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### Question:
Given a r.v. $Y \sim U]0;1[$ and $ \mu $ a probability measure on $B( \mathbb{R})$. Build a new r.v. $X$ of law $ \mu $ by composing $Y$ with a well chose function.
### Answer:
1- We define $0<t = F_X(x)=P(X<x)= \mu (] -\infty; x]) < 1 $
2- Our objective is to get $ F_X(F^{-1}_X(t)) = t... |
The original question was: Differentiate the given function with respect to x:
$$
f(x) = \dfrac{ax^2+bx+c}{\sqrt{x}}
$$
I simplified till
$$
\dfrac{\sqrt{x}\left(2ax+b\right)-\frac{ax^2+bx+c}{2\sqrt{x}}}{x}
\tag1$$
then the next step simplified it into
$$
\dfrac{2ax+b}{\sqrt{x}}-\dfrac{ax^2+bx+c}{2x^\frac... |
How does $ \frac{\sqrt{x}\left(2ax+b\right)-\frac{ax^2+bx+c}{2\sqrt{x}}}{x} $ become $ \frac{3ax^2+bx-c}{2x^{3/2}} $? |
You are essentially *inlining* the ubiquitous [proof](https://math.stackexchange.com/a/3289264/242) that ideals in a Euclidean domain are principal, generated by any minimal element. This implies $\,I = (g)\,$ so $\,(g)\supsetneq (f)\,\Rightarrow\, g\mid f\,$ *properly*, hence $f$ irreducible $\Rightarrow \,g\,$ is a u... |
Can a oblique antiprism be constructed?
Intuitively, it would seem oblique antiprism exist: Take any right antiprism and translate one of the parallel faces within it's plane.
I'm baffled, though, because Google shows countless models and pictures of (right) antiprisms, but [not a single oblique antiprism](https:... |
No.
You know that if $a$ divides $b$ in $\mathbb{Z}[\sqrt{-5}]$, then $N(a)$ divides $N(b)$ in $\mathbb{Z}$.
However:
1. You don't have an $N(a)$ and an $N(b)$. You have $N(a)$, $N(2+\sqrt{-5})$, and integers $3$ and $6$, and note that $N(a)$ divides $18$ but does not divide $3$ nor $6$. So? You have not eve... |
If I can Taylor expand $f(x) = 2+ x+ x^2/2 +....$ and $g(x) = x/2 + x^2/3 + ....$ and I want to find the limit as $x\to0$, why is it that I can write
$$
\frac {f(x)}{g(x)} = \frac {2+ x+ x^2/2 + O(x^3)}{ x/2 + x^2/3 + O(x^3)} = \frac {2+ x+ x^2/2 }{ x/2 + x^2/3} + O(x^3) \to 2.
$$
I can obviously see why it is... |
How to justify expanding both f and g to find the limit of f/g? |
I am looking at a variation of this [question][1]). I am interested in a solution that can be solved using basic concepts (rather than Stirling numbers). Hence, I have attempted to implement the comment by Alex.
> There are $m$ male and $f$ female students in a class (where $m$ and $f$ are each less than 365) What ... |
You are essentially *inlining* the ubiquitous [proof](https://math.stackexchange.com/a/3289264/242) that ideals in a Euclidean domain are principal, generated by any minimal element. This implies $\,I = (g)\,$ so $\,(g)\supsetneq (f)\,\Rightarrow\, g\mid f\,$ *properly*, hence $f$ irreducible $\Rightarrow \,g\,$ is a u... |
> There are $m$ male and $f$ female students in a class (where $m$ and $f$ are each less than 365) What is the probability that a male student shares a birthday with a female student?
I have attempted the method suggested by Alex in his comment on the linked [question][1].
Using 365 stars(dates) and $m$ bars(ma... |
The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. [this][1] MathSE post):
\begin{align}
&\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y)
\quad \Rightarrow \quad
\phi(x) = \frac{-1}{4 \pi |x-y|} \qquad
\\
&\mathbb{R}^2: \qquad \nabla^2 \phi(x) = \delta^2(x-y)
\quad ... |
### Question:
Given a r.v. $Y \sim U]0;1[$ and $ \mu $ a probability measure on $B( \mathbb{R})$. Build a new r.v. $X$ of law $ \mu $ by composing $Y$ with a well chose function.
