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Is it true that the probability of n hyperplanes with a maximum of K-2 dimensions intersecting in K-dimensional ambient space is 0? |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
For back-ground, I'm not well-schooled in higher-dimensional geometry, but I'm currently learning statistical data-science in which many methods rely on the properties of higher-dimensional space.
In Support-Vector Machine (SVM) clustering, the goal is to find a hyper-plane which will cleanly divide the data-points ... |
For back-ground, I'm not well-schooled in higher-dimensional geometry, but I'm currently learning statistical data-science in which many methods rely on the properties of higher-dimensional space.
In Support-Vector Machine (SVM) clustering, the goal is to find a hyper-plane which will cleanly divide the data-points ... |
While playing a city builder game, I noticed that the area of the buildings I can place on the game map is lower than the area of the map itself, since I need to place the roads that connect all buildings to a central block, and they occupy space themselves.
That brings the question: Is it possible to predict, for a... |
Given a manifold $M$ of dimension $\mathbb{R}^{3D}$, we can construct an alternating trilinear map $\alpha(a, b, c) = \int_M a \cup b \cup c$ where $a, b, c \in H^D(M)$.
The question is the realization question: Given an arbitrary alternating trilinear map, is there a manifold $M$ that generates such a trilinear map... |
For back-ground, I'm not well-schooled in higher-dimensional geometry, but I'm currently learning statistical data-science in which many methods rely on the properties of higher-dimensional space.
In Support-Vector Machine (SVM) clustering, the goal is to find a hyper-plane which will cleanly divide the data-points ... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
**Problem**: A renovation of an arena proposes to give the seats colors from a color scheme with 5 different colors. In each row, all 5 colors must be used at least once. In how many different ways can this be done for a row of 20 seats?
**Answer**: 5^20 - (5c1)*4^20 + (5c2)*3^20 - (5c3)*2^20+(5c4)*1^20
(Where "5... |
Let $\Lambda$ and $\mathcal{C}$ be categories and $\mathcal{W}$ a subset of morphisms of $\mathcal C$ such that there exists the localization $L: \mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$. Denote
$$ \mathcal{W}^{\Lambda} = \{\theta \in \mathrm{Mor}(\mathcal{C}^{\Lambda}) \,|\, \theta_\lambda \in \mathcal{W}, \for... |
How to prove a matrix $M$ has incoherence property? |
Let $\Lambda$ and $\mathcal{C}$ be categories and $\mathcal{W}$ a subset of morphisms of $\mathcal C$ such that there exists the localization $L: \mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$. Denote
$$ \mathcal{W}^{\Lambda} = \{\theta \in \mathrm{Mor}(\mathcal{C}^{\Lambda}) \,|\, \theta_\lambda \in \mathcal{W}, \for... |
**Hint**: The singular value decomposition of the rank $r = 1$ matrix $M = x\vec{1}^T$ is $\sigma_1u_1v_1^T$ where $u_1 = \dfrac{x}{\|x\|}$, $v = \dfrac{\vec{1}}{\|\vec{1}\|} = \dfrac{1}{\sqrt{n}}\vec{1}$, and $\sigma_1 = \|x\| \cdot \|\vec{1}\| = \sqrt{n}\|x\|$.
So, the rank-$1$ projection matrices are $P_U = u_1u... |
[![][1]][1]
I was reading *On discrete subgroups of the two by two projective
linear group over $\mathfrak{p}$ -adic fields* by Ihara, and I would like know why the set $\lbrace Ug :g \in G_l\rbrace$ has $q^l-q^{l-1}$ elements.
First the definition of double coset ring it's not clear to me. The only reference that... |
It should be clear that $5^{20}$ is the total number of ways to assign a color to each of the $20$ seats. We can instead answer the question
> How many colorings do not use at least one color?
and subtract the answer to that question from $5^{20}$ to get the desired answer.
