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**Question**: What is the probability that the coefficients of a quadratic equation form the sides of triangle given that it has real roots? Assume that the coefficients are uniformly distributed and positive.
Note that it is sufficient to assume that the coefficients are uniformly distributed in $(0,1)$ since we ca... |
How many pairs $(a,b)$ such that $\frac{x}{a}+\frac{y}{b}$ is in set $S=\{x \in \mathbb{R} : 0 \le x \le 1\}$? |
I am trying to understand the proof of this statement.
> Any Hausdorff topology on a finite-dimensional vector space with
> respect to which vector space operations are continuous is equivalent
> to the usual one.
Source: https://personal.math.ubc.ca/~cass/research/pdf/TVS.pdf
In the proof it is stated that:... |
Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, then it is standard that $R$ is also a normal domain. This leads me to ask the following question:
Let $X, Y$ be Noe... |
Assume an invertible matrix $C$ is in the form $\left[ \begin{matrix}A \\ B \end{matrix}\right]$, where the linear subspace generated by the rows of $A$ is orthogonal to the linear subspace generated by the rows of $B$. Denote the Moore–Penrose pseudoinverses of $A$ and $B$ by $A^{+}$ and $B^{+}$, respectively.
How... |
**Question**
Suppose a stock pays 2 discrete dividends $d_1, d_2$ at times $t_1, t_2$ respectively, where $ t < t_1 < t_2 < T.$ Assume the risk-free rate, $r$, is a positive constant. Given that
1. The lower and upper bounds for an American call, $C_t$, is trivially determined as
$$\max{ \left( S_t - D_t - Ke^{-... |
I was writing a simple program to help visualize inequalities based on 2 variables. The test inequality that I was using was this: $$\sin\left(0.1(x^2+y^2)\right)\geq0$$
Regions that satisfy the inequality are coloured in white.
Here's the result, for a bounding box of $(-25,-25),(25,25)$:
[![enter image descrip... |
Consider a function $g : [-1,1] \to \mathbb{R}$ and $c_n$ denotes the $nth$ Chebyshev coefficient of the function $g$. Moreover, $$c_n = \frac{2}{\pi}\int_{-1}^{1} T_n(x) g(x) \frac{1}{\sqrt{1-x^2}} dx = O(n^{-m}),\quad n\ge 1,$$ for some natural number $m$. Define $f$ as the restriction of $g$ on closed interval $[a,... |
"It is false that books are made completely out of water molecules."
The above sentence has a connective in it: 'It is false that'. So, the above sentence is a composite sentence. Thus, the denial of any simple statement must be a compound statement, wouldn't it? |
Is this a composite sentence? |
$$ y'' + \lambda y = 0, \quad y'(0) = 0 \quad |y(x)| < \infty \quad \text{for all } y \text{ in } (0, \infty),$$
I have tried numerous Sturm Liouville Boundary Value problems, but never done problems involving boundary conditions where there are inequalities and infinities. Is this a regular SLBVP or does it classif... |
You're correct; we should add σ-Locally Finite Base + Weakly Lindelöf => Second Countable to the pi-Base. https://github.com/pi-base/data/issues/584
First, I'll confirm that your claim that every locally finite (or even locally countable) collection in a weakly Lindelöf space is countable. So a σ-locally finite base... |
The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences.
"It is false that books are made completely out of water molecules."
The above sentence has a connective in it: 'It is false that'. So, the above sentence is a composite senten... |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences.*
> It is false that books are made completely out of water molecules.
The above sentence has a connective in it: 'It is false that'. So, the above sentence is a composite sent... |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
> It is false that books are made completely out of water molecules.
