instruction stringlengths 12 30k |
|---|
Problem: Find a nice simple closed curve other than circle which passes through the points $(0,0)$ and $(1,0)$ on the Cartesian plane and whose length is $L\pi$.
If the given condition is not the loop length but the loop area, it is easy to find a nice curve. The ellipse
$$(2x-1)^2+\frac{y^2}{4A^2}=1$$
has the are... |
An amorphous set is a set that cannot be partitioned into $2$ disjoint infinite sets.
If $A$ is an amorphous set then it has an equal number of finite subsets and infinite subsets, with the bijection $x \mapsto x^c$. Is the opposite also true? That is, given a nonempty set $x$ with an equal number of infinite and fi... |
Let $G$ be a compact topological group and $f:G\to G$ a continuous homomorphism. Assume there is a dense subset $X\subset G$, such that that for all $x\in X$, we have that $f(x)$ is conjugate to $x$. Is it then true that $f(g)$ is conjugate to $g$ for all $g\in G$?
If $G$ were a metric space, we could make a sequenc... |
Could anyone point to some literature regarding the construction of multivalued functions as holomorphic functions on a differentiable manifold with complex structure? For example with the function $f(z) = \sqrt{z}$. One considers the sheets $F_1$ and $F_2$ respectively given by $f(z)$ and $-f(z)$. Taking as a branch f... |
I have a Lie algebra that breaks up into as subalgebra $B$ and its complement $\overline B$ (dividing the set of Lie generators into mutually exclusive subsets $B$ and $\overline B$ such that $B$ is a subalgebra but $\overline B$ is not). Then I find the property:
$$
\forall L_a \in B, \forall J_b \in \overline B:
... |
If have a point on an arc and other points related to it. I now want to move the point on the arc to a given target and rotate everything so that the arc-based-point has a vertical angle as the new position (like the if the arc-center would now be just below (on y-axis) the target.
Example: I have an arc-center $R(0... |
Calculate the volume of solid consisting of the cylinder $x^2+y^2\leq 4, 0 \leq z \leq 2$ and by cone $x^2+y^2\leq z^2, 2\leq z \leq 5.$
I tried to draw the figure on geogebra and I'm trying to use cylinder coordenate but I could not.
My attempt: put $x=r\cos(\theta), y=r \sin(\theta)$ and $z=z.$
We have $... |
"Let $C\subset\mathbb{R}^2$ be a regular curve and let $\alpha:I\rightarrow C$ be its natural parametrization. Consider $p=\alpha(s_0)$ and define $\varphi_p:I\rightarrow C$ as $\varphi_p(s)=\parallel\alpha(s)-p\parallel^2 \forall s\in I$. Prove that f is differentiable (I mean $C^\infty$) in a neighborhood of $s_0$ ... |
For $u,v\in \mathcal{S}(\mathbb R^d)$, $s,s_1,s_2\in\mathbb R$,
$$
\|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... |
Does this proof for product estimate in Sobolev spaces work? |
$$\lim_{(x,y)->(0,0)}\frac{-x^6y^1(x^2+1)}{(x^6+y^2)\sqrt{x^2+y^2}}$$
I need to find out if the limit exists. First, I approached the limit from $y=0$, and the result was $\frac{0}{x^7}$. Then, I approached from $x = 0$, and the result was $\frac{0}{y^3}$. This made me assume that the limit does not exist. However, ... |
I have been asking to myself for a while now why $2$ has such an exceptional behaviour in algebraic number theory. For example, the Kronecker-Weber Theorem proof was completed for all cases but that of number fields of degree a power of 2 by Kronecker, and a full proof was found by [Neumann/Hilbert only after $40/50$ y... |
Why is $2$ so troublesome a prime? |
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 V.V. 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 satis... |
"Let $C\subset\mathbb{R}^2$ be a regular curve and let $\alpha:I\rightarrow C$ be its natural parametrization. Consider $p=\alpha(s_0)$ and define $\varphi_p:I\rightarrow C$ as $\varphi_p(s)=\parallel\alpha(s)-p\parallel^2 \forall s\in I$. Prove that f is differentiable (I mean $C^\infty$) in a neighborhood of $s_0$ ... |
> since each right ideal of a can be considered an algebra over $$
An algebra over $A$? In what sense? Algebras are usually not defined over noncommutative rings, and that is what your context allows for.
