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I have daily close prices of two stocks, A and B. Suppose that we long stock A and short stock B. Assume that we do the long-short every day and hold that portfolio for some days. How to calculate each day's holding period return of this long-short strategy?
My idea is that we first calculate the holding period ret... |
How to calculate holding period return of a long-short strategy? |
Examples of statements that could be proven using strong induction without checking a special case? |
I am an engineer who is working with some linear equation problems.
In my application, I found out that in having the symbolic form of such matrix inverse actually speed things up (for example ${A^{ - 1}} = \frac{1}{{\det \left( A \right)}}\left[ {\begin{array}{*{20}{c}}
d&{ - b}\\
{ - c}&a
\end{array}} \right]$... |
Complexity of symbolic computation of matrix inverse? |
When we define a sigma algebra $F$ on some set $\Omega$, we will require that the measure is a nonadditive set function, that is
- $\mu(A) \geq \mu(\emptyset)$
- $\mu(\bigcup A_i) = \sum \mu(A_i)$ for all sets of disjoint events $\{A_i\}$ $\subseteq F$
My questions revolves around whether or not it is always po... |
Let $A$ be a $n \times n$ matrix over $\mathbb{R}$ or $\mathbb{C}$.
Let $\text{adj}(A)$ denote the classical adjoint matrix of $A$ (or adjugate matrix of $A$).
It is known that
$A \cdot \text{adj}(A) = \text{det}(A) I_n = \text{adj}(A) \cdot A$.
Thus, if $A$ is invertible, then $\text{det}(A) \neq 0$, and s... |
If a $n \times n$ matrix $A$ is not invertible, then is it possible that the classical adjoint matrix of $A$ is invertible? |
> Let $G$ be a group with the following property: for every $H, K \leq G$, either $H \subseteq K$ or $K \subseteq H$. Show that there exists a prime number $p$ such that the order of every element of $G$ is a power of $p$.
It is straightforward to show the result above when $G$ is finite. Indeed, fix $g \in G$ and... |
I am totally confused about a detail in computing group cohomology. Let $G$ be a group, $A$ a $\mathbb{Z}[G]$-module. $H^i(G, -)$ is the $i$th right derived functor of $\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}, -)$, i.e. $\operatorname{Ext}_{\mathbb{Z}[G]}^i(\mathbb{Z}, -)$. Here, $G$ acts trivially on $\mathbb{Z}... |
I am totally confused about a detail in computing group cohomology. Let $G$ be a group, $A$ a $\mathbb{Z}[G]$-module. $H^i(G, -)$ is the $i$th right derived functor of $\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}, -)$, i.e. $\operatorname{Ext}_{\mathbb{Z}[G]}^i(\mathbb{Z}, -)$. Here, $G$ acts trivially on $\mathbb{Z}... |
First, note next definition.
> **Definition 1.** Let $X$ be a topological space.
(1) Let $Z \subseteq X$ be a closed irreduiclbe subset. The *codimension* $\operatorname{codim}(Z,X)$ of $Z$ in $X$ is the supremum of the legnth of chains of irreduicible closed subsets $Z_0 \supsetneq Z_1 \supsetneq \cdots \su... |
When we define a sigma algebra $F$ on some set $\Omega$, we will require that the measure is a nonadditive set function.
My questions revolves around whether or not it is always possible to find a logically consistent set of probabilities $p_i \in [0,1]$, that we can assign to each singleton, such that it is logical... |
I have encountered a rather curious question.
Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n-$dimensional subspace of $\mathbb R^N$. We can generate $A$ as $VV'$ where $V\in\mathbb R^{N\times n}$ is a partial Haar orthogonal ma... |
When we define a sigma algebra $F$ on some set $\Omega$ and we would like to talk about the probabilities associated, we will require that the measure is a nonadditive set function.
My questions revolves around if we define a measure $\mu$ on $(\Omega, F)$, then can we always find another measure $\mu'$ such that $\... |
Suppose we have some $\Omega$ and $F$ which is a sigma algebra on $\Omega$.
My questions revolves around if we define a measure $\mu$ on $(\Omega, F)$, then can we always find another measure $\mu'$ such that $\mu'(\{\omega\})$ is defined $\forall \omega \in \Omega$ and $\mu(A_i) = \sum_{\omega_j \in A_i} \mu'(\{\om... |
Suppose we have some $\Omega$ and $F$ which is a sigma algebra on $\Omega$.
