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I'm trying to prove the inequality shown in this [post](https://math.stackexchange.com/questions/1136050/let-a-b-c0-such-that-abc-1-prove-that-fraca-bcabc-fracb-acba?noredirect=1), but I'm confused on how OP concluded that, $$\dfrac{a-bc}{a+bc}+\dfrac{b-ac}{b+ac}+\dfrac{c-ab}{c+ab}\le \dfrac32$$
Implies, $$\dfrac{bc... |
What is the manipulation used to transform $\sum_{cyc} \dfrac{a-bc}{a+bc}$ into $\sum_{cyc}\dfrac{bc}{a+bc}$? |
I'm trying to prove that $n^2 < 2^n$. I see this proof question from the example of 2.5.3 [here][1].
I understand that when you prove an inductive case where $P(k+1)$ is true, you have first to assume the base case where $k^2 < 2^k$ is true.
Based on the base case assumption, you have to consider the case where ... |
How can you prove an inductive case of $n^2 < 2^n$? |
For k-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is (5,1,3) a 3-cycle? Or the elements in the rotation must satisfy a law of order? i.e. 3-cycle just are (1, 3, 5) or (5, 3, 1).
For k-cycle, do the elements in them need to be in order? |
For k-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order? |
For k-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is (3,1,5) a 3-cycle? Or the elements in the rotation must satisfy a law of order? i.e. 3-cycle just are (1, 3, 5) or (5, 3, 1).
For k-cycle, do the elements in them need to be in order? |
For reference, eigendecomposition of a matrix $A$ $\in R^{n \times n}$ is defined as:
$A = P \Lambda P^{-1}$
where $P$ is a matrix whose columns are the eigenvectors of $A$, and $\Lambda$ is a diagonal matrix whose entries are the corresponding eigenvalues. This is only possible when $P$ is invertible, which is t... |
For k-cycle $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5)$ a 3-cycle? Or the elements in the rotation must satisfy a law of order? i.e. 3-cycle just are $(1, 3, 5)$ or $(5, 3, 1)$.
For k-cycle, do the elements in them need to be in order? |
What is a complete geometric interpretation of Eigendecomposition of matrices? |
For $k-cycle$ $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5)$ a 3-cycle? Or the elements in the rotation must satisfy a law of order? i.e. 3-cycle just are $(1, 3, 5)$ or $(5, 3, 1)$.
For $k-cycle$, do the elements in them need to be in order? |
What is a complete geometric interpretation of the eigendecomposition of matrices? |
For $k-cycle$ $(1, 2, ..., k)$, do the elements in it need to be taken in order? |
For reference, eigendecomposition of a matrix $A$ $\in R^{n \times n}$ is defined as:
$A = P \Lambda P^{-1}$
where $P$ is a matrix whose columns are the eigenvectors of $A$, and $\Lambda$ is a diagonal matrix whose entries are the corresponding eigenvalues. This is only possible when $P$ is invertible, which is t... |
How to determine the range of `a` in this inequation formula? |
For $k-cycle$ $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5,2,4)$ a 3-cycle? Or the elements in the rotation must satisfy a law of order? i.e. 3-cycle just are $(1,2,3,4,5)$ or $(5,4,3,2,1)$.
For $k-cycle$, do the elements in them need to be in order? |
For $k-cycle$ $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5,2,4)$ a 5-cycle? Or the elements in the rotation must satisfy a law of order? i.e. 5-cycle just are $(1,2,3,4,5)$ or $(5,4,3,2,1)$.
For $k-cycle$, do the elements in them need to be in order? |
For $k-cycle$ $(1, 2, ..., k)$, do the elements in it need to be taken in order?
Is $(3,1,5,2,4)$ a $5-cycle$? Or the elements in the rotation must satisfy a law of order? i.e. 5-cycle just are $(1,2,3,4,5)$ or $(5,4,3,2,1)$.
For $k-cycle$, do the elements in them need to be in order? |
Let $R \subset S$ a flat, injective extension of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M \otimes_S 1_R \subset N \otimes_R 1_S \subset N \otimes S$ as $R$-submodules.
**Question:** Is there an immediate argument to show that $M \otimes_R 1_S = (N \otimes_R 1_S) \cap M \otimes_R... |
This question may do some matter in finding chips for CDMA encryption,or cracking it's encryption.