> However by definition of minimility of both ideals we get that $=_$
You could only get equality if one ... |
Is the following statement true?
>Let $k\geq 1$ be an integer and $N=3\cdot 2^k+1$. Then $N$ is a prime prime if and only if $2^{N-1}\equiv 1 \pmod {N}$
One implication is simply a Fermat's Little Theorem, so the difficult part is proving or disproving that $2^{N-1}\equiv 1 \pmod {N}$ implies $N$ is a prime. Or i... |
I'm a beginner in complex geometry and I want to ask a question about the definition which I did not find explicit formula.
What's the meaning of **'a connection preserve the complex structure'**?
$$
\nabla J=0.
$$
Does it mean that for local coordinate $\left(U ; x^i, y^i\right)$, for any two $v_1$ and $v_2$ in... |
We work in $\sf ZF$.
An amorphous set is a set that cannot be partitioned into $2$ disjoint infinite sets.
If $A$ is an amorphous set then it has an equal number of finite subsets and infinite subsets, with the bijection $x \mapsto x^c$. Is the opposite also true? That is, given a nonempty set $x$ with an equal n... |
50% don't smoke, 20% light smokers, 30% heavy smokers.
heavy smokers are twice as likely to die prematurely as light smokers, light smokers are twice as likely to die prematurely as nonsmokers.
What's the probability of being a heavy smoker given person died prematurely?
I'm sure this would be a Bayes' theore... |
I have a question regarding quasiconvexity and its usage in CVXPY.
I have the following optimization problem.
\begin{equation*}
\begin{aligned}
\min_{x} \quad & \sqrt x\\
\textrm{subject to:} \quad & 1 \leq x \leq 2\\
\end{aligned}
\end{equation*}
The solution is trivial $x^* = 1, \ p^* = 1$ but I want to... |
I have an equation of $y = \lfloor 400(x-6)^{1.1} \rfloor$ where x is equal to or greater than 6 and increases by an increment of 1.
I want to calculate what the sum of the equations added up together might be.
For example: $\lfloor 400([x+1]-6)^{1.1} \rfloor + \lfloor 400([x+2]-6)^{1.1} \rfloor + \lfloor 400([x+... |
Given the ANOVA model $Y_{ij} = \mu_i + \varepsilon_{ij}, \varepsilon_{ij}\sim N(0, \sigma^2)$, $i = 1, 2, \ldots , I, \space j = 1,2, \ldots, n_i$, I am trying to find the MLE's $\hat\mu_1, \hat\mu_2, \ldots , \hat\mu_n, \hat\sigma^2$.
I have that the likelihood function is
$L = \prod_{i=1}^{I}\prod_{j=1}^{n_i} ... |
I've been studying the slope of the sawtooth waveform $$\sum_{n=1}^\infty \frac{sin(nx)}{x}$$ And found that, for $x \in (0, 2\pi)$, its graph seems to line up perfectly with the line $$y = \frac{x}{2}+\gamma+1$$ Where $\gamma$ is the Euler-Mascheroni constant, $\approx.57721566...$ What is this constant doing here? |
Connection between the sawtooth Fourier series and the Euler-Mascheroni constant? |
How to prove that ${\displaystyle \, \, \lim_{r \to 0} \, \sup_{x \in \mathbb R^n} \, r^{-\lambda} \int_{B(x,r)} |f_\alpha(y)|^p \, dy = 0}$? |
A faster method would be assuming 6 is also cashing out, then dividing by 2 (because 6 and 5 are identical other than the fact that 6 multiplies your score by 0). you have a 1/3 chance of cashing out each turn, meaning each outcome is expected to be rolled "half a time". the calculation $\frac{\frac{1+2+3+4}{2}}{2} =2.... |
I am trying to calculate this expression:
$\int dp \text{exp} ;\ [-Aip^2+iBp]$
where A and B are real. How can I visualize this expression and will it yield a sensible result? |
I am trying to calculate this expression:
$\int dp \text{exp} [-Aip^2+iBp]$
where A and B are real. How can I visualize this expression and will it yield a sensible result? |
I am trying to calculate this expression:
$\int dp \text{exp} [-Aip^2+iBp]$
where A and B are real. How can I visualize this expression and will it yield a sensible result? |
> Find the Green's function for the BVP $$y''-\frac1xy=0 \ \ ; \ \ y(0)=y(1)=0$$
Clearly the operator is not self-adjoint, so the equivalent self-adjoint equation is $$\left(\frac{y'}{x}\right)'=0$$ Therefore the Green's function could be taken as $$G(x,t)=\begin{cases}A+Bx^2 & \text{if} \ \ 0\leq x<t \\ C+Dx^2 & \t... |
I want to find **and proof** a closed formula for the following sum $$\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\dots +\sqrt{1+\frac{1}{k^2}+\frac{1}{(k+1)^2}}$$
I have found a closed formula but I have problems with proofing it and... |
This is called a [Riemann surface][1], in the case of functions of one variable.
