instruction stringlengths 12 30k |
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We wish to show that the list $(1,2-4),(7,-5,6)$ is linearly independent in $\mathbb{F}^3$ but is not a basis of $\mathbb{F}^3$ because it does not span $\mathbb{F}^3$
Then for some $v\in\mathbb{F}^3, \hspace{0.5em}v = (a,b,c), \hspace{0.5em} \forall a,b,c\in\mathbb{F}$.
We know that if we write the vectors as colu... |
Let $X,H$ be Hilbert spaces and let $f\colon H \to L(H,X)$, where $L(H,X)$ is the space of linear bounded operators. In this case it is natural to define the adjoint function $f^*\colon H \to L(X,H)(=L(X^*,H^*))$ by
$$f^* (x)=[f(x)]^* \quad \forall x \in H,$$
where $*$ denotes the adjoint of the operator $f(x) \in L... |
Let $f\colon H \to L(H,X)$ strongly continuous ($X,H$ Hilbert spaces). Is the adjoint function $f^*\colon H \to L(X,H)$ strongly continuous? |
For a set $A$, do $A\to 2^{A}$ or its elements have canonical names?
I know the term "set-valued", but I want to express specifically that the same set appears on both sides.
I'd also be interested in domain specific terminology, e.g., when $A$ is a topological space or lattice or when specific conditions like $f... |
Does $A\to 2^{A}$ have a name? |
#### Question
I would like to find a closed-form, or determine that none exists, for the ODE $$\frac{d\mathbf{x}}{dt} = \cos(\mathbf{c}^T \mathbf{x})\mathbf{a} - \sin(\mathbf{c}^T \mathbf{x})\mathbf{b}$$ where $a,b,c,\mathbf{x} \in \mathbb{R}^d$. When $d=1$, the solution is given in closed form [here][1].
My ques... |
I know and can show that
\begin{equation*}
{}_{1}F_{1}\left(1,s,z\right)=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right)
\end{equation*}
From here, I get:
\begin{equation*}
\begin{split}
{}_{1}F_{1}\left(1,s+1,z\right)&= \frac sz\left\{{}_{1}F_{1}\left(1,s,z\right)-1\right\} \\
& = \frac sz\left[\frac {s-1... |
Let $(M,d_{M})$ and $(N,d_{N})$ be metric spaces and let $f:M\longrightarrow N$ be a function;
**Metric space definition of continuity:** $f$ is continuous at a point $x\in M$ if for every $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that for all $y\in M$
\begin{align*}
d_{M}(x,y)<\delta(\epsilon)\Longr... |
Left and right continuity in metric spaces only for $f:\mathbb{R}\longrightarrow\mathbb{R}$? |
I would like to find a closed-form, or determine that none exists, for the ODE $\frac{d\mathbf{x}}{dt} = \cos(\mathbf{c}^T \mathbf{x})\mathbf{a} - \sin(\mathbf{c}^T \mathbf{x})\mathbf{b},$ where $a,b,c,\mathbf{x} \in \mathbb{R}^d$. When $d=1$, the solution is given in closed form [here][1].
My question is: does such... |
Hint. Let $Y_i = 1$ if there is a run of $k$ ones starting at position $i$ ($i=1, 2 \cdots n$), $Y_i=0$ otherwhise.
Then $X = \sum_i Y_i$. This should give you $E[X]$ directly.
For the variance, compute $E[X^2]$ (this is a little more convolved, but not much).
Added:
$$\begin{align}E[X^2] &= E[ (\sum_{i=1}... |
Why can $P_1 β P_2 β P_3 β \ldots β Pn$ be true when not all the $P$βs have the same truth value?
---
For example: If P1 = T P2 = T P3 = F P4 = F would this be true?
T(P1) β T(P2) β F(P3) β F(P4) =
T(P1&2) β F(P3) β F(P4) =
F(P1,2,3) β F(P4) =
True
If so how can an evaluation for a series of compound... |
Why can a compound biconditional statement whose individual statements don't all have the same truth values be true? |
Using [Lagrange reversion](https://en.wikipedia.org/wiki/Lagrange_reversion_theorem): $$x=a+bx^x\implies x=a+\sum_{n=1}^\infty\frac{b^n}{n!}\frac{d^{n-1}}{da^{n-1}}a^{na}$$ Now use a [Stirling number of the first kind](https://reference.wolfram.com/language/ref/StirlingS1.html) formula, possible to derive from [this pa... |
For a set $A$, do $A\to 2^{A}$ or its elements have canonical names?