The above sentence has a connective in it: 'It is false that'. So,... |
Let $G$ be a countable group, and $l^\infty(G)$ the collection of functions $f:G \rightarrow \mathbb{R}$ such that $\sup |f(x)| < \infty$. I know that $G$ is amenable if and only if for all $\epsilon > 0$ and for all finite sets $H \subset G$ there exists a nonempty finite set $K \subset G$ such that $$ \frac{|Kx \Delt... |
For example in this pdf https://bpb-ca-c1.wpmucdn.com/sites.uoguelph.ca/dist/8/175/files/2021/04/Fib_lin_alg.pdf
The author created a matrix from the output of Fibonacci sequence in group of 2, and take 2 of the group
Fibonacci sequence 0 1 1 2 3 5 8 13 21 34
Group of 2
fi 0 1 1 2 3
fi+1... |
Whats the intuitive explanation that matrix multiplication solves linear recurrence relationship? |
Whats the intuitive explanation that matrix multiplication of recurrence output solves linear recurrence relationship? |
Whats the intuitive explanation that matrix multiplication of first 4 recurrence value solves linear recurrence relationship? |
I am trying to show that Mat$_{n\times k}^k\to G_k(\mathbb{R}^n)$ is a fibration where Mat$_{n\times k}^k$ denotes the full-rank matrices of rank $k$. I know that $V_k(\mathbb{R}^n)\to G_k(\mathbb{R}^n)$ is a fibration and composition of fibrations is a fibration by https://math.stackexchange.com/questions/2188538/comp... |
As title states, I'm a tiny bit confused as to how the standard extension of a language $\mathcal{L}$ into $\mathcal{L}(\mathfrak{A})$ over a given structure $\mathfrak{A}$ works exactly.
As I've seen it (and this seemed totally intuitive to me up till now), $\mathcal{L}(\mathfrak{A})$ simply is built off $\mathcal{... |
How does the extended language $\mathcal{L}({\mathfrak{A}})$ over a structure $\mathfrak{A}$ work exactly? |
For $\mu \in \mathcal P(\mathbb R)$, let $m(\mu)$ be the median of $\mu$, defined as the smallest of all medians of $\mu$ as follows:
$$
m(\mu) = \inf \left\{ x \in \mathbb R \,\middle|\, \mu((-\infty,x]) \ge \frac12 \right\}.
$$
Assume that $\mathcal P(\mathbb R)$ is endowed with the weak convergence topology. I... |
This was confusing me when learning about curvature and smoothness. The condition for smoothness on interval $I$ is given as:
1. $\mathbf r'$ is continuous;
2. $\mathbf r'(t) \neq \mathbf 0$.
In this particular example, the space curve of this parametrization is just a straight line passing through the origin, bu... |
Is the vector function $\mathbf r(t) = \langle t^3, t^3 \rangle$ smooth at $t = 0$? |
For example in this pdf https://bpb-ca-c1.wpmucdn.com/sites.uoguelph.ca/dist/8/175/files/2021/04/Fib_lin_alg.pdf
The author created a matrix from the output of Fibonacci sequence in group of $2$, and take $2$ of the group
Fibonacci sequence 0 1 1 2 3 5 8 13 21 34
Group of 2
fi 0 1 1 2 3
... |
I was writing a simple program to help visualize inequalities based on 2 variables. The test inequality that I was using was this: $$\sin\left(0.1(x^2+y^2)\right)\geq0$$
Regions that satisfy the inequality are coloured in white.
Here's the result, for a bounding box of $(-25,-25),(25,25)$:
[![enter image descrip... |
I tried to prove the standard identities of the Dedekind eta function
$$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n)$$
, where $q=\exp(2\pi i \tau)$ for some complex number $\tau$, but encountered some challenges.
