case that $K \subseteq K_i$ for $i$ sufficiently large?
I... |
**TASK**: Let $(X_1, X_2)$ is a random i.i.d. sample from $N(0, \sigma ^2)$ distribution. Find the unbiased estimator for the parameter $\sigma$.
**SOLUTION**: The estimator $\theta$ is unbiased if $E\theta = \theta$.
In this case $\theta = \sigma ^2$, and $\sigma^2 = \frac{1}{2}( X_1^2 + X_2^2$).
So $E\sigma^2... |
I'm working on a pet project and I've reached a sticking point, I'll describe the problem, then I'll describe the direction I've been trying.
We have a sphere of radius $R$, filled with $N$ points. I don't think it matters, but $N$ is very high. Each point has probability of an event $E$ occuring that depends on it'... |
Let $R$ and $S$ be two rings with unity. If $R$ and $S$ are Morita equivalent (the category of $R$ modules and $S$ modules are equivalent), does this mean $R[t]$ and $S[t]$ are Morita equivalent, for some indeterminate $t$? Let's also assume $t$ commutes with elements in $R$ and $S$. If not, does anyone know conditions... |
Is Morita equivalence preserved when adjoining a variable? |
Let's say I have a stochastic optimization problem of the type ($x \in \mathbb{R}^n$):
$$J(x,t)= \max_u E\bigg [\int_t^{+\infty} g(x,u)e^{-\rho s}ds\bigg]$$
The SHJB equation for this is ($J$ is the value function, $\mu (x,t,u)$ is the drift, I'm assuming constant $\sigma$ and $\nu = \sigma \sigma'$).
$$\rho J = \... |
In a [comment](https://math.stackexchange.com/posts/comments/10429851) elsewhere, @Justauser pointed out that, for separably closed fields, your claim follows from the general machinery that algebraic representations carry semisimple (respectively, nilpotent) elements to semisimple (respectively, nilpotent elements). ... |
The Fourier transform of a function is given by $$f(ξ) = \int_{-∞}^{∞}
f(x)
e^{-2πixξ}dx$$
the paper I was reading from says that the test function $e^{2πixξ}$ is a periodic with period $2\pi/ξ$ and it continues by saying ("so integrating $f$ against this test function gives
information about the extent to which... |
In a [comment](https://math.stackexchange.com/posts/comments/10429851) elsewhere, @Justauser pointed out that, for separably closed fields, your claim follows from the general machinery that algebraic representations carry semisimple (respectively, nilpotent) elements to semisimple (respectively, nilpotent elements). ... |
I am recently reading about Fourier transforms and convolutions. It was a surprise to me that it takes quite several paragraphs to prove the measurability of innocent looking $f(x-y)$ (reference: https://math.stackexchange.com/questions/508158/proof-that-hatfx-y-fx-y-is-measurable-if-f-is-measurable-stein) . It makes m... |
In a [comment](https://math.stackexchange.com/posts/comments/10429851) elsewhere, @Justauser pointed out that, for separably closed fields, your claim follows from the general machinery that algebraic representations carry semisimple (respectively, nilpotent) elements to semisimple (respectively, nilpotent elements). ... |
(EDIT: I’ve updated this question to feature a new cubic. Sorry for asking so many similar questions. The responses that I’ve received so far have been extremely helpful.)
>Consider a cubic in $P^2(\mathbb{C})$ defined by a conic and its tangent. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds t... |
I need help with the last part of the following question:
>The number density of a population of amoebae is $n(\mathbf{x}, t)$. The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration $a(\mathbf{x}, t)$. The equations governing $n$ and $a$ are
$$
\begin{aligned... |
What type of perturbation am I considering here? |
Let's call an *algebraic invariant* (under the action of $\text{GL}_n(\mathbb{C})$) a tensor of order $N$ to an algebraic combination of the components of that tensor. In other words, a homogeneous polynomial in the components of the tensor.
* For $N = 2$ algebraic invariants are well known: given a second-order tens... |
Suppose $\{Q_{nj}\in\mathbb{R}^{p\times p}\}_{n, j\in\mathbb{N}\cup\{0\}}$ are invertible, $Q_{nj}^{\sf{T}} = Q_{jn}$ and every $Q_{nn}$ is positive-definite and $\{s_n\in\mathbb{R}^p\}_{n\in\mathbb{N\cup\{0\}}}$ are bounded (with respect to some norm, say, the $\ell^2$-norm). I wish to find the solution (and its prope... |
# Theorem 9.12
For the second paragraph of the proof, I do not see how it is obvious that $\theta^{(p)}$ is the unique, maximal extension of all such integral curves. Without appealing to a Zorn's Lemma argument.