Assume for the sake of simplicity th... |
**Problem**: A renovation of an arena proposes to give the seats colors from a color scheme with $5$ different colors. In each row, all $5$ colors must be used at least once. In how many different ways can this be done for a row of $20$ seats?
**Answer**: $5^{20} - \binom{5}{1} (4^{20}) + \binom{5}{2} (3^{20}) - \bi... |
The obvious problem here is that because $c > 1$, then $c^n \to \infty$ for $n \to \infty$. So we have:
$$\| (c \cdot T)^n x \| \le \underbrace{|c^n|}_{\to \infty}\cdot \underbrace{\|T^nx\|}_{\to 0}$$
So the only possible way for $\left( |c^n| \cdot \| T^n x \| \right) \to 0$ is, if $\|c^n|$ will grow to $\infty$... |
Let $T$ be a contraction in a Hilbert space $H$. If $\| T^n x\| \to 0$ for $n \to \infty$, does $\| (c \cdot T)^n x \| \to 0$, where $c > 1$? |
${\rm Aut }\left({\mathbb R}/{\mathbb Q}\right)$ is trivial, which implies there is no intermediate field between the field of rationals and field of real numbers, but we know there are infinitely many intermediate fields of $\mathbb R$ containing $\mathbb Q$.
Why I am getting an contradictions? Please explain whe... |
This integral: $\int\left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$ seems to be complicated even [Wolfram Alpha][1] can't solve it, but it can be easily solved if one know the formula: $$\tan(a+b+c)= \frac{\tan(a)+ \tan(b)+ \tan(c ) - \tan(a) \tan(b)\tan(c)}{1-\tan(a)\tan(b)-\tan(... |
Assume I have a scalar function $\phi(p): p \in \Omega \subset \mathcal{R}^3 \to \mathcal{R}$. I would like to use it to represent a 2D surface. For example, if $\phi(\cdot)$ is a signed distance function (SDF), then $\phi(p)=0$ represents a 2D surface.
Now, I would like to define the surface area over $\phi(\cdot)... |
What is the scalar field the integral of which gradient norm is equal to the surface area of the surface that the scalar field represent? |
I am tasked with finding whether this function can be developed into a laurent series around $z = 1$, and if so what is the radius of convergence and what is the residue?
So far what I got is:
- The function has an isolated singularity at $z = 1$ so indeed it can be developed into a laurent series at that point.
-... |
I am tasked with finding whether the function $f(z) = \cos(z/(1-z))$ can be developed into a Laurent series around $z = 1$, and if so what is the radius of convergence and what is the residue?
So far what I got is:
- The function has an isolated singularity at $z = 1$ so indeed it can be developed into a Laurent s... |
For practice I am trying to use the standard definition of volume for a set $X \subset \mathbb{R}^n$ given by:
$$vol(X) := \int_X dx_1 \ \ldots \ dx_n$$
to compute the area of the triangle spanned by the points $(0,0,1), (0,2,0)$ and $(1,1,1)$ in $\mathbb{R}^3$. (The result should be $1.22$.)
If I am not mis... |
I think I managed to solve this integral. However, it would be good if someone could double-check my solution.
First, a couple of observations. The function $z^z/z$ seems to be continuous. I checked it using the [Desmos][1] graphing calculator. The picture is given below:
[![Desmos][2]][2]
Second, the integral... |
[![][1]][1]
I was reading *On discrete subgroups of the two by two projective
linear group over $\mathfrak{p}$ -adic fields* by Ihara, and I would like know why the set $\lbrace Ug :g \in G_l\rbrace$ has $q^l-q^{l-1}$ elements.
