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Given a (smooth but not holomorphic) surface bundle over surface $S_g\to M\to S_h$ (where $g,h>1$, there is an induced surface group representation
$$\pi_1(S_h)\to \text{SP}(2g,\mathbb R).$$
I was told that there is an associated equivariant holomorphic map
$$\tilde{S_h}=\mathbb H^2\to \text{SP}(2g,\mathbb R)/U(g)... |
Given a (smooth but not holomorphic) surface bundle over surface $S_g\to X\to S_h$, then there are two associated vector bundles $H^1(S_g,\mathbb C)\to E\to S_h$ and $H^{1,0}(S_g,\mathbb C)\to E_1\to S_h$, s.t. the Toledo invariant of $E$ is $\deg{E_1}$.
I think I know why we have the vector bundle $H^{1,0}(S_g,\mat... |
sorry if this question seems basic but my professor asked me this and it kind of left me confused, so I'm asking this here for some clarity on the matter.
I have a simple question that boils down to the basic definitions of probablity.
We could define probablity of an event A, denoted as ```P(A)```, as the ratio of... |
I have a simple question that boils down to the basic definitions of probablity.
We could define probablity of an event A, denoted as ```P(A)```, as the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space.
Sample space is a collection or a set of possible outcomes of... |
With reference to the Series and Product Development chapter of Ahlfors' complex analysis text (Page 201-205), after deriving Stirling's formula $$\Gamma(z)=\sqrt{2\pi}z^{z-\frac{1}{2}}e^{-z}e^{J(z)},$$ where $$J(z)=\frac{1}{\pi}\int_{0}^{\infty}\frac{z}{\eta^2+z^2}\log\frac{1}{1-e^{-2\pi\eta}}d\eta,$$ it is noted that... |
> How to prove the Bessel modified equation $$x^2y''+xy'-(x^2+n^2)y=0$$
> has no oscillatory solution (preferably using comparison theorem)?
For $x>0$ we have one zero of any solution of $$x^2y''+xy'-n^2y=0$$ lies between any two consecutive zeroes of solutions of the former one. But the latter equation has solutio... |
There are a lot of hyperbolic trigonometric identities derived in Poincare disk model that resemble similar identities from the Euclidean geometry. For example, the analogue of the Pythagoras theorem is cosh(c) = cosh(a)cosh(b), the analogue of the sine law is sin(A)/sinh(a) = sin(B)/sinh(b)=sin(C)/sinh(c) and so on.<b... |
I'm studying the book *Matrix Differential Calculus with Applications in Statistics and Econometrics* by Magnus and Neudecker (1988). Within there (Chapter 5, Section 4), the authors gave the following definition of differentiability of vector-to-vector function:
**Definition 5.2:** Let $f:S\rightarrow \mathbb{R}^m$... |
I was studying **Fourier Analysis: An Introduction** by *Stein and Shakarchi*.
I came across this theorem in Section 2.2, page number 39
> **Theorem 2.1** Suppose that $f$ is an integrable function on the circle with $\hat f(n) = 0$ for all $n\in\mathbb{Z}$. Then $f(\theta_0) = 0$ whenever f is continuous at the ... |
Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the of $$\frac{d}{dz}E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$
My idea is to use the property $E(XY)=E(Y(E(X|Y))$ for two random variables $X$ and $Y$.Does this help me?
My question is, how to determine the... |
I have come across an integral that involves a hypergeometric function, which can be expressed as follows:
$$I = \int_0^1 x^{1/2}(1-x)^{\epsilon-1} {_{2}F_1}(\frac{1}{2}+\epsilon,1+\epsilon;\frac{3}{2};x) dx.$$
Here, $\epsilon$ is a small complex quantity where $|\epsilon|\ll1$.
I found an integral formula i... |
Recently in my research I stumbled upon this splitting of a periodic continued fraction. I wondered whether there is any closed form or literature on this topic.
# Visualizing continued fractions by rectangles
It is known that a continued fraction $a = [n_0, n_1, ...]$ can be visualized by filling in squares of a r... |
Recently in my research I stumbled upon this splitting of a periodic continued fraction. I wondered whether there is any closed expression or literature on this topic.