My questions revolves around if we define a measure $\mu$ on $(\Omega, F)$, then can we always find another measure $\mu'$ such that $\mu'(\{\omega\})$ is defined $\forall \omega \in \Omega$ and $\mu(A_i) = \sum_{\omega_j \in A_i} \mu'(\{\om... |
Suppose we have some $\Omega$ and $F$ which is a sigma algebra on $\Omega$.
My questions revolves around if we define a measure $\mu$ on $(\Omega, F)$, then can we always find another measure $\mu'$ such that $\mu'(\{\omega\})$ is defined $\forall \omega \in \Omega$ and $\mu(A_i) = \sum_{\omega_j \in A_i} \mu'(\{\om... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from i \to i{\infty}.
the contour encloses only two poles, nam... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from i \to i{\infty}, and another one from -i \to -i{\infty}.
... |
It is well known that $SL_2(\mathbb{C})$ is the universal cover of $SO^+(1,3)$, see for example the Wikipedia page on the Lorentz group [1]. The map goes like this (some of this is not standard notation):
First, establish a bijection between $\mathbb{R}^{1,3}$ and the set $M_{s.a.}(2,\mathbb{C})$ of self-adjoint 2... |
Consider the nonlinear Fredholm integral equation of the second kind:
$$
\varphi(x) = f(x) + \lambda \int_a^b F(x, t, \varphi(t)) \, dt
$$
where
* $(f)$ and $(K)$ are given functions,
* $(a, b)$ are constants, and
* $(\varphi)$ is the unknown function.
The associated nonlinear Fredholm operator for this i... |
What are the conditions for solution of nonlinear Fredholm equations with Banach fixed point theorem? |
In Vakil's book *Foundations of Algebraic Geometry*, he defines a linear series on a $k$-scheme $X$ as a vector space $V$, an invertible sheaf (a line bundle) $X$, and a linear map $\lambda: V\to\Gamma(X, \mathscr{L})$.
He doesn't define what it means for a linear series to be basepoint free, although he does say t... |
HINT:
Consider the function
$$3(a^2 + b^2 +c^2 + d^2)-(a+b+c+d)^2$$
It is strictly convex in each variable separately ( being quadratic with leading coefficient $>0$). Such a function, on a product of closed intervals, achieves its maximum only at a vertex.
In this case the values at vertices are
$$3(1+... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from i $\to$ i${\infty}$, and another one from -i $\to$ -i${\inft... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from i $\to$ i${\infty}$, and another one from -i $\to$ -i${\inft... |
First, note next definition.
> **Definition 1.** Let $X$ be a topological space.
(1) Let $Z \subseteq X$ be a closed irreduiclbe subset. The *codimension* $\operatorname{codim}(Z,X)$ of $Z$ in $X$ is the supremum of the legnth of chains of irreduicible closed subsets $Z_0 \supsetneq Z_1 \supsetneq \cdots \su... |
I am totally confused about a detail in computing group cohomology. Let $G$ be a group, $A$ a $\mathbb{Z}[G]$-module. $H^i(G, -)$ is the $i$th right derived functor of $\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}, -)$, i.e. $\operatorname{Ext}_{\mathbb{Z}[G]}^i(\mathbb{Z}, -)$. Here, $G$ acts trivially on $\mathbb{Z}... |
Suppose that $A$ is a finite subset of a group $G$. Let $L$ ($R$) denote the set of all those group elements that fix $A$ from the left (right): $$L:=\{g\in G\colon Ag=A\},\quad R:=\{g\in G\colon gA=A\}.$$ It is easily seen that $L$ and $R$ are subgroups of $G$, and also that $L=R$ in the commutative case. Is it true t... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from $i \to i{\infty}$, and another one from $-i \to -i{\infty}$.... |
Using some mathematical software like `mathematica`, one can find that
$$\int_0^1\frac{dx}{\sqrt{4x^5+1}}=\frac{3+\sqrt5}{2}\int_1^{+\infty}\frac{dx}{\sqrt{4x^5+1}}=0.84984011984360712818935\cdots. $$
The equality is accurate up to at least $100$ digits. This form looks like some kind of elliptic integral, but the po... |
$(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$
According to a proof by contradiction,
Why is the proof of $0\le\sqrt{a}$ not needed?