First the definition of double coset ring it's not clear to me. The only reference tha... |
>**[Tom Leinster, Exercise 1.3.31]** A permutation of a set $X$ is a bijection $X \rightarrow X$. Write **Sym**$(X)$ for the set of permutations of $X$. A total order on a set $X$ is an order $\leq$ such that for all $x, y \in X$, either $x \leq y$ or $y \leq x$; so a total order on a finite set amounts to a way of pla... |
Let $(\mathcal{M}, \otimes, \mathbb{k})$ be a symmetric closed monoidal category, which in my application is the category of $dg$-modules over some commutative ring. Let $A$ be a bialgebra/bimonoid in $\mathcal{M}$ whose underlying object is dualizable. Then $A^{\vee}$ is also a bialgebra, and we have an isomorphism $A... |
Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$.
If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get some properties about $\sum_{m=1}^\infty \phi_{m}z^m$ and $\sum_{m=1}^\infty \psi_{-m}z^{-m}$ from this equation $P(z)... |
[pictured graph][1]
[1]: https://i.stack.imgur.com/KzmF1.png
Based on this position graph of a particle, how am I supposed to know when it is speeding up or slowing down? I know that with a velocity graph, when velocity and acceleration have the same sign the object is speeding up and when they differ it is slo... |
I think I managed to solve this integral. However, it would be good if someone could double-check my solution.
First, a couple of observations. The function $z^z/z$ seems to be continuous. I checked it using the [Desmos][1] graphing calculator. The picture is given below:
[![Desmos][2]][2]
Second, the integral... |
[![enter image description here][1]][1]
**Question:** Based on this position graph of a particle, how am I supposed to know when it is speeding up or slowing down?
I know that with a velocity graph, when velocity and acceleration have the same sign the object is speeding up and when they differ it is slowing dow... |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
> Let $f$ be holomorphic on $D(0,1)$, continuous on $\overline{D}(0,1)$
> s.t $f(z)=0\quad\forall |z|=1$ and $0\leq\arg z\leq\dfrac{2\pi}{n}$
> for some $n\in\mathbb{N}^*.$
>
> a) Let
> $g(z)=\prod\limits_{k=0}^{n-1}f\left(e^{\frac{2k\pi i}{n}}z\right)$. Prove
> $g(z)=0\quad\forall |z|=1$.
>
> b) Prove that $... |
How would you solve this?
[enter image description here][1]
[1]: https://i.stack.imgur.com/XHy0n.png
I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, f(t)=3+t^2/2, does not satisfy the integral equation, nor d... |
How would you solve this?
[![enter image description here][1]][1]
I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, f(t)=3+t^2/2, does not satisfy the integral equation, nor does it satisfy the initial conditions. I... |
So I have a polynomial $p$ of order $2n$, which I want to factor into two polynomials $q$ and $r$, each of order $n$, while minimizing the expression
$$\frac{|\sum_{k} a_k |}{\sum_{k} | a_k |} \frac{|\sum_{k} b_k |}{\sum_{k} | b_k |}$$
with $a_k$ and $b_k$ being the coefficients of $q$ and $r$ respectively. How wo... |
I came up with this feature about 3 months ago while using GeoGebra
If we have a triangle, and we draw the three circles that touch its sides from the outside, then the six points of contact lie on one conic section.
[![enter image description here][1]][1]
Is this property already known?
How do we prove it anywa... |
Do the tangent points of the circles that are tangent to the sides of a triangle lie on a conic section? |
I am working on part (b) of exercise 1 in Chapter 4 of Brian C. Hall's book. It states:
$\textbf{Exercise 1}$:
A $\textbf{locally integrable}$ function $\psi(x,t)$ satisfies the free Schr$\ddot{\textrm{o}}$dinger equation in the weak (or distributional) sense if for each smooth compactly supported function $\chi$, ... |
Here is the question I am trying to see the solution of part (a) in it:
Fix $\mathbb F_p$ a field with $p$ elements for $p$ prime. Consider $f(X) = X^{p^n} - X + 1$ over $\mathbb F_p$ and let $E$ be a splitting field of $f.$
$(a)$ Prove that $E$ has a subfield isomorphic to $\mathbb F_{p^n}.$
**My questions ... |
How would you solve this?