The question can be expressed mathematically by following:
> How many n-dimensional vector sets containing $n$ vectors with elements of $-1$ and $1$, orthogonal to each other pairwise, are there?
when $n=4$, a s... |
> Find the Laurent expansion of $\dfrac{e^\frac{1}{z}}{z+1}$ in the
> domain $|z|>0$.
My attempt:\
Since we have to work on domain $|z|>0$, let $z=\dfrac{1}{t}$, so I have to find the Laurent expansion of $\dfrac{te^t}{t+1}$ at $t=0$. Particularly, I have to find the Laurent expansion of $\dfrac{e^t}{t+1}$ at $t=0... |
O.K. I think that I solved the problem. I leave an answer as a record.
For simplicity, let $X:=\operatorname{Spec}A$ be an affine scheme with $x:=\mathfrak{p} \in X$. By the hungerford, Algebra, p.147, Theorem 4.11-(i), there exists a bijection :
$$\psi : \{ \operatorname{prime ideals} Q\subseteq A \operatorna... |
In the book "Fourier-Mukai Transforms in Algebraic Geometry", the author has been using the following terminology quite a few times in the first two chapters (Proof of Lemma 1.58, Definition 2.72, etc.), and I can't make sense of it. After digging further, I think here is the correct way to phrase the question:
Fix ... |
In the book "Fourier-Mukai Transforms in Algebraic Geometry", the author has been using the following terminology quite a few times in the first two chapters (Proof of Lemma 1.58, Definition 2.72, etc.), and I can't make sense of it. After digging further, I think here is the correct way to phrase the question:
Fix ... |
O.K. I think that I solved the problem. I leave an answer as a record.
For simplicity, let $X:=\operatorname{Spec}A$ be an affine scheme with $x:=\mathfrak{p} \in X$. By the hungerford, Algebra, p.147, Theorem 4.11-(i), there exists a bijection :
$$\psi : \{ \operatorname{prime ideals} Q\subseteq A \operatorna... |
This is very difficult to explain, but what I'm looking for, is kind of like sigma but instead I want the middle to do something like this "x=n+1" instead of doing (n+1)+(n+1)+(n+... etc. etc.
So, as an explanation, I am in short trying to recreate the square root symbol function following these requirements;
- ... |
what symbol allows you to start at a number, repeat an expression a curtain amount of times, but just sets a variable instead of summing the answer? |
How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix}
0&I_6&0\\I_6&0&0\\0&0&I_{16}\end{bmatrix} \tag{2}$$
I know that a symmetric matrix has $28 \times 29/2=406$ free parameters and I know that the final ans... |
I am looking for an explanation for the terminology "artificial smoothing" and "artificial diffusion" in the context of numerical schemes.
For instance, consider the upwind scheme where we estimate a derivative of $f$ at $x$ by
$$
\frac{f(x)-f(x-h)}{h}
$$
where $h>0$ is our mesh-size. This scheme is said to crea... |
Let $Y\to X$ be a Galois cover of complex varieties (or only consider complex algebraic curves) with the ramification data $(p_\bullet,\eta_\bullet)$, namely branch points $p_\bullet$ with a *preferred generator* $\eta_i$ of the stabilizer $G_{g_i}$ for each $p_i$. (To be precise, the preferred generator is the one cor... |
This is very difficult to explain, but what I'm looking for, is kind of like sigma but instead I want the middle to do something like this "x=n+1" instead of doing (n+1)+(n+1)+(n+... etc. etc.
So, as an explanation, I am in short trying to recreate the square root symbol function following these requirements;
- ... |
In the topic of divisibility. Suppose I starts with $a \mid n$, then I manipulate it into $a \mid m$, then $a \mid s$, and finally $a \mid t$. Should I write my train of thought as
$$
\begin{align}
a \mid n
&\iff a \mid m \\
&\iff a \mid s \\
&\iff a \mid t.
\end{align}
$$
Or can I just shorthanded it (in ... |
Can I shorthand chain divisibility in the same manner as equality? |
Is there a concept of natural splliting? |
I'm trying to calculate the standard matrix for orthogonal projection using the formula for the matrix P=$A(A^T A)^ {-1} A^T$.
My approach has diverged from the supplied answer to this question:
[![my_attempt][2]][2]
[![supplied][3]][3]
It seems that I've done the working for the inner part ($A^T A$) correctly ... |
In calculus, if $\frac{dy}{dx}$ is not in fact a fraction, is the equation below for geometric brownian motion technically incorrect? |
Let $f:\mathbb R^n \to V$ with $V$ be a Banach space (over $\mathbb R$) suppose that $f$ is frechet differentiable in $\mathbb R^n$ then there is a way to define the Jacobian of $f$?
