instruction stringlengths 12 30k |
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Let $g:[0,T] \to \mathbb{R}^n$, $T>0$ and let $f:\mathbb{R}^n \to [0,+\infty)$.
I know that $g(\cdot)$ is absolutely continuous, and I want to know under what assumptions on $f(\cdot)$ the composition $f \circ g$ is a measurable function? (the underlying measurable space is Lebesgue measurable).
Thanks for any ... |
What regularity on a function when composed with an absolutely continuous makes the composition measurable? |
Suppose I have a distribution $D$, a function $f$, and another distribution $D_f$. Suppose I want to transform $D$ to a similar distribution $D'$ such that for $X \sim D'$ it holds that the distribution of $f(X)$ is $D_f$. If $D$ is discrete, I may define $D'$ as follows
$$
D'(x) = E_{y \sim D_f}[P_D[x | f(x) = y]]
... |
So I'm currently doing an advanced linear algebra module and I'm confused on two statements from two different lectures. I'll attach a section of two lectures and highlight the statements that confuses me. (I apologize for poorly typing the equations, I don't know yet how to type formulas on this site)
My understand... |
I am trying to self-study the textbook "An Introduction to Convexity, Optimization, and Algorithms" by Walaa M. Moursi. In part 3.1 she states that:
> $f$ is called *proper* if $dom(f) \neq \emptyset$ and every $x \in X$ satisfies $-\infty < f(x)$.
Keep in mind that, in this book, $X$ stands for Euclidean space. ... |
Why every function $f: X \rightarrow \mathbb{R}$ is proper? |
From https://math.stackexchange.com/questions/2350219/how-to-show-orthogonality-of-associated-laguerre-polynomials or https://math.stackexchange.com/questions/92612/finding-a-generating-function-for-the-laguerre-polynomials it appears that the expression, $L^q_p(x)$ is that of an associated Laguerre polynomial and defi... |
N > 1 is a natural number.
There are two vectors of N dimension, A and B:
$A = (a_1, a_2, ..., a_N)$ where $a_1>a_2>...>a_N>0$
$B = (b_1, b_2, ..., b_N)$ where $1>b_1>b_2>...>b_N>0$
Let $\lVert A \rVert_p$ denote a $p$-norm of vector A, $\lVert A \rVert_p\equiv\left(\sum_{k=1}^N a_k^p\right)^{1/p}$.
Th... |
>Suppose functions $y$ and $f$ are non-negative and continuous on the interval $<a,b>$ (where $<$ can be $( $ or $[ $ and $>$ can be $] $ or $) $), and let $\lambda > 0$. If the inequality
$
y(x) \leq f(x) + \lambda \Big| \int_{x_0}^x y(s) \, ds \Big|, \quad x, x_0 \in (a,b)
$
holds, then
$
y(x) \leq f(x) + \lam... |
I am given the following definition of the right endpoint $x_F$ of the distribution function $F$:
$x_F = sup\{x \in \mathbb R: F(x) < 1\}$
I know that the right endpoint of a discrete uniform distribution $\mathcal U[1,5]$ is $x_F = 5$. However, this does not make sense to me with respect to the definition above.... |
I am trying to understand the proof to Lemma 5.6 (original paper here: https://arxiv.org/pdf/2303.05423.pdf), which reads as follows(I include lemma 5.5 as well, which is referenced):
Lemma 5.5. The interior of the complement of a proper convex set $C \subset \mathbb{R}^n$ is non-empty.
Lemma 5.6. Every proper $... |
Is there any prime $p$ such that $6x^3 − p^2 − y^2 = 0$ has an integer solution? |
From https://math.stackexchange.com/questions/2350219/how-to-show-orthogonality-of-associated-laguerre-polynomials or https://math.stackexchange.com/questions/92612/finding-a-generating-function-for-the-laguerre-polynomials it appears that the expression, $L^q_p(x)$ is that of an associated Laguerre polynomial and defi... |
I would approach using Lagrange multiplier method. Let the vector $k = [0,0,1]^T$, and let $ r = [x,y,z]^T $, then you cone and ellipsoid equations are
$ r^T Q_c r = 0 $ (for the cone), and
$r^T Q_e r = 1 $ (for the ellipsoid)
And you want to maximize $z = k^T r $, so define
$ G(r, \lambda_1, \lambda_2) = ... |
I am given a smooth connected manifold $M$ with two smooth metric tensors $g_1$ and $g_2$. Let us denote by $d_1$ and $d_2$ the distance they induce on $M$, respectively. I have two ways of "mixing" these two distances on $M$: for points $z_0, z_1 \in M$ we can define
$$d'(z_0,z_1):= \inf \left \{ \int_0^1 \left (|... |
My question is :
How many solutions are there to the equation $x_1 + x_2 + x_3 + x_4 +x_5= 10$ where $x_1, x_2, x_3, x_4, x_5$ are positive integers and $x_1$ is an odd number?