[![enter image description here][1]][1]
I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, f(t)=3+t^2/2, does not satisfy the integral equation, nor does it satisfy the initial conditions. I... |
So going off of Kurt G, for $\alpha = 1$ on the boundaries, I should get:
$$\oint_C 1\nabla \phi_i \cdot \hat{\textbf{n}} ds = \int\int_D \nabla \cdot \nabla\phi_i dA$$
And computing the double integral, I should get:
$$-2(n\pi)^2\int_0^1 \int_0^1 \sin(n\pi x)\sin(n \pi y)dxdy = \\ 2n\pi \int_0^1 \Big(\cos(n\p... |
Here is a proof of Kirszbraun theorem for a case where E is bounded and convex :
- If we assume that E is a bounded convex set of R^n, then the proof of Kirszbraun's theorem becomes easy:
- We start by setting K = closure of E, which is a compact convex set.
- We extend f to K by uniformly continuous extension... |
What is the role of the hypothesis of E being convex and bounded in Kirszbraun's theorem? |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
I am working on part (b) of exercise 1 in Chapter 4 of Brian C. Hall's book. It states:
$\textbf{Exercise 1}$:
A $\textbf{locally integrable}$ function $\psi(x,t)$ satisfies the free Schr$\ddot{\textrm{o}}$dinger equation in the weak (or distributional) sense if for each smooth compactly supported function $\chi$, ... |
What am I doing wrong when finding the Fisher information of a binomial distribution? |
Since being symmetric is the property of an operator, not just its associated matrix, let me use $\mathcal{A}$ for the linear operator whose associated matrix in the standard basis is $A$. Arturo and Will proved that a real symmetric operator $\mathcal{A}$ has real eigenvalues (thus real eigenvectors) and that eigenvec... |
Let $\langle A,R \rangle$ and $\langle B,S \rangle$ be two well-orders, and let $\text{pred}(A,x,R) := \{y \in A \;|\; yRx\}$ and similarly for $\text{pred}(B,z,S)$.
It is claimed that one of the following must hold:
$1$. $\langle A,R \rangle \cong \langle B,S \rangle$.
$2$. $\exists y \in B\left(\langle A,R... |
I am working on part (b) of exercise 1 in Chapter 4 of Brian C. Hall's book. It states:
$\textbf{Exercise 1}$:
A $\textbf{locally integrable}$ function $\psi(x,t)$ satisfies the free Schr$\ddot{\textrm{o}}$dinger equation in the weak (or distributional) sense if for each smooth compactly supported function $\chi$, ... |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
Let's say I have 7 independent variables (inputs) and 1 dependent variable (output). I don't know the function that relates the inputs to the output. All I can do is define each variable, and the "blackbox" will spit out an output. As you can imagine, there are infinite amounts of combinations, and I can't enter each c... |
How can I maximize my output through a "blackbox" function? |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
I'm wondering how can i make sense of a partial solution to a two-system of polynomial equations.
Let $\{f,g\}\subseteq K[x,y]$ be polynomials in a polynomial ring over the field $K$. Given the equations:
$$f(x,y)=a$$
$$g(x,y)=b$$
Every solution is contained in the affine variety $\mathcal{V}(I_0)$ of the ideal
$... |
Are solutions to polynomial equations subideals in a polynomial ring? |
I am trying to find the Fisher information of a binomial distribution where $n=2$ and $n=\theta$. I have the second partial derivative of the log-likelihood function as $$\frac{-\sum^{n}_{x=1}x_i}{\theta^2}-\frac{2n-\sum^{n}_{x=1}x_i}{(1-\theta)^2}$$
When I go to take the negative expectation of this, I get $$\frac{... |
The following exercise appears in Ridgway Scott's Numerical Analysis:
[![ex][1]][1]
Where $\omega_n(x)$ is the Chebyshev Polynomial of the first kind, that is $$\omega_{n+1}(x)=2^{-n}\cos((n+1)\cos^{-1}(x)$$
I have seen (and proven) that these polynomials are orthogonal w.r.t the weight function $1/\sqrt{1-x... |
Chebyshev polynomials orthogonal with respect to different weight function? |
I know this question sounds like a joke. And it probably is:).