I try to found literature abou this but i cant found anything.
|
In $\Delta ABC$, such that $\cos^2{A} + \cos^2{B} + \cos^2{C} = \sin^2{B}$, then $\tan{A} \cdot \tan{C} =$? |
During my research, I was analyzing a linear system with characteristic polynomial of six order, $p(x)=x^6+3ax^5+3a^2x^4+a^3x^3 - a^3b_1b_3b_3$. I used Mathematica and it found all roots as the following:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/Y4ps0.png
I was quite impressed ... |
**My attempt:**
Define the collection of set $\mathcal{I}:=\{(a,b):a,b\in\mathbb{Q},a<b\}$ in $\mathbb{R}$. Hence, $\mathcal{I}\subset\mathbb{Q}\times\mathbb{Q}$, so there is injective function, namely indentity function $id:\mathcal{I}\to\mathbb{Q}\times\mathbb{Q}$. We also know that $\mathbb{Q}\times\mathbb{Q}\... |
I'm studying this series
\begin{align*}
a_1=\mathrm{i}, \quad a_{n+1}=\mathrm{i} + \frac{\mathrm{i}}{a_n},
\end{align*}
where $\mathrm{i}$ is the imaginary unit, in order for me to see clearly the structure of this sequence, I separate the real part and the imaginary part as a sequence, as follows.
\begin{align*}
... |
Suppose $J \subseteq \mathbb{C}[x_1,\ldots,x_n]$ is an ideal, and $I = \sqrt{J}$ is its radical. How large can the smallest integer $e$ for which $I^e \subseteq J$ be?
Is anything known in this regard? Any upper bound in terms of the maximal degree(s) of the Groebner basis of $J$ (and maybe also that of $I$), and $n... |
What should be a general approach towards solving these kind of questions algebraically?
I tried to code this up and find a solution but I don't know whether it's correct or not. As I've used a scipy package and don't know the theory of this problem that well.
from scipy.optimize import linprog
#li... |
Suppose $\mathcal V\subset \mathbb{R}^n$ is a subspace and $\vec{x}\in\mathbb{R}^n$. I want to show there is a unique way to write $\vec{x}$ as a sum of two vectors $\vec{x}=\vec{x}^{\parallel}+\vec{x}^{\perp},$ the first of which lies in $\mathcal{V}$ and the second of which is orthogonal to every member of $\mathcal{... |
This is a special case of *Hewitt-Marczewski-Pondiczery theorem*, see e.g. Theorem 2.3.15 in Engelking's General Topology:
> If $d(X_s)\leq \alpha\geq\aleph_0$ for every $s\in S$ and $|S|\leq 2^\alpha$, then $d(\prod X_s)\leq\alpha$.
The $d(X)$ denotes the *[density](https://en.wikipedia.org/wiki/Cardinal_functio... |
> Classify the singularities (removable, pole, essential) of
> $f(z)=\dfrac{e^z}{1+z^2}$.
I already pointed out $z=\pm i$ are pole points with order $1$.\
Since $\lim\limits_{z\to\infty}f(z)=\infty$, I can conclude $\infty$ is also a pole point. But I don't know its order? Can someone help me to find the order of ... |
> Classify the singularities (removable, pole, essential) of
> $f(z)=\dfrac{e^z}{1+z^2}$.
I already pointed out $z=\pm i$ are poles with order $1$.\
Since $\lim\limits_{z\to\infty}f(z)=\infty$, I can conclude $\infty$ is also a pole point. But I don't know its order? Can someone help me to find the order of $\inft... |
I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions:
If $x(t)=y(t)+o(t)$ for $t$->$0$ then $x=y$ ($x$ and $y$ are numbers which corresponds for Cauchy sequence for functio... |
How to choose correct number for Cauchy sequence for infinitesimal functions? |
Let $\cos(A-C)=k\cos(A+C)=-k\cos B$
$\implies k=\dfrac{\cos(A-C)}{\cos(A+C)}\iff\dfrac{k-1}{k+1}=\cdots=\tan A\tan C$
From the given condition,
$$\cos^2A-\sin^2C+2\cos^2B=0$$
$$\iff\cos(A-C)\cos(A+C)+2\cos^2B=0$$
$$\iff-\cos B(\cos(A-C)-2\cos B)=0$$
If $\cos B\ne0,k=-\dfrac{\cos(A-C)}{\cos B}=-2,\tan A\... |
Let $n = 2^5 \times 3^3$ and let $a$ be a positive integer that is relatively prime to $n$. I am trying to find the maximum order of $a$ modulo $n$.