I know the term "set-valued", but I want to express specifically that the same set appears on both sides.
I'd also be interested in domain specific terminology, e.g., when $A$ is a topological space or lattice or when specific conditions like $f... |
Hint. Let $Y_i = 1$ if there is a run of $k$ ones starting at position $i$ ($i=1, 2 \cdots n$), $Y_i=0$ otherwhise.
Then $X = \sum_i Y_i$. This should give you $E[X]$ directly.
For the variance, compute $E[X^2]$ (this is a little more convolved, but not much).
Added:
$$\begin{align}E[X^2] &= E[ (\sum_{i=1}... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
This may be more of a pun than anything else, as the two usages of the word "natural" are unrelated, but I was wondering if there was any nontrivial sense in which the natural logarithm could be viewed as an example of naturality.
Many things seem to be reframeable in the context of category theory. In the category ... |
Can the natural logarithm be meaningfully "natural" in the category theory sense? |
I am attempting to solve equations with Galois theory. I find the following method works:
For an equation, such as $x^3+x-1=0$, assume it splits as $(x-x_1)(x-x_2)(x-x_3)=0$ in $\mathbb Q(x_1,x_2,x_3)$.
Put $\theta_1=(x_2x_3^2-x_1x_3^2)+(x_3x_1^2-x_2x_1^2)+(x_1x_2^2-x_3x_2^2)$, one calculates that $\theta_1^2=-27... |
$$\sum_{n=1}^{k}\frac{1}{n^2+n}=\frac{k}{k+1}$$
I don't think that this summation requires too much context as this is a Q&A site, but I was just wondering why the summation is evaluated so nicely. I don't have too much knowledge and creativity in evaluating sums, so is there a bunch of equations that will make this... |
I've gone through many math courses thinking that a function $f(x)=x^2$ could mean $f(anything) = anything^2$, however I realized that that isn't always true.
For example, suppose we have a function $f(x) = kx$ where $f$ represents the opposing force. We can't just substitute $t$ for time into that equation, as tha... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
I am examining the density of Primes in other sets than the naturals.
E.g. we want to have the density of Mersenne Primes.
From the prime number theorem I know that in the naturals we have
$$
\frac{\pi(n)}{\frac{n}{\log(n)}} \rightarrow 1.
$$
But what if I have a set
$M^{\leq n} := \left\{m\in\mathbb N: m\leq... |
I am attempting to solve equations with Galois theory. I find the following method works:
For an equation, such as $x^3+x-1=0$, assume it splits as $(x-x_1)(x-x_2)(x-x_3)=0$ in $\mathbb Q(x_1,x_2,x_3)$.
Put $\theta_1=(x_2x_3^2-x_1x_3^2)+(x_3x_1^2-x_2x_1^2)+(x_1x_2^2-x_3x_2^2)$, one calculates that $\theta_1^2=-27... |
Using Newton's Method, solve the following equation:
$\displaystyle 0 = \frac{1}{ 2}+ \frac{1}{4}x^2- xsin(x)- \frac{1}{2}cos(2x) $
iterate until you achieve an accuracy of $\displaystyle 10^{β5}$ with:
$ \displaystyle p_0 = \frac{\pi}{3}
\\
p_0 = 3\pi$
How do I do the calculations to get the precision... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
(I am reading Huybrechts's Complex Geometry)
I also looked some classic sources, but I didn't find something anything addressing the below.
---
**Proposition $B.0.35$**
Let $$0\longrightarrow \mathcal{F}^0\longrightarrow \mathcal{F}^1\longrightarrow \mathcal{F}^2\longrightarrow 0$$
be a short exact seque... |
A group G has sixteen elements:
{e, r, r^2
, . . . , r^
7
, s, rs, r^
2
s, . . . , r^
7
s}
where r and s satisfy the relations
r^
8 = e, s^
2 = e, sr = r^
3
s.
