instruction stringlengths 12 30k |
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Do you need to enforce the condition $\max f(x)=1$? Otherwise, any function of the form
$$f(x)=\sum_{j=1}^{n}c_j (1-x)^{a_j}(1+x)^{b_j}\tag{1}$$
with $a_j, b_j, c_j > 0$ will have the desired property.
Moreover, by adjusting the values of $n$, $a_j$'s, $b_j$'s, $c_j$'s, functions of the form $\text{(1)}$ can a... |
I am doing some self-study from Enderton's *Elements of Set Theory* and he leaves Theorem 3E below to the reader to prove:
> **Theorem 3E.** (1) For a set $F$, dom $F^{-1} =$ ran $F$ and ran $F^{-1} =$ dom $F$. (2) For a relation $F$, $(F^{-1})^{-1} = F$.
The definition of the inverse of a set was given earlier a... |
Why do I need that $F$ a relation to show $(F^{-1})^{-1} = F$? |
Suppose we have an operator $T$ such that $||Tf||_s \leq C_s ||f||_s$ for all $f \in H^s(M)$ and $s \in \mathbb{N} \cup \{0\}$, where $M$ is a Riemannian manifold. Can we say from here that $||Tf||_s \leq C_s ||f||_s$ for all $f \in H^s(M)$ and $s \geq 0$ ?
I was trying to approach it in the following two ways but ... |
Studying Gabriel Navarro's book "Character Theory and the McKay Conjecture", I've come across the following problem. First, let's fix some notation: $G$ will be a finite group, $R$ will denote the algebraic integers of $\mathbb{C}$ over $\mathbb{Q}$, $p$ will be a prime, $M$ will be a maximal ideal of $R$ containing $p... |
you have some normal distribution with mean 0
and variance $^2$
and that you want the interval [,]
such that the uniform distribution on the interval has mean 0
and variance $^2$
?
This is my variance equation when the numbers are drawn from a normal distribution $$2/(n_{in}+1/n_{out})$$
The answer for the... |
Since it is possible to [approximate](https://math.stackexchange.com/questions/1699366/can-we-know-the-first-few-digits-of-chaitins-constant) [Chaitin's constant](https://en.wikipedia.org/wiki/Chaitin%27s_constant) ($\Omega$) for some systems, is it known whether these numbers are somehow related to another when consid... |
Is there some shared structure between Chaitin's constants of similar systems? |
I'm confused by the B distribution growing exponentially -- which I think means a histogram of the values would be increasing exponentially -- and yet there is an upper cap oon the values such that they are all in the range of [0,1]. Could it be that they are frequencies, i.e. normalized counts? For example the origin... |
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... |
Assume that a quadratic function $f_{c}(z)=z^{2}+c$ have attractive cycle with period $p$. Is the number of external rays landed at the fixed point of $f_{c}$ exactly $p$? Does the Julia set is locally connected? I have searched the book by Milnor and Gamelin and many papers by Milnor, but none of them give similar sta... |
I've encountered a problem with combining uncertainties for later use in uncertainty propagation formula with some measurements of the same quantity having different uncertainties.
Having made direct voltage and current measurements, I'm trying to indirectly calculate the electrical resistance's uncertainty via the ... |
graphs of homomorphism of $(\mathbb{R}^{n},+)$ are minimal submanifolds and viceversa.
The same holds for ($S^{1},\times$).
Indeed geodesic on the torus which are graphs are precisely the graphs of $S^{1}$ homomorphism.
Are there generalizations of these statements?
|
I'm working with a sequence of probability spaces. Here are some details:
I roll a six sided die. My sample space is: $\Omega = \{1,2,3,4,5,6\}$.
Let $P$ be the uniform probability on $\Omega$. Then I have this probability space:
$(\Omega,\mathcal{P}(\Omega), P)$.
If I roll the die $n$ times, then my sample s... |
Sequence of probability spaces: Is it a thing? |
Let $(\frac{a}{p})$ denote the Legendre symbol, and let $\psi(q)=\sum_{n\geq 0} q^{n(n+1)/2}$.
We define Ramanujan's general theta function $f(a,b)$ for $\mid ab \mid <1$ as
$$
f(a,b)=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}.
$$
Let the $p$-dissection of $\psi(q)$:
$$
\psi(q)=\sum_{k= 0}^{(p-3)/... |
How to show that the set of minimizers for a convex function over a convex set is closed? |
I am doing some self-study from Enderton's *Elements of Set Theory* and he leaves Theorem 3E below to the reader to prove:
> **Theorem 3E.** (1) For a set $F$, dom $F^{-1} =$ ran $F$ and ran $F^{-1} =$ dom $F$. (2) For a relation $F$, $(F^{-1})^{-1} = F$.