First,
$$\eta(-\frac{1}{\tau}) =\sqrt{-i\tau}\eta (\tau) $$
I think there's something... |
I noticed that **sum of digits (upto a single digit) of a prime number** (except 3) **raised to an even power is always either 1,4 or 7** (3 is an exception. **3 raised to any power always gives the sum of digits upto single digit as 9**). *The primes which have sum of their digits upto single digit as 1 or 8 always gi... |
Let $G$ be a countable group, and $l^\infty(G)$ the collection of functions $f:G \to \mathbb{R}$ such that $\sup |f(x)| < \infty$. I know that $G$ is amenable if and only if for all $\epsilon > 0$ and for all finite sets $H \subset G$ there exists a nonempty finite set $K \subset G$ such that $$ \frac{|Kx \Delta K|}{|K... |
What are useful tricks to solve quantifiers questions? Like for example here
Let P(x,y) be the statement "x has taken y" and Q(x) be the statement "& knows Java", where the domain for x consists of all students and the domain for y consists of all Math courses. The statement "No student has taken any Math course or k... |
What are helpful tricks to solve quantifiers questions in discrete mathematics? |
I tried to prove the standard identities of the Dedekind eta function
$$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n),$$
where $q=\exp(2\pi i \tau)$ for some complex number $\tau$, but encountered some challenges.
First,
$$\eta(-\frac{1}{\tau}) =\sqrt{-i\tau}\eta (\tau) $$
I think there's something ... |
Is there a framework where a derivative of a singular function makes sense? |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
> It is false that books are made completely out of water molecules.
The above sentence has a connective in it: 'It is false that'. Is ... |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
> It is false that books are made completely out of water molecules.
Is the above sentence a composite sentence? If so, the denial of a... |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
> It is false that books are made completely out of water molecules.
1. Is the above sentence a composite sentence?
2. If so,... |
Suppose we have a cumulative distribution function $F(x)$
Then by Lebesgue decomposition theorem, we may decompose $F$ into:
absolutely continuous part, singular part, piecewise step part.
As for the absolutely continuous function, traditional notion of derivative works.
As for the piecewise step part, we ca... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 15.19 ( or Lemma 14.18 in the second edition ) and stuck at some statement. I've searched for math stack exchnage, but there doesn't seem to be any related questions, so I'm asking this.
**Q.1** Let $X$ be a reduced scheme with finitely many irreducible ... |
Assume an invertible matrix $C$ is in the form $\left[ \begin{matrix}A \\ B \end{matrix}\right]$, where the linear subspace generated by the rows of $A$ is orthogonal to the linear subspace generated by the rows of $B$. Denote the Moore–Penrose pseudoinverses of $A$ and $B$ by $A^{+}$ and $B^{+}$, respectively.
How... |
I tried to prove the standard identities of the Dedekind eta function
$$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n),$$
where $q=\exp(2\pi i \tau)$ for some complex number $\tau$, but encountered some challenges.
First,
$$\eta(-\frac{1}{\tau}) =\sqrt{-i\tau}\eta (\tau) $$
I think there's something ... |
Suppose we have a cumulative distribution function $F(x)$
Then by Lebesgue decomposition theorem, we may decompose $F$ into:
absolutely continuous part, singular part, piecewise step part.
As for the absolutely continuous function, traditional notion of derivative works.
As for the piecewise step part, we ca... |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
> It is false that books are made completely out of water molecules.
1. Is the above sentence a composite sentence?
2. If so,... |
Suppose there is a single-elimination bracket style tournament where the number of players *n* = 2<sup>*k*</sup> where *k* > 1. Each player is assigned a seeding of 1 through *n*, and during each round the matchups are uniformly chosen at random. The lower seed always wins in this fixed tournament but the matchups for ... |
Your procedure of calculating a sum of digits of $n$ over and over, until one digit is left, gives just the remainder of dividing $n$ by $9$ (where $0$ is substituted by $9$). This fact follows from a well-known divisibility test: the remainder of a number after dividing by $9$ is equal to the reminder of the sum of it... |
I noticed that **sum of digits (upto a single digit) of a prime number** (except 3) **raised to an even power is always either 1, 4 or 7** (3 is an exception. **3 raised to any power always gives the sum of digits upto single digit as 9**).