You are incorrect about the result of stitching those two spaces together. The Riemann surface domain of $\sqrt{\cdot}$ is diffeomorphic to $\mathbb C\setminus\{0\}.$
In general, if $g:U\to\mathbb C$ is holomorphic, then the inverse ... |
I am trying to calculate this expression:
$\int dp \ \exp(-Aip^2+iBp)$
where A and B are real. How can I visualize this expression and will it yield a sensible result? |
I am trying to calculate this expression:
$\int dp \ \exp(-Aip^2+iBp)$
where A and B are real. I integrate p over the reals from -infinity to infinity. How can I visualize this expression and will it yield a sensible result? |
I'm working on a problem which asks to calculate the value of $\lim_{h\rightarrow 0} \theta$,where $\theta$ comes from the mean value theorem $f(x+h)-f(x)=hf'(x+\theta h)$,and $f$ is first order continuous differentiable. It's quite easy if $f$ is second order continuous differentiable by comparing the coefficient of T... |
How can I evaluate the limit of \theta in the expression f(x+h)-f(x)=h f(x+\theta h)? |
Since it seems that you haven't understood how 2-complement works, I'll explain it first, then I'll explain how right shift works for 2-complement negative numbers.
How 2-complement works
----------------------
Let $N$ be the number of binary digits used for storing a number. For every $n\in\mathbb{Z},N\in\mathb... |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonia... |
In [this][1] paper (bottom of p. 4), the authors state the following (I've added all definitions below):
> Given the deterministic mapping $z=g_{\phi}(\epsilon, x)$ we know that: $q_\phi (z|x)\Pi_i dz_i = p(\epsilon)\Pi_i d\epsilon_i$. Therefore, $$\int q_\phi(z|x)f(z)dz = \int p(\epsilon)f(z)d\epsilon=\int p(\epsil... |
Weird change of variables (?) I would like to understand formally |
This is called a [Riemann surface][1], in the case of functions of one variable.
You are incorrect about the result of stitching those two spaces together. The Riemann surface domain of $\sqrt{\cdot}$ is diffeomorphic to $\mathbb C\setminus\{0\}.$
In general, if $g:U\to\mathbb C$ is holomorphic, then the inverse ... |
After using the Log map, as defined in this paper [Riemannian geometry applied to BCI classification][1], to project points from the manifold onto the tangent space $T_P(M)$, do we use the Euclidean metric or the Riemannian metric to measure distances and norms of the vectors in the tangent space?