Edit March 28: The Zorn's Lemma argument is unnecessary because we can view the integral curves as a... |
Stabilization problem is a fundamental problem in control theory. There are many literature to achieve stabilization but fewer results are related in computational complexity. Consequently, we consider the computational complexity of stabilization problem.
Given a linear Boolean control network (BCN)
$$x(t+1)=Ax(t)... |
I have stumbled upon two different definitions of a submodule and I cant see why they are equivalent, let me state them ($M$ is a module and $R$ a commutative ring):
First Definition: A set $N \subset M$ is called a submodule of $M$ if $$an + bm \in N$$ for all $a,b \in R$ and all $m,n \in N$
Second definit... |
I'm going through Euclid's Elements (Book 1) (Proposition 4). We're given two triangles (ABC, DEF) where two sides of the first triangle are equal to the two other sides (say AB=DE and AC=DF) of the second triangle. The angle between these two sides are also equal so (A=D).
In the proof, we place vertex A onto verte... |
Given two triangles, if the vertices of the first triangle coincide with the vertices of the second triangle, are the triangles are equal? |
I came across the following statement:
For any finite $S \subset \mathbb{Z}^{d}$, there exists $i \in \lbrace 1,\cdots,d\rbrace$ such that
\begin{align*}
|P_{i}\left(S\right)| \geq |S|^{\frac{2}{3}}.
\end{align*}
A well-known statement in this direction would be the "Discrete Loomis and Whitney Inequality". Tr... |
(EDIT: I’ve updated this question to feature a new cubic. Sorry for asking so many similar questions. The responses that I’ve received so far have been extremely helpful.)
>Consider a cubic in $P^2(\mathbb{C})$ defined by a conic and its tangent. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds t... |
I study from "ِ sequences and series in Banach spaces" by j.Diestel
A series $\sum_n x_n$ is said to be weakly unconditionally Cauchy (wuC) if, given
any permutation $\pi$ of the natural numbers, $(\sum_{k=1}^n x_{\pi(k)})$ is a weakly Cauchy sequence; alternatively, $\sum_n x_n$ is wuC if and only if for each $x^* ... |
I came across the following statement:
For any finite $S \subset \mathbb{Z}^{d}$, there exists $i \in \lbrace 1,\cdots,d\rbrace$ such that
\begin{align*}
|P_{i}\left(S\right)| \geq |S|^{\frac{2}{3}}.
\end{align*}
where $P_{i}$ be the projeciton on the hyperplane $H_{i}(0) := \lbrace \sum_{i \neq j} k_{j}e_{j} ... |
This question is a response to the answer given in [this SE post][1], concerning the use of the Vitali covering lemma to prove the Hardy-Littlewood maximal inequality in general dimension. For reference, the covering lemma is
> **Lemma (Vitali).** Let $B_1,\ldots,B_n$ be a finite collection of open balls in $\mathbb... |
I came across the following statement:
For any finite $S \subset \mathbb{Z}^{d}$, there exists $i \in \lbrace 1,\cdots,d\rbrace$ such that
$\begin{align*}
|P_{i}\left(S\right)| \geq |S|^{\frac{2}{3}}.
\end{align*}$ where $P_{i}$ be the projeciton on the hyperplane $H_{i}(0) := \lbrace \sum_{i \neq j} k_{j}e_{j} :... |
I'm working on a pet project and I've reached a sticking point, I'll describe the problem, then I'll describe the direction I've been trying.
We have a sphere of radius $R$, filled with $N$ points. I don't think it matters, but $N$ is very high. Each point has probability of an event $E$ occurring that depends on it... |
I'm working through *Introduction to Lie Algebras* by Erdmann and Wildon. In Chapter 3, Exercise 3.9 asks the reader to develop the semidirect product of Lie algebras. I got very stuck on the verification of the Jacobi identity, but the link [here](https://math.stackexchange.com/questions/1684193/verifying-the-jacobi... |
I am preparing data for spatial correlation analysis. In one column I have integer values '<5' and values ranging from 5 to 2321.