### Answer:
1- We define $0<t = F_X(x)=P(X<x)= \mu (] -\infty; x]) < 1 $
2- Our objective is to get $ F_X(F^{-1}_X(t)) = t... |
Is every subspace of a infinite-dimensional vector space an intersection of hyperspaces? |
The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. [this][1] MathSE post):
\begin{align}
&\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y)
\quad \Rightarrow \quad
\phi(x) = \frac{-1}{4 \pi |x-y|} \qquad
\\
&\mathbb{R}^2: \qquad \nabla^2 \phi(x) = \delta^2(x-y)
\quad ... |
Can $4\cdots41$ (with odd number of $4$s) be a Square Number? |
I would like to prove the following statement:
>Let $g:\mathbb{R}^n\to\mathbb{R}^n$ be contraction mapping. If function $f:\mathbb{R}\to\mathbb{R}^n$ satisfies:
$$f(t)+f'(t)=g(f(t))$$
then the limit $\lim_{t\to\infty}f(t)$ exists.
Below I write a sketch of my solution to this problem. My question is: is there som... |
I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is defined as:
$$ F(x) = x^T(wQ_1 + (1-w)Q_2)x $$
where $x$ is the vector of decision variables subject to $x^Tx = 1$, $Q... |
How to Compute the Second Derivative of a Quadratic Maximization Problem with Respect to the Weight Parameter? |
Let $A\in M_n(\mathbb{R})$. I know that $A$ has a real Jordan form which is obtained from the complex Jordan form by using the usual Jordan blocks for the real eigenvalues and by associating to the complex conjugate eigenvalues $\lambda=a+bi$ and $\overline{\lambda}=a-bi$ block diagonal matrices formed by $2\times 2$ ... |
Is the real Jordan form unique? |
The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. [this][1] MathSE post):
\begin{align}
&\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y)
\quad \Rightarrow \quad
\phi(x) = \frac{-1}{4 \pi |x-y|} \qquad
\\
&\mathbb{R}^2: \qquad \nabla^2 \phi(x) = \delta^2(x-y)
\quad ... |
> There are $m$ male and $f$ female students in a class (where $m$ and $f$ are each less than 365) What is the probability that a male student shares a birthday with a female student?
I have attempted the method suggested by Alex in his comment on the linked [question][1].
Using 365 stars(dates) and $m$ bars(ma... |
I'm reading the proof of the cone theorem from Koll\'ar-Mori and I find step 3 to be very unclear. In particular below is a simple proof of this step which must be wrong somehow. I am wondering where I went wrong.
The setup for this step is that we have the closed cone of curves $\overline{NE}$ and we define a close... |
So I had an assignment due yesterday, regarding the probability of the attackers second highest dice roll, being strictly bigger than the defenders second highest dice roll, in the game Risk.
In the game, the attacker has three dices and the defender has two. The highest dice of the attacker is compared with the hi... |
I understand that there are multiple ways to calculate the knot determinant, one is through the Alexander polynomial, the other is by creating another matrix which uses the linesections and crossings, the last method is explained at this page. https://www.dtubbenhauer.com/slides/geotop2022/4-knot-determinant.pdf
My... |
From [Wikipedia](https://en.wikipedia.org/wiki/Shattered_set):
>The class [of sets] $C$ shatters the set $A$ if for each subset $a$ of $A$, there is some element $c \in C$ such that $a = c\cap A$.
In other words, $C$ shatters $A$ if for every subset $S\subset A$, it's possible to find a set $c\in C$ which cover... |
### Question:
Given a r.v. $Y \sim U]0;1[$ and $ \mu $ a probability measure on $B( \mathbb{R})$. Build a new r.v. $X$ of law $ \mu $ by composing $Y$ with a well chose function.