Proof:
If
>$ (\sqrt{a}\ge\sqrt{b})$
then
>$a = (\sqrt{a})^2 \ge (\sqrt{b})^2 = b$
A contradiction |
Prove $(0\le a\lt b)\Rightarrow (0\le\sqrt{a}\lt\sqrt{b})$. Why is the proof of $0\le\sqrt{a}$ not needed? |
For an integral domain, if every irreducible is prime, then every non-zero element that factors into irreducibles does so uniquely, up to units and ordering. (It is not necessarily true that every non-zero element has such a factorization).
I am looking for examples that disprove the converse. What are some example... |
I'm currently reading "Tensors, Differential forms and Variational Principles" By D. Lovelock and H. Rund. This is from page 2 :
[![enter image description here][1]][1]
and then on page 7, concerning equation 1.1
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/0AdVv.png
[2... |
Why is $\textbf{A} = A_i \textbf{e}_i$ not independant of choice of coordinates? |
In Landau's book on quantum mechanics an expression of the form
$$
(x-a)^{-1/4}
$$
is analytically continued around the upper half-plane along the semicircle from $0$ to $\pi$. The result stated is
$$(a-x)^{-1/4}e^{-i\pi/4}$$
How does one derive this result? Landau writes $(x-a)=\rho e^{i\phi}$ and lets... |
Analytic continuation of $(x-a)^{-1/4}$? |
Why is $\textbf{A} = A_i \textbf{e}_i$ not independent of choice of coordinates? |
I've been tasked to calculate/forecast the weighted exposure of a financial product.
I work with bunker prices and we have access to bunker future prices everyday. They look similar to this in an excel sheet:
```
Prices of futures on 1st Feb 2024
Feb 2024 - $120
Mar 2024 - $110
Apr 2024 - $108
...
Jan 2027 -... |
# Effect of Basis Change on Vector Lengths in Different Vector Spaces
## Does a change in the vector space of a matrix affect the length of vectors?
For instance, if in basis B, $$[v_1]_B = 3[v_2]_B$$, would it be possible in basis C for $$[v_1]_C ≠ 3[v_2]_C$$
for example:
$$[v_1]_C = 4[v_2]_C$$
given that B ... |
Consider a minimax problem $$\inf_{x\in X} \sup_{y\in Y} L(x,y)$$ where $L$ is convex in $x$ and concave in $y$. We also assume that $$\inf_x \sup_y L(x,y) = \sup_y \inf_x L(x,y)$$
holds.
Now let $z=(x,y)$ and $Z^*$ denote the solution set of $z$ for the minimax problem $L(z)=L(x,y)$.
Let $X^*$ denote the se... |
I've been tasked to calculate/forecast the weighted exposure of a financial product.
I work with bunker prices and we have access to bunker future prices everyday. They look similar to this in an excel sheet:
```
Prices of futures on 1st Feb 2024
Feb 2024 - $120
Mar 2024 - $110
Apr 2024 - $108
...
Jan 2027 -... |
**The problem**
Let $X \sim Binomial(l, p)$ and $\sum_{i=1}^{m} X_i = n$, where $l$, $m$, $n$ and $p$ are all known constants. Find:
$$Var\left(\sum_{i=1}^{m} X_i^2 \right)$$
**What I tried**
My first attempt was using $Var\left(\sum_{i=1}^{m} X_i^2 \right) = \sum_{i=1}^{m} Var(X_i^2)$, but this only holds ... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from $i \to i{\infty}$, and another one from $-i \to -i{\infty}$.... |
Consider a minimax problem $$\inf_{x\in X} \sup_{y\in Y} L(x,y)$$ where $L$ is convex in $x$ and concave in $y$. We also assume that $$\inf_x \sup_y L(x,y) = \sup_y \inf_x L(x,y)$$
holds.
Now let $z=(x,y)$ and $Z^*$ denote the solution set of $z$ for the minimax problem $L(z)=L(x,y)$.
