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Let $(X,d_X), (Y,d_Y)$ be two metric spaces.
Let $\mathcal{F}\subset C(X,Y)$ be a family of continuous functions, each of them continuous at $x_0\in X.$ I'd like to get a sufficient condition on $\mathcal{F}$ to guarantee that $\mathcal{F}$ is equicontinuous at $x_0\in X.$
One immediate condition is that $\mathca... |
Is there a way to check that a given family $\mathcal{F}$ of continuous functions at $x_0$ is equicontinuous at $x_0?$ |
Let $G$ be a group, and a $H$ a subgroup, and for $a,b \in G$ let $a\sim b$ if $aH = bH$. Suppose I have a measure $\mu$ on $G/\sim$, the left cosets of $H$, and suppose that $G$ is equipped with a sigma algebra $\Sigma_G$ fine enough to contain all the cosets of $H$. How can I define a measure $\nu$ on $G$ that satisf... |
I am a noob at math and I had to solve this equation for an exam:
The curve Y is the intersection of the plane $x=-1$ and the cone $z^2=x^2+y^2$, find the curvature at a point $P(-1,2,\sqrt{5})$
This is how I solved it:
**1.** The intersection of the plane and the cone is giving the function $f(x,y,z)=1+y^2-z^2$... |
Let
$$f(z)=\frac{z}{1+|z|},\ z\in\mathbb{C}.$$ What's the image of $f$? (note that f(z) lies in the unit disc)
Also prove that $f$ is a homeomorphism. |
Let $(X,d_X), (Y,d_Y)$ be two metric spaces.
Let $\mathcal{F}\subset C(X,Y)$ be a family of continuous functions, each of them continuous at $x_0\in X.$ I'd like to **get a sufficient condition on the family $\mathcal{F}$** to guarantee that $\mathcal{F}$ is equicontinuous at $x_0\in X.$
One immediate condition i... |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
$|z-a|$ represents the distance of $z$ from $a$
<br>
hence,
<br>
$|z-2(1+i)|$ represents the distance of $z$ from $2(1+i)$ it is true for all others too
<br>
so, if you see the points $(2,2);(-1,5);(6,-2)$ {it's my way to represent complex no. on the argand plane (real part, imaginary part)}
<br>
so these point... |
I am a noob at math and I had to solve this equation for an exam:
The curve Y is the intersection of the plane $x=-1$ and the cone $z^2=x^2+y^2$, find the curvature at a point $P(-1,2,\sqrt{5})$
This is how I solved it:
**1.** The intersection of the plane and the cone is giving the function $f(x,y,z)=1+y^2-z^2$... |
I am an engineer, so maybe this question is naive.
I study equivalence relations and equivalence classes. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. Any number is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b, and b = c, then a=c (transitive)... |
How change an non-equivalency relation to equivalency relation? |
Which is greater : $66^{70}$ or $70^{66}?$
I have solved it with the help of Logarithmic table.Is there any rigorous method?
Any help would be highly appreciated!
Thank you! |
Which is greater : $66^{70}$ or $70^{66}?$ |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
Let $G$ be a group, and a $H$ a subgroup, and for $a,b \in G$ let $a\sim b$ if $aH = bH$. Suppose I have a measure $\mu$ on $G/\sim$, the left cosets of $H$, and suppose that $G$ is equipped with a sigma algebra $\Sigma_G$ fine enough such that its projection is contained in the sigma algebra of $G/\sim$. How can I def... |
Which is greater : $66^{70}$ or $70^{66}?$
My approach:
Consider $x^y > y^x$.
This is true if and only if $\ln x^y > \ln y^x$,
$y \ln x > x \ln y$
$\frac{\ln x}{\ln y} > \frac{x}{y}$.
Therefore
$66^{70} > 70^{66}$,
since
$\frac{\ln 66}{\ln 70} \approx 0.98615027281$
and
$\frac{66}{70} = \frac{33}{35} \ap... |
Find the Automorphism group of direct product of $\mathbb{Z}(\operatorname{mod} m)$ and $\mathbb{Z}(\operatorname{mod} n)$.