First by the Euler-Phi function, $\phi(n)= \phi(2^5) \times \phi(3^3) = 288$ is number of $a$ that is relatively prime with n.
The goal is to find $x$ such that $a^x ... |
I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions:
If $x(t)=y(t)+o(t)$ for $t$->$0$ then $x=y$ ($x$ and $y$ are numbers which corresponds for Cauchy sequence for functio... |
How to correctly choose number for Cauchy sequence for infinitesimal functions? |
If $I\subset\Bbb{R}$ is an open interval and $f:I\to\Bbb{R}$ is convex and differentiable, then for all $a,b\in I$, we have the inequality
\begin{align}
0\leq f(a)-f(b)-f’(b)[a-b].\tag{$*$}
\end{align}
Apply this with $a=g(h(t)-c,t)$ and $b=g(h(t),t)$.
_____
Why is the inequality $(*)$ above true? Well, for a... |
Given a topological spaces
$(X,\tau)$ and a subset $S\subseteq X$, we say that $X$ is $S$-contractible if
$$X\simeq X/R$$
Where $$R= S^2 \cup ( \Delta_X/ \Delta_S )$$
The idea is that pulling $S$ to a single point doesn't change $X$. I have two questions:
1. Is it possible to classify all spaces $X$ with... |
I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions:
If $x(t)=y(t)+o(t)$ for $t$->$0$ then $x=y$ ($x$ and $y$ are numbers which corresponds for Cauchy sequence for functio... |
My question is: How to correctly choose number for Cauchy sequence for infinitesimal functions $x=y=[sin(t)]$ or $x=y=[t]$? |
How to correctly choose number for Cauchy sequence for infinitesimal functions $x=y=[sin(t)]$ or $x=y=[t]$? |
If $I\subset\Bbb{R}$ is an open interval and $f:I\to\Bbb{R}$ is convex and differentiable, then for all $a,b\in I$, we have the inequality
\begin{align}
0\leq f(a)-f(b)-f’(b)[a-b].\tag{$*$}
\end{align}
Geometrically, $a\mapsto f(b) +f’(b)[a-b]$ is the equation of the tangent line to the graph of $f$ at the point $b... |
I have a question in my exam which is:
> Let :v = [3,1] and *T* : R2 -> R2 denote the transformation such that
> T(x) is the vector in span{v} nearest to x
>
> Let :0<a<pi/2 denote the angle between the positive x-axis and the
> vector v. Let *R*: R2->R2 denote the transformation such that R(x) is
> the vector... |
Given a topological spaces
$(X,\tau)$ and a subset $S\subseteq X$, we say that $X$ is $S$-contractible if
$$X\simeq X/R$$
Where $$R= S^2 \cup ( \Delta_X/ \Delta_S )$$
Where $\Delta$ is the diagonal.
The idea is that pulling $S$ to a single point doesn't change $X$. I have two questions:
1. Is it possi... |
does this expression look familiar (natural log with exponential)? |
I've stumbled on this [nice formula][1] to compute the barycenter $\bar{c}$ of an arbitrary (but not self-intersecting) polygon
\begin{equation}
\bar{c} \triangleq \sum_{i=1}^n \frac{A_i}{A} \bar{c}_i
\qquad
A \triangleq \sum_{i=1}^n A_i
\qquad
\bar{c}_i \triangleq \frac{V_i + V_{i+1}}{3}
\qquad
A_i \triang... |
I've stumbled on this [nice formula][1] to compute the barycenter $\bar{c}$ of an arbitrary (but not self-intersecting) polygon
\begin{equation}
\bar{c} \triangleq \sum_{i=1}^n \frac{A_i}{A} \bar{c}_i
\qquad
A \triangleq \sum_{i=1}^n A_i
\qquad
\bar{c}_i \triangleq \frac{V_i + V_{i+1}}{3}
\qquad
A_i \triang... |
Regarding linear regression, in the (informal) [solution manual][1] of T. Hastie's "The Elements of Statistical Learning" book, there is a paragraph (p. 16 in the manual) that goes:
"... Recall that we assume that each coordinate of the vector y is independent, and distributed as a normal random variable, centered a... |
Suppose that you initially have a container which has the shape of a paraboloid with equation
$ z = a x^2 + b y^2 $ where $ 0 \le z \le h $
Now you tilt this paraboloid by rotating it about any point (let's say the origin) about a horizontal axis whose unit vector is $v = (\cos \varphi, \sin \varphi, 0)$. Let t... |
I am studying differential geometry.