The definition of the inverse of a set was given earlier a... |
During the resolution of a physics problem, I have encountered the sum:
$$\sum_{i=1}^{10} i(i+1)$$
For practical purposes, I evaluated it using a Python script, but a calculator would have been enough since the sum stops at $i=10$. However, I wonder what I could do mathematically with that sum if I did not have acc... |
How can I find the sum $\sum_{i=1}^{10} i(i+1)$? |
For full context, I work on a complex manifold $M$ of dimension $n$ where I have a Hermitian $(1,1)$-form $\Omega$ and $(1,1)$-form $\tau$ which is negative semidefinite at a point. I am trying to see why it is that at the point where $\tau$ is negative semidefinite:
$$
\frac{(\Omega+\tau)^n}{\Omega^n} \leq 1
$$
Wh... |
The question was to decide wthether $\varphi$ is a linear tarsnformation – I've checked and properly shown that it is infact a linear transformation. Now I'm trying to decide whether it's isomorphic.
I've checked for injection - everything checks out but I got stuck at evaluating surjection.
$\varphi : \mathbb{... |
Is $\varphi : \mathbb{R}_{2x2}\rightarrow \mathbb{R}^{4}$ onto? |
Let $(\frac{a}{p})$ denote the Legendre symbol, and let $\psi(q)=\sum_{n\geq 0} q^{n(n+1)/2}$.
We define Ramanujan's general theta function $f(a,b)$ for $\mid ab \mid <1$ as
$$
f(a,b)=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}.
$$
Let the $p$-dissection of $\psi(q)$:
$$
\psi(q)=\sum_{k= 0}^{(p-3)/... |
The question was to decide wthether $\varphi$ is a linear tarsnformation – I've checked and properly shown that it is infact a linear transformation. Now I'm trying to decide whether it's isomorphic.
I've checked for injection - everything checks out but I got stuck at evaluating surjection.
$\varphi : \mathbb{... |
Is $\varphi : \mathbb{R}_{2 \times 2}\rightarrow \mathbb{R}^{4}$ onto? |
The question was to decide whether $\varphi$ is a linear transformation – I've checked and properly shown that it is in fact a linear transformation. Now I'm trying to decide whether it's isomorphic.
I've checked for injection - everything checks out but I got stuck at evaluating surjection.
$\varphi : \mathbb{... |
I am calculating a bound for a random variable. Using the Chernoff bound, I get to the following:
$$
\sum_{s=1}^{\frac{n}{2}} n^{ -2s\big[ \frac{\log(s)-1}{\log (n)} \pm 2 o_n(1) \big]}
$$
Can someone help me to find the asymptotic behavior of the sum?
P.S.: $o_n(1)$ goes to zero as $n \to \infty$. |
In the maximum modulus therem we have two sets
$E_1=\{z \in E:|f(z)|<M\} \quad E_2=\{z \in E:|f(z)|=M\}$ I know that the set $E_1$ is open because pre image of an open set. But should't be also $E_2$ closed because pre image of only a point ? |
we define
$ l_p=\{x=(x_n) : \sum_{i=1}^\infty |x_n|^p < \infty\} $
I tried to prove that
$$\left\|x\right\|=\left(\sum_{i=1}^{n}\left|x_i\right|^p\right)$$ is a p norm.
I tried 1. and 2. conditon but I can't prove triangle inequality for $p$-norm.
I tried to use minkowski inquality but I can't use it because th... |
How to prove triangle inequality this p- norm? |
I'm interested in the following problem:
>If $A,B,C,D$ are four distinct points in $\Bbb R^3$ satisfying $\angle ABC=\angle BCD=\angle CDA=\angle DAB$,\
then is it must be true that $|AB|=|BC|=|CD|=|DA|$?
For $|AB|=|BC|=|CD|=|DA|$ I have an example:\
$A = (0, -1, -a)$\
$B = (-1, 0, a)$\
$C = (0, 1, -a)$\
$D ... |
I'm interested in the following problem:
>If $A,B,C,D$ are four distinct points in $\Bbb R^3$ satisfying $\angle ABC=\angle BCD=\angle CDA=\angle DAB$,\
then is it must be true that $|AB|=|CD|,|BC|=|DA|$?