We know Automorphism of direct product of $\mathbb{Z}(\operatorname{mod} p)$ and $\mathbb{Z}(\operatorname{mod} p)$ is isomorphic to $\operatorname{GL}_2(\mathbb{Z}(\operatorname{mod} p))$. Then... |
This is a proposition in my textbook：
**Let $\ x=A|B,x'=A'|B'$ be Dedekind cuts in Q. Define
$$x+x'=(A+A')|rest\ \ of\ \ Q $$
Then the union of A+A' and B+B' may not equal to Q.(Here we define $\ A+A'=\{y\ |\ \exists\ x \in A,\exists x' \in A',y=x+x'\} $\)**
My question is under what circumstances are they not... |
This is a proposition in my textbook：
**Let $\ x=A|B,x'=A'|B'$ be Dedekind cuts in Q. Define
$$x+x'=(A+A')|rest\ \ of\ \ Q $$
Then the union of A+A' and B+B' may not be Q.(Here we define $\ A+A'=\{y\ |\ \exists\ x \in A,\exists x' \in A',y=x+x'\} $\)**
My question is under what circumstances are they not equal... |
Let $G$ be a group, and a $H$ a subgroup, and for $a,b \in G$ let $a\sim b$ if $aH = bH$. Suppose I have a measure $\mu$ on $G/\sim$, the left cosets of $H$, and suppose that $G$ is equipped with a sigma algebra $\Sigma_G$ that is closed under right multiplication by $H$ and whose projection is contained in the sigma a... |
I’m trying to understand the transformation law for the exterior connection form given in [Wikipedia][1]. If $\omega$ is a connection form with components in frame $\mathbf e$ given by $\omega_\alpha^\beta = \omega_\alpha^\beta(\mathbf e)$ and $g$ is a matrix representing a change of basis $\mathbf e’ = \mathbf e g$, w... |
Let $M:L:K$ be normal and seperable field extensions, then we know by the fundamental theorem of Galois theory that
$$
\frac{\text{Gal}(M:K)}{\text{Gal}(M:L)} \cong \text{Gal}(L:K)
$$
I was wondering if the following is true
Let us fix a basis for $M$: $x = \sum_i x_i\alpha_i \quad \forall x\in M$.
Can $\... |
What is Corr(X+Y, Z) in terms of Corr(X, Z) and Corr(Y, Z)? If there is no direct formula, is there any relationship in terms of > or <? Also, the same thing for rank correlation and Kendall's Tau? |
For a finite field $\mathbb{F}_q$, let $\mathcal{R}_k(N,M)$ be the number of matrices of shape $N\times M$ with entries in $F$ and rank exactly $k$.
Is there anything known about the numbers $\mathcal{R}_k(N,M)$, either exactly
or asymptotically? |
I am looking into repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
Fo... |
Let $G$ be a group, and a $H$ a subgroup, and for $a,b \in G$ let $a\sim b$ if $aH = bH$. Suppose I have a measure $\mu$ on $G/\sim$, the left cosets of $H$, and suppose that $G$ is equipped with a sigma algebra $\Sigma_G$ that is closed under right multiplication by $H$ and whose projection is contained in the sigma a... |
Assume a static (time-independent) random variable $r$ for which we know its probability distribution $P(r)$. Consider this to be a Gaussian distribution, such that $\langle r \rangle=0$ and $\langle r^2 \rangle=r_0$, etc.
Consider now the following differential equation: $\frac{d}{dt}X(t)=r A X(t)$, where $X=(x_1,x... |
[In this question][1], why did we turned |tan$^{-1}$θ| to (-)tan$^{-1}$θ [in this step][2]
why is there a negative sign? i first thought that the negative sign was because π/2 < θ < π , so tan is negative on this domain. But if tan is negative, why do we put a (-) and make it positive?
in a similar question [(qu... |
Let $F$ be a cdf with associated pdf $f$ on support $[0,b]\subseteq R_+$. Suppose $f$ is log-concave, which implies that $F$ and $1-F$ are also log-concave. Specifically, we have that $\frac{f'}{f}$ and $\frac{1-F}{f}$ are both decreasing functions, and $\frac{F}{f}$ is an increasing function.
Is $\frac{F}{f}\cdot\... |
I am looking into repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
Fo... |
I am not sure about the following derivative:
$ \frac{db^T e \Omega r r^T \Omega^T e^T b}{d r} = ?$
where $b \in \mathbb{R^N}$, $e \in \mathbb{R^{N \times L}}$, $\Omega \in \mathbb{R^{L \times K}}$, $r \in \mathbb{R^{K}}$.