I tried to solve it using Stars and bars, by getting to this formula $x_1=2y_1, x_2=y_2+1,x_3=y_3+1,x_4=y_4+1,x_5=y_5+1.$
which equals to... |
How many solutions are there to the equation $x_1 + x_2 + x_3 + x_4 +x_5= 10$ where $x_1, x_2, x_3, x_4, x_5$ are positive integers? |
Sudeep has given a nice hint in the comments. First observe that for $n=3$, you require 2 moves in the worst case, so $C\le 0$. By "determine" I mean deduce the type of coin.
Next, note that we can divide the $n$ coins into $n/3$ parts or $\lfloor n/3\rfloor+1$ parts depending on $n \text{ mod } 3$. In the second ca... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
A Cunningham Chain is defined as a sequence of prime numbers $p_i$ such that $p_{i+1} = 2p_i +1$ (or $p_k = 2^k (p_0 + 1) - 1$). The conjecture is that there infinitely many Cunningham Chains of any given length $k$. One way this conjecture could be false, is if there exist positive integer $N$ such that for every $0≤n... |
Let $\mathrm{Ai}$ denote the Airy function.
Is it possible to compute explicitly (or bound)
$$
\int_0^{K} \frac{\mathrm{Ai}(z)}{\mathrm{Ai}(-z)} dz
$$
for $K >0$ (or also for $K = \infty$)?
|
Let $\mathrm{Ai}$ denote the [Airy function][1].
Is it possible to compute explicitly (or bound)
$$
\int_0^{K} \frac{\mathrm{Ai}(z)}{\mathrm{Ai}(-z)} dz
$$
for $K >0$ (or also for $K = \infty$)?
[1]: https://en.wikipedia.org/wiki/Airy_function |
A Cunningham Chain is defined as a sequence of prime numbers $p_i$ such that $p_{i+1} = 2p_i +1$ (or $p_k = 2^k (p_0 + 1) - 1$). The conjecture is that there infinitely many Cunningham Chains of any given length $k$. One way this conjecture could be false, is if there exist positive integer $N$ such that for every $1≤n... |
I am given a smooth connected manifold $M$ with two smooth metric tensors $g_1$ and $g_2$. Let us denote by $d_1$ and $d_2$ the distance they induce on $M$, respectively. I have two ways of "mixing" these two distances on $M$: for points $z_0, z_1 \in M$ we can define
$$d'(z_0,z_1):= \inf \left \{ \int_0^1 \left (|... |
I'm trying to derive a general formula for the number of distinguishable ways to paint the sides regular $n-$gon using $m$ colors.
Two rules:
We don't have to use all of the $m$ colors and we have to account for rotations.
My progress:
We have to divide whatever we end up getting by $n$ to account for rota... |
> Prove the polynomial $$T(\theta)=a_0+\sum\limits_{k=1}^{n}(a_k\cos
k\theta+b_k\sin k\theta)$$ has no more than $2n$ distinct roots on
> $[0,2\pi).$
My attempt:\
From $\cos k\theta=\dfrac{e^{ik\theta}+e^{-ik\theta}}{2}$ and $\sin k\theta=\dfrac{e^{ik\theta}-e^{-ik\theta}}{2i}$, $T$ can be rewritten as $T(\th... |
Consider the ODE
$$h'(t) = -k \sqrt{h(t)} \,.$$
Suppose we also know that $h(0)=16$ and $h(30)=4$. How do we find the function $h(t)$?