On one hand, I fe... |
This is from Munkres' Topology page 431:
> **Theorem 70.2** (Seifert-van Kampen theorem, classical version). Assume the hypotheses of the preceding theorem. Let
>
> $$ j : \pi_1(U, x_0) * \pi_1(V, x_0) \longrightarrow \pi_1(X, x_0) $$
>
> be the homomorphism of the free product that extends the homomorphisms
... |
In [this][1] paper (bottom of p. 4), the authors state the following (I've added all definitions below):
> Given the deterministic mapping $z=g_{\phi}(\epsilon, x)$ we know that: $q_\phi (z|x)\Pi_i dz_i = p(\epsilon)\Pi_i d\epsilon_i$. Therefore, $$\int q_\phi(z|x)f(z)dz = \int p(\epsilon)f(z)d\epsilon=\int p(\epsil... |
The following is an order 9 [Room square][1]. Copying from Wikipedia,
1. Each cell of the array is either empty or contains an unordered pair from the set of symbols.
2. Each symbol occurs exactly once in each row and column of the array.
3. Every unordered pair of symbols occurs in exactly one cell of the ar... |
Is there a 9x9 Sudoku Room Square? |
When considering log-concave distributions within the range [0,1], if I utilize the beta distribution with parameters $\alpha, \beta \geq 1$, does this choice encompass all potential shapes generated by the log-concave distributions? Or are there shapes that would be overlooked by this assumption?
Thanks |
Is the beta distribution a good representative of the family of log-concave distributions? |
This is from Munkres' Topology page 431:
> **Theorem 70.2** (Seifert-van Kampen theorem, classical version). Assume the hypotheses of the preceding theorem. Let
>
> $$ j : \pi_1(U, x_0) * \pi_1(V, x_0) \longrightarrow \pi_1(X, x_0) $$
>
> be the homomorphism of the free product that extends the homomorphisms
... |
I've seen [this post][1] which addresses the question of exponentiating a vector. I was wondering if there's a well-defined notion of exponentiating a rank $r$ tensor? For instance, if I have a rank 3 tensor $A_{ijk}$, can I compute something like $\mathrm{Exp}[A_{ijk}]$? If you know of any references/insights on this ... |
Minimax Estimation: What's the difference between Minimax, Sharp Minimax, First-Order Sharp Minimax and Second-Order Sharp Minimax Estimator? |
We need to show that the map
$$\theta:\mathbb{C}[z_0,\ldots,z_n]_k\to H^0(\mathbb{P}^n,\mathcal{O}(k)) $$
Is an isomorphism.
Huybrecht's describes the map $\theta$ right before the proposition.
***This is what I understand from the description*:**
Given a homogeneous polynomial of degree $k$ we can ... |
**So your question is:** For a nonempty set A of real numbers with its elements as cuts, does the least upper bound always exist as an element in the set?
**The short answer is:** Yes. If A does have an upper bound, it will definitely have a least upper bound in the subset, and the least upper bound is the union of ... |
I'm working on a problem which asks to calculate the value of $\lim_{h\rightarrow 0} \theta$,where $\theta$ comes from the mean value theorem $f(x+h)-f(x)=hf'(x+\theta h)$,and $f$ is first order continuous differentiable. It's quite easy if $f$ is second order continuous differentiable by comparing the coefficient of T... |
How can I evaluate the limit of $\theta$ in the expression $f(x+h)-f(x)=h f'(x+\theta h)$? |
## Question: Find the number of solution of equation $f(x)=2^x-x^2+x+\cos x=0$ ##
----------
Given answer is $1$
*My Attempt*
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 this gr... |
Since it seems that you haven't understood how 2-complement works, I'll explain it first, then I'll explain how right shift works for 2-complement negative numbers.
How 2-complement works
----------------------
Let $N$ be the number of binary digits used for storing a number. For every $n\in\mathbb{Z},N\in\mathb... |
**does there exist a finite algebraic structure on {0,1} whose quasi-equational theory is not finitely based?**<p>
No, there does not exist a $2$-element algebra in a finite signature that has a nonfinitely axiomatizable quasiequational theory. This is the content of the main result of<p>
V. A. Gorbunov<br>
Qua... |
Is there a 9×9 Sudoku Room Square? |
Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective elliptic partial differential operator of order $2$ on $\overline{M}$.