My guess is that the derivative is
$e \Omega^T e^T b b^T e \Omega$
Is this corre... |
Let $F$ be a cdf with associated pdf $f$ on support $[0,b]\subseteq R_+$. Suppose $f$ is log-concave, which implies that $F$ and $1-F$ are also log-concave. Specifically, we have that $\frac{f'}{f}$ and $\frac{1-F}{f}$ are both decreasing functions, and $\frac{F}{f}$ is an increasing function.
Is $\frac{F}{f}\cdot\... |
In this question, why did we turn |tan$^{-1}$θ| to (-)tan$^{-1}$θ in the highlighted step?
[![enter image description here][1]][1]
Why is there a negative sign? I first thought that the negative sign was because π/2 < θ < π , so tan is negative on this domain. But if tan is negative, why do we put a (-) and make ... |
[![Question Statement][1]][1]
I tried to understand the solution from the book for this question, but I am having hard time to understand it.
[![A Solution][2]][2]
How do we know that there exist two terms $x_{j}$ and $x_{j+1}$ such that $x_{j}\gt0$,$x_{j+1}\le0$ by taking indices modulo $k+1$?
[1]: https... |
This question was completely restated to make it well-posed and understandable.
Let $\mathcal{C}$ be the algebra of random variables on $\mathbb{R}$ with finite expectation. By a random variable on $\mathbb{R}$ with finite expectation I mean a Borel-measurable probability distribution $\mu$ on $\mathbb{R}$, such th... |
ThHow can one characterize all linear functionals on random variables on $\mathbb{R}$ with finite expectation? |
Let $f_c : z \rightarrow z^2 +c$ and $Q_a: x \rightarrow ax(1-x)$, I have to show that for $c \in [-2, \frac{1}{4}]$ there is an $a\in[1,4]$ such that $f_c$ is conjugate to $Q_a$
Unfortunately, I'm uncertain about how to proceed. My initial thought was to explore periodic or fixed points, given their preservation un... |
We have a bead sliding along a smooth wire in the shape of a cycloid with equations,
$$
x=a(\theta-sin\theta),\space y=a(1+cos\theta),
$$where $0\leq\theta\leq2\pi$.
To find the number of coordinates required to describe the motion of the particle (call it $N$) we can use $3n-c=N$ where $n$ is the number of particl... |
For the following setup, what is the second holonomic constraint to limit the bead to one dimension? |
Let $F$ be a cdf with associated pdf $f$ on support $[0,b]\subseteq R_+$. Suppose $f$ is log-concave, which implies that $F$ and $1-F$ are also log-concave. Specifically, we have that $\frac{f'}{f}$ is a decreasing function, and $\frac{F}{f}$ is an increasing function. Is $\frac{F}{f}\cdot\frac{f'}{f}$ necessarily a de... |
We have a bead sliding along a smooth wire in the shape of a cycloid with equations,
$$
x=a(\theta-\sin\theta),\space y=a(1+\cos\theta),
$$where $0\leq\theta\leq2\pi$.
To find the number of coordinates required to describe the motion of the particle (call it $N$) we can use $3n-c=N$ where $n$ is the number of parti... |
This question was completely restated to make it well-posed and understandable.
Let $\mathcal{C}$ be the algebra of random variables on $\mathbb{R}$ with finite expectation. By a random variable on $\mathbb{R}$ with finite expectation I mean a Lesbegue-measurable probability distribution $\mu$ on $\mathbb{R}$, such... |
How can one characterize all linear functionals on random variables on $\mathbb{R}$ with finite expectation? |
I am studying Discrete Maths on my own, and I need help proving that there exists a set $S \subseteq P(\mathbb{R})$ - the powerset of $\mathbb{R}$ with the following 3 conditions:
1) $S \sim \mathbb{R}$
2) if $X, Y \in S$ and $X \neq Y$, then $X \cap Y = \emptyset$
3) if $X \in S$, then $X \sim \mathbb{R}$
Re... |
Let $F$ be a cdf with associated pdf $f$ on support $[0,b]\subseteq R_+$. Suppose $f$ is log-concave, which implies that $F$ and $1-F$ are also log-concave. Specifically, we have that $\frac{f'}{f}$ is a decreasing function, and $\frac{F}{f}$ is an increasing function. Is $\frac{F}{f}\cdot\frac{f'}{f}$ necessarily a de... |
What are the applications of stochastic differential equations to number theory? |
What should be a general approach towards solving these kind of questions algebraically?