(Note that G is not a dihedral group.)
</p>
a) Make a list of the cyclic subgroups of G. Make sure to state the
elements of each cyclic s... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
I'm studying matchings in graph theory. The definition of matching that I have in my textbook states: "A set M of independent edges in a graph G = (V, E) is called a matching.".
Later, the definition of an augmenting path (referring to bipartite graphs with
bipartition { A, B }) is given: "A path in G which starts ... |
Can a matching be empty and an augmenting path of length 1? |
(I am reading Huybrechts's Complex Geometry)
I also looked some classic sources, but I didn't find anything addressing the below.
---
**Proposition $B.0.35$**
Let $$0\longrightarrow \mathcal{F}^0\longrightarrow \mathcal{F}^1\longrightarrow \mathcal{F}^2\longrightarrow 0$$
be a short exact sequence of she... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
Let $S$ be the set of points whose coordinates $x,$ $y$ are integers that satisfy $0\le x\le3,$ $0\le y\le4$. Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is?
My approach:
I tried finding the probability of the points ***not... |
<p>I want to transform the graph $y=\sin{x}$ by the matrix $\begin{bmatrix}3 & 4 \\-2 & 8\end{bmatrix}$, which roughly judging would give a clockwise rotation and a stretch.</p>
<p>
My solution is to first parametrize the graph to get $x=t$ and $y=\sin{t}$. Then, I computed
$$\begin{bmatrix}3 & 4 \\-2 & 8\end{bmatr... |
I am currently engaging with the following hypergeometric function as a result of attempting to find [a solution for this probability problem for $n$ number of dice][1]:
$$_2F_1\left (\frac{n+k}{2}, \frac{n+k+1}{2};k+1;\frac{4b(d-e+1)}{d^2}\right ); \{n,k,b,d,e\}\in \mathbb{N}^+, z\in \mathbb{R},b < e \leq d$$
I'... |
In this post, I want to ask the community about an issue regarding an analytical approach to dealing with the integer factorization problem.
The problem definition is as follows:
**1. Sets**
Let $C=\{x^2:x\in\mathbb{Z}\}$.
We define the sets $A_n=C\cap \{x-n:x\in C\}$ and $B_n=C\cap\{x+n:x\in C\}$ to repr... |
We have shown that for an affine cone of a nonempty subset $X$ of $\mathbb P^n$, i.e $C(X)$, that the irreducible decomposition of $C(X)$ given by $Y_1 \cup \dots \cup Y_m$ has that $Y_i = C(X_i)$ for some closed $X_i \subseteq \mathbb P^n$.
I wanted to know, what exactly is $X_i$ in this case? I was perhaps thinkin... |
Based on my limited understanding, the precise formulation of [Lowenheim-Skolem (LS) theorem](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) is stated as follows:
> Let $\Gamma$ be a set of sentences from a countable language $\mathcal L$. Let $\mathcal M$ be an infinite model for $\Gamma$, i.e... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. The set $Y=\alpha(A)$, together with the map $\alpha$, constitute what is called parametrized-manifold, of dimension $k$. We ... |
Why do we say this is a reparametrization? ("Analysis on Manifolds" by James R. Munkres.) |
Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then the law of total variance states that:
$$ V[X] = V[E[X|Z]] + E[V[X|Z]] $$
Then what if I want to apply the law of total variance on the following conditional variance: $V[X|Y]$? The goal is to deco... |
(I am reading Huybrechts's Complex Geometry)
I also looked some classic sources, but I didn't find anything addressing the below.
---
**Proposition $B.0.35$**
Let $$0\longrightarrow \mathcal{F}^0\longrightarrow \mathcal{F}^1\longrightarrow \mathcal{F}^2\longrightarrow 0$$
be a short exact sequence of she... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. The set $Y=\alpha(A)$, together with the map $\alpha$, constitute what is called parametrized-manifold, of dimension $k$. We ... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. The set $Y=\alpha(A)$, together with the map $\alpha$, constitute what is called parametrized-manifold, of dimension $k$. We ... |
I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from its probability distribution, or I observe a sample of values $x_1,...,x_n$ drawn from that distribution and I infer something about the distribution of the r.v. $X$. I have a conceptual problem ... |
>A group $G$ has sixteen elements:
>
>$$\{e, r, r^2, \dots , r^7, s, rs, r^2s, \dots , r^7s\},$$
>
>where $r$ and $s$ satisfy the relations $r^8 = e, s^2 = e, sr = r^3s$.