For $k\geq 0$, let $u \in H^{2+k}$, the $L^2$-Sobol... |
The following is an order 9 [Room square][1]. Copying from Wikipedia,
1. Each cell of the array is either empty or contains an unordered pair from the set of symbols.
2. Each symbol occurs exactly once in each row and column of the array.
3. Every unordered pair of symbols occurs in exactly one cell of the ar... |
>Let $\vec{OA}=\vec{a}, \vec{OB}=\vec{b}$ and $\vec{OC}=\vec{c}$ and $|\vec{a}|=3, |\vec{b}|=2$ and $|\vec{c}|=4$. We also know that $\measuredangle(\vec{a},\vec{b})=\measuredangle(\vec{a},\vec{c})=\measuredangle(\vec{b},\vec{c})=\dfrac{\pi}{3}.$ Given the affine coordinate system $O{\vec{a}\vec{b}\vec{c}},$ find the c... |
Let $\tau$ be the most coarse topology on $\mathbb{R}$ that contains the euclidean topology and all subsets of $\mathbb{Q}$. We already know that $\tau_{Eucl} \cup P(\mathbb{Q}$ is a base for $\tau$.
We also have a metric $d$ on $\mathbb{R}$ defined as the following. Let $(r_k)_{k\in \mathbb{Z}_{>0}}$ be a sequence of... |
Let $\tau$ be the coarsest topology on $\mathbb{R}$ that contains the euclidean topology and all subsets of $\mathbb{Q}$. We already know that $\tau_{Eucl} \cup P(\mathbb{Q}$ is a base for $\tau$.
We also have a metric $d$ on $\mathbb{R}$ defined as the following. Let $(r_k)_{k\in \mathbb{Z}_{>0}}$ be a sequence of di... |
Faced with the following problem, I do not understand how to solve this problem:
>Random vector $(X, Y)$ has a uniform distribution on the unit circle. Will its components be independent?
It is not very clear to me how to approach such tasks, is it necessary to look for a vector distribution function here? But it... |
After using the Log map, as defined in this paper [Riemannian approaches in Brain-Computer Interfaces: a
review][1] (Section III. A page 2&3), to project points from the manifold onto the tangent space $T_P(M)$, do we use the Euclidean metric or the Riemannian metric to measure distances and norms of the vectors in t... |
Is my solution to count the integer solutions for a linear inequality with bounds correct? |
Faced with the following problem, I do not understand how to solve this problem:
>Random vector $(X, Y)$ has a uniform distribution on the unit circle. Will its components be independent?
It is not very clear to me how to approach such tasks, is it necessary to look for a vector distribution function here? But it... |
My question is the following:
Let $T$ be a topos and $X\to e$ be a covering of the final object $e\in T$ with respect to the canonical topology.
We have a morphism of topoi $f\colon T/X\to T$ where $f^\star F=F\times X$ for $f\in T$.
Now Martin Olsson claims in *Algebraic Spaces and Stacks* in Lemma 2.4.18 that to... |
After using the Log map, as defined in this paper [Riemannian approaches in Brain-Computer Interfaces: a
review][1] (Section III. A page 2&3), to project points from the manifold onto the tangent space $T_P(M)$, do we use the Euclidean metric or the Riemannian metric to measure distances and norms of the vectors in t... |
> Find the Green's function for the BVP $$y''-\frac1xy'=0 \ \ ; \ \ y(0)=y(1)=0$$
Clearly the operator is not self-adjoint, so the equivalent self-adjoint equation is $$\left(\frac{y'}{x}\right)'=0$$ Therefore the Green's function could be taken as $$G(x,t)=\begin{cases}A+Bx^2 & \text{if} \ \ 0\leq x<t \\ C+Dx^2 & \... |
I wish to compute an analytic solution to
$\int_{-\inf}^{\inf}dx e^{-x^2} erf(ax +b)$
I gave integration by parts a go but essentially ended up with the same integral. Anyone know if this can be solved? |
Consider a continuous time biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the left. The probability $p_n(t)$ of being at location $n$ at time $t$ is then described by
$$\frac{\mathrm{d}}{\mathrm{d}t}p_n(t)=k_\mathrm{R}p_{n-1}(t)+k_\math... |
I have a nonlinear system of ODEs with constant coefficients $A, B, C, D, E, M$:
\begin{align}
&\dot{n}(t)=-An(t)+Bm(t)n(t)+C \\
&\dot{m}(t)=-Bm(t)n(t) + (M-m(t))R_0 \sin{\omega t} +Dm(t)+E,
\end{align}
with driving amplitude $R_0$. I know from numerical simulations that solutions of $n$ and $m$ are periodic wit... |
Is it possible to find a solution to ODEs assuming the solution is periodic with known period? |
For me you proof is very confusing and it is wrong. It is wrong because the equation $|f(x)-f(2)| < \frac{\delta}{\sqrt{3}+\sqrt{2}}$ does not hold. $|f(2.1)-f(2)|\approx 0.034$, but $0.1/(\sqrt(2)+\sqrt(3)\approx 0.032$.