\[
\begin{align*}
x_1 - 2x_2 + 3x_3 - 4x_4 + \dots + 9x_9 &\geq 1 \\
9x_1 - 8x_2 + 7x_3 - 6x_4 + \dots + x_9 &\geq 2 \\
-x_1 + x_2 - x_3 + x_4 - \dots - x_9 &\geq -0.2
\end{align*}
\]
I tried to code this up and find ... |
Is the provided solution complete? Or is there any aspect left to prove? |
This question was completely restated to make it well-posed and understandable.
Let $\mathcal{C}$ be the algebra of random variables on $\mathbb{R}$ with finite expectation. By a random variable on $\mathbb{R}$ with finite expectation I mean a Lesbegue-measurable probability distribution $\mu$ on $\mathbb{R}$, such... |
I have seen the following statements of Gödel's Incompleteness Theorems:
> **Gödel's First Incompleteness Theorem *(v1)*** If $T$ is a recursively axiomatized consistent theory extending PA, then $T$ is incomplete.
> **Gödel's Second Incompleteness Theorem *(v1)*** If $T$ is a recursively axiomatized consistent t... |
In this [answer][1], the choices for the distinct birthdays of single people are counted as if order matters:
$$363\times 362...$$
But the choices for the birthdays of the two pairs(four people) that have a birthday on the two different days are considered as if order does not matter:
$${365 \choose2 } {n \choose ... |
Out of interest, I wanted to develop some basic understanding of formal logic, but I am having trouble understanding the truth table of implications.
Especially examples such as this:
A: I want a pizza
B: I go shopping
A --> B: If I want a pizza, then I will go shopping
Why is it, for example, that assu... |
Consider $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and the following unimodular matrix
\begin{equation}
A=\begin{bmatrix}
2& 1\\
1& 1
\end{bmatrix}.
\end{equation}
We know $F\colon\mathbb{R}\rightarrow\mathbb{R}$, $F(\mathbf{v})=A\mathbf{v}$ induces a well-defined map $f\colon\mathbb{T}^2\rightarrow\mathbb{T}^2$... |
Assume a static (time-independent) random variable $r$ for which we know its probability distribution $P(r)$. Consider this to be a Gaussian distribution, such that $\langle r \rangle=0$ and $\langle r^2 \rangle=r_0$, etc.
Consider now the following differential equation: $\frac{d}{dt}X(t)=r A X(t)$, where $X=(x_1,x... |
I was reading an article about the number of group morphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_m$ for $m,n>0$ and it turns out it's the same number of cosets $[x]\in\mathbb{Z}_m$ that satisfy $nx\equiv 0 \mod m$; it also said that the number of solutions is $ \gcd(m,n)$ but it didn't explain why. Any explanation to th... |
The Boolean function $g$ of the variables $x_1 , . . . , x_5 , y_1 , . . . , y_5$ , is given by the formula:
$$
\bigwedge_{i=1}^{5} (x_i \ \vee \ y_i)
$$
**Mission:** find the dnf($g$). (the minimum number of conjuncts in the DNF of function $g$)
**Some clarifications:**
The
smallest number of conjuncts in ... |
First, take n^2 common from the denominator and out of the nth root. n^2 in numerator gets cancelled with the one taken common in the denominator
Next, we assume the value of limit to be L and take log (ln) on both sides. We get to see a standard form of integration of the limit of sum.
Now, on solving it we get ... |
Does anyone have any alternative solutions for this exercise. I'm self studying this book and I don't find many answers online, and don't even know if mine is correct, I find it a bit long. I would like to see how anyone else reasons through this exercise. I find it curious that you can get every function $Y^X$ from $... |
I'm having some trouble understanding how to calculate how many trials are needed before you expect to see a given event, I'm not sure what I've misunderstood yet.
