instruction stringlengths 12 30k |
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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... |
Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is strictly increasing with respect to $t$ on $(0,1)$, where
$$h(t)=\frac{1-t+F(t)+t(1-f(t))}{(\phi-t)(1-f(t))}-\frac{F(1)}{... |
Let $P(n)$ be the largest prime factor of $n$, and let $\Psi(x,B) = |\{ n \mid n \leq x \wedge P(n) \leq B\}|$. (This is a well-studied function in analytic number theory.) Define $\Psi'(x,B) = | \{ n \mid n \leq x \wedge P(n) \leq B \wedge \mbox{$n$ odd}\}|$.
The answer to [this post][1] shows how to prove that $\P... |
I know the result should be true, but I saw this very interesting proof [here](https://math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf):
The idea is to find a category (the twisted arrow category), having adjunctions to $\mathcal{C}$ and $\mathcal{C}^{\mathrm opp}$, then by some abstract homotopy theory we have the eq... |
**Background**
I've recently been formulating a game that incorporates elements from <a href="https://en.wikipedia.org/wiki/Egyptian_fraction">Egyptian fractions</a>, <a href="https://en.wikipedia.org/wiki/Magic_square">magic squares</a>, and <a href="https://en.wikipedia.org/wiki/Tessellation">tilings</a>. It is a ... |
Motivation for the problem
https://mathoverflow.net/questions/154913/a-hard-integral-identity-on-mathse
Is it possible to evaluate $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}$$? Or numerical value?
I attempted to use Wolfram Alpha, but I didn't receive any... |
Is it possible too evaluate $\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}$? Or numerical result? |
**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 ... |
I just started learning myself some basic algebraic geometry and I have some trouble doing these rather elementary exercises. I am basically misunderstanding something fundamental so it's causing me trouble to progress further.
**Some definitions:**
$X\subset k^n$ is an affine (algebraic) variety if $X=\mathbb{V}(... |
I was looking for a simple explanation of why the Gaussian random variable can be the only distribution appearing in the Central limit theorem.
From the statement of the Central limit theorem, it is pretty natural to expect that if $X$ is the limiting random variable in the Central limit theorem, then $X$ must be in... |
I have a level surface in $(x,y,z)$ space. For concreteness, let's say
$\frac{1}{r^2}x^2 + \left(\frac{\sin \theta}{r}\right)^2y^2 + \left(\frac{\cos \theta}{r}\right)^2z^2 + \frac{2 \cos \theta \sin \theta}{r^2} xy = 1$
which is describes a circle with radius $r$ initially in the $xz$ plane that is tilted by ang... |
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... |
In the literature I've found two notions of "$\pi$-cocartesian morphism" in a category, and to my knowledge they're not equivalent.
The first and I think most common one is the following :
> Let $\pi : D\to C$ be a functor. A morphism $\phi:X\to Y$ of $D$ is said to be $\pi$-cocartesian if for any $\psi:X\to Z$... |
Prove the following inequality for $a,b,c, \in (1,\infty)$ , such that $abc = 10$ $$\log_a 10 + \log_b 10 + \log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$$
I will transform logarithms to base 10, so we have to prove that $$\frac{1}{\log_{10} a}+\frac{1}{\log_{10} b}+\frac{1}{\log_{10} c} \ge \sqrt{\frac{3... |
Good evening to everyone. I am new to Manifold Theory so I am trying the last weeks to study some chapters from the book of John M. Lee 's "Introduction to smooth manifolds''. I was trying to understand this evening the concept of the Grassmannian manifold, which he introduced on pages 23-24 in the book. More specifica... |
So I was messing around and realized that the geometric series and exponential series are really similar when written as a power series.
$$\sum_{n=0}^\infty \frac{x^n}{(n!)^0}=\sum_{n=0}^\infty {x^n}=\frac{1}{1-x}$$
$$\sum_{n=0}^\infty \frac{x^n}{(n!)^1}=\sum_{n=0}^\infty \frac{x^n}{n!}=e^x$$
By adding a simple ... |
Is there a way to calculate a power series between the geometric series and exponential series? |
I'm reading Riehls' Category Theory in Context and came across the following confusion.