[![Example][1]][1]
[1]: https://i.stack.imgur.com/hD21G.png
The values represent counts of cases that I intend to use with the population column to get a case rate. What is the a... |
Let $K$ be a $(p,q)$ torus knot on the torus $T_1$. Via the map
\begin{equation*}
H=\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
\end{equation*}
$K$ becomes a $(q,p)$ torus knot on $T_2=H(T_1)$. If we fill in both these tori to become solid tori $D_1$ and $D_2$, then
\begin{equation*}
S^3 = D_1\cu... |
Does a torus knot give a Seifert fibering of the 3-sphere? |
Suppose I know that for some values $x,a,b$ that
$$
x = 0\mod a
$$
$$
x = k\mod b.
$$
Is there a simple expression that I can use to get the value of $x$ $\mod ab$ assuming that $a,b$ are coprime?
I know from prior threads that I can use the [extended euclidean algorithm][1] to find this value, however I ... |
If $x=0\mod a$ and $x=k\mod b$, is there a simple expression for $x \mod ab$? |
If $x\equiv0\pmod a$ and $x\equiv k\pmod b$, is there a simple expression for $x \pmod {ab}$? |
Let's say I know the position $e \in \mathbb{R}^N$ of a local extremum point of a non-convex function $f: \mathbb{R}^N \mapsto \mathbb{R}$. Is there an efficient method for finding $K$ closest "neighbouring" extremum points or all extremum points within a specific radius from the extremum point $e$? One option would be... |
Can a function be converted into a Voronoi diagram of its local extremum points and basins of convergence? |
Suppose I know that for some values $x,a,b$ that
$$
x \equiv 0\pmod a
$$
$$
x \equiv k\pmod b.
$$
Is there a simple expression that I can use to get the value of $x$ $\pmod{ab}$ assuming that $a,b$ are coprime?
I know from prior threads that I can use the [extended euclidean algorithm][1] to find this val... |
This is a question on one part from Tao's note: https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/. Refer to Prediction $3$ (Riemann hypothesis).
Construct a random model ${{\mathcal P}'}$ of the primes ${{\mathcal P}}$ to be a random subset of th... |
During my studying linear algebra, I came across the following problem:
> Let $A$ be a square matrix with real entries, $k$ be a positive integer, and assume $\operatorname{rank}(A^k)=\operatorname{rank}(A^{k+1})$.
Then, prove $\operatorname{rank}(A^k)=\operatorname{rank}(A^m)$ for any integer $m \ge k$.
Below i... |
If one were to take the Fibonacci numbers and arrange them so that the units digit of the nth Fibonacci number is in the $(n+1)th$ decimal place and sum these numbers up they would get the reciprocal of the eleventh number $\frac{1}{89}$ or 0.011235955. This might look something like:
$$ 0.01 + .0001 + 0.0002 + 0.00... |
Why does the sum of the set of Fibonacci numbers arranged where the units digit of the nth Fibonacci number is in the (n+1)th place equal 1/89? |
If one were to take the Fibonacci numbers and arrange them so that the units digit of the nth Fibonacci number is in the $(n+1)th$ decimal place and sum these numbers up they would get the reciprocal of the eleventh number $\frac{1}{89}$ or 0.011235955. This might look something like:
0.01 + .0001 + 0.0002 + 0.00003... |
If $A$ is an unital associative $\mathbb C-$algebra, and it admits a decomposition
$$
A = U V \; ,
$$
where $U,V$ are subalgebras of $A$ and $U \cap V= \mathbb C \textbf 1$, then can we show that
$$
A \cong U \otimes V \; ?
$$
If we consider the map
$$
\begin{matrix}
\varphi: & U \otimes V & \longrightarr... |
Let's say I know the position $e \in \mathbb{R}^N$ of a local extremum point of a non-convex function $f: \mathbb{R}^N \mapsto \mathbb{R}$. Is there an efficient method for finding $K$ closest "neighbouring" extremum points or all extremum points within a specific radius from the extremum point $e$? One option would be... |
For a matrix $A$, by $||A||$, I mean the matrix-norm induced by the $\ell^2$-norm. Let $A\in\mathbb{R}^{m\times m}$, $B\in\mathbb{R}^{m\times p}$, $C\in\mathbb{R}^{n\times m}$ with $||A|| < 1$. Define,
$$
N_k :=
\begin{cases}
CA^{k - 1}B, &\text{if }k \geq 1\\\\
0, &\text{otherwise}.