**Attempt:** we could try integrating the ODE between $t=0$ and $t=30$:
\begin{align*}
\int_0^{30} h'(s) ds &= -k \int_0^{30} \sqrt{h(s)} ds
\end{align*}
the left side of... |
Help solving a simple ODE? |
Can anyone help with this matrix problem? |
I am working on understanding the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. As part of their definition of equations for a 3D surface they use a function $F(ξ, \bar{ξ})$, where $ξ$ is "the local complex coordinate on the unit 2-sphere in $^3$ obtained by stere... |
I found in Encyclopedia of math
https://encyclopediaofmath.org/wiki/Abelian_surface
there is a claim that:
"A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of a smooth projective curve $H$ of genus $2$, and $λ=θ$ is the class of the theta divisor, or $A$ is the product of t... |
A Cunningham Chain is defined as a sequence of prime numbers $p_i$ such that $p_{i+1} = 2p_i +1$ (or $p_k = 2^k (p_0 + 1) - 1$). The conjecture is that there infinitely many Cunningham Chains of any given length $k$. One way this conjecture could be false, is if there exist positive integer $N$ such that for every $1≤n... |
2n knights around a table with namecards, is it possible that for every rotation there is exactly one person with a correct namecard? |
Define a non trivial additive character $\psi: \mathbb F_q \rightarrow \mathbb C$ s.t $\psi(x)= \xi^{Tr(x)}$ where $\xi=e^{2\pi i /p}$, where $Tr(x):\mathbb F_q \rightarrow \mathbb F_p$ .\
First question: is it correct that $Tr(x)= x+ x^p + ... + x^{p^{r-1}}$, with $q=p^r$?\
Define now a multiplicative character $\c... |
Consider the ODE
$$h'(t) = -k \sqrt{h(t)} \,,$$
where $k$ is a positive constant. Suppose we also know that $h(0)=16$ and $h(30)=4$. How do we find the function $h(t)$?
**Attempt:** we could try integrating the ODE between $t=0$ and $t=30$:
\begin{align*}
\int_0^{30} h'(s) ds &= -k \int_0^{30} \sqrt{h(s)} d... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
I would approach this problem by using the Lagrange multiplier method. Let the vector $k = [0,0,1]^T$, and let $ r = [x,y,z]^T $, then you cone and ellipsoid equations are
$ r^T Q_c r = 0 $ (for the cone), and
$r^T Q_e r = 1 $ (for the ellipsoid)
And you want to maximize $z = k^T r $, so define
$ G(r, \lam... |
Consider the ODE
$$h'(t) = -k \sqrt{h(t)} \,,$$
where $k$ is a positive constant. Suppose we also know that $h(0)=16$ and $h(30)=4$. How do we find the function $h(t)$?
**Attempt:** we could try integrating the ODE between $t=0$ and $t=30$:
\begin{align*}
\int_0^{30} h'(s) ds &= -k \int_0^{30} \sqrt{h(s)} d... |
(Summary of the comments:)
The input of Turaev-Viro model is a spherical fusion 1-category $\mathcal{C}$, and it produces a **fully extended** 3d TQFT in the Morita 3-category of tensor categories c.f. [DSPS](https://arxiv.org/abs/1312.7188) whose value on the circle $S^1$ gives you the Drinfeld center $\mathcal{Z}(\m... |
This is a bit of a soft question, but why is the Monster group called that? Does it have any connection with monster models in model theory? I would be interested to learn who came up with that terminology. |
Why is the Monster group called that? |
Theorem 3 in the paper of Bary-Soroker et al. (https://arxiv.org/pdf/2007.14567.pdf) states the following about probability of a polynomial over $\mathbb{Z}$ being irreducible:
> Let $H \ge 3$ and $n \ge 3$ be integers, and let $\mu$ be a
> probability measure on $\mathbb{Z}$ such that:
> * (support not too large)... |
Let $C_i\sim C$ and $R_i \sim R$ denote the durations (in ticks) of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$... |
I am given a smooth connected manifold $M$ with two smooth metric tensors $g_1$ and $g_2$. Let us denote by $d_1$ and $d_2$ the distance they induce on $M$, respectively. I have two ways of "mixing" these two distances on $M$: for points $z_0, z_1 \in M$ we can define
$$d'(z_0,z_1)^2:= \inf \left \{ \int_0^1 \left ... |
In [this MathOverflow link][1] -originally posted here in MSE- I have placed 4 fast hypergeometric-type series for logarithm constants. I could prove 3 of them by using a Beta function approach transforming the series to integrals that are evaluated straightwise. One of them, Eq.(2), remains un-proven.
Applying the ... |
How to prove this difficult double integral identity? |
As I understand it, if a region contains the origin, Green's Theorem cannot be applied. However, one question in my issued lecture notes appears to contradict this and another one appears to follow this rule.