I've followed these explanations so far:
https://www.cut-the-knot.org/Probability/LengthToFirstSuccess.shtml
https://www.geeksforgeeks.org/expect... |
We toss a fair coin $12$ times.
1. Determine the probability that as many heads fall out as tails.
2. Determine the probability that as many heads came out as tails, with the $12$th toss being the first toss after which the numbers of heads and tails became equal.
The first point is easy since we can see that ... |
In 3D, we tend to define the angle between two planes as the minimum angle of rotation required to turn one plane into the other. Rarely does this make sense in 4D. Let's see why.
In 4D we must get more specific about what a rotation is. In 3D, there is no distinction between a planar rotation, that is, a rotation w... |
My abstract algebra textbook as the following part of a theorem that it almost treats as trivial, and I want to prove this myself:
**Theorem.** Let $G$ be a group, $a \in G$ and $i, j \in \mathbb{Z}$. If $a \in G$ has infinite order, then $a^i = a^j$ if and only if $i = j$.
I tried to prove the first direction, b... |
First, take $n^2$ common from the denominator and out of the $n$-th root. $n^2$ in numerator gets cancelled with the one taken common in the denominator
Next, we assume the value of limit to be $L$ and take $\log$ $(\ln)$ on both sides. We get to see a standard form of integration of the limit of sum.
Now, on sol... |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with an infinite order generator and integer points $(-1,\pm1)$. The
[elliptic divisibility sequence](
https://en.wikipedia.org/wiki/Ellipti... |
**Background**
I've recently been formulating a game that incorporates elements from <a href="https://en.wikipedia.org/wiki/Egyptian_fraction">Egyptian fractions</a>, <a href="https://en.wikipedia.org/wiki/Magic_square">magic squares</a>, and <a href="https://en.wikipedia.org/wiki/Tessellation">tilings</a>. It is a ... |
**Question**
> Define $\mathbf{a} = (a_1, \ldots, a_p)$ where $p$ is a positive
> integer and the $a_l$ are i.i.d $\text{Uniform}(-1,1)$ random
> variables. Fix a unit vector $w \in \mathbb{R}^p$. Consider the
> quantity $S = \mathbf{a}^T w = \sum_{j=1}^p a_j w_j$. Prove the
> following bound on the cumulant gener... |
I have two lines:
$$\frac{x-1}{3}=\frac{y+2}{-2}=\frac{z}{1} = L_1$$
$$ \frac{x+1}{4} = \frac{y-3}{1}=\frac{z}{\alpha} =L_2$$
How can I find the value of $\alpha$ for which these two lines lie on the same plane?
Just notice that this is not a homework question. I took it from an exam held a few years ago, a... |
How to check if the given lines are coplanar? |
I have a set of $n$ vectors that form a basis for a subspace, $v_i$. Suppose I take a linear combination $w = \sum^n_{i=0} v_i \alpha_i, \quad \alpha_i \in \mathbb{R}$ such that for a set of vectors, $\sigma_1, ...\sigma_m$, $w^T \sigma_i > 0 \quad \forall i\lt m$. I'm scratching my head trying to think of some sort of... |
There are two sets of four 2D points.
There is a 1 to 1 correspondence between the two sets.
The points in both sets are close to the following nominal positions, only deviate with a random value in both direction: (-1.5, 0.0), (-0.5, 0.0), (0.5, 0.0), (-1.5, 0.0).
I would like to find a transformation (translatio... |
Given an arbitrary scheme $X$, there is a notion of its reduction $X_{\operatorname{red}} \to X$.
A well-known fact says that a scheme is reduced if and only if it satisfies Serre's condition R0 and S1. Hence the reduction is an R0 + S1-ification of a scheme.
Now let's say I don't want to make the scheme fully re... |
Is there a notion of an S1-ification of a scheme? |
Let $G$ be a finite group.