The primes which have sum of their digits upto single digit as 1 or 8 alway... |
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... |
*The following question is not on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
A sentence such as:
> Antarctica lies to the south of the Indian Ocean and the Indian Ocean lies to the south of India.
is a composi... |
*The following question is ***not*** on my homework. It is a conceptual question that followed when I saw the definition of composite sentences in a chapter on mathematical reasoning.*
A sentence such as:
> Antarctica lies to the south of the Indian Ocean and the Indian Ocean lies to the south of India.
is a c... |
How do I find the third solution of the system of differential equations, when I was able to find two vector solutions with the eigenroot 5? |
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 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... |
Milk is an element.
=> not a composite sentence.
Not(Milk is an element.).
=> not a composite sentence.
"Not" is just a synonym of "It is entirely untrue that".
Hence:
It is entirely untrue that milk is an element.
=> Why would this be a composite sentence? |
Given the function $H:]0,2[\times S^1\rightarrow \mathbb{R}$, $H(r,\theta)=\frac{\sqrt{r-(1-r^2/2)^2}}{\sqrt{2}}\cos(\theta)+\frac{4}{5}(1-\frac{r^2}{2})^2$ I wanted to compute the volume of the open set $H^{-1}(]\frac{4}{5},\infty[)$ with respect to the form $rdr\wedge d\theta$.
I tried using Mathematica to get va... |
Good morning, my answer is related to the one introduced here
https://math.stackexchange.com/questions/3142753/convergence-of-images-of-a-sequence-of-converging-continuous-functions/4878876#4878876
but with stronger hypotheses. Suppose you have a sequence of invertible functions $f_n$ that converges uniformly to ... |
Good morning, my answer is related to the one introduced here
https://math.stackexchange.com/questions/3142753/convergence-of-images-of-a-sequence-of-converging-continuous-functions/4878876#4878876
but with stronger hypotheses. Suppose you have a sequence of invertible functions $f_n$ that converges uniformly to ... |
Let $A$ be a non-symmetric matrix of real numbers, $A\in\mathbb{R}^{n\times n}$.
If the quadratic form $x^\intercal Ax$ is positive definite in the sense that $x^\intercal Ax>0$ $\forall x\neq 0\in\mathbb{R}^n$, then the real part of all eigenvalues of $A$ is positive, $\Re(\lambda_i)>0$ for $i=1,\dots,n$.
Howeve... |
Good morning, my answer is related to the one introduced here
https://math.stackexchange.com/questions/3142753/convergence-of-images-of-a-sequence-of-converging-continuous-functions/4878876#4878876
but with stronger hypotheses. Suppose you have a sequence of invertible functions $f_n$ that converges uniformly to ... |
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)(x-x_4)=0$ in $\mathbb Q(x_1,x_2,x_3,x_4)$.
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 $\th... |
Galois Theory. Why does this construction work? |
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... |
I am wondering about the existence of projections from a multivariate polynomial $p\in\mathbb{R}[x_1,...x_n]$ to a polynomial $q\in\mathbb{R}[t]$ in the sense that there is a map $\phi:\mathbb{R}\to\mathbb{R}^n$ such that for all $t\in\mathbb{R}$ we have
$$q(t)=p(\phi(t)).$$
Basically, this amounts to finding a path ... |
I am attempting to solve equations with Galois theory. I find the following method works:
For an equation, such as $x^3+x-1=0$, assume it splits as $(x-x_1)(x-x_2)(x-x_3)=0$ in $\mathbb Q(x_1,x_2,x_3)$.
Put $\theta_1=(x_2x_3^2-x_1x_3^2)+(x_3x_1^2-x_2x_1^2)+(x_1x_2^2-x_3x_2^2)$, one calculates that $\theta_1^2=-27... |
I am attempting to solve equations with Galois theory. I find the following method works:
For an equation, such as $x^3+x-1=0$, assume it splits as $(x-x_1)(x-x_2)(x-x_3)=0$ in $\mathbb Q(x_1,x_2,x_3)$.