\end{cases}
$$
Define, $Q_{... |
Is this infinite sum of matrices convergent? |
In Veritasium's new video about 37 there is brought up something interesting about it's multiples. \
For any multiple of 37 reverse it and put a 0 between all of its digits and the new number will be a multiple for 37
Example :
$37 \rightarrow703 ,703=19*37$\
$74\rightarrow407,407=11*37$\
Why does this happen can ... |
I am trying to understand measure concentration on a unit sphere, using this text book.
Lectures on Discrete Geometry
by Jiří Matoušek https://link.springer.com/book/10.1007/978-1-4613-0039-7
On page 331, there is a theorem (Measure concentration for the sphere) as follows.
Let $A\subseteq S^{n-1}$ be a meas... |
Given a positive integer $n$ and a primitive $m^{th}$ root of unity $\omega_m$ over $\Bbb Q$, how could one determine if $\sqrt{n}$ lies in $\Bbb Q[\omega_m]$?
In the case of $n=p>0$ being an odd prime, the question is known by some algebraic number theory. Indeed primes ramified in $\Bbb Q\left[\sqrt p\right]$ are ... |
When is $\sqrt n$ in $\Bbb Q[\omega_m]$? |
If $A$ is an unital associative $\mathbb C-$algebra, and it admits a decomposition
$$
A = U V \; ,
$$
where $U,V$ are subalgebras of $A$ and $U \cap V= \mathbb C \textbf 1$, then can we show that
$$
A \cong U \otimes V \; ?
$$
If we consider the map
$$
\begin{matrix}
\varphi: & U \otimes V & \longrightarr... |
Let $A$ be an invertible opeartor, $B$ being another operator, is \|ABA^{-1}\| \leq \|B\|? |
Given a positive integer $n$ and a primitive $m^{th}$ root of unity $\omega_m$ over $\Bbb Q$, how could one determine if $\sqrt{n}$ lies in $\Bbb Q[\omega_m]$?
In the case of $n=p>0$ being an odd prime, the question is known by some algebraic number theory. Indeed primes ramified in $\Bbb Q\left[\sqrt p\right]$ are ... |
Let's consider a system of $n$ equations and $n$ real unknowns $(x_1, \dots, x_n)$:
$$\sum_{k=1}^n A_{ik} x_k = B_i, \qquad \text{with } i\in\{1,\dots,n\}$$
The solution of which exists as long as the determinant $A = \det(A_{ij})$ is not zero. However, the seemingly innocent second-order system given by:
$$\s... |
What is the simple reason why there is no simple solution to a system of quadratic equations? |
**An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?**
*PS: Thanks everyone for your comments and interesting answers showing how currently the logarithm function is numerically calculated, but so far* **nobody is answering the question I am asking for,** *which is relat... |
In Veritasium's new video about 37 there is brought up something interesting about its multiples. \
For any multiple of 37 reverse it and put a 0 between all of its digits and the new number will be a multiple for 37
Example :
$37 \rightarrow703 ,703=19*37$\
$74\rightarrow407,407=11*37$\
Why does this happen can s... |
Can someone solve this limit? |
I read these
1-introduction to analytic number theory by Tom M.Apostol
2-the prime number theorem by G.J.O.Jameson
3-the distribution of prime numbers by A.E.Ingham
4-the paper "Special Values of the Polygamy Functions" by M.RAM MUTRY
5-the paper improved explicit estimate for zeta function" in 2... |
This is a question on one part from Tao's note: https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/. Refer to Prediction $3$ (Riemann hypothesis).
Construct a random model ${{\mathcal P}'}$ of the primes ${{\mathcal P}}$ to be a random subset of th... |
I was reading McGuire's [paper][1] on why the minimum number of clues in a Sudoku puzzle is 17 when I came across a curious comment:
> n 2008, a 17-year-old girl submitted a proof of the nonexistence of a 16-clue sudoku puzzle as
an entry to Jugend forscht (the German national science competition for high-school st... |
What was the gap in Ariane Papke's proof that the minimum number of sudoku clues is 17? |
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