### Answer:
1- We define $0<t = F_X(x)=P(X<x)= \mu (] -\infty; x]) < 1 $
2- Our objective is to get $ F_X(F^{-1}_X(t)) = t... |
When solving one specific PDE to find $f(x,y)$, I came across the following equation through which I would find the eigenvalues $\lambda_l$:
$a*tanh^{-1}(\frac{\lambda}{b}) - \lambda = 0$,
in which $a$ and $b$ are real constants.
I found out graphically that when $a < b$, the values of $\lambda_l$ are real, but... |
When solving one specific PDE to find $f(x,y)$, I came across the following equation through which I would find the eigenvalues $\lambda_l$:
$$a\tanh^{-1}(\frac{\lambda}{b}) - \lambda = 0,$$
in which $a$ and $b$ are real constants.
I found out graphically that when $a < b$, the values of $\lambda_l$ are real, but w... |
Is it possible that one eigenvalue of a PDE equals 0? |
Is it possible that an eigenvalue of a PDE equals 0? |
So I had an assignment due yesterday, regarding the probability of the attackers second highest dice roll, being strictly bigger than the defenders second highest dice roll, in the game Risk.
In the game, the attacker has three dices and the defender has two. The highest dice of the attacker is compared with the hi... |
> There are $m$ male and $f$ female students in a class (where $m$ and $f$ are each less than 365) What is the probability that a male student shares a birthday with a female student?
I have attempted the method suggested by Alex in his comment on the linked [question][1].
The number of ways to allocated dates ... |
### Question:
Given a r.v. $Y \sim U]0;1[$ and $ \mu $ a probability measure on $B( \mathbb{R})$. Build a new r.v. $X$ of law $ \mu $ by composing $Y$ with a well chose function.
### Answer:
1- We define $0<t = F_X(x)=P(X<x)= \mu (] -\infty; x]) < 1 $
2- Our objective is to get $ F_X(F^{-1}_X(t)) = t... |
This is my starting equation for complex number, $ \hat{k} $
$$ \hat{k}^2 = \mu \epsilon \omega - \mu \sigma \omega^2 i $$
I assumed $ \hat{k} = \alpha + \beta i $ where $ \alpha , \beta \in \mathbb{R} $
I then derived the following:
$$ \alpha^4 - \mu \epsilon \omega \alpha^2 - \frac{1}{4} \mu^2 \sigma^2 \... |
Assume that $\sqrt{n} (\bar{X}_n - \mu) \stackrel{d}{\to} N(0,\sigma^2)$ and assume that $\mu=0$. I wish to apply the second order delta method, to find the asymptotic distribution of
$$
\sqrt{n} ( \bar{X}_n^2 ).
$$
I have shown that for any twice continuously differentiable function $f$ (in some neighborhood of $... |
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results as answers.
If possible, it would be helpful to link to proofs of the results. These can be external p... |
During a differential equations class, we were given the example:
$\frac{d^2y}{dx^2} -6\frac{dy}{dx} + 9y = 10e^{3x}$.
Solving the auxiliary equation gives: $(Ax+B)e^{3x}$.
When finding a value for the particular solution, we picked y = $Cx^2e^{3x}$, so $\frac{dy}{dx} = 2Cxe^{3x} + 3Cx^2e^{3x}$ and $\frac{d^2y}{d... |
Proof of resonance giving a valid solution for an ODE? |
The electric field at some point $(x,y,z)$ generated by a ring of radius $R$ and uniform linear density $\lambda$ is
$$\mathbf E=\frac{\lambda R}{4\pi\varepsilon_0}\int_0^{2\pi} \frac{[(x-R\cos\phi)\mathbf{\hat x}+(y-R\sin\phi)\mathbf{\hat y}+z\mathbf{\hat z}]\text d\phi}{[x^2+y^2+z^2+a^2-2a(x\cos\phi+y\sin\phi)]^{\fr... |
I have the 1D heat equation $u_t= \alpha u_{xx}$ on $x \in [0,1]$.
It has homogeneous Dirichlet boundary conditions.