In M. Isaacs "Character theory of finite groups", chap. 2, we cover the fact that $\mathbb{C}[G] = \bigoplus_{i = 1}^{k} M_i(\mathbb{C}[G])$, where $M_i(\mathbb{C}[G])$ is the sum of all submodules of $\mathbb{C}[G]$ that are isomorphic to an irreducible $\mathbb{C}[G]$-module $M_i$. The rea... |
Your last sentence is the key. Setting aside the issue of optimizing for *strength* (e.g. replacing $\mathsf{PA}$ by a weaker theory of arithmetic such as $\mathsf{Q}$), the following is a language independent presentation:
> (G1IT) Suppose $S$ is any consistent recursively axiomatizable theory which interprets $\ma... |
The notation $R^c_{acb}$ is ambiguous: does it mean $g^{cd}R_{dacb}$, or $g^{cd}R_{adcb}$, $g^{cd}R_{acdb}$, or $g^{cd}R_{acbd}$? In other words, which index position corresponds to the raised index $c$? The various choices differ by a sign. For that reason, it's important to maintain both the horizontal and vertical p... |
This isn't exactly a math question (apologies!), but it could prevent many potential misunderstandings I might otherwise encounter in the near future (and also prevent many dumb questions I will post here otherwise). What is a common reference for linear, time-varying systems?
I saw two references: **Linear System T... |
Any common reference for linear, TIME-VARYING systems? |
A locally finite measure $\mu$ is a measure such that for all $x \in \mathbb{R}$, there exists $\epsilon > 0$ such that $\mu(]x-\epsilon;x+\epsilon]) < + \infty$.
My goal is to show that a measure $\mu$ of $(\mathbb{R}, B(\mathbb{R}))$ which is locally finite is a Stieltjes measure.
I started by showing that fo... |
Let $K$ be a field. I would like to prove that any algebraic extension $L$ of $K$ is separable iff ($\textrm{char} K = 0$ or $K = K^p$ in the case that $\textrm{char} K = p > 0$.
I have already proven the backward direction. In remains to show the forward direction. Thus, let $K$ be a field such that any algebraic e... |
Let $X$ be some measure space (esp probability space), $Y$ some metric space and we consider $g:X\to Y$ as an estimator for $f:X\to Y$. It seems convenient to consider an estimation as 'good' in case for any small $\epsilon > 0$ the measure $ \mu(\{d(f,g) > \epsilon\}) $ is small. Using Markov's Inequality we have for... |
Considering IVP:
$\dot{x} = \cos(x) +1, \; x(0) = 0.$
I should compute the omega and alpha limit of the given initial point.
So I first found equilibria, which in this case is: $ x = k \pi, \; k \in \mathbb{Z}$, and this equilibrium is not hyperbolic.
But I am not sure how to continue further?
|
The problem:<br>
> Suppose $AB, CD, EF, GH$ are four equally inclined (angle of $45^\circ$) chords of a circle concurrent at an interior point $P$ of the circle, dividing the circle into 8 parts. Show that the sum of the areas of sector-like regions $HPA, CPE, GPB, DPF$ is equal to the sum of the areas of the remain... |
The notation $R^c_{acb}$ is ambiguous: does it mean $g^{cd}R_{dacb}$, or $g^{cd}R_{adcb}$, or $g^{cd}R_{acdb}$, or $g^{cd}R_{acbd}$? In other words, which index position corresponds to the raised index $c$? The various choices differ by a sign. For that reason, it's important to maintain both the horizontal and vertica... |
Let $a\in\mathbb R^n$ be a tuple of length $n\in\mathbb Z_{>0}$. Let $X=\{a_i:1\le i\le n\}$ be the set of elements of $a$ for $x\in X$ let
$$i(x)=\min\{j:a_j=x\}$$
be the first occurence of $x$ in $a$. Now, for $x,y\in X$ let $x<y$ if $i(x)<i(y)$.
**QUESTION**: Can you point me to a reference where this order is ... |
Your last sentence is the key. Setting aside the issue of optimizing for *strength* (e.g. replacing $\mathsf{PA}$ by a weaker theory of arithmetic such as $\mathsf{Q}$), the following is a language independent presentation:
> (G1IT) Suppose $S$ is any consistent recursively axiomatizable theory which interprets $\ma... |
Prove that $\mathbb F_{p^m} \subseteq \mathbb F_{p^n}$ iff $m$ divides $n$. Here, $\mathbb F_{p^m}$ is the finite field of $p^m$ elements.