Let $X^*$ denote the se... |
Let $\text{circ}(x, y; R, c)$ be the characteristic function on the plane of a disk of radius $R$ and centre $c=(c_1,c_2)$:
$$\text{circ}\left(\frac{\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}}{R}\right)=\begin{cases}1 & \text{if }\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}\le R, \\ 0 & \text{otherwise.} \end{cases} $$
In the case where $... |
Let $F$ be the zero locus of $x^2-y$
The origin is a simple point of $F$
Thus, the set of rational functions of $x$ and $y$ on $F$ defined at the origin is a discrete valuation ring $R$
Therefore, there exists an irreducible element (uniformizing parameter) $f(x,y)$ in this ring with the following property:
... |
For two functions $f$ and $g$, if $\nabla f(x) = \nabla g(x)$, $f = g + c$ for some constant $c$. Does the same hold if the gradient is replaced by the (convex) subdifferential, ie $\partial f(x) = \partial g(x)$ for all $x$ ?
And, as a stronger result, can we characterize pairs $(f, g)$ for which $\partial f(x) \c... |
When do two functions have the same subdifferentials? |
I often see field $\mathbf{Z}/\mathbf{2Z}=\{0,1\}$. Without other indication we might see elements of field $\mathbf{Z}/\mathbf{2Z}$ as a subset of $\mathbf{Z}$. Operations such as $2+1=3=1$ are also given as examples:
[![enter image description here][1]][1]
[Source][2].
I understand this field is the one def... |
What are 0 and 1, the elements of Z/2Z? How do they relate to Z? |
First, note next definition.
> **Definition 1.** Let $X$ be a topological space.
(1) Let $Z \subseteq X$ be a closed irreduiclbe subset. The *codimension* $\operatorname{codim}(Z,X)$ of $Z$ in $X$ is the supremum of the legnth of chains of irreduicible closed subsets $Z_0 \supsetneq Z_1 \supsetneq \cdots \su... |
I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)... |
Does the exponential of a function converge? What can we do with it? |
Let $f$ be a function a $C^\infty(\mathbb{R}^n)$ function such that for all $x_2,...,x_n\in\mathbb{R}$, $f(.,x_2,...,x_n)$ goes to zero at infinity. By Rolle's theorem there exists $y\in\mathbb{R}$ such that $\partial_1f(y,x_2,...,x_n)=0$.
The thing is that $y$ depends on $x_2,...,x_n$, is there a technique to show th... |
I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)... |
Consider a minimax problem $$\inf_{x\in X} \sup_{y\in Y} L(x,y)$$ where $L$ is convex in $x$ and concave in $y$. Assume that $$\inf_x \sup_y L(x,y) = \sup_y \inf_x L(x,y)$$ and all the inf's and sup's are attainable(if they are not infinity). I also want to add that $X$, $Y$ are typically closed convex set in $\mathbb{... |
We denote the $(A|B):=-{\rm tr}(AB)$ as the inner product of the skew-hermitian matrix, which is easy to prove by the definition. I'm stuck by the problem when I read the Lie algebra of matrix group, finding it's a calculation problem of matrix .
> $A\in{{\rm SU}(2)}$
So $A$ can be written as
> $$A=\left(\begin... |
You've done IMO most part of the work, with a few technical details missing and a relatively straightforward argument missing at the end to get at the subproblem a) and then b) and c).
As has been alluded to in the comments, the intermmediate goal is to show that
$$M_n = \{a \in \mathbb N\,|\, a > 0, (a-1)^2 \le... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
What is the quantitative relationship between $\angle BAF$ and $\angle CHG$ and the equivalent line of $AF$? |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
What is the equivalent line of $AF$? |
I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)... |
Using some mathematical software like `mathematica`, one can find that
$$\int_0^1\frac{dx}{\sqrt{4x^5+1}}=\frac{3+\sqrt5}{2}\int_1^{+\infty}\frac{dx}{\sqrt{4x^5+1}}=0.84984011984360712818935\cdots. $$
The equality is accurate up to at least $100$ digits. This form looks like some kind of elliptic integral, but the po... |
I started off with this contour(I apologize for the software i used but I am new to this all).
> i used this contour https://dochub.com/m/shared-document/madhavasthana/1XEpyxzwN5JN1vZVQZGd38/null-1-png?dt=wSzeX2E6NohcehhbdSE6
there is a branch cut from $i \to i{\infty}$, and another one from $-i \to -i{\infty}$.... |
Let $A$ and $B$ be complete Boolean algebras and $f:A\to B$ a homomorphism. Recall that $f$ is complete if $f$ it preserves arbitrary (including possibly infinite) unions.