Riehl says (page 66):
> For a contravariant functor $F: C^\text{op} \rightarrow \text{Set}$, the category of elements $\int F$ is defined to be the opposite of the category of elements of $F$, regarded as a covariant functor o... |
Why isn't the definition of the category of elements of a contravariant functor dual to the definiton of the category of elements of a covariant one? |
I'm wondering if it's possible to do this integral with contour integration: $$\int_0^\infty dr \frac{r\cos(ax)}{(r^2+1)(1+2\sqrt{r^2+1})^2}. $$
Having integrated this numerically, I'm sure this integral converges. I thought of computing this as the real part and changing $\cos(ax)$ to $\text{Re}(e^{iax})$, but then I... |
Is it possible to do this integral with contour integration? |
There is a *N* * *N* diagonal matrix *A* with an infinite cycle with a period of 3, and an infinite cyclic matrix *B*, as follows. Assume $\Phi$ is a matrix of eigenvectors of $A^{-1}B$, where $\Phi^{T}A\Phi=I$, *I* is the *N* * *N* identity matrix.
Currently, a *N* * *N* diagonal matrix *C* has been constructed as fo... |
Question on Sparse Cyclic Diagonal Matrices: How to Prove Regularity of Eigenvector Structures Numerically? |
There is a $N \times N$ diagonal matrix *A* with an infinite cycle with a period of 3, and an infinite cyclic matrix *B*, as follows. Assume $\Phi$ is a matrix of eigenvectors of $A^{-1}B$, where $\Phi^{T}A\Phi=I$, *I* is the $N \times N$ identity matrix.
Currently, a $N \times N$ diagonal matrix *C* has been construc... |
This is more like a coding question, but I thought I'll post it here because of its mathematical foundation.
Two ODEs first degree can describe a Hamiltonian System. The connection between the ODEs and the Hamiltonian System is given by the canonical equations which are:
$
\dot{x}=\frac{\partial H}{\partial y}
... |
Given a cycle, how can I break it into simple cycles? I know there are general DFS algorithms to find cycles. I want to know if there are any advantages I can use to simplify the calculation/algorithm, considering that this is already a cycle.
For example, I have the cycle below:
cycle_edges = [(4, 5), (3, 5), ... |
Break non-simple cycle to simple cycles? |
Let us assume a linear mixed model of the form
$$
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{Z}\boldsymbol{b} + \boldsymbol{\epsilon}
$$
where $\boldsymbol{X}$ is the fixed-effect design matrix, $\boldsymbol{\beta}$ the fixed-effect coefficient vector, $\boldsymbol{Z}$ a block-diagonal rand... |
Good evening to everyone. I am new to Manifold Theory, so I am trying the last weeks to study some chapters from the book of John M. Lee 's *Introduction to Smooth Manifolds*.
I was trying to understand this evening the concept of the Grassmannian manifold, which he introduced on pages 23–24 in the book. More speci... |
I just started learning myself some basic algebraic geometry and I have some trouble doing these rather elementary exercises. I am basically misunderstanding something fundamental so it's causing me trouble to progress further.
**Some definitions:**
$X\subset k^n$ is an affine (algebraic) variety if $X=\mathbb{V}(... |
Perhaps a naive notation question. Let us say we have some PDE/function defined on $u(x, t)$ where $x \in [0, 1]$ and $t \in \mathbb{R}^+$ (positive reals). Assume the function is continuous with respect to the second derivative of $x$ and only continuous with the respect to the first derivative of $t$. So we can say $... |
[Here][1], the Ramanujan Master Theorem is proven in the following way:
$$
\begin{split}
\int_0^\infty x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} \lambda(n)x^n
& = \int_0^\infty x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} E^n\lambda(0)x^n \\
& = \int_0^\infty x^{s-1}e^{-Ex}dx \;\lambda(0) = \frac{\Gamma(s)}{E... |
I am learning about omega limit sets, and even though I understand the definition, I am struggling with explicitly determining them.