Put $\theta_1=(x_2x_3^2-x_1x_3^2)+(x_3x_1^2-x_2x_1^2)+(x_1x_2^2-x_3x_2^2)$, one calculates that $\theta_1^2=-27... |
I am looking at the proof of Proposition 2.5. from Convex Functions, Monotone Operators and Differentiability, which asserts that the subdifferntial map $x \to \partial f(x)$ of a continous convex function $f: D\to \mathbb R$ on Banach ($D \subset E,$ convex open)space is $||\cdot||-w^*$ upper semicontinous.
The pro... |
Study the punctual, absolute and uniform convergence of the following series of functions:
$$\sum_{n=1}^\infty\left[\arcsin\left(\frac{2x^2-1}{x}\right)\right]^n$$
I have done this. Is it correct?
**1. Pointwise Convergence**:
For each fixed $x$, we need to check if the sequence of functions converges. Let's ... |
The following link in chess.stackexchange https://chess.stackexchange.com/questions/27259/has-anyone-attempted-to-characterize-chess-mathematically is a quick sketch I made of some math areas for chess. Something might catch your interest - please ask me if you want to know more.
In the most recent issue of the Span... |
Suppose that $A \in \mathbb{R}^{n \times n}$ is symmetric positive definite. In this case, solving $Ax = b$ with $x,b \in \mathbb{R}^{n}$ is equivalent to find
\begin{align}
\underset{x \in \mathbb{R}^{n}}{\arg\min}\ f(x) = \frac{1}{2} x^{\mathrm{T}}Ax - b^{\mathrm{T}}x.
\end{align}
So, I can use graident descent (... |
Gradient descent for solving complex-valued $Ax = b$? |
Let $X$ be a topological vector space with a countable local base. Now a seminorm $p_i$ can be associated with each element of local base. From rudins book on functional analysis we get that this space is metrizable by the following metric..
$$
d(x,y)= \text{max}_{i\geq 1} \frac{c_i p_i(x-y)}{1+p_i(x-y)}
$$
whe... |
Let $X$ be a topological vector space with a countable local base. Then how does the following metric attain maximum? |
is this true that if $f\in C_c(X)$, then $f\bigg[\partial\Big(supp(f)\Big)\bigg]=0$?
if $X$ be a topological space and $f\in C_c(X)$, then we know $f\bigg[\Big(supp(f)\Big)^c\bigg]=0.$
now, if $x\in \partial \Big(supp(f)\Big) $, since $$\partial \Big(supp(f)\Big)\subseteq\overline{ \Big(supp(f)\Big)^c } $$ there ... |
These days I am stuck with the definition of triangulated categories where the auto-isomorphism is replaced by auto-equivalence. For example, by the axiom TR2, when we have a distinguished triangle $X\rightarrow Y \rightarrow Z \rightarrow TX$, then so is $T^{-1}Z \rightarrow X\rightarrow Y\rightarrow Z$. But the right... |
I am attempting to solve equations with Galois theory. I find the following method works:
For an equation, such as $x^3+x-1=0$, assume it splits as $(x-x_1)(x-x_2)(x-x_3)=0$ in $\mathbb Q(x_1,x_2,x_3)$.
Put $\theta_1=(x_2x_3^2-x_1x_3^2)+(x_3x_1^2-x_2x_1^2)+(x_1x_2^2-x_3x_2^2)$, one calculates that $\theta_1^2=-27... |
**Context :**
We perform measurements of a set of parameters $(p_1,p_2,...)$ to retrieve the experimental mean of each parameters and the covariance matrix between them. These parameters are combined to construct the answer $\epsilon = f(p_1,p_2,...)$. The question is, how can we propagate the uncertainty stored in ... |
if $X$ be a topological space and $f\in C_c(X)$, then we know $$f\bigg[\Big(supp(f)\Big)^c\bigg]=0\quad\quad \text{and} \quad\quad f\bigg[\Big(supp(f)\Big)^{\circ}\bigg]\neq 0 $$
>is this true that if $f\in C_c(X)$, then $f\bigg[\partial\Big(supp(f)\Big)\bigg]=0$?