I intend to use the forward Euler numerical scheme.
$$\frac{u^{n+1}-u^{n}}{k} = \alpha u^{n}_{xx} $$
The discrete energy of the heat equation is
$$ E^{n} = \int_0^1 (u^{n})^{2} dx $$
*... |
**Hints.**
- If $x\in K$ is not zero, what can you say about $x^{q-1}$ (Subhint: Lagrange) ?
- Deduce that every element of $K$ is a root of $P=z^q-z$
- Conclude. |
Let $X$ be a real algebraic surface with a unique $A_1$ point at $p \in X$ (that is, the germ of $X$ at $p$ is isomorphic to the germ of $\{x^2+y^2=z^2\}$ at the origin). Let $Y \to X$ be the blow up of $X$ at $p$. Is it true that $X$ is homotopy equivalent to the connected sum of $X$ with $\mathbb{RP}^2$?
We know ... |
Consider two functions $f\in\mathcal{S}(\mathbb{R}^2)$ and $\phi\in C^\infty(\mathbb{R}^2)$ satisfying that $\nabla\phi(x_0,y_0)=0$ and $\mathrm{det}\text{ }\mathrm{Hess}\text{ }\phi(x_0,y_0)\neq 0$, and that this is the only critical point of $\phi$. Consider $$
I(\lambda)=\iint_{\mathbb{R}^2} e^{i\lambda\phi(x,y)}f... |
If $F=y+y'$, then Euler-Lagrange equation $\frac{\partial F}{\partial y}-\frac{d}{dx}(\frac{\partial F}{\partial y'})=0$ is not satisfied. Hence, we may conclude that there is no extremal for the functional concerned.
However, since $F$ does not contain $x$ explicitly, we can use the Beltrami identity $F-y'\frac{\pa... |
About Euler-Lagrange equation and Beltrami identity. What is the error in the process? |
The electric field at some point $(x,y,z)$ generated by a ring of radius $R$ and uniform linear density $\lambda$ is
$$\mathbf E=\frac{\lambda R}{4\pi\varepsilon_0}\int_0^{2\pi} \frac{[(x-R\cos\phi)\mathbf{\hat x}+(y-R\sin\phi)\mathbf{\hat y}+z\mathbf{\hat z}]\text d\phi}{[x^2+y^2+z^2+R^2-2R(x\cos\phi+y\sin\phi)]^{\fr... |
I'm trying to understand the following proposition.
> IF $\nabla$ and $\nabla'$ are two connections on a vector bundle $E$, then $\nabla-\nabla'$ is a $C^\infty(M)$-linear and can, therefore be considered as an element in $\mathcal{A}^1(M,\text{End}(E))$. If $\nabla$ is a connection on $E$ and $a \in \mathcal{A}^1(M... |
A paper by Erdős ("Extremal problems in number theory", Matematika 11:2 (1967); [pdf](https://static.renyi.hu/~p_erdos/1965-02.pdf)) has a very nice short argument for this. It's a more sophisticated version of the remark by Ma Ming in comments on the question.
Take any $\alpha$ and consider $B_\alpha=\{\,a_i\,:\,\f... |
What are the most general conditions for which the commutation relations $[A,BC] = 0$ and $[AB,C] = 0$ equivalent? Alternatively, can it at least be shown to hold if $A$, $B$, and $C$ are all involutant? If $A$, $B$, and $C$ are involutant permutation matrices? I am particularly interested in cases for which none of th... |
Suppose $\mathrm{E}$ is a spectral measure for some normal operator $N$ and $A\in\mathcal{B}(\sigma(N))$. I am unable to find an expression of the spectral measure of $A^c:=\sigma(N)\setminus A$ in terms of $\mathrm{E}(A)$.
My conjecture is that the correct expression would be
$$\mathrm{E}(A^c) ~=~ I-\mathrm{E}(A)... |
Consider the following boundary value problem
$$
\begin{cases}
y''(x)=p(x)y'(x)+q(x)y(x)+r(x)\\
y(0)=\alpha,\quad y(1)=\beta.
\end{cases}
$$
for continuous $p(x)$, $q(x)$, $r(x)$ and positive $q(x)>0$.