My classmate suggested that I count the dimensions, considering both of them as vector spcaes over the base field $\mathbb F_p$. However, I am not sure if counting dimensions wor... |
If I have a subset $A \subseteq \mathbb{R}$, a function $f: A \to \mathbb{R}$, and a point $a \in A$, then (according to the definitions I came across) for $f$ to be differentiable at $a$, then there must exist some open interval $I$ so that $a \in I$ and $I \subseteq A$.
I tried understanding why this requireme... |
For every line in the 2d plane, we can construct a shape with an "inside and outside" (often a circle) such that the shape is cut by the line into two symmetrical parts.
Does this property of lines extend to all curves?
My idea is if you "zoom in" on a curve, if you get small enough, it might become a line segme... |
I *do* believe quaternions are the most direct approach to it. That is to say: to computing the rotation that corresponds to a given point on $\mathbb{S}^3/\sim_A$. Showing that this correspondence is bijective is a different story, but you said you already understood that part.
You know that elements of $SO(3)$ are... |
In this question, why did we turn $|\tan^{-1}\theta|$ to $-\tan^{-1}θ$ in the highlighted step?
[![enter image description here][1]][1]
Why is there a negative sign? I first thought that the negative sign was because $\frac\pi2 < \theta < \pi$, so $\tan\theta$ is negative on this domain. But if $\tan\theta$ is ne... |
Why do we turn $|\tan^{-1}\theta|$ to $-\tan^{-1}θ$? |
If I have a subset $A \subseteq \mathbb{R}$, a function $f: A \to \mathbb{R}$, and a point $a \in A$, then (according to the definitions I came across) for $f$ to be differentiable at $a$, then there must exist some open interval $I$ so that $a \in I$ and $I \subseteq A$.
I tried understanding why this requirement e... |
Let $a\in\mathbb R^n$ be a tuple of length $n\in\mathbb Z_{>0}$. Let $X=\{a_i:1\le i\le n\}$ be the set of elements of $a$ for $x\in X$ let
$$i(x)=\min\{j:a_j=x\}$$
be the first occurence of $x$ in $a$. Now, for $x,y\in X$ let $x<y$ if $i(x)<i(y)$.
This order is known as *order of first occurrence*.
Now, let $x$ ... |
Prove the following inequality $$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$$ where $x,y,z \ge 0$ and $x+y+z=3$
I don't really know how to approach such an inequality. I tried the inequality of means and Cauchy's inequality, but it doesn't seem like I'm getting close to anything... |
We're interested in $L(s):=\int_0^\infty\tanh t\exp(-st)dt$. The integrand is $\sim t$ for small $t$, and $\sim\exp(-st)$ for large $t$, so $L$ converges for all $s>0$. For such $s$,$$\begin{align}L(s)&=\int_0^\infty\frac{1-\exp(-2t)}{1+\exp(-2t)}\exp(-st)dt\\&\stackrel{\ast}{=}\sum_{k\ge0}(2-\delta_{k0})(-1)^k\int_0^\... |
Here is the term: $M M C$, where $M = S (S S) S$, which involves the combinators $C = λxλyλz((xz)y)$ and $S = λxλyλz((xz)(yz))$. What does $M M C$ reduce to, under β-equivalence? Or ... if its β-reduction sequences are all infinite (as they appear to be) ... what infinitary combinator expression does it unfold into, in... |
A normal form for a chaotic combinator term? |
This question is about a gap in a proof I'm writing. I understand intuitively what the gap is - but can't find the correct or standard phrasing.
**Context**
The exercise is as follows (edited from ex 3.4.5 Terence Tao Analysis I 4th ed):
> Let $f : X \to Y$ be a function from one set $X$ to another set $Y$ .
... |
This question is about a gap in a proof I'm writing. I understand intuitively what the gap is - but can't find the correct or standard phrasing.
**Context**
The exercise is as follows (edited from ex 3.4.5 Terence Tao Analysis I 4th ed):
> Let $f : X \to Y$ be a function from one set $X$ to another set $Y$ .
... |
This question was posited to me by a coworker. I was able to get the answer, but I'm unsatisfied with how I did it.
1. $n < 354$
$n^5 = 133^5 + 110^5 + 84^5 + 27^5 < (133+110+84+27)^5$
$n < 354$
2. $n \equiv 0 \pmod 2$
133 and 27 are odd, 110 and 84 are even. Thus $133^5 + 110^5 + 84^5 + 27^5$ is the s... |
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