If $A,B,C,D$ are coplanar then $ABCD$ is a rectangle, so $|AB|=|CD|,|BC|=|DA|$.\
If $A,B,C,D$ are not copla... |
Randomize two random integers between 1 and 100 (including 1 and 100) and pick the lowest of the two, what is the expected value of this integer? I believed it is (1-1/sqrt(2))*100 since the chance of getting 100 would be 1/100 * 1/100 = 1/10,000 and chance of getting 99 is 3/10,000 since there are three possible ways ... |
If you randomize two numbers from 1-100 and pick the lowest number, what is the expected value of that number? |
[Theorem with Diagram][1]
Can anyone help me with this diagram?
[1]: https://i.stack.imgur.com/KxQPz.png |
I have seen the phrase "weak-star weak-star continuous" many times. But I'm don't know what it means for a function to be weak-star weak-star continuous. I just assumed it means that the function is continuous in the weak topology.
Could someone provide a formal definition and an example on this?
Thank you in ad... |
My professor gave the class a hint to try to find a continuous extension for this problem. In our textbook, *Elementary Analysis: The Theory of Calculus* by Ross, one of the theorems states "A real-valued function $f$ on $(a,b)$ is uniformly continuous on $(a,b)$ if and only if it can be extended to a continuous functi... |
Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior of its closure), bounded and simply connected set. Let $\gamma : [0,1] \rightarrow \partial U$ be a simple curve such that $\gamma(0) \neq \gamma(1)$.
Is $\partial U \setminus \gamma((0,1))$ still connected ?
I think this should be true.... |
I'm studying multivariable calculus, and I was wondering:
Is there a way to characterize all sets in $\mathbb{R}^2$ that are continuously parametrizable? What I mean with a continuously parametrizable set $A$ is that there exists a continuous function $f:[0, 1] \rightarrow \mathbb{R}^2$ onto $A$.
I know that *cur... |
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of spaces over $BG$.
Is there some category with the notion of homotopy $\mathscr X(A)$ one may assign to a homotopy commu... |
$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$ |
WIll my furniture fit through a corridor and door? |
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of spaces over $BG$.
Is there some category with the notion of homotopy $\mathscr X(A)$ one may assign to a homotopy commu... |
A friend came up with what looks like a very simple and beautiful construction for mapping an uncountable number of dimensions into the plane ($\mathbb{R}^2$).
Specifically, each dimension is indexed by a number $\theta \in [0,2\pi)$. And each location $x$ along dimension $\mathbb{R}_\theta$ gets mapped to the polar... |
The semisimple Lie algebra $\mathfrak{g}:=sl(2,\mathbb{R})$ has the Casimir operator
$$\Omega:=(1/2)h^2+ef+fe$$
with respect to the standard basis $(h,e,f)$. For an irreducible unitary representation $\pi$ of $G:=SL(2,\mathbb{R})$ the global character $\Theta_\pi$ satisfies the differential equation
$$\Omega.\Theta_... |
Explicit differential equation of the global character of UIR's of SL(2,R)? |
This is an exercise from my General Relativity course, so Einstein summation is implied.
On a Riemannian manifold with metric $g$, show that if for a vector field $X^{\mu}$ and a geodesic with tangent $T^{\nu}$ the quantity $g(T,X)\equiv T_\mu X^\mu$ is conserved along the geodesic, then $X$ satisfies *Killing's equ... |
A friend came up with what looks like a very simple and beautiful construction for mapping an uncountable number of dimensions into the plane ($\mathbb{R}^2$).
Specifically, each dimension is indexed by a number $\theta \in [0,2\pi)$. And each location $x$ along dimension $\mathbb{R}_\theta$ gets mapped to the polar... |
graphs of homomorphism of $(\mathbb{R}^{n},+)$ are minimal submanifolds and viceversa minimal submanifolds which are graphs are graphs of homomorphisms.
The same holds for ($S^{1},\times$).
Indeed geodesic on the torus which are graphs are precisely the graphs of $S^{1}$ homomorphism.
Are there generalizations of th... |
A friend came up with what looks like a very simple and beautiful construction for mapping an uncountable number of dimensions into the plane ($\mathbb{R}^2$).