But now to the confusion. You state
> Givens: $\varepsilon > 0$ and $\delta > 0$ and $|x-2|... |
Find all natural numbers n for which: in a triangle with sides $|AB| = 33$, $|AC| = 21$ and $|BC| = n$ we can place point $D$ on $AB$ and point $E$ on $AC$ such that $|AD| = |DE| = |EC| = m$, where $m$ is also a natural number.
Attempt: I saw that $m=21$ satisfied the condition, so $n$ can be any natural number betw... |
As mentioned here https://math.stackexchange.com/questions/1999381/a-dominated-convergence-theorem-for-the-conditional-expectation-e-cdot-mid
>Given a sequence $(\mathcal{F}_n)_n$ of $\sigma$-Algebras with $\mathcal{F}_{n+1} \subset \mathcal{F}_n$ and defining $\mathcal{F}_\infty := \bigcap_n \mathcal{F}_n$. If $Y_... |
For me you proof is very confusing and it is wrong. It is wrong because the equation $|f(x)-f(2)| < \frac{\delta}{\sqrt{3}+\sqrt{2}}$ does not hold. $|f(2.1)-f(2)|\approx 0.034$, but $0.1/(\sqrt(2)+\sqrt(3)\approx 0.032$.
But now to the confusion. You state
> Givens: $\varepsilon > 0$ and $\delta > 0$ and $|x-2|... |
I've been trying to work on a solution for this equation
$$
(1+x{ ²})y''+y=0
$$
using power series around $x_{0}=0$. So far, I have reached the following recursion relation:
$$
a_{n+2}=\frac{-a_{n}(n^{2}-n+1)}{(n+2)(n+1)}
$$
but I have been struggling to find a generic form for the coefficients. $a_{0}$... |
I am taking a first course on group theory.
I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$
How can I use this fact to show that
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_2\times\mathbb{Z}_6$ ?
More in general, it looks like (but I am not sure if this is true) th... |
After using the Log map, as defined in this paper [Riemannian approaches in Brain-Computer Interfaces: a
review][1] (Section III. A page 2&3), to project points from the manifold onto the tangent space $T_P(M)$, do we use the metric defined by the ambient Euclidean space or the Riemannian metric to measure distances ... |
For me you proof is very confusing and it is wrong. It is wrong because the equation $|f(x)-f(2)| < \frac{\delta}{\sqrt{3}+\sqrt{2}}$ does not hold. $|f(2.1)-f(2)|\approx 0.034$, but $0.1/(\sqrt(2)+\sqrt(3)\approx 0.032$.
But now to the confusion. You state
> Givens: $\varepsilon > 0$ and $\delta > 0$ and $|x-2|... |
$$S :=\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$$
$$= \sum_{n=1}^k\sqrt{\frac{n^2(n+1)^2+(n+1)^2+n^2}{n^2(n+1)^2}}$$
$$= \sum_{n=1}^k{\frac{\sqrt{n^4+2n^3+n^2+n^2+2n+1+n^2}}{n(n+1)}}$$
$$= \sum_{n=1}^k{\frac{\sqrt{n^4+2n^3+3n^2+2n+1}}{n(n+1)}}$$
The radicand has no rational roots, but maybe we can fact... |
50% of people don't smoke, 20% are light smokers & 30% are heavy smokers.
Heavy smokers are twice as likely to die prematurely as light smokers, light smokers are twice as likely to die prematurely as nonsmokers.