# Visualizing continued fractions by rectangles
It is known that a continued fraction $a = [n_0, n_1, ...]$ can be visualized by filling in squares ... |
Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.
Does there exist an example of the following?
* A commutative ring with unity $R$
* A mapping $\operatorname{d}: R\operatorname{-Mod} \to \overline{\mathbb{R}}_{\geq 0}$ with the following properties:
* If $A \cong B$, $\operatorname{... |
From [wikipedia](https://en.wikipedia.org/wiki/Radon_measure), Radon measure is defined as
>Radon measure is a measure on the $\sigma$-algebra of Borel sets of a Hausdorff topological space $X$ that is finite on all compact sets, outer regular on all **Borel sets**, and inner regular on **open sets**.
I am readi... |
Following this question:https://math.stackexchange.com/questions/145222/f-is-integrable-prove-fx-int-inftyx-ft-dt-is-uniformly-continuo
Our question is that let $f\in L_1(\mathbb{R})$. Prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.
---
I have the following Lemma:
> For every $\epsilon>0... |
Using local Lipschitz continuity of $f(\cdot)$:
\begin{align}
f(\mathbf{a}) &\leq f(\mathbf{b})+ L_0 \lVert{\mathbf{a}-\mathbf{b}}\rVert
\end{align}
In the FedProx Paper (https://arxiv.org/pdf/1812.06127.pdf, Equation 9 of the Theorem 4 proof (page no. 13), the authors have used a relation between local Lip... |
Let $M_{n\times n}$ be the vector space of real square matrices of size $n\times n$. Does there exist a matrix norm $\left\lVert\cdot\right\rVert_M$ such that if $\left\lVert A\right\rVert_M \leq \left\lVert B\right\rVert_M$, then for all $v\in\mathbb{R}^n$, $\left\lVert Av\right\rVert_2\leq \left\lVert Bv\right\rVert_... |
Let X and Y be independent random variables on $\mathbb{R}$ equally distributed with a probability measure which is singular with respect to the Lebesgue measure. What is P(X=Y)? |
Let $X$ and $Y$ be independent random variables on $\mathbb{R}$ equally distributed with a probability measure which is singular with respect to the Lebesgue measure. What is $P(X=Y)$? |
<br>
For context, my question stems from a claim of [Waldhausen](https://link.springer.com/chapter/10.1007/BFb0074449) (page 342) where from a map
$$|wS.\mathcal{A}|\wedge |wS.\mathcal{B}|\to |wwS.S.\mathcal{C}|.$$
He claims it induces a map
$$\Omega|wS.\mathcal{A}|\wedge \Omega|wS.\mathcal{B}|\to \Omega^2|wwS.S... |
## Problem
> Let $M\in\mathbb{M}^{loc}_C$, i.e. $M$ is an (a.s.) continuous local martingale with $M_0=0$. Show that $M$ is constant on an interval $[a,b]$ with $0\leq a < b$ if $\langle M\rangle$ is constant on $[a,b]$.
## My Attempt
Let $\left\langle M\right\rangle$ be constant on $[a,b]$, i.e. $\forall t\in... |
<br>
I call Segal subdivision the endofunctor of simplicial objects in a category $\mathcal{C}$ induced by the doubling endofunctor of $\Delta^{op}$ sending $x_0<\cdots<x_n$ to $x_0<\cdots <x_n<x_n'<\cdots<x_0'$. I am calling this Segal subdivision as in Weibel's K-book chapter IV exercise 3.10. Here is Segal's orig... |
I'm reading Munkres Topology and I'm stuck in lemma 68.5 as you can see he uses the theorem 68.4 in order to imply that there is a isomorphism between $G$ and $G'$, but in order for this theorem to be applied we must have that $\{i_{\alpha}(G_{\alpha})\}$ and $\{i'_{\alpha}(G_{\alpha})\}$ generate $G$ and $G'$ which fo... |
Given $A^TA$ = \begin{pmatrix}
3& 4& 0\\
4& 10& -3\\
0& -3&
3\end{pmatrix}
(a) Solve the homogeneous system $Ax = 0$. Explain how you derive your
answer.