For example, many of our exercises contain initial value problems and ask for omega limit sets. In the case of the following initial value problem:
$$\dot{x} = x^2 +x^3, \; x(0) = -\... |
Does the category of $\text{List}$-algebras have an initial algebra? |
Do you have a probabilistic interpretation of this monster? If $Y$ is a random variable with density $g(y)=f'(y)$ I found that
$$h(t)=\frac{1-\int_0^tyg(y)dy}{(\phi-t)\int^1_tg(y)dy}-\frac{1-E(Y)}{\int^1_t(y-t)g(y)dy + (\phi-t)\int^1_tg(y)dy}.$$ This suggests to check the property when $g(y)dy$ is replaced by the Di... |
Hint: What are the equilibrium points for this differential equation? What does a phase portrait look like? |
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... |
Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run the simulation interactively in real-time.
Goal: Find the global minimum of a univariate, nonnegative rational function... |
I'm wondering if it's possible to do this integral with contour integration: $$\int_0^\infty dr \frac{r\cos(ar)}{(r^2+1)(1+2\sqrt{r^2+1})^2}. $$
Having integrated this numerically, I'm sure this integral converges. I thought of computing this as the real part and changing $\cos(ar)$ to $\text{Re}(e^{iar})$, but then I... |
In a card game with a standard deck of 52 cards, dealing stops when either two kings appear or at least one king and one ace appear. What is the expected number of cards that will be dealt? |
How many cards until 2 kings or a king and an ace? |
In a card game with a standard deck of 52 cards, dealing stops when either two kings appear or at least one king and one ace appear. What is the expected number of cards that will be dealt?
I know that the expected number of cards to see the first king/ace is 44/9. But I don't know how to proceed to the next step.
... |
I am reviewing Fisher information and saw that one of the requirements is that the distribution of the data, say $f(x|\theta)$, involves a parameter $\theta$ that is unknown but lies within a given open interval $\Omega$ of the real line.
Does the mean that if I have a distribution where the parameter is the natura... |
To get a good initial idea, play with unbelievably simple examples.
For example, maybe I'll look at $k = \mathbb{R}$, choose polynomials in $R = \mathbb{R}[X, Y]$, and look at resulting ideals. (Later it will turn out that there are complications from not using an algebraically closed ring $k$, in which case you mig... |
I'm wondering if it's possible to do this integral with contour integration: $$\int_0^\infty dr \frac{r\cos(ar)}{(r^2+1)(1+2\sqrt{r^2+1})^2}. $$
Having integrated this numerically, I'm sure this integral converges. I thought of computing this as the real part and changing $\cos(ar)$ to $\text{Re}(e^{iar})$, but then I... |
I am having problem with the following question:
>If $A=\{\text{sin }x^2 | x \in [0,1]\cap\mathbb{Q}\}$, then is $A$ measurable?
---
***My work so far:***
> Let $f:[0,1]\to\mathbb{R}$ be defined as $f(x) =\text{sin }x^2\quad$ $\forall x\in[0,1]\cap\mathbb{Q}$
Then, $f$ is absolutely continuous, and $m([0,1]\... |
Is the set A measurable? |
I am very confused with questions such as:
\begin{equation*}
\text{Differentiate}~~f(x)=x~~\text{with respect to }x^2
\end{equation*}
I have seen solutions for many similar questions, but they always just apply the usual rules for differentiation such as the chain rule. I am still confused on the "actual meani... |
What does "differentiate with respect to g(x)" actually mean intuitively? |
https://math.stackexchange.com/questions/153123/hitting-probability-of-biased-random-walk-on-the-integer-line?noredirect=1&lq=1
In the response to the above question (question and response below), Did mentions that the probability that we **eventually** reach 0 from 1 is labeled as r. This is distinct from the proba... |
Random Walk Intuition: why do we multiply probabilities? |
**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 am very confused with questions such as:
\begin{equation*}
\text{Differentiate}~~f(x)=x, ~x\in\mathbb{R}~~\text{with respect to }x^2
\end{equation*}
I have seen solutions for many similar questions, but they always just apply the usual rules for differentiation such as the chain rule. I am still confused on ... |