if $x\in \partial \Big(supp(f)\Big) $, si... |
Let consider the interval $I=[0,1]$ and we work in base-$3$, then every $x\in I$ can be represented as $$x=\sum_{k=1}^\infty\frac{a_k}{3^k},\quad a_k\in\{0,1,2\}.$$
I must prove that if $x\in I_{0,1}=(\frac{1}{3},\frac{2}{3})$, then the first digit of $x$, $a_1=1$. I reasoned as follows. I observe that $$\frac{a_1}{... |
I'm dealing with a question that request to calculate the mutual information of two normal random vectors, this is the description:
If $\mathbf X\sim \mathcal N(\mu_X,\Sigma_X),~\mathbf Y\sim \mathcal N(\mu_Y,\Sigma_Y)$ are random vectors with $N$ elements, the covariance matrix of $\mathbf X$ and $\mathbf Y,$ $\Sig... |
In one of my recent answers to an optimization question (see [here][1]), I came across the following equation:
$$\sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}=1$$
where $a_j>0, \lambda_j \ge 0, j=1,...,n$. After doing some numerical checks, I realized that this equation always has **two real roots** (try it yourse... |
Let consider the interval $I=[0,1]$ and we work in base-$3$, then every $x\in I$ can be represented as $$x=\sum_{k=1}^\infty\frac{a_k}{3^k},\quad a_k\in\{0,1,2\}.$$
I must prove that if $x\in I_{0,1}=(\frac{1}{3},\frac{2}{3})$, then the first digit of $x$, $a_1=1$. I reasoned as follows. I observe that $$\frac{a_1}{... |
I am analyzing the asymptotics of the following function
$$E:=\Big(\overbrace{-2C_0-(n-2)C_1}^A+(4+(n-2)\overbrace{(2C_1+C_2)}^B)\cdot p-(n-2)\overbrace{(C_1+C_2-C_3)}^Cp^2\Big)^2\\$$
where $p=p(n)$ is a function of $n$, and it is asymptotically equivalent to
$$p(n)\sim \frac{2C_1+C_2}{2(C_1+C_2-C_3)}-\frac{\... |
Long ago, perhaps 40-50 years ago, I read a paper or book that contained advice about how to write and publish mathematics that contained a certain quip (below). Perhaps it was something published by the AMS, or perhaps it was written by Paul Halmos. Anyway, although I've tried, I've not been able to find it again.... |
I am faced with the following problem. Consider $v$ random in $\mathbb{R}^n$, say $\sim \mathcal{N}(0,I_n)$. Then I know probably (for $n$ large), $||v||_\infty = O(\sqrt{n \log n}) $, as that is the maximum of $n$ standard Gaussians. However, now consider picking an orthonormal basis $B = (\frac{v}{||v||}, e_2,..e_n)$... |
What is the expected $L^\infty$ norm of a random affine subspace? |
Conditions:
$f \in C^{\infty}(\mathbb{R}^2 \times [0;+\infty[,\mathbb{R})$.
Is the function $\displaystyle t \longrightarrow \frac{\partial^2 f}{\partial x_1 \partial x_2} (x_1,x_2,t)$ continuous on $[0;+\infty[$ for any $(x_1,x_2) \in \mathbb{R}^2$?
I don't find any counter-example, but I am not sure. And I hav... |
Is $t \mapsto \frac{\partial^2 f}{\partial x_1 \partial x_2}(x,t)$ continuous if $f \in C^{\infty}(\mathbb{R}^2 \times [0;+\infty[,\mathbb{R})$? |
The second fundamental theorem of Calculus says if a function is continuous (hence, Riemann integrable) then $\int_a ^x f(t) dt$ is a primitive of $f$. Ok this is for continuous functions.