According to Section 8.1 of "A friendly introduction to numerical analysis" by Brian Bradie solution exists ... |
I'd appreciate if somebody could check whether my proof (mathematical induction) is correct.
$$(\forall n \in \mathbb{N}, n\geq 5) 2^{n+3} < (n+1)!$$
(i) let $n=5$
$2^{5+3} = 2^{8} < (5+1)!=6!$
(ii) Let's assume that the statement is true for $k \in \mathbb{N}, k\geq 5$.
Our inductive hypothesis (IH) is t... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
## Problem
Consider $N$ random variables $X_1,\, \dots,\, X_N \in \mathbb{R}^d$. They are assumed zero-mean, i.e, $\text{E} \left(X_i\right) = 0$, and their covariances are denoted $\text{E}\left[X_i X_i^\intercal\right] = P_i$.
Howerver, the cross-covariances $P_{i,j} = \text{E}\left[X_i X_j^\intercal\right]$ are ... |
I want to decompose a wave into non-sinusoidal waves $u(nx+t)$, where $n$ is the frequency and $t$ is the phase.I found that if we fourier transform a $\sin(x+\frac{\pi}{4})$ we get $\frac{\sqrt{2}}{2}\cos(x)+\frac{\sqrt{2}}{2}\sin(x)$, but i want to give me just $\sin(x+\frac{\pi}{4})$ and when I was wondering about t... |
In the Wikipedia about [Larent polynomial][1], there is this result:
The ring of Laurent polynomials over a field is Noetherian.
I was wondering if would it be enough for us to have Laurent polynomial over a Noetherian ring? For example, I think $\mathbb Z[X,X^{-1}]$ is Noetherian.
**Question**: Let $R$ ... |
Given an function $f:R^n \rightarrow R^n$, is it always possible to find a vector field $V:R^n \rightarrow R^n$ such that f is a first integral of the vector field? If not, when is it possible?
I read from [this question][1] that for every differentiable vector field, there exists a function that is constant along t... |
When is an analytic function a first integral of a vector field? |
Hatcher's Algebraic Topology states on page 108 that singular homology is a special case of simplicial homology:
> Though singular homology looks so much more general than simplicial homology,
it can actually be regarded as a special case of simplicial homology by means of the
following construction. For an arbitr... |
What is the intuition behind exponentials in the frequency domain signifying delay in a signal? |
**Premises to the answer**
The question you are asking is not an easy one as it is a version of the ***inverse problem of the calculus of variation*** (see [1] and [3] for some bibliographical references). In the form of your operator equation, the problem is hardly solvable, but if you leave the classic formulation... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
I'm really in trouble trying to understand Itô integral. I can work with it without any problem, but I don't understand what is it. And why is it an integral? How can we interpret $$I=\int_0^T f(s,B_s)dB_s \ \ ?$$
- Would it be: we consider the Brownian Motion on $[0,T]$ and a function $f:[0,\infty )\times \mathbb R... |
Let $a_n = \frac1{nx_n}$ it is not hard to prove that, $$a_{n+1} = \frac{n}{n+1} a_n + \frac1{n+1}\times \frac1{a_n}$$
Let $b_2 = a_2$ and $$b_{n+1} = \frac{n}{n+1} b_n + \frac1{n+1}$$
Using very simple induction for $n \ge 2$, one can prove that $$1 \le a_n \le b_n.$$
It remains to compute the limit of $b_n$.... |
Assume that $\sqrt{n} (\bar{X}_n - \mu) \stackrel{d}{\to} N(0,\sigma^2)$ and assume that $\mu=0$. I wish to apply the second order delta method, to find the asymptotic distribution of
$$
\sqrt{n} ( \bar{X}_n^2 ).
$$
I have shown that for any twice continuously differentiable function $f$ (in some neighborhood of $... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
>Let $(X,Y)$ have joint mass function
>
>$$p(k, n) = \frac{C \cdot 2^{-k}}{n}, \quad \text{for } k = 1, 2, \ldots \text{ and } n = 1, \ldots, k$$
>
>and suitable constant $C$. Keeping in mind $X$ and $Y$ are not independent. Find $E[X \mid Y=y]$
My attempt was to use the conditional probability definition; find ... |
How does one propery perform induction on integers in both directions? |
How does one perform induction on integers in both directions? |
Given an function $f:\Bbb R^n \rightarrow \Bbb R^n$, is it always possible to find a vector field $V:\Bbb R^n \rightarrow \Bbb R^n$ such that f is a first integral of the vector field? If not, when is it possible?