In this [other question](https://math.stackexchange.com/q/4838311/909869) I found an approximation as follows: $$\ln(x+1) \approx \dfrac{\ln(x)}{1-x^{-\frac{1}{\ln(2)}}}$$
By differentiating both sides, since just one logarithmic terms remains you could find the following approximation:
$$\ln(x)\approx \dfrac{\ln(2... |
I am reading the paper "Entropy and Equilibrium States in Classical Statistical Mechanics" by Oscar E. Lanford III (published in 1973). The image below shows an excerpt where the symbol $\lor$ is used in a way that I don't understand (in the equation boxed in red). This is the first instance of the $\lor$ symbol in the... |
Let $A$ be $\mathbb Z_{\geq 0}$ graded commutative ring. Now, in almost every resource I read, we define the structure a scheme on $\operatorname{Proj} A$, by taking the distinguished basis $D_+(f)$ of $\operatorname{Proj}A$, identifying $D_+(f)$ with $\operatorname{Spec}(A_f)_0$ and gluing the sheafs $\mathcal{O}_{\op... |
https://math.stackexchange.com/questions/153123/hitting-probability-of-biased-random-walk-on-the-integer-line?noredirect=1&lq=1
In the response to the above question (question and response below), Did mentions that the probability that we **eventually** reach 0 from 1 is labeled as r. This is distinct from the proba... |
https://math.stackexchange.com/questions/153123/hitting-probability-of-biased-random-walk-on-the-integer-line?noredirect=1&lq=1
In the response to the above question (question and response below), Did mentions that the probability that we **eventually** reach 0 from 1 is labeled as r. This is distinct from the proba... |
In so many math books we find definitions like this:
> Let $A(2,-3)$ be a point...
Why the sign '$=$' is omitted for points? Where this habit comes from? Shouldn't we write 'Let $A=(2,-3)$ be a point' ?
I find the usual notation a bit awkward, if you ask me.
|
Why points are defined without =? |
Recall that a [monoidal monad](https://ncatlab.org/nlab/show/monoidal+monad) on a monoidal category $(\mathcal{C}, \otimes, I)$ is a monad $(M, \eta, \mu)$ on $\mathcal{C}$ such that $M$ is also equipped with the structure of a lax monoidal functor in a way that makes $\eta$ and $\mu$ monoidal natural transformations.
... |
Is there a non-symmetric monoidal monad? |
In the response to [this question][1] (question and response below), Did mentions that the probability that we **eventually** reach $0$ from $1$ is labeled as $r$. This is distinct from the probability that given you are at $1$, on the next step, you go to $0$, which is $p = \tfrac13$.
To my understanding, Did's lo... |
If I have two numbers $x$ and $y$, how do I find the smallest value of $ax + by$, where $a$ and $b$ are integers, such that $ax + by$ is still larger than another number $z$?
For example, imagine I have two types of drug vials, one holds 20mg and the other 50mg. How do I find the combination of these vials so that I... |
If I have two numbers $x$ and $y$, how do I find the smallest value of $ax + by$, where $a$ and $b$ are integers, such that $ax + by$ is still larger than another number $z$?
For example, imagine I have two types of drug vials, one holds 20mg and the other 50mg. How do I find the combination of these vials so that I... |
If I have two numbers $x$ and $y$, how do I find the smallest value of $ax + by$, where $a$ and $b$ are integers, such that $ax + by$ is still larger than another number $z$?
For example, imagine I have two types of drug vials, one holds 20mg and the other 50mg. How do I find the combination of these vials so that I... |
If $M$ is a Riemannian manifold and $E \to M$ a Riemannian bundle, then the Riemannian metric $g$ can be viewed as a section of the bundle $\bigwedge^2 T^*M$, which means that it is a $(0,2)$-tensor field.
If we instead consider a complex vector bundle $E$ over $M$ with a hermitian metric $h$, [this][1] wikipedia ar... |
It is not meant to be a real answer. Probably only a proof of the theorem would clarify what's going on.
**Question 1**: I see another assumption of the theorem. The random variables $\xi_i$ are taking values on a *Polish space* $\mathcal{X}$.