(b) Suppose $(A^T A)^{−1}A^T$ = \begin{pmatrix}
\frac{3}{5}& \frac{-4}{5}& 0& \frac{3}{5}& \frac{1}{5}\\
\frac{-1}{5}& \frac{3}{5}& 0& ... |
$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s) $$
This has a double pole at $s=0$ , from $\zeta(s+1)$ and $\Gamma(s)$ respectively, and $\lim_{s\to0}s^2f(s) = -1/2$
Then, the next step, I have difficulty with understanding the following (from my book without any explanation)
$$ \lim_{s\to0} s(f(s) - \frac{-1/2}{s^2}) =... |
Following this question:https://math.stackexchange.com/questions/145222/f-is-integrable-prove-fx-int-inftyx-ft-dt-is-uniformly-continuo
Our question is that let $f\in L_1(\mathbb{R})$. Prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.
---
I have the following Lemma:
> If for every $\epsilo... |
There are a lot of hyperbolic trigonometric identities derived in Poincare disk model that resemble similar identities from the Euclidean geometry.
For example, the analogue of the Pythagoras theorem is $cosh(c) = cosh(a)cosh(b)$, the analogue of the sine law is $\frac{sin(A)}{sinh(a)}$ = $\frac{sin(B)}{sinh(b)}=\f... |
There are a lot of hyperbolic trigonometric identities derived in Poincare disk model that resemble similar identities from the Euclidean geometry.
For example, the analogue of the Pythagoras theorem is $\cosh(c) = \cosh(a)\cosh(b)$, the analogue of the sine law is $\frac{\sin(A)}{\sinh(a)}$ = $\frac{\sin(B)}{\sinh... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(r)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $S^2... |
$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s) $$
This has a double pole at $s=0$ , from $\zeta(s+1)$ and $\Gamma(s)$ respectively, and $\lim_{s\to0}s^2f(s) = -1/2$
Then, the next step, I have difficulty with understanding the following (from my book without any explanation)
$$ \lim_{s\to0} s\left(f(s) - \frac{-1/2}{s^... |
A [linearly-ordered topological space](https://topology.pi-base.org/properties/P000133/) or LOTS is one whose topology admits a basis generated by open intervals of a total ordering of its points.
A [well-based](https://topology.pi-base.org/properties/P000174/) space is one which admits a local basis of each point t... |
Are linearly ordered topological spaces well-based? |
I saw this Corollary on page 21 of [this book](https://u.cs.biu.ac.il/~plotkin/resources/MFAtiyah_IGMacDonald_IntroToCommAlgebra.pdf) *"Introduction to commutative algebra"* by Atiyah and Macdonald.
> Corollary2.5. Let $M$ be a finitely generated $A$-module and let $a$ be an ideal of $A$ such that $aM=M$. Then the... |
Is this Corollary a historical or modern product? |
I saw this Corollary on page 21 of [this book](https://u.cs.biu.ac.il/~plotkin/resources/MFAtiyah_IGMacDonald_IntroToCommAlgebra.pdf) *"Introduction to commutative algebra"* by Atiyah and Macdonald.
> Corollary2.5. Let $M$ be a finitely generated $A$-module and let $a$ be an ideal of $A$ such that $aM=M$. Then the... |
In my University for a project they gave us this question:
We denote by the number of vertices and by the number of edges of a graph. Let be a graph with ≥ 5 and ≥ 3 − 5. Let be a set of vertices of with || = 2, such that the graph that results after the removal of from (i.e., the − ) is non-connected. In ... |
Not asking for the distribution of $m$th order statistic, for it has been well documented and covered and in fact well-known. What I am rather confused, perhaps basic but failing to see, is the approach using the argument of symmetry in the book, _An Introduction to Order Statistics_:
Let $f_{m:n}$ be the pdf of the... |
Question：
A, B, and C students will participate in a badminton competition, and the agreed format is as follows:
Those who have accumulated two losses are eliminated;
Draw lots before the competition to determine the first two contestants, with the other person taking turns;
The winner of each game will pla... |
Let's start with a definition of an ellipse that states: An ellipse is a set of points where each point's sum of distances from two focal points is equal to a constant value.