What's the probability of being a heavy smoker given person died prematurely?
I'm sure this would... |
> Suppose $Y(f)$ denotes the number of mutually incongruent solutions
> $(x,y)$ of the congruence $y^2\equiv f(x)\pmod{p}$ ($p$ an odd prime),
> where $f(x)$ is a polynomial with integral coefficients. Prove that
> $$Y(f)=p+\sum_{n=0}^{p-1}\left(\dfrac{f(n)}{p}\right).$$ Note: Two
> solutions $(x_1,y_1)$ and $(x_2,... |
I do not understand how
P(A|B,C)=P(A|C)
is derived from
P(A∩B|C)=P(A|C)P(B|C)
on this [site][1].
I'm stuck on the statement:
> By conditioning on C, we obtain
> P(A|B,C)=P(A∩B|C)/P(B|C)
Should the left hand side of the equation be P(A|B|C)?
**Update**
From this [site][1]:
Definition
>... |
As mentioned here https://math.stackexchange.com/questions/1999381/a-dominated-convergence-theorem-for-the-conditional-expectation-e-cdot-mid
>Given a sequence $(\mathcal{F}_n)_n$ of $\sigma$-Algebras with $\mathcal{F}_{n+1} \subset \mathcal{F}_n$ and defining $\mathcal{F}_\infty := \bigcap_n \mathcal{F}_n$. If $Y_... |
What you mentioned is usually called the substitution rule. And it's fine for logic in general. But if you want to get into categorical logic you need something that fits better with categorical notions.
In this case you get that the equality is left adjoint to contraction. Let me explain a bit further.
---
As... |
I'm trying to solve a geometry problem involving intersecting lines and angles in an equilateral triangle. Could you please explain how to find the measure of the angle marked with a question mark, given that the second marked angle is 71 degrees?
In the sketch, an equilateral triangle is intersected by a pair of p... |
How to solve this geometry problem? Must measure the marked angle. |
>**problem statement:** Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$ but the Fourier series of $f^2(x)$ is uniformly convergent on $[0, 2\pi]$.
**Some context**
The background of the problem. The following statement is a special... |
I've found myself in a situation where I need to prove a PBW theorem for a certain quotient of an algebra $A/I$ where $A$ is a filtered algebra given by generators and relations. Furthermore $A$ has a PBW theorem in the sense that $grA \cong U(\mathfrak{g})$ for some infinite dimensional Lie algebra $\mathfrak{g}$.(For... |
Assume that $u(x)$ is the classical solution solving
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
on $\mathbb{R}^n$ for some smooth enough coefficients and uniformly elliptic $a_{ij}$. I am looking for the gradient bound of $u$ explicitly on the behavior of the coefficients.
I found that in Gi... |
I am trying to compute the following quantity:
$$ \sum_{k=-\infty}^{\infty} 2^{k \frac{1}{2}}\left\{\int_{\Gamma_k}|\widehat{f}(x)|^2 d x\right\}^{\frac{1}{2}}$$
where: $ \Gamma_k=\left\{x \in \mathbb{R}: 2^{k-1}<\left|x\right|<2^k\right\}$ and:
$$ f(x) = \delta^{-1} x^{1/4} 1_{(\delta,M)}(x)$$
where $\delt... |
Let $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (I will denote this by just $(\mathcal{C},\otimes)$) be a tensor-category (in P. Deligne & J.S. Milne's vocabulary, see https://www.jmilne.org/math/xnotes/tc2018.pdf), or we could call it a symmetric monoidal category.
Furthermore, assume that $(\mathcal{C},\otimes)$ ... |
Is a symmetric monoidal category ("tensor-category" in P. Deligne & J.S. Milne's vocabulary) neccessarily locally small? |
Let $k$ be a perfect field of characteristic $p$. Let also $G$ be a (affine) group scheme over $k$, and $V$ be a representation of $G$.
Given a scheme $X$ over $k$, one can define its *Frobenius twist* $X^{(1)}$. (See, for instance, [here][1].) This construction is functorial (and symmetric monoidal): In particular,... |
Is the Frobenius twist of a representation a fully faithful functor? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.