**Question 2**: Why do we ask the random variable $U$ to be an $\mathr... |
Most of my exposure to modules has been over the various forms of category algebras, such as group algebras, path algebras, and incidence algebras. As such, I don't have a lot of exposure to modules over arbitrary rings. Right now, I am trying to understand modules over rings of formal power series $k[\![X]\!]$, and I'... |
Are modules for the ring of formal power series representations of a certain object? |
I just learned about the dirac comb
$$ Ш_T(t) = \sum_n \delta(t-nT) $$
and wanted to use it and the convolution theorem to understand the spectrum of a train of sinc pulses
$$ x(t) = \text{sinc}(t)* Ш_1(t) $$
However, my considerations for the fourier transformed lead to an apparently incorrect result:
T... |
If $M$ is a Riemannian manifold and $E \to M$ a Riemannian bundle, then the Riemannian metric $g$ can be viewed as a section of the bundle $\bigotimes^2 T^*M$, which means that it is a $(0,2)$-tensor field.
If we instead consider a complex vector bundle $E$ over $M$ with a hermitian metric $h$, [this][1] wikipedia a... |
I am learning about omega limit sets, and even though I understand the definition, I am struggling with explicitly determining them.
For example, many of our exercises contain initial value problems and ask for omega limit sets. In the case of the following initial value problem:
$$\dot{x} = x^2 +x^3, \; x(0) = -\... |
Could someone tell me the definition of the **separable closure** of a field $K$?
Furthermore, I would like to know whether it is a Galois extension of $K$. Also, why is this construction useful?
|
Let $P(n)$ be the largest prime factor of $n$, and let $\Psi(x,B) = |\{ n \mid n \leq x \wedge P(n) \leq B\}|$. (This is a well-studied function in analytic number theory.) Define $\Psi'(x,B) = | \{ n \mid n \leq x \wedge P(n) \leq B \wedge \mbox{$n$ odd}\}|$. Is there a good estimate for $\Psi'(x,B)$, or for the ratio... |
I read on a scientific paper (*) the following equations:
$$
z = \dot{\Theta}^2 sgn(\dot{\Theta})
$$
and then:
$$
\dot{\Theta} = sgn(z) \sqrt{\left| z \right|}
$$
Could you tell me how to pass from the first equation to the second one please?
(*) Dynamics and stability of a rimless spoked wheel: a s... |
**Background**
When I met the integral
$$\int_0^1 \frac{\left(x^\phi-1\right)^2}{\ln ^2 x} d x\\$$
where $\phi$ is the golden ratio: $\phi^2= \phi+1, $
I was surprised by its simple and decent value though it is hard to tackle. I had tried some methods such as substitutions, integration by parts etc. and fai... |
Hi I have the following question:
**Given that $P(x)=4x^4-4x^3y-7x^2y^2+4xy^3+3y^4$ can be expressed a product of 4 linear factors $ax+by$, where $a$ and $b$ are constants. Find the expression.**
First I thought that $P(x)=(ax+by)^4$ and expanded it to match up the coefficients to find $a$ and $b$. However, it wa... |
$\def\sgn{\mathrm{sgn}}$
Since $z = \dot{\Theta}^2 \sgn(\dot{\Theta})$ we have $|z| = \dot{\Theta}^2$ and then $\dot{\Theta} = \pm \sqrt{|z|}$
Which sing we choose? Well, from $z = \dot{\Theta}^2 \sgn(\dot{\Theta})$ we have $\sgn(z) = \sgn(\dot{\Theta})$, so $\dot{\Theta} = \sgn(z) \sqrt{|z|}$. |
I read on a scientific paper (*) the following equations:
$$
z = \dot{\Theta}^2 \operatorname{sgn}(\dot{\Theta})
$$
and then:
$$
\dot{\Theta} = \operatorname{sgn}(z) \sqrt{| z |}
$$
Could you tell me how to pass from the first equation to the second one please?