Does there exist a non-complete endomorphism of $2^{\mathbb{N}}$ that preserves aribitrary **disjoint** unions?
|
Let $A$ and $B$ be complete Boolean algebras and $f:A\to B$ a homomorphism. Recall that $f$ is complete if $f$ it preserves arbitrary (including possibly infinite) joins.
Does there exist a non-complete endomorphism of $2^{\mathbb{N}}$ that preserves aribitrary **disjoint** unions?
|
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
What is the quantitative relationship between $\angle BAF$ and $\angle CHG$ and the equivalent line of $AF$? |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Using some mathematical software like `mathematica`, one can find that
$$\int_0^1\frac{dx}{\sqrt{4x^5+1}}=\frac{3+\sqrt5}{2}\int_1^{+\infty}\frac{dx}{\sqrt{4x^5+1}}=0.84984011984360712818935\cdots. $$
The equality is accurate up to at least $100$ digits. This form looks like some kind of elliptic integral, but the po... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is an action $G\times \text{Conf}_n(\mathbb R^k)\to \text{Conf}_n(\mat... |
Quotient of configuration space is a manifold? |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is an action $G\times \text{Conf}_n(\mathbb R^k)\to \text{Conf}_n(\mat... |
If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?
It seems closed under complement since for all $x$ in the union, $x$ has to belong to some $\mathcal{F}_i$, and so must its complement.
It seems closed under count... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.
For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers are defined. You need to understand what it ... |
Let $I$ be an index set and $\mathcal{X}_i$ for $i\in I$ be a standard measurable space (or say a Polish space or $\mathbb{R}$). Suppose that the probability measure $P(X_I)$ on the product space $\mathcal{X}_I:=\prod_{i\in I}\mathcal{X}_i $ can be decomposed into independent components, i.e., $P(X_I)=\bigotimes_{i\in ... |
Let $I$ be an index set and $\mathcal{X}_i$ for $i\in I$ be a standard measurable space (or say a Polish space or $\mathbb{R}$). Suppose that the probability measure $P(X_I)$ on the product space $\mathcal{X}_I:=\prod_{i\in I}\mathcal{X}_i $ can be decomposed into independent components, i.e., $P(X_I)=\bigotimes_{i\in ... |
The important thing to understand is the convergence of the series for $\exp(t)$.
We'd like to find out if the sum
$$\sum_{n=0}^\infty \frac{t^n}{n!}$$
is defined for all real $t$ (I'm just going to focus on reals here).
Dealing with infinite sums can be tricky. If you're unsure about whether a particular serie... |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.
For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers work. You need to understand not only what i... |
I have two sets $A = \{a1,...,a_n\}, B = \{b_1,...,b_m\}$, for some $n,m$ and with $m \leq n$. I want to map some elements from $A$ to $B$, based on the following assumptions:
1. each item of $A$ can only have one corresponding item of $B$ as a mapping;
2. there is the possibility that some items of $A$ could not... |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet. If anyone can provide a proof or disproof then that would be helpful.
The proof may depend on the type of the input. If $t$ is a real/complex number, then convergence criteria like the [ratio test](https://en.wikipedia.or... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.
For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers work. You need to understand not only what i... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Is this quotient of configuration space a manifold? |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is a natural smooth action $G\times \text{Conf}_n(\mathbb R^k)\to \tex... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Question:
> In isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in
> turn on the extension line of $BC$, and take point $BD$ on the extension
> line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the
> extension line of $DH$ at point $G$, connect $AF$, and ... |
Assume that we have a function:
$$
y=f\left(x,z(x)\right)
$$e take the derivative of this function with respect to $x,$ we
get:
$$
\frac{\partial y}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\times\frac{\partial z}{\partial x}
$$
My question is this: while $f\left(.\right)$ is an un... |
The derivative of an unknown function is additive? |
I often see field $\mathbf{Z}/\mathbf{2Z}=\{0,1\}$. Without other indication we might see elements of field $\mathbf{Z}/\mathbf{2Z}$ as a subset of $\mathbf{Z}$. Operations such as $2+1=3=1$ are also given as examples:
[![enter image description here][1]][1]
[Source][2].
I understand this field is the one def... |
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