However what happens for not necessarily continuous functions ? The first fundamental theorem of Calculus says if $f$ is Riemann... |
Are there other kind of primitives, other than $\int_a ^x f(t)dt +k$ ? For example for non-continuous functions. |
Suppose that $X$ is a Polish space. We know that for each $\epsilon>0$, $X$ can be partitioned into a countable number of non-empty Borel subsets, $\{B_n\}$, whose diameters do not exceed $\epsilon$. Given a strictly positive measure $\mu$ on $(X,\mathcal{B}(X))$ (i.e., $\operatorname{supp}(\mu)=X$), can we assume that... |
I am faced with the following problem. Consider $v$ random in $\mathbb{R}^n$, say $\sim \mathcal{N}(0,I_n)$. Then I know probably (for $n$ large), $||v||_\infty = O(\sqrt{n \log n}) $, as that is the maximum of $n$ standard Gaussians. However, now consider picking an orthonormal basis $B = (\frac{v}{||v||}, e_2,..e_n)$... |
> How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
I am a little unclear what you are trying to ask here, but I suspect it is: how can $P1 ⇔ P2 ⇔ P3, ... ⇔ Pn$ be true if it is not the case that all $P$’s are true, nor the case that all $P$’s are false? In other words... |
I am faced with the following problem. Consider $v$ random in $\mathbb{R}^n$, say $\sim \mathcal{N}(0,I_n)$. Then I know probably (for $n$ large), $||v||_\infty = O(\sqrt{n \log n}) $, as that is the maximum of $n$ standard Gaussians. However, now consider picking an orthonormal basis $B = (\frac{v}{||v||}, e_2,..e_n)$... |
Suppose that $X$ is a Polish space. We know that for each $\epsilon>0$, $X$ can be partitioned into a countable number of non-empty Borel subsets, $\{B_n\}$, whose diameters do not exceed $\epsilon$. Given a strictly positive measure $\mu$ on $(X,\mathcal{B}(X))$ (i.e., $\operatorname{supp}(\mu)=X$), can we assume that... |
> How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
I am a little unclear what you are trying to ask here, but I suspect it is: how can $P1 ⇔ P2 ⇔ P3, ... ⇔ Pn$ be true if it is not the case that all $P$’s are true, nor the case that all $P$’s are false? In other words... |
Let $G$ denote a Lie group and $\mathfrak{g}$ its Lie algebra. $P$ is a smooth principal bundle.
Given a smooth $\mathfrak{g}$-valued one-form $\omega_p: TP_p \to \mathfrak{g}$, which fulfils the properties listed below, is there an easy way to see that $p \mapsto ker(\omega_p) =: H_p$ is smooth?
1. For all $p \... |
Going through my notes on predicate logic, I read the following inductive definitions:
**Definition:** An _atomic term_ is either a variable or a constant. If $f$ is an $n$-place function symbol and $t_1, . . . , t_n$ are _terms_, then $f(t_1, . . . , t_n)$ is also a _term_.
**Definition:** An _atomic formula_ is... |
Is $\forall x\exists x(x < x)$ a sentence? |
> How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
Your ‘P1 =/= P2 =/= P3 =/= Pn’ is a little unclear, but I suspect your question is: how can $P1 ⇔ P2 ⇔ P3, ... ⇔ Pn$ be true if it is not the case that all $P$’s are true, nor the case that all $P$’s are false? In oth... |
M. Schottenloher in his book "A mathematical introduction to conformal field theorey" in Remark 4.3 says that: For every central extension of Lie algebras $0\to \mathfrak{a}\to \mathfrak{h}\overset{\pi}{\to} \mathfrak{g}\to 0$, there is a linear map $\beta: \mathfrak{g}\to \mathfrak{h}$ with $\pi \circ \beta = id_{\mat... |
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