Specifically, each dimension is indexed by a number $\theta \in [0,2\pi)$. And each location $x$ along dimension $\mathbb{R}_\theta$ gets mapped to the polar... |
Let $A$ be real $n$ by $n$ matrix with positive entries, and $I_n$ be the identity matrix of the same dimension.
Assume the spectral radius/Perron eigenvalue of $A$ is stricly less than 1.
Then one can prove $0$ is not an eigenvalue of $I_n-A$, hence $I_n-A$ is invertible, and this implies that the equation with ... |
Again Perron Forbenius or not? |
This is a follow-up to my [previous question][1]. I now make my question more precise to avoid the counter-example given there.
Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior of its closure), bounded and simply connected set. In that situation, $U$ has a connected boundary but it is not... |
> Let $X$ be a set. Consider a sequence of subsets
\begin{align}
\big\{S_n\subseteq X\,\big|\,S_i\cap S_j=\varnothing\ \forall i\ne j \big\}.
\end{align}
Then $\lim_{n\to\infty}S_n$ exists and is equal to $\varnothing$.
My proof :
... |
You're completely correct that $\delta$-at-$0$ in 3D is _some_ sort of "product" of the one-dimensional $\delta$'s in any choice of (maybe best to be orthogonal...?) coordinates.
But, although this is probably unsatisfying, the "decomposition" of it is as a tensor _product_ $\delta_x\otimes \delta_y\otimes \delta_z$... |
> Consider the CG method for the iterative solution of the system of
equations $A \mathrm{x}=\mathrm{b}$ with a positive definite and
symmetric matrix $A \in \mathbb{R}^{n \times n}$ and $b \in
\mathbb{R}^{n}$. Show that to reduce the initial error by the factor
> $\varepsilon$ at most $t(\varepsilon) \leq \f... |
Let's say I have a question asking me to find the eigenvectors associated with lambda = 2 for A (a 3x3 matrix)
I find that the eigenvectors associated with lambda = 2 are all vectors spanned by u1 and u2 (a plane)
can I call u1 and u2 the eigenvectors associated with lambda = 2, even though there is a whole plane... |
How to find the original matrix given its basis for column & null space? |
graphs of homomorphism of $(\mathbb{R}^{n},+)$ are minimal submanifolds and viceversa minimal submanifolds in $\mathbb{R}^{2n}$ which are graphs are translations of graphs of homomorphisms.
The same holds for ($S^{1},\times$).
Indeed geodesic on the torus which are graphs are precisely the translations of graphs of $... |
> Let $X$ be a set. Consider a sequence of subsets
\begin{align}
\big\{S_n\subseteq X\,\big|\,S_i\cap S_j=\varnothing\ \forall i\ne j \big\}.
\end{align}
Then $\lim_{n\to\infty}S_n$ exists and is equal to $\varnothing$.
My proof :
... |
How do I calculate an integral $\int_{-\infty}^{\infty} \frac{z-a}{\sinh(z-a)}\frac{z+a}{\sinh(z+a)} dz$ for $a>0$?
Expanding the integrand and integrating term-by-term (a-la polylog) gives rise to an ugly looking **double** series. |
graphs of homomorphism of $(\mathbb{R}^{n},+)$ are minimal submanifolds.
The same holds for ($S^{1},\times$).
Are there generalizations of these statements?
|
Consider the following statement of Cauchy's theorem
**Cauchy**: If $\Omega \subset \mathbb{C}$ is open and simply connected, and $f$ is holomorphic on $\Omega$, then for $\gamma$ smooth, closed curve, $\int_{\gamma} f d\gamma = 0$.
In the following proof, let us use Green's theorem, stated as follows:
**Green... |
Let's say we have $\mathbb{Z}_3[X]\mathbin{/}(7x^2 + 1)$, such that we take the quotient of the polynomial ring $\mathbb{Z}_3[X]$ with that specific ideal.
Is this still a ring, and does it make sense to ask what the result of $(x^2 + 3x^3) \cdot (13 + 7x)$ is in it? What's the procedure for it? |
Let $H$ be the orthocentre of an acute $\Delta ABC$, as in [figure][1]. Let $X$ be the reflection of $H$ over $\overline{BC}$ and $Y$ be the reflection of $H$ over the midpoint of $\overline{BC}$. Show that $X$ and $Y$ lie on the circumcircle of $\Delta ABC$.