In the first question from the lecture notes, Green's Theorem can be used in this question (and is used in t... |
I am interested in a particular 16-dimensional representation of $A6$, the alternating group on 6 things. I first construct an amalgam, gamma, of two copies of SymmetricGroup(4):
F:=FreeGroup(["s1","s2","s3","t1","t2","t3"]);
AssignGeneratorVariables(F);
rel:=Union(
[s1^2,s2^2,s3^2,s1*s3*s1^-... |
Since you have a basis of eigenvectors you have a have a unique decomposition of vectors on the distinct eigenspaces. That is, for all $v \in V$ there is a unique decomposition
$$ v = \sum_{\lambda \in \sigma(A)} v_\lambda $$
where $v_\lambda \in V_{(\lambda)} := N(A - \lambda I)$. This defines the projections
... |
We want to know if a smooth surface under a $C^{\infty}$ diffeomorphism (infinitely differentiable with a infinitely differentiable inverse) map to a smooth surface too?
Further, it would be nice to understand what happens if the diffeomorphism was $C^k$ for some $k\geq 1$.
So far I have looked at these books:
... |
Does a $C^{\infty}$ diffeomorphism map a smooth hypersurface to a smooth hypersurface? |
(Summary of the comments:)
The input of Turaev-Viro model is a spherical fusion 1-category $\mathcal{C}$, and it produces a **fully extended** 3d TQFT in the Morita 3-category of tensor categories c.f. [DSPS](https://arxiv.org/abs/1312.7188) whose value on the circle $S^1$ gives you the Drinfeld center $\mathcal{Z}(\m... |
Suppose $A$ is a fixed densely defined self-adjoint operator on a complex Hilbert space $\mathfrak{h}$.We are given a function $f \in C_{0}^{\infty}([0,\infty))$ with compact support and smooth such that $0 \le f \le 1$, $f(x) = 1$ if $0 \le x \le 1$ and $f(x) = 0$ if $x \ge 2$.
Let $\lambda \in \mathbb{R}$ be nonz... |
How should I interpret this operator equality? |
I have a bunch of signed distance functions representing non-overlapping objects in 3d space. I would like to find boundaries between the these signed distance functions such that the boundary marks the surface that is maximally distant to its own sdf while not being closer to any other sdf present in the volume. What ... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
As I understand it, if a region contains the origin, Green's Theorem cannot be applied. However, one question in my issued lecture notes appears to contradict this and another one appears to follow this rule.
In the first question from the lecture notes, Green's Theorem can be used in this question (and is used in t... |
This is a follow up to https://math.stackexchange.com/questions/2978806/a-martingale-converging-in-distribution-but-not-a-s-or-in-probability?noredirect=1&lq=1. Suppose we have a sequence of integer-valued RVs that satisfies
(1) $P(X_n=a~i.o.)=1,a=-1,0,1$
(2) $\sup_n|X_n(w)|<\infty$ a.s.
(3) For some $p\in(0,1... |
Why do these three conditions imply divergence in probability? |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
We want to know if a smooth surface under a $C^{\infty}$ diffeomorphism (infinitely differentiable with a infinitely differentiable inverse) map to a smooth surface too?
Further, it would be nice to understand what happens if the diffeomorphism was $C^k$ for some $k\geq 1$.
So far I have looked at these books:
... |
$2n$ knights around a table with namecards, is it possible that for every rotation there is exactly one person with a correct namecard? |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
The question asked
> How does one compute the constant
$$K=\prod_{n=0}^\infty \frac{1}{\sqrt{1+2^{-2n}}}$$
to an arbitrary number of decimal places?
Define the sequence of finite products
$$ K_n(q) := \prod_{k=0}^n \frac1{\sqrt{1+q^k}}. \tag1 $$
If $\,|q|<1,\,$ this sequence converges to the limit
$$ K(q... |
I have difficulty on the part a)5 and b) 2
### Question
Let a linear mappimg $\phi : E \rightarrow F$ between two normed vector space $(E; \| . \|_E)$ and $(F; \| . \|_F)$.
a) Prove that in $ \mathbb{R}^+ \cup \infty $. We have:
$$ \| \phi \| =_{def} Sup \{ \frac{ || \phi(x) ||_E}{ \| x \|_F} : x \in E -... |
The question asked
> How does one compute the constant
$$K=\prod_{n=0}^\infty \frac{1}{\sqrt{1+2^{-2n}}}$$
to an arbitrary number of decimal places?