>(Note that $G$ is not a dihedral group.)
>
>a) Make a list of the cyclic subgroups of $G$. Make sure to state the elements of each cyclic su... |
Say we have $f : X \dashrightarrow Y$ as a dominant rational map between two affine varieties. Is it necessarily true that $\dim Y \leq \dim X$?
$f$ being dominant means that we must have that $f(\text{dom}(f)) = Y$, however I don't quite see how this may give the desired inequality. |
Letting $(\ * \ )$ denote convolution, I have an ODE of the form
\begin{align}
\dot{r} &= -7.4r -1.6f - 8.8 (f(t)*e^{-t}) - 10.4(f(t)*(te^{-t})) \\
\dot{f} &= 0.25r
\end{align}
with initial conditions $r(0) = f(0) = 0.$
I would like to linearize about $[r=0, f=0]^t$. I realized that in general I can linear... |
How to linearize time-variant ordinary differential equation (ODE), in particular, if there is a convolution term? |
Let $f \in L^1(\mathbb{R}^n)$ and $T$ is a distribution. Show that $T_{D_j^hf} \rightarrow \partial x_j T_f$ as $h \rightarrow 0$ in the sense of distributions, where $$D_j^hf(x) := \dfrac{f(x + e_jh) β f(x)}{ h}, \quad h>0.$$
I try to prove that, for any $\phi \in C^{\infty}_c(\mathbb{R}^n$
$lim_{h \rightarrow... |
I am currently engaging with the following hypergeometric function as a result of attempting to find [a solution for this probability problem for $n$ number of dice][1]:
$$_2F_1\left (\frac{n+k}{2}, \frac{n+k+1}{2};k+1;\frac{4b(d-e+1)}{d^2}\right ); \{n,k,b,d,e\}\in \mathbb{N}^+, z\in \mathbb{R},b < e \leq d$$
I'... |
Based on my limited understanding, the precise formulation of [Lowenheim-Skolem (LS) theorem](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) is stated as follows:
> Let $\Gamma$ be a set of sentences from a countable language $\mathcal L$. Let $\mathcal M$ be an infinite model for $\Gamma$, i.e... |
Based on my limited understanding, the precise formulation of [Lowenheim-Skolem (LS) theorem](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) is stated as follows:
> Let $\Gamma$ be a set of sentences from a countable language $\mathcal L$. Let $\mathcal M$ be an infinite model for $\Gamma$, i.e... |
So, I'm trying to see if my approach can show that the set $S$ is not affine and is convex with a similar argument for both cases
$S = \left\{ \alpha \in \mathbb{R}^3 : \alpha_1 + \alpha_2 e^{-t} + \alpha_3 e^{-2t}\right\} \\ \text{where } t \ge 1 $
My first observation is that, given $x \in S$ I can write it a... |
How to prove that the set $S = \left\{ \alpha \in \mathbb{R}^3 : \alpha_1 + \alpha_2 e^{-t} + \alpha_3 e^{-2t}\right\} $ is convex but not affine? |
Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both dense in $Y$.
- Show that $f$ is not a pointwise limit of continuous functions $X \to
Y$.
- Show that in the above s... |
Using Newton's Method, solve the following equation:
$\displaystyle 0 = \frac{1}{ 2}+ \frac{1}{4}x^2- x\sin(x)- \frac{1}{2}\cos(2x) $
iterate until you achieve an accuracy of $\displaystyle 10^{β5}$ with:
$ \displaystyle p_0 = \frac{\pi}{3}
\\
p_0 = 3\pi$
How do I do the calculations to get the precisi... |
I have it in the notes of an old project that $\cos^{2n}\left(\frac{x}{\sqrt{n}}\right) \to e^{-x^2}$ pointwise as $n\to \infty$. The proof I gave at the time sounds bogus, but it's quite close empirically for $n\ge 10$.