[1]: https://i.stack.imgur.com/Rwodb.png
To prove th... |
I am currently reading Carmo's book about differential forms. I tried excercise 6 from chapter 1, but I am a bit confused. Here is the excercise:
Let $f: U\subset \mathbb{R}^m \rightarrow \mathbb{R}^n$ be a diff. map. Assume that m<n and let $\omega$ be a k-form in $\mathbb{R}^n$, with k>m. Show that $f^*\omega=0$.
... |
Is it *problematic* to allow for expressions such as
$\forall x\exists x (x < x)$
to count as well formed sentences? No (as others have explained). You can set up your formal first-order logical languages that way, while keeping everything working systematically in an acceptable way.
Is is *necessary* to allow fo... |
I am currently reading Carmo's book about differential forms. I tried excercise 6 from chapter 1, but I am a bit confused. Here is the excercise:
> Let $f: U \subset \mathbb{R}^m \to \mathbb{R}^n$ be a diff. map. Assume that $m<n$ and let $\omega$ be a $k$-form in $\mathbb{R}^n$, with $k>m$. Show that $f^*\omega=0$.... |
Suppose, $A,B\in \mathbb{C}^{d\times d}$, $A$ is normal, and $\text{dim ker}(AB-BA)\ge (1-\delta)d$. Does then follow that $\text{dim ker}(A^*B-BA^*)\ge (1-\delta)d$, possibly for different $\delta$?
This is known to hold for $\delta=0$ because there is polynomial $P$ such that $A^*=P(A)$. But here, if $v\in \text{k... |
If $A$ is normal and $A$ and $B$ almost commute, do $A^*$ and $B$ almost commute? |
**"Direct" problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where the RV $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and the RV $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we... |
Map between RVs: how to practically deal with the PDF "inverse" problem? |
Let $h \in L^1(\mathbb{R}).$
Consider the Hardy-Littlewood maximal function defined as
$$h^*(b) = \sup_{t > 0}\frac{1}{2t}\int_{b-t}^{b+t}h.$$
Does it hold that $\{b\in \mathbb{R} : h^*(b) = \infty\}$ is a closed set? |
Let $A$ be real $n$ by $n$ matrix with positive entries, and $I_n$ be the identity matrix of the same dimension.
Assume the spectral radius/Perron eigenvalue of $A$ is strictly less than 1.
Then one can prove $0$ is not an eigenvalue of $I_n-A$, hence $I_n-A$ is invertible, and this implies that the equation with... |
Again Perron Frobenius or not? |
Let's say I have a question asking me to find the eigenvectors associated with $\lambda = 2$ for $A$ (a $3 \times 3$ matrix).
I find that the eigenvectors associated with $\lambda = 2$ are all vectors spanned by $\mathbf{u}_1$ and $\mathbf{u}_2$ (a plane).
Can I call $\mathbf{u}_1$ and $\mathbf{u}_2$ the eigenvec... |
**"Direct" problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where the RV $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and the RV $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we... |
> Let $X$ be an RV, $\mathcal{F}$ a $\sigma$-algebra, $\phi$ the characteristic function of a distribution $\nu$. Assume that for the conditional characteristic function
>
> $$\phi_\mathcal{F}(t) := \mathbb{E}[\exp(itX) \mid \mathcal{F}]$$
>
> holds $$\phi_\mathcal{F}(t) = \phi(t) \quad a.s.$$
>
> and show that... |
Given a probability space $(Ω, \mathcal F, P)$ and an integrable random variable $ξ: Ω \to \Bbb R$, what kind of entity is the expectation of $ξ$ i.e. $E(ξ)=\int_Ω ξdP$? My sense is that we have a functional (or an operator) $RV\mapsto\Bbb R$, where $RV\subseteq F:=\{f|f \space \mathrm {function}\}$ is the set of rando... |
What kind of entity is the expectation of a random variable? |
**Note:** I'm using the terms "covariant" and "contravariant" a bit loosely in this question.
The standard function notation seems to be naturally codomain-covariant and domain-contravariant. And we have seen examples of this since the very first algebra class. In order to translate a function upwards, we add to the... |
A domain-covariant notation for functions? |
I'm learning the basics of Fourier Analysis. Reading a book, I've found this inequality:
" If $x\geq -3$ and $|\eta | >2$ we have $|x + 3\eta^2| \geq |x| + |\eta|^2$. I've tried to solve this inequality applying triangle one but I failed. Can you help me, please? |
I'm struggling with the separable closure problem and
I don't understand some points.