Define the sequence of finite products
$$ K_n(q) := \prod_{k=0}^n \frac1{\sqrt{1+q^k}}. \tag1 $$
If $\,|q|<1,\,$ this sequence converges to the limit
$$ K(q... |
I originally wanted to compute the following quantity
$$
<v_{max}^2>_{S_d}=< \frac{\| x\|_{\infty}^2}{\|x\|_2^2}>_{\mathcal{N_n(0,1)}}
$$
which is the average value of the squared ratio between the max norm and the 2 norm, where $$
x_i \sim \mathcal{N(0,1)}$$
This is basically the average value of the max of a v... |
What is the average of the squared ratio between the 2 norm and the max norm of a vector of iid gaussian variables? |
Let $X$ be a mean-zero random variable whose moment generating function is bounded by a symmetric exponential function, e.g:
$$
\mathbb{E}[e^{\lambda X}] \leq \exp(c|\lambda|)
$$
with $c > 0$.
**Question**:
Is it true that $X$ must be **bounded**?
I asked one of my professors this and he told me it was the... |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y$$
I cannot pursue it because of the floating point in comp... |
I need to find the missing point of a tetrahedron side length x-- being v1=(0,0,0) v2=([x/2],0,{[sqrt3]/2}x) v3=(x,0,0). I can't seem to find the 4th point. Can anyone help? I just need the equation |
I am studying a course on Stochastic Processes and encountered the following proof exercise on Poisson Processes:
> If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k \leq n$ are non-negative integers, then prove the following:
$$ P(N(s) = k | N(t) = n) = \binom{n}{k} \Big{(} \frac{s}... |
Suppose you have a matrix $A$. Is there a "standard"/mathematical elegant way to denote all members of the matrix as a set?
For example: suppose there is a matrix $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] $ then I would like to know how to define a function $set( \cdot)$ such that we have:
... |
How to concisely denote all the elements of a matrix as a set? |
I need to find the missing point of a tetrahedron side length $x$, with three points being
$$v_1=(0,0,0) \quad v_2=\left(\frac{1}{2}x,0,\frac{\sqrt3}{2}x\right) \quad v_3=(x,0,0)$$
I can't seem to find the 4th point.
Can anyone help? I just need the equation. |
Is there a name for this property of sequences? |
I am reading "Introduction to Topology" by Gamelin and Greene (2nd edition) and I am confused by what appears to be an inconsistency in notation for subsets and strict subsets.
The image below shows page 61 from this book. From the solid red highlighted expressions, it appears that $A \subset B$ means $A$ is a subse... |
A set is $C$ is convex if $tC+(1-t)C\subset C$ where $0\leq t \leq 1$.
how do we manage to extend it for n vectors?
I manage to get $t^2 C + t(1-t)C + t(1-t)C + (1-t)^2 C \subset C$ but induction through this process only gives convexity definition for 2,4,8.. vectors only.
There is an alternate derivation for t... |
I have the following question in real analysis course:
\
let $p(x)=x^{4}+a_{3}x^{3}+a_{2}x^{x}+a_{1}x+a_{0}$. Prove that if $x_{0}$ is a root of P, meaning $p(x_{0})=0$ and $p'(x_{0})\neq0$, then p has at least two roots.
\
approach: I tried to make several things, like first of looking at $p(-x_{0})$ and denote $... |
This [question](https://math.stackexchange.com/questions/4817128/fundamental-group-of-a-simplicial-space-with-trivial-0-skeleton) asks the same as mine, but it was unsuccessful in getting an answer, so I try again. <br>
For context I am reading Weibel's K-book and am struggling with proposition 8.4 which computes $... |
I've seen the axiom of choice as $\forall X\left[\varnothing \notin X\implies \exists f\colon X\rightarrow \bigcup _{A\in X}A\quad \forall A\in X\,(f(A)\in A)\right]$.
But isn't there a version which doesn't need the definition of a function and of the union? I tried numerous times to come up with one but unsuccessf... |
What is the full axiom of choice (with "$\in$" as the only atomic formula)? |
I've seen the axiom of choice as $\forall X\left[\varnothing \notin X\implies \exists f\colon X\rightarrow \bigcup_{A\in X}A\quad \forall A\in X\,(f(A)\in A)\right]$.