The bogus argument is basically that
$$\cos^{2}\left(\frac{x}{\sqrt{n}}\right)\approx \left(... |
I currently have the correlation/covariance matrix for a set of variables and the results of regression analysis but lack access to the raw dataset. Under these constraints, is it feasible to conduct a thorough analysis of endogeneity? I welcome any guidance on methodologies or statistical techniques that could be util... |
Endogeneity Analysis without the access of raw data? |
Firstly, here are two relevant definitions:
>Let $a \lt b$. A **partition** of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$.
>Suppose $f$ is bounded on $[a,b]$ and $P=\{t_0,t_1, \cdots, t_n\}$ is a partition of $[a,b]$. Let \begin{align} m_i=\inf... |
I have it in the notes of an old project that $\cos^{2n}\left(\frac{x}{\sqrt{n}}\right) \to e^{-x^2}$ pointwise as $n\to \infty$. The proof I gave at the time sounds bogus, but it's quite close empirically for $n\ge 10$.
The bogus argument is basically that
$$\cos^{2}\left(\frac{x}{\sqrt{n}}\right)\approx \left(... |
Problem
==============
I encountered this [statement](https://ncatlab.org/nlab/show/weakly+Lindel%C3%B6f+spaces+with+countably+locally+finite+base+are+second+countable) on nLab, which says that **weakly LindelΓΆf spaces with a $\sigma$-locally finite basis are second-countable**. The original proof given below the sta... |
Using $x'$ and $y'$ for the transformed points, from your method 1,
$$\begin{align*}
\begin{bmatrix}3 & 4 \\-2 & 8\end{bmatrix} \begin{bmatrix} t \\ \sin{t} \end{bmatrix} &= \begin{bmatrix} x' \\ y' \end{bmatrix}\\
\begin{bmatrix} t \\ \sin{t} \end{bmatrix} &= \begin{bmatrix}3 & 4 \\-2 & 8\end{bmatrix}^{-1} \begin... |
Consider the random variable $\delta_1$ with the following probability distribution, where we have:
$\mathbb{P}(\delta_1=0) = 1 - 2 \delta$, $\mathbb{P}(\delta_1=1)= \delta$, $\mathbb{P}(\delta_1=-1)= \delta$, where $\delta \ll 1$ is a small number.
Let $(\delta_i)_{1 \leq i \leq n}$ by an i.i.d. collection of r... |
Let $x\in R^d$ and $A\in R^{d\times d}$ positive definite. Is the map
$$
B
\mapsto x^T B^{1/2} A B^{1/2} x
$$
always concave? One known result that gives a little hope is the Lieb inequality
(cf. https://arxiv.org/abs/quant-ph/0404126 for instance) on the concavity of $trace[A^T \sqrt B A \sqrt B]$. |
Based on my limited understanding, the precise formulation of [Lowenheim-Skolem (LS) theorem](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) is stated as follows:
> Let $\Gamma$ be a set of sentences from a countable language $\mathcal L$. Let $\mathcal M$ be an infinite model for $\Gamma$, i.e... |
Firstly, here are two relevant definitions:
>Let $a \lt b$. A **partition** of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$.
>Suppose $f$ is bounded on $[a,b]$ and $P=\{t_0,t_1, \cdots, t_n\}$ is a partition of $[a,b]$. Let \begin{align} m_i=\inf... |
Firstly, here are two relevant definitions:
>Let $a \lt b$. A **partition** of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$.
>Suppose $f$ is bounded on $[a,b]$ and $P=\{t_0,t_1, \cdots, t_n\}$ is a partition of $[a,b]$. Let \begin{align} m_i=\inf... |
I'm trying to improve my understanding of complexity classes by reading the complexity zoo here, https://complexityzoo.net/Complexity_Zoo, and a number of other resources. I'm having an argument with a friend about a certain problem which I am hoping to resolve.
Suppose you have a machine, call it a 'mystery box'. I... |
Guess the number in the box - which complexity class does this belong in? |
In a monoidal category $\mathcal{C}$, Does any $f\in \operatorname{Hom}_{\mathcal{C}}(X\otimes \mathbf{1},Y\otimes \mathbf{1})$ can be expressed as $f=g\otimes \operatorname{Id}_{\mathbf{1}}$, where $g\in\operatorname{Hom}_{\mathcal{C}}(X,Y)$?