(*) Dynamics and stability of a rimless... |
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... |
Most of my exposure to modules has been over the various forms of category algebras, such as group algebras, path algebras, and incidence algebras. As such, I don't have a lot of exposure to modules over arbitrary rings. Right now, I am trying to understand modules over rings of formal power series $k[\![X]\!]$, and I'... |
From classical algebraic geometry we learn that maximal ideals in polynomial rings correspond to points, in the sense that if $K$ is an algebraically closed field, and $\mathfrak a$ is a maximal ideal in $K[x_1,\dots,x_n]$, then there is a $p\in\mathbb A^n$ such that $\mathfrak a=I(\{p\})$, and hence $V(\mathfrak a)=\{... |
What can be said about an ideal whose vanishing locus consists of a single point? |
$\def\sgn{\mathrm{sgn}}$
Since $z = \dot{\Theta}^2 \sgn(\dot{\Theta})$ we have $|z| = \dot{\Theta}^2$ and then $\dot{\Theta} = \pm \sqrt{|z|}$
Which sign we choose? Well, from $z = \dot{\Theta}^2 \sgn(\dot{\Theta})$ we see that $\sgn(z) = \sgn(\dot{\Theta})$, so $\dot{\Theta} = \sgn(z) \sqrt{|z|}$. |
How to determine in polynomial time if a number is a product of two consecutive or close primes? |
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... |
In an economic model I am working with I would like to establish that the difference of two special convex functions is convex. I know that in general the difference of convex functions need not be convex, see [here][1], but I was hoping someone can point out if the properties of my functions suffice or can provide a ... |
In an economic model I am working with I would like to establish that the difference of two special convex functions is convex. I know that in general the difference of convex functions need not be convex, see [here][1], but I was hoping someone can point out if the properties of my functions suffice or can provide a ... |
From Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 25, p 87.
>25. A crime is committed by one of two suspects, A and B. Initially, there is equal
evidence against both of them. In further investigation at the crime scene, it is
found that the guilty party had a blood type found in 10%... |
$$Form \dashv Lang : Hyp \rightarrow FOL$$
---
We consider an object of $FOL$ to be a signature made of sorts, function symbols and relation symbols, together with a set of axioms of first order logic.
We consider an object of $Hyp$ to be a first order hyperdoctrine formed by a cartesian category $C$ and a fun... |
In an economic model I am working with I would like to establish that the difference of two special convex functions is convex. I know that in general the difference of convex functions need not be convex, see [here][1], but I was hoping someone can point out if the properties of my functions suffice or can provide a ... |
In an economic model I am working with I would like to establish that the difference of two special convex functions is convex. I know that in general the difference of convex functions need not be convex, see [here][1], but I was hoping someone can point out if the properties of my functions suffice or can provide a ... |
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:X... |
I am learning about connections using Lee's Intro to Riemannian Manifolds, and I am confused regarding the coefficients of the pullback connection. For the setting, let $(M,g) $ be a Riemannian manifold and let $\nabla$ be a connection on $M$. From Prop 4.7 in Lee's book, if we have two local frames $(E_i)$ and $(\tild... |
I recently learned that if $M$ and $N$ are both manifolds, then for a smooth map $F:M\to N$, the preimage $F^{-1}(q)$ of a point $q\in N$ is always a closed subset of $M$. How about the other direction? If we are given a closed subset $S\subset M$ and a point $q\in N$, can we define a smooth map $F:M\to N$ such that $S... |
I know the result should be true, but I saw this very interesting proof [here](https://math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf):
The idea is to find a category (the twisted arrow category), having adjunctions to $\mathcal{C}$ and $\mathcal{C}^{\mathrm opp}$, then by some abstract homotopy theory we have the eq... |
Hint: What are the equilibrium points for this differential equation? What does a phase portrait look like?
EDIT: You are correct, the equilibrium points are $0$ and $-1$. A very important fact is that (because of the uniqueness theorem) two solutions can never meet or cross. So since your solution starts between... |
Denote
$$
\operatorname{atan2}(y,x) = \arctan z, \quad \mbox{where} \quad z = \dfrac{y}{x}.
$$
Note that (in degrees) if $z>1$, then
$$
\arctan z = 90^\circ - \arctan \dfrac{1}{z},
$$
and that
$$
\arctan(-z) = -\arctan ~ z.
$$
So, it is enough to write approximation for $z\in [0,1]$.
This simple ap... |
My question stems from trying to solve the following problem:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/OBbBz.png
My goal is to calculate the Herbrand quotient of $\mathcal{O}_L^\times$ by hand, so only using elementary calculations in $\mathbb{Q}_p$. If I'm not mistaken it has... |
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