I read from [this question][1] that for every differentiable vector field, there exists a function that... |
Let $a_n = \frac1{nx_n}$ then,
\begin{align}
a_{n+1} &= \frac1{n+1} \times \frac1{x_{n+1}}\\
&= \frac1{n+1} \left(\frac{1+nx_n^2}{x_n}\right)\\
&= \frac1{n+1} \left(\frac{n}{nx_n} + nx_n\right)\\
&= \frac{n}{n+1} a_n + \frac1{n+1}\times \frac1{a_n}
\end{align}
Let $b_2 = a_2$ and $$b_{n+1} = \frac{n}{n+1} b... |
I do not undersand how to take the time derivative of the following Hamiltonian $\hat{H}(t) = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2(\hat{x}-a(t))^2$ where $a(t) = v_0t$. For instance how does $\hat{p}$ change when taking the time derivative of it? And the same for $\hat{x}$. |
Given an function $f:\Bbb R^n \rightarrow \Bbb R^n$, is it always possible to find a non vanishing vector field $V:\Bbb R^n \rightarrow \Bbb R^n$ such that f is a first integral of the vector field? If not, when is it possible?
I read from [this question][1] that for every differentiable vector field, there exists a... |
Let $G$ be a Hausdorff topological group; let $X$ be a locally compact Hausdorff space; let $A$ be a C*-algebra.
Define an action of $G$ on $X$ to be a continuous map $G \times X \to X, \ (g,x) \mapsto gx$ such that $ghx = g(hx), \ ex = x$ for every $g,h \in G$; define an action of $G$ on $A$ to be a group homomorph... |
Let $a_n = \frac1{nx_n}$ then,
\begin{align}
a_{n+1} &= \frac1{n+1} \times \frac1{x_{n+1}}\\
&= \frac1{n+1} \left(\frac{1+nx_n^2}{x_n}\right)\\
&= \frac1{n+1} \left(\frac{n}{nx_n} + nx_n\right)\\
&= \frac{n}{n+1} a_n + \frac1{n+1}\times \frac1{a_n}
\end{align}
Let $b_2 = a_2$ and $$b_{n+1} = \frac{n}{n+1} b... |
I am currently studying three classic topology books: (1) "Introduction To Topology" (Gamelin & Greene, 1975), (2) (same title by Bert Mendelson, 1958), and (3) "General Topology" (John Kelley, 1955). Of these, only the latter by Kelley even mentions the term Moore-Smith Convergence, also covering related topics such a... |
There are several errors in the assertion:
> "I understand that I have to show that if $I$ is an ideal of $R$, then for some fixed prime $p$, there exists a smallest $t\gt 0$ such that $p^t\in I$."
First, $p$ is fixes *throughout*. It is part of the definition of $R$. Saying "for some fixed prime $p$" after sayi... |
Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts for another 12,000 to 180,000 ticks, etc. This cycle repeats forever. ... |
Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts for another 12,000 to 180,000 ticks, etc. This cycle repeats forever. ... |
Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts for another 12,000 to 180,000 ticks, etc. This cycle repeats forever. ... |
Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts for another 12,000 to 180,000 ticks, etc. This cycle repeats forever. ... |
I intend to dumb down a bit modified version of the argument presented by @Anon [here][1].
Let $\mathbb F$ be an arbitrary field and we pick matrices $A$ and $B$ from $\mathbb F^{n\times n}$.