I found it kind of annoying, but also interesting, to call $-\Delta=-\sum_{j=1}^n\partial^2_{jj}$ "the positive Laplacian" as it is the positive operator with respect to standard $L^2$-pairing. But in the mean time it is also regarded as the "negative of... |
**Question**
>The work done by the force ${\vec{F}}=(x^{2}-y^{2})\hat{i}+(x+y)\hat{j}$ in moving a particle along the closed path $C$ containing the curves $x+y=0,x^{2}+y^{2}=16$ and $y = x$ in the first and fourth quadrant is?
___________
The path given is [![this the path on which the particle has moved][1]][1]
... |
A disjoint union of open balls is of course disconnected. [Here](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/spaces-which-cannot-be-written-as-a-countable-disjoint-union-of-closed-subsets/D24FAD263DF49239740F1A548826CA1B) it is proved that a locally compact, connected, Hausdorff space... |
What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order.
Examples:
1. Quadratic Diophantine equations are always decidable, but cubic ones are not.
2. Time independent second order ODEs cannot admit chaotic solutions, but ... |
I am trying to read G.E. Sacks's book on higher recursion theory and he refers to the notation $\{ e \}^{f}$ as the $e$th element in an enumeration of partial recursive functions in $f$, where $f$ is a total function.
Am I right to interpret that $f$ is actually an oracle ? (as commonly referred to in other recursi... |
Is my interpretation of this notation correct ? (G.E. Sacks book) |
Let $(R, m_R, \kappa_R) $ a discrete valuation ring with quotient field $K$, and $K \subset L$ a finite field extension. Let $ B$ be the integral closure of $R$ in $L$, which is in general not a dvr, let denote by $\frak{P}_1, \frak{P}_2, ..., \frak{P}_m$ it's maximal primes lying over $m_R$. Then we obtain ramificat... |
To preface, I am currently following eigenchris' series on tensor calculus in an effort to learn GR. I believe I understand the sense in which differential operators representing derivatives are vectors, but I may be wrong here too.
The way eigenchris explains it, $\frac{\partial \vec R}{\partial x_i}=\vec e_{x_i}$ ... |
In what sense are differentials from calculus covectors in differential geometry? Why would there be a covector present in integration? |
I'm wondering how can i make sense of a partial solution to a two-system of polynomial equations.
Let $\{f,g\}\subseteq K[x,y]$ be polynomials in a polynomial ring over the field $K$. Given the equations:
$$f(x,y)=a$$
$$g(x,y)=b$$
Every solution is contained in the affine variety $\mathcal{V}(I_0)$ of the ideal
$... |
Consider two orthonormal sets $\mathcal{B}$ and $\mathcal{B}'$ of the same size $|\mathcal{B}|=|\mathcal{B}'|=n$. Is it possible to label the vectors in $\mathcal{B}=\{1,...,n\}$ and $\mathcal{B}'=\{1',...,n'\}$ so that each pair has nonzero overlap, i.e., $\langle i|i'\rangle \ne 0$ for all $i$? |
I have a system of equations.
$$
\begin{aligned}
(1 - p_0)u_0 + (q_0 - 1)v_0 + u_1 - v_1 = r_0 - c t_0 \\\\
(1 - p_1)u_1 + (q_1 - 1)v_1 + u_0 - v_0 = r_1 - c t_1 \\\\
(1 - p_2)u_2 + (q_2 - 1)v_2 + u_3 - v_3 + u_4 - v_4 = r_2 - c t_2 \\\\
(1 - p_3)u_3 + (q_3 - 1)v_3 + u_2 - v_2 + u_4 - v_4 = r_3 - c t_3 \\\\
(1... |
Can a (bounded) linear least-squares problem include a scale factor in its solution? |
This is from the exercise problems(not the live contest) for regional math competition.