But isn't there a version which doesn't need the definition of a function and of the union? I tried numerous times to come up with one but unsuccessfu... |
$\textbf{Ordinary Differential Equation:}$
Let $x(\cdot) \in C([0,1];\mathbb{R}^n)$ (C($\cdot$) denotes the set of continuous functions) be the trajectories satisfying the following differential equation
$$\begin{gather}\label{eq:ode}
\dot{x}(t) = f(t,x(t))\quad x(0) = x_0\in \mathbb{R}^n
\end{gather}
$$
such... |
Is every convex cone a subset of a half space? |
I wonder, is there any subgroups of $\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}$ isomorphic to $\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}$? It seems like the answer is no since $\mathbb{Z}_{16}$ is not isomorphic to $\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}$, but I want to know more
explicit and accurate proof. Or may be there is some c... |
### Question
Let a linear mappimg $\phi : E \rightarrow F$ between two normed vector space $(E; \| . \|_E)$ and $(F; \| . \|_F)$.
a) Prove that in $ \mathbb{R}^+ \cup \infty $. We have:
$$ \| \phi \| =_{def} Sup \{ \frac{ || \phi(x) ||_E}{ \| x \|_F} : x \in E - \{ 0\} \} = Sup \{ || \phi(x) ||_F : \| x \... |
I originally wanted to compute the following quantity
$$
\langle v_{\max.}^2 \rangle_{S_d} = \left\langle \frac{\| x\|_{\infty}^2}{\|x\|_2^2} \right\rangle_{\mathcal N_n(0,1)}
$$
which is the average value of the squared ratio between the max norm and the 2-norm, where $$
x_i \sim \mathcal N(0,1)$$
This is basic... |
I am studying numerical methods from the textbook [Monte Carlo Methods in Financial Engineering][1] by Paul Glasserman and have encountered the following exercise:
> I want to simulate from the stochastic processes satisfying the following SDEs
>
> $$dX_i = X_i (\mu dt + \sigma_{(1)} dB^{(1)}_i) \tag{1}$$
> $$... |
One way to solve $x^2y''+2xy'+2y=0$ is using the substitution $x=e^t$. This time I'm asked to use power series (Fröbenius method). Assuming $y=\sum_{n\geq 0}a_nx^{n+r}$, the diff eq reduces to
$$\sum_{n\geq 0}(n+r)(n+r-1)a_nx^{n+r}+\sum_{n\geq 0}2(n+r)a_nx^{n+r}+\sum_{n\geq 0}2a_nx^{n+r}=0$$
$$\sum_{n\geq 0}\{(n+r)(n... |
Is there any non-surjective linear operator that is open?
By the open mapping theorem, $T:X \to Y$ is open iff $T$ is surjective. Is it possible then to have a non-surjective linear operator that is open?
I have been thinking of $T:\mathbb{R}^2 \to \mathbb{R}^2,\ (x, y) \to (x, 0)$
Here T is not surjective but... |
Non-surjective linear operator that is open? |
I have the following question in real analysis course:
\
let $p(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$. Prove that if $x_{0}$ is a root of P, meaning $p(x_{0})=0$ and $p'(x_{0})\neq0$, then p has at least two roots.
\
approach: I tried to make several things, like first of looking at $p(-x_{0})$ and denote $... |
I saw this problem a while ago and a found the same answer as the textbook, yet I wasn’t truly sure if that was correct. Here was the problem:
>You have a wheel with 4 equal sections, red, yellow, green and blue. Another wheel has 3 equal sections: red, yellow, and green. Knowing that exactly 1 wheel landed green, wh... |
Suppose there is a sequence of$N$ numbers $x_1, x_2, x_3, ... x_N$.
There are then gaps $|x_i - x_j|$, and the minimum gap:
$\delta (N) = \text{min}_{i \ne j \le N} \{ | x_i -x_j | \}$.
Let the mean gap be normalized to $1/N$.
If the sequence of numbers $x_1, x_2, x_3, ... x_N$ is random, the minimum ga... |
I am interested in undirected, planar, connected graphs where every edge is in a 3-cycle. If there are 4 vertices then, up to isomorphism, there are two such graphs.
[![enter image description here][1]][1]
How many are there for 5 vertices?
[1]: https://i.stack.imgur.com/ZbGrf.png |
If $m$ is the Lebesgue measure on $\mathbb{R}^n$, for any $E \subseteq \mathbb{R}^n$ we can define the inner and outer Lebesgue measure as:
$$ m^*(E) = \inf { m(U) : E \subseteq U, \text{ $U$ is open } \} $$
$$ m_*(E) = \sup { m(F) : F \subseteq E, \text{ $E$ is closed } \} $$
I know that if $E$ is measurable th... |
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