Definition of surface is
S subset of $R^3$ is surface if for every $p \in S$,\
exist open set U in $R^2$, open neighborhood V of p in S, differentiable map
\begin{equation}
X:U \rightarrow R^3
\end{equation}
s.t.
1. X(U) = V
2. X is homeomorphism
3. for every q in U, ... |
We have the optimization problem $$\begin{align}
\min_{[a_1,a_2,\ldots,a_N]}&\prod_{i=1}^{N}\left[1-{A\over\prod_{n=1}^{R}\left(\lambda_n+{\sum_{u\ne i}|C_i\cdot h_u|^2a_u+B\over a_{i}}\right) }\right]\end{align}$$
and constraints
$$P_\max>a_1>a_2>\cdots>a_N>0,$$
and
$$\gamma_{i}={|C_i\cdot h_i|^2a_{i}\over\sum_{u... |
Let set $S$ be open, meaning that $\forall s \in S$, $\exists \varepsilon > 0$ such that $(s-\varepsilon,s+\varepsilon) \subset S$. In other words, there is always some small $\varepsilon$ that allows for the neighborhood of $s$ to belong to $S$ (correct my wording if I'm wrong). If it were closed, then $S^c$ would be ... |
What is the difference between the definition a closed set and a non-open set? |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Theorem 8.2. Let $A$ be open in $\mathbb{R}^n$; let $f:A\to\mathbb{R}^n$ be of class $C^r$; let $B=f(A)$. If $f$ is one-to-one on $A$ and if $Df(x)$ is non-singular for $x\in A$, then the set $B$ is open in $\mathbb{R}^n$ and the inverse function $g:B\to A... |
For a better vision, see this image:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/PSC8B.png
We have the optimization problem $$\begin{align}
\min_{[a_1,a_2,\ldots,a_N]}&\prod_{i=1}^{N}\left[1-{A\over\prod_{n=1}^{R}\left(\lambda_n+{\sum_{u\ne i}|C_i\cdot h_u|^2a_u+B\over a_{i}}\righ... |
How to effectively determine that a N-dim point from a set is a vertex of the convex hull of that set without calculating the entire hull? |
I am studying measure theory, topology, and functional analysis in mathematics. Let $X$ be a compact topological space. We assume that there exists a countable family $\{f_n\: X \to \mathbb C: n\in\mathbb{N}\}$ of continuous complex-valued functions on $X$ that {\em separates points} in the following sense: for a... |
How to effectively det in N-dim space that a point from a set is a convex hull vertex of that set, without calculating the entire hull? |
Let V be the following vector space.
> $V = \{ f : \mathbb{R} \longrightarrow \mathbb{R} : \text{f = infinitely differentiable at all x, } f^{(k)}(0) = 0 \}$.
How would one show that this vector space is infinite dimensional?
----------
To show that a vector space is infinite dimensional, one needs to... |
Is a Set of Infinitely Differentiable Functions Linearly Independent? |
The statement is as follow:
> Suppose $(S_n)$ is a 2d simple random walk, and $(S^1_n)_n, (S^2_n)_n$ are two independent random walks of the same type. Let $R_n = \sum_{j=1}^n\mathbf{1}\{S_j = 0\},$ and $ L_n = \sum_{j=1}^n \mathbf{1}\{ S^1_j = S^2_j\}$. Show that
$$
L_n = R_{2n} \enspace \text{ in distribution}.... |
Suppose $\mathcal V\subset \mathbb{R}^n$ is a subspace and $\vec{x}\in\mathbb{R}^n$. I want to show there is a unique way to write $\vec{x}$ as a sum of two vectors $\vec{x}=\vec{x}^{\parallel}+\vec{x}^{\perp},$ the first of which lies in $\mathcal{V}$ and the second of which is orthogonal to every member of $\mathcal{... |
As you can see (page 44) the *inverse* of $F$ is defined in general for a set $F$ whatever: but what $F^{−1}$ is when $F$ is not a relation? i.e. it is not a set of ordered pairs?
Recall that (page 40) $xFy$ is a shorthand for $\langle x,y \rangle \in F$; if not, $F^{−1}=\text {dom}F=\text{ran}F=\emptyset$.