Short remark: I am reading a lecture note about tensor category by P. Eti... |
Suppose we define a relation $R$ in the natural set $\mathbb N$ which says: $$(x,y)\in R\iff x^2-4xy+3y^2=0$$ and we would like to find which of the following properties does $R$ satisfy.<br><br>
My book gives four options:<br>
(a) reflexive and transitive<br>
(b) reflexive and symmetric<br>
(c) symmetric and trans... |
Based on my limited understanding, the precise formulation of [Lowenheim-Skolem (LS) theorem](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) is stated as follows:
> Let $\Gamma$ be a set of sentences from a countable language $\mathcal L$. Let $\mathcal M$ be an infinite model for $\Gamma$, i.e... |
Assuming $B$ is a symmetic bilinear form over a vector $\mathcal V$, about the null space of $B$, I found two different definition:
(1) $v\in \mathcal V$ is the element of null space, if $B(v,w)=0$ for all $w\in \mathcal V$.
(2) $v\in \mathcal V$ is the element of null space, if $B(v,v)=0$.
I want to know whet... |
Suppose $f$ is a continuous function and $\lim\limits_{x\to\infty} f(x)$ does not exist. How do I prove that $\int_a^\infty f(x) \dx$ diverges? |
*Note: This is an [tag:alternative-proof] question, since I know how to prove the result but I'm asking for a **particular kind** of proof. Why? For the fun of it!*
Motivation:
---
I've been trying to give a reason behind the statement in [this question](https://math.stackexchange.com/q/4879328/104041) from a ... |
Suppose $f$ is a continuous function and $\lim\limits_{x\to\infty} f(x)$ does not exist. How do I prove that $\int_a^\infty f(x) \,dx$ diverges? |
Shouldn't the product of importance sampling weight and likelihood equal 1 in PSIS? |
Firstly, here are two relevant definitions:
>Let $a \lt b$. A **partition** of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$.
>Suppose $f$ is bounded on $[a,b]$ and $P=\{t_0,t_1, \cdots, t_n\}$ is a partition of $[a,b]$. Let \begin{align} m_i=\inf... |
I have the general solution $y(x) = Ee^x + De^{-x} + Fxe^x$, $E,D,F$ are reals arbitary constants. I need to find a 3rd order, linear, homogeneous ODE by this.
I know there are a few things I can do, for example, finding the first, second and third derivatives.
$ y'(x) = Ee^x-De^{-x}+F\left(e^x+e^xx\right)$
... |
***Background:***
-----------------
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
For primes in the range $1<p<100$, these are all $P$:
$$2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89$$
Here the proportion of $P$s out of all primes in $1<p<100$ is $\frac{14}{2... |
There's a problem that's bothering me lately: What's the area of a section of a unit sphere the shape of a non-euclidian triangle given that, when connecting the vertices of the triangle to the sphere's center, the angles formed by the three circular sectors of length $1$ are $\alpha$, $\beta$ and $\gamma$? My guess is... |
I'm not following this proof of what Bools,Burgess,Jeffrey call the *productivity function*. The proof states that the value from the *productivity function* is not computable. They call it $s$, and it computes a score for a $k$-state Turing machine, $s(k)$. They construct a new machine with one more state ($k + 1$ sta... |
***Background:***
-----------------
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
I will call a special prime like this $P$.
For primes in the range $1<p<100$, these are all $P$:
$$2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89$$
Here the proportion of $P$s ... |
Consider the following equation:
$$\left (A + \text{sgn}(x)\text{sgn}(x)^T \right )x= y \, x $$
where $A$ is an $n \times n$ **symmetric** matrix. The variables are $y \in \mathbb R, x \in \mathbb R ^n$ with $\| x \|_2=1$. Note this can be equivalently written as
$$Ax +\|x\|_1 \text{sgn}(x) =y \, x.$$
T... |
In AtCoder Regular Contest 172, the problem [E. Last 9 Digits](https://atcoder.jp/contests/arc172/tasks/arc172_e) asks to solve $n^n \equiv X \pmod{10^9}$ for $n$ (the smallest positive $n$, more precisely), given $X$ coprime to $10^9$.