Let triangle $ABC$ satisfy $\angle C > 90^\circ$ and let its incenter be $I$. Let $\ell$ be the line parallel to $BC$ passing through $I$, and call $\ell \cap AB = D$, $\ell \cap BC = E$. We know that $AI = 3$ and the radius of ... |
This integral: $\int\left( \frac{x^2}{1+x^2} \right)\frac{1+x\tan(x)+ \tan(x)-x}{1 +x\tan(x)-\tan(x)+x}dx$ seems to be complicated, but it can be easily solved if one know the formula: $$\tan(a+b+c)= \frac{\tan(a)+ \tan(b)+ \tan(c ) - \tan(a) \tan(b)\tan(c)}{1-\tan(a)\tan(b)-\tan(a)\tan(c)-\tan(b)\tan(c)}$$. so $$\int\... |
The product of elements in the main diagonal is $100^7$. Estimate the sum of all other summands in the determinant: notice, that every other permutation contains $100$ at most five times. How many permutations are there all together? It is $7!=5040$. But $100^7>100^5*5040$, which means that all the other permutations c... |
I have a system of equations.
$$
\small
\begin{aligned}
(1 - p_0)u_0 + (q_0 - 1)v_0 + u_1 - v_1 = r_0 - c t_0 \\\\
(1 - p_1)u_1 + (q_1 - 1)v_1 + u_0 - v_0 = r_1 - c t_1 \\\\
(1 - p_2)u_2 + (q_2 - 1)v_2 + u_3 - v_3 + u_4 - v_4 = r_2 - c t_2 \\\\
(1 - p_3)u_3 + (q_3 - 1)v_3 + u_2 - v_2 + u_4 - v_4 = r_3 - c t_3 ... |
I have a system of equations.
$$
\small
\begin{aligned}
(1 - p_0)u_0 + (q_0 - 1)v_0 + u_1 - v_1 = r_0 - c t_0 \\\\
(1 - p_1)u_1 + (q_1 - 1)v_1 + u_0 - v_0 = r_1 - c t_1 \\\\
(1 - p_2)u_2 + (q_2 - 1)v_2 + u_3 - v_3 + u_4 - v_4 = r_2 - c t_2 \\\\
(1 - p_3)u_3 + (q_3 - 1)v_3 + u_2 - v_2 + u_4 - v_4 = r_3 - c t_3 ... |
Let $A = \{a+bi \in \mathbb{C} : a > 4\}$ and $B = \{ a+bi \in \mathbb{C} : a \geq 4\}$ and let $d(z,w)$ be the metric on $\mathbb{C}$ defined as,
$$d(z,w) = \begin{cases} 0 &, \mathrm{if}\, z =w\\
|z| +|w| &, \mathrm{if}\, z\neq w \end{cases}$$
My questions are:
1. Are $A$ and $B$ open in $(\mathbb{C},d_E)$... |
The product of elements in the main diagonal is $100^7$. Estimate the sum of all other summands in the determinant: notice, that every other permutation contains $100$ at most five times. How many permutations are there all together? It is $7!=5040$. But $100^7>100^5\cdot 5040$, which means that all the other permutati... |
Let $X$ and $Y$ be two continuous positive random varianbles, that are not independent.
We know that $ E(X)^{-1} \neq E(1/X)$ unless we look at trivial cases.
Is there some way to simplify
$$ E\left\{ \frac{1}{E \left(Y|X \right)}
\right\} = \int \left\{ \int y f_{Y|X} \left(x \right) \right\}^{-1} dx ?$$... |
I'm wondering how can i make sense of a partial solution to a two-system of polynomial equations.