>Let $y,z$ be indeterminates and let $u=y^{12}$ and $v={y^2}{z^{18}}$. Describe the separable closure of $\mathbb F_3(u,v)$ in $\mathbb F_3(y,z)$.
The separable closure contains $y^3$.
Then it must contain $z^{54}$ and hence... |
I would like to prove the following statement.
>Let $E/F$ be a normal field extension of finite degree and $K=\mathrm{Fix}(\mathrm{Gal}(E/F))$. Then $K/F$ is purely inseparable. (By the way, $E/K$ is Galois, for which I have a proof).
I started like this:
If $F=K$, this statement is trivial. So let $F \neq K$... |
I recently saw this old sketch of the Gamma function that was done by hand and tools, and I thought that it was really cool. I realized before computer simulations and graphics, if we wanted to visualize a function, we had to accurately draw it ourselves.
It's a question for something very eclectic, but I want to l... |
I am looking for references containing results on the minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite.
If we denote as $k(n)$ this minimum $k$ for some $n$, $k$ is indeed always equal or greater than $n+2$, and grows exponentially as $n$ grows. I attach the graph of $k(n)$ for... |
Nowadays it is common to see PDEs defined as either closed embedded submanifolds of a jet bundle (of some appropriate order) or else, if some further conditions are satisfied, as a fibered submanifold. In some earlier literature, though (e.g., Quillen's dissertation), a PDE is defined as a *subbundle* of a jet bundle.... |
When is a PDE a subbundle of a jet bundle (as opposed to a fibered submanifold or just a closed embedded submanifold)? |
> Let $(X_n)_n$ be a sequence of square-integrable RVs and let $\mathcal{F}$ be given by $\mathcal{F}_n = \sigma (X_1, \ldots, X_n)$. Suppose that $S_n := \sum_{i=1}^n X_i$ is an $\mathcal{F}$-martingal. Show that
>
> $$Cov[X_m, X_n] = \mathbb{E}[X_m X_n]$$
I know that by definition we have
$$Cov[X_m, X_n] = ... |
In Liu's *Algebraic Geometry and Arithmetic Curves*, the author says that an $\mathcal{O}_X$-module is *generated by its global sections at $x \in X$* if
> the canonical homomorphism $\mathcal{F}(X) \otimes_{\mathcal{O}_X(X)} \mathcal{O}_{X,x} \to \mathcal{F}_x$ is surjective.
I assume that what is meant by "the... |
I know of a number of techniques for computing the condition number that are either slow (doing a whole singular value decomposition) or fast (ratio of norm of matrix and its inverse obtained via LU decomposition) but maybe unstable. I just want to determine whether a matrix is non-singular or not (i.e. whether its con... |
Are there any numerically stable methods for computing the condition number of a near-singular matrix that are fast? |
Intuition about the rules of integration? |
> Lagrange Inversion Formula: Let $A(u) = \sum_{k \ge 0} a_k z^k$ be a power series in $\mathbb{C}[[z]]$ with $a_0 \ne 0$. Then the equation
>
>$$B(z) = zA(B(Z)) \qquad (1)$$
>
> has a unique solution $B(z) \in \mathbb{C}[[z]]$ such that $b_n = \frac{1}{n}[z^{n-1}]A(z)^n$.
>
> Consider what happens if $a_0 = 0$... |
In **Example 5.1.13** of Analysis I by Terrance Tao, we are asked to show that the infinite sequence
$$1, -2, 3, -4, 5, -6, \ldots$$
is unbounded by using the following result and definition:
**Proposition 4.4.1** (Interspersing of integers by rationals). Let $x$ be a rational number. Then there exists an int... |
Would it help if you took expectations w.r.t. to the conditioning variable on both sides of the second equation and then applied the [law of total expectation](https://en.wikipedia.org/wiki/Law_of_total_expectation)?
P.S. I'm a little unsure of what you mean by having a $\sigma$-algebra in the conditioning side in y... |
How can we prove if $A = M - N$ (non-singular) and if the induced matrix norm $\|M^{-1}N\| > 1$ then the iteration error increases at some point? |
In **Example 5.1.13** of Analysis I by Terence Tao, we are asked to show that the infinite sequence
$$1, -2, 3, -4, 5, -6, \ldots$$
is unbounded by using the following result and definition:
**Proposition 4.4.1** (Interspersing of integers by rationals). Let $x$ be a rational number. Then there exists an inte... |
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