The intended solution is, first solve it in reducing $10^2$, then lift the solut... |
There's a problem that's bothering me lately: What's the area of a section of a unit sphere the shape of a non-euclidian triangle given that, when connecting the vertices of the triangle to the sphere's center, the angles formed by the three circular sectors of length $1$ are $\alpha$, $\beta$ and $\gamma$? My guess is... |
In AtCoder Regular Contest 172, the problem [E. Last 9 Digits](https://atcoder.jp/contests/arc172/tasks/arc172_e) asks to solve $n^n \equiv X \pmod{10^9}$ for $n$ (the smallest positive $n$, more precisely), given $X$ coprime to $10^9$.
The intended solution is, first solve it in reducing $10^2$, then lift the solut... |
Problem 9(i) of Spivak's *Calculus* (Chpt 22) asks for the following limit:
>$\displaystyle \lim_{n \to \infty}\frac{\sqrt[n]{e^1}+\sqrt[n]{e^2}+\cdots\sqrt[n]{e^n}}{n}$
The terse solution manual's argument reads as follows:
>$\displaystyle \int_0^1 e^x dx = e-1$
**Note: My book has not yet covered series**... |
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$ in $[0, 1)$, the por... |
Consider the random variable $\delta_1$ with the following probability distribution, where we have:
$\mathbb{P}(\delta_1=0) = 1 - 2 \delta$, $\mathbb{P}(\delta_1=1)= \delta$, $\mathbb{P}(\delta_1=-1)= \delta$, where $\delta \ll 1$ is a small number.
Let $(\delta_i)_{1 \leq i \leq n}$ by an i.i.d. collection of r... |
I was solving this question related to ellipses but got stuck, please help me in angle chasing:
Could not even move further :
> Consider the ellipse $4x^2+y^2-8x+4y+4=0.$ Pair of tangents $PT_1$ and $PT_2$ are drawn to the ellipse from $P(-8,-2).$ Let $S_1'$ be the image of $S_1$ about $PT_1$ and $S_2'$ be the i... |
I realized that **Q.2** is wrong in general. If we assume that $Y$ is irreducible, it is still true though.
Your attempt to answer **Q.2** works in the case that $B$ is an integral domain, because then $l \varphi(a) = 0$ implies $l = 0$ or $\varphi(a) = 0$. But $l \in B \setminus \mathfrak p$, so $l \neq 0$.
Let'... |
There are so many defects in the formulas given on that page that it is hard to figure out what formulas they meant to write there.
> $\theta = \cos^{-1}(x / r)$
If you solve this for $x$, it says that $x = r \cos\theta$.
But if you look at the diagram at the top of the page that supposedly says what the angles ... |
Why can $P_1 β P_2 β P_3 β \ldots β P_n$ be true when not all the $P$βs have the same truth value?
---
For example: If P1 = T P2 = T P3 = F P4 = F would this be true?
T(P1) β T(P2) β F(P3) β F(P4) =
T(P1&2) β F(P3) β F(P4) =
F(P1,2,3) β F(P4) =
True
If so how can an evaluation for a series of compoun... |
Disclaimer: This isn't very elegant and just brute-forces through the solution.
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You're trying to find $$P(a+b\ge c,a+c\ge b,b+c\ge a|a,b,c\sim U(0,1),b^2-4ac\ge0)$$ which is equal to $$\frac{P(a+b\ge c,a+c\ge b,b+c\ge a,b^2-4ac\ge0|a,b,c\sim U(0,1))}{P(b^2-4ac\ge0 | a,b,c\sim U(0,1))}$$
Since $a,b,c$ are c... |
I was solving this question related to ellipses but got stuck, please help me in angle chasing:
Could not even move further :
> Consider the ellipse $4x^2+y^2-8x+4y+4=0.$ Pair of tangents $PT_1$ and $PT_2$ are drawn to the ellipse from $P(-8,-2).$ Let $S_1'$ be the image of $S_1$ about $PT_1$ and $S_2'$ be the i... |
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