**Case 1**: $B$ is invertible. <br>
Check that $AB=B^{-1}(BA)B$ and hence $AB\sim BA$. Now use the fact that similar mat... |
Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts for another 12,000 to 180,000 ticks, etc. This cycle repeats forever. ... |
If we solve a LPP with let's say 2 constraints with all slack starting variables with non negative right hand side. We get the optimal table. Now suppose we change the constraint coefficient of a basic variable and compute the optima using post optimality analysis using a combination of dual simplex and simplex method.... |
In (first-order) logic, I understand that there are two notions of the truth of a sentence $\phi$ in a theory $T$:
1. Syntactic truth: $T\vdash \phi$ if $\phi$ is provable from $T$,
2. Semantic truth: $T\models \phi$ if $\phi$ is true in every model $M$ of $T$.
Naturally, both the notion that "$\phi$ is provable... |
How do we escape infinite regress when discussing truth in mathematical logic? |
I want to show that
$$ \underset{Q \in \mathcal{P}^0}{\inf} \ \{\mathbb{E}_{x \sim Q} \sin(1/x)\} > \inf_{x \in \mathbb{R}}\{ \sin(1/x)\} = -1$$
holds where $\mathcal{P}^0$ is the set of all probability distributions supported on $\mathbb{R}$.
Is this possible? I don't have a good intuition of "inf" over the set... |
I intend to dumb down a bit modified version of the highly abstract proof presented by @Anon [here][1].
Let $\mathbb F$ be an arbitrary field and we pick matrices $A$ and $B$ from $\mathbb F^{n\times n}$.
**Case 1**: $B$ is invertible. <br>
Check that $AB=B^{-1}(BA)B$ and hence $AB\sim BA$. Now use the fact tha... |
Given an function $f:\Bbb R \rightarrow \Bbb R^n$, is it always possible to find a non vanishing vector field $V:\Bbb R^n \rightarrow \Bbb R^n$ such that f is a first integral of the vector field? If not, when is it possible?
I read from [this question][1] that for every differentiable vector field, there exists a f... |
Given an function $f:\Bbb R^n \rightarrow \Bbb R$, is it always possible to find a non vanishing vector field $V:\Bbb R^n \rightarrow \Bbb R^n$ such that f is a first integral of the vector field? If not, when is it possible?
I read from [this question][1] that for every differentiable vector field, there exists a f... |
Problem: Let $X$ be a separable complete metric space and let the borel measure be inner and outer regular, and sigma finite, and $F$ a closed subset of $X$. Assume $f\in L^p(X)$ such that $f|_{F}=0$. Then show that there for every $\epsilon>0$, there exists a continuous function $g\in L^p(X)$ such that $g|_{F}=0$ and ... |
Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts for another 12,000 to 180,000 ticks, etc. This cycle repeats forever. ... |
There are many questions from students about (presumably) real functions of (presumably) real variables. I say presumably, because information is often scarce. Now with real functions there are often issues with domain. Examples are $\sqrt{1−x}$, $\log(1+x)$ and $\tanh^{-1}(x)$. One may manipulate these functions for a... |
Let $R$ be a ring with $1$ and $M$ be a right $R$-module. A homomorphism $f:P\to M$ is said to be a *projective cover* of $M$ if $P$ is projective and $f$ is an epimorphism and $\ker f \ll P$. Recall that $N$ is a *small* submodule of $K$ (written $N \ll K$) if $K=N+L \Longrightarrow K=L$.
I need to prove that the ... |
So
- $e^{iz} = e^{ix-y} = \frac{e^{ix}}{e^y} = \frac{\cos(x)+i\sin(x)}{e^y} = \frac{\cos(x)}{e^y} +\frac{i\sin(x)}{e^y} = -1 + i$
Then
- $\frac{1}{e^y} \cos(x) = -1$
- $\frac{1}{e^y} \sin(x) = 1$
With some basic manipulation, we get $\tan(x) = -1$ which gives us $x = \frac{3\pi}{4} + k$, for some integer ... |
In his *Mathematical Analysis I*, Zorich says the following after introducing the reals axiomatically:
> In relation to any abstract system of axioms, at least two questions arise immedi- ately.
First, are these axioms consistent? That is, does there exist a set satisfying all the conditions just listed? This is the ... |
Is consistency sufficient for existence? |
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