Let $\{f,g\}\subseteq K[x,y]$ be polynomials in a polynomial ring over the field $K$. Given the equations:
$$f(x,y)=a$$
$$g(x,y)=b$$
Every solution is contained in the affine variety $\mathcal{V}(I_0)≠\emptyset$ of th... |
Let $X$ and $Y$ be two continuous positive random varianbles, that are not independent.
We know that $ E(X)^{-1} \neq E(1/X)$ unless we look at trivial cases.
Is there some way to simplify
$$ E\left\{ \frac{1}{E \left(Y|X \right)}
\right\} = \int \left\{ \int y f_{Y|X} \left(x \right) \right\}^{-1} f_{x}(... |
$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ \ \ x>0. $$
Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $(0,\infty)$ that has these properties
1. $f(x+1)=xf(x)$
2. $f(1)=1$
3. $f(x)$ is a continuous
4. $f(x)$ is log convex
My question is... |
Why do we want these properties to be in the extended factorial function? |
What is the range of following function f(x)=3|sinx|?
I tried multiplying by 3 with the range of sin(x) function which is [-1,1] which gave me the outcome [-3,3]. and I feel like it's not the correct range of it. how to get the range of it? |
How do you calculate the range of trigonometric function? |
Let $X$ and $Y$ be two continuous positive random varianbles, that are not independent. In addition, all of the stated expectations are bounded away from zero. We know that $ E(X)^{-1} \neq E(1/X)$ unless we look at trivial cases.
Is there some way to simplify
$$ E\left\{ \frac{1}{E \left(Y|X \right)}
\right\... |
I'm dealing with the question proving the Bowen metrics induce the same topology. More specifically, given $(X,T)$ be a topological dynamical systems equipped with a metric $d$ ($X$ is compact, $T$ is continuous). For each $n \ge 1$, define
$$d_n(x,y) = \max \{d(T^k(x),T^k(y)) : k = 0,1,\ldots,n-1 \}.$$
We call $d_n$... |
What is the range of following function $f(x) = 3|\sin x|$?
I tried multiplying by $3$ with the range of $\sin x$ function which is $[-1, 1]$ which gave me the outcome $[-3, 3]$, but I feel like it's not the correct range of it. How to get the range of it? |
Let $U(x)$ be a positive, strictly increasing, strictly convex $C^2$ function in $x$, is it generally true that $U^{-1}(U(x)+U(y))$ is a convex function in $x,y$ ? For $U(x)=e^x$, it is well known that the log-sum-exponential function is convex. The statement is also true for power functions $U(x)=x^{1+\beta}$. What ab... |
Is $U^{-1}(U(x)+U(y))$ a convex function in general? |
First time posting here so please forgive any lack of adherence to best practices.
sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result. In particular, sin(q) (in degrees) or sin(qπ) (in radians) will always be algebraic.
I can reverse-engineer a tr... |
Is my trigonometric proof of the Pythagorean theorem correct? |
First time posting here so please forgive any lack of adherence to best practices.
sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result. In particular, sin(q) (in degrees) or sin(qπ) (in radians) will always be algebraic.
I can reverse-engineer a tr... |
First time posting here so please forgive any lack of adherence to best practices.
sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result. In particular, sin(q) (in degrees) or sin(qπ) (in radians) will always be algebraic.
I can reverse-engineer a tr... |
**Let G be a simple n-vertex triangle free graph with (G) ≥ 3. Prove that if n ≤ 11, then G is 3-edge-connected. Show that such a graph could be not 2-(vertex)-connected.**
Hint: Using Mantel's Theorem can help.
Mantel's Theorem: Let f(n) be the maximum number of edges in a simple n-vertex graph with no triangles... |
First time posting here so please forgive any lack of adherence to best practices.
sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result. In particular, sin(q) (in degrees) or sin(qπ) (in radians) will always be algebraic.
I can reverse-engineer a tr... |
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