Thus... |
I'm currently reading some codes about conical projection.
Let the viewed target position point be $P$, and the observer point be $Q$, the local coordinate system's $x,y$ plane is perpendicular to $\vec{PQ}$. $z$ axis along $\vec{PQ}$. Let the coordinates of the $x,y,z$ axis's directional unit vector be denoted as ... |
How to calculate trangent vector with conical projection? |
Why is $P_{\rho}$ is a probability measure on the Borel subsets of $H\ $? |
In Brent and Cohen's [paper][1] about odd perfect numbers, they show this inequality.
$N \ge p^a\sigma(p^a) \gt p ^ {2a}$ where a is even.
I understand the next second half of this: $p^a\sigma(p^a) \gt p ^ {2a}$. The component * sum of its divisors is always larger than the square of the component, , since the su... |
Why is this inequality in Brent and Cohens paper on odd perfect numbers true? |
I am given the function $f:\mathbb{C}\setminus \{0, -1, -2, \ldots \}\to\mathbb{C}$ that is holomorphic on its domain, such that $f(1)=1$ and $f(z+1)=zf(z)$. I want to show that $0, -1, -2, \ldots$ are simple poles of $f$ and that the residues are given by $(-1)^n/n!$ at $z=-n$. Any hints? I am thinking of using a Laur... |
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ and $n\geq 2$ is an integer, then is it true that $$(I(R:_{Q(R)} I))^n=\sum_{f\in (R:_{Q(R)} I)} (fI)^n$$ ?
I think... |
How do we read x∈[0,50]?
Should we read it as "x is an element of the set 0 to 50"?
I tried to search the answer but I couldn't find how to read the set.
Thank you. |
How do we read x∈[0,50]? |
First I made this question for myself just out of quriosity. But got stcuk in the middle of solving it.
The question:
Say we have a probability density function of a continuous stochastic variable X as
$$f_X(x)=\begin{cases}
\frac{2}{\pi^2}x & 0\leq x\leq\pi \\
0 & otherwise
\end{cases}$$
If w... |
Could someone find how to solve this question? I am stuck in the middle. |
How do we read $x∈[0,50]$?
Should we read it as "$x$ is an element of the set $0$ to $50$"?
I tried to search the answer but I couldn't find how to read the set.
Thank you. |
How do we read $x∈[0,50]$? |
How to calculate tangent vector with conical projection? |
First I made this question for myself just out of quriosity. But got stcuk in the middle of solving it.
The question:
Say we have a probability density function of a continuous stochastic variable X as
$$f_X(x)=\begin{cases}
\frac{2}{\pi^2}x & 0\leq x\leq\pi \\
0 & otherwise
\end{cases}$$
If w... |
Let $a_{i}$ and $b_{i}$, $i = 1,\dotsb,n$ be real numbers. Denote $S_{pq}:=\sum_{i=1}^{n}a_{i}^{p}b_{i}^{q}$.
I want to show that,
$$ (S_{20}S_{02} - S_{11}^2)S_{00} - (S_{10}S_{02} - S_{01}S_{11})S_{10} + (S_{10}S_{11} - S_{01}S_{20})S_{01} \geq 0. $$
and find its equivalence condition.
I thought about it in s... |
A sort of extension of Cauchy-Schwarz inequality? |
First I made this question for myself just out of curiosity. But got stuck in the middle of solving it.
**The question:**
Say we have a probability density function of a continuous stochastic variable X as
$$f_X(x)=\begin{cases}
\frac{2}{\pi^2}x & 0\leq x\leq\pi \\
0 & otherwise
\end{cases}$$
... |
I have two tensors - $A_{k}$ and $B_{lm}$. How can I find wedge product of them? Wedge product of two tensors $\alpha$ $(p, 0)$ and $\beta$ $(q, 0)$ is defined as $$\alpha \wedge \beta = \frac{(p + q)!}{p! q!} \text{Alt} (\alpha \otimes \beta)$$, where $Alt$ is antisymmetrisation of tensor.
The result I got is $A_k ... |
Let set $S$ be open, meaning that $\forall s \in S$, $\exists \varepsilon > 0$ such that $(s-\varepsilon,s+\varepsilon) \subset S$. In other words, there is always some small $\varepsilon$ that allows for the neighborhood of $s$ to belong to $S$ (correct my wording if I'm wrong). If it were closed, then $S^c$ would be ... |
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