Now, we have got a two focal points and a given constant that creates an ellipse. Then we make those focal points further and further away. In... |
The little-known [Pappus's area theorem](http://users.math.uoc.gr/~pamfilos/eGallery/problems/Pappus.html) in Euclidean plane geometry can “be thought of as a generalization of the Pythagorean theorem”. However, the Problem 34 in Nicholas Donat Kazarinoff's _Geometric Inequalities_ stated that
this Pappus' Theorem ma... |
Is there any rigorous proof of the generalized Pappus's area theorem in ℝ³? |
I find it difficult to demonstrate a certain proposition about how Seshadri’s constants vary "as a family". This is **Example 5.1.11** from Lazarfeld's book "*Positivity in Algebraic Geometry I*", page 272 :
> Let $X$ be an irreducible projective variety, $T$ an other variety, $L$ a divisor on $X$ and $p : X \to T$ ... |
A vector function $F$ is monotonic in $S_i$ if
$$
(F(x)-F(y))^T (x-y)\geq 0, \forall x,y\in S_i.
$$
However, I am unsure if this property holds for continuous functions when considering the union of the sets, i.e., if a continuous vector function is monotonic in $S_1, S_2, \dots, S_n$ respectively, is it monotoni... |
Is monotonicity hold of a continuous vector function across union of sets? |
>In a survey of 480 students at a university. The following information was collected. Of those surveyed, 200 students liked the subjects that involved mathematics, 260 students liked the subjects that involved programming,170 students disliked subjects involving physics, and 250 of them liked subjects involving two ... |
I have a physics background with very little knowledge of graph theory, but a question arose related to directed acyclic graphs (DAGs) and I'm hoping to get hints towards an answer.
Assume we have a system where:
1. The interactions between components are described by a DAG.
2. The inputs to the system are random ... |
Can a directed acyclic graph exhibit chaotic behavior? |
Let $(\Omega,\mathcal{F},p)$ be a probability space, and let $X_n$ be a sequence of random variables.
Suppose that for a random variable $X$ we have that for all events $F\in\mathcal{F}$,
$$
\int_F X_n dp = \int_F X dp .
$$
Can we conclude that $X_n\to X$ almost surely? |
>A triangle is circumscribed to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and two of its vertices lie on the directrices, such that one lies on each directrix. Then, the locus of the third vertex of the triangle is? <br>
I tried using an affine transform to simplify the situation and began working with $x^2+y^... |
Let $(\Omega,\mathcal{F},p)$ be a probability space, and let $X_n$ be a sequence of random variables.
Suppose that for a random variable $X$ we have that for all events $F\in\mathcal{F}$,
$$
\lim_{n\to\infty}\int_F X_n \, dp = \int_F X \, dp .
$$
Can we conclude that $X_n\to X$ almost surely? |
After working this out and writing it down, I found that it was already done in 1969: [Eric Langford, The probability that a random triangle is obtuse, Biometrika (1969) 56 (3): 689-690][1].
For this kind of problem, it's important to make good use of the symmetries and choose convenient coordinates. A significant ... |
I am considering $2^n\times 2^n$ complex matrices and a general anti-involution $A^{**}=A$, $(AB)^*=B^* A^*$. To my knowledge all anti-involutions are a combination of transpose, field involution and similarity transform (is that correct?).
What are the possible dimensions of the space created by all matrices which ... |
What should $n$-transfors be? |
I was studying some measure theory and upon searching for additional exercises on the internet I came upon one that said: given $H$ a Borel set subset of the real numbers, prove that if $\lambda(H)=0$ (its Lebesgue measure) then there exist a number $\alpha$ such that $\alpha + H$ , the set of all the reals of the form... |
Let $B$ be a Banach space. Let $x\in B.$ Then by Hahn-Banach there exists $f\in B^\ast$ such that $\|f\| =\|x\|$ and $f(x) = \|x\|.$
Question: can we strengthen this result so that now, for each $\epsilon>0$ we can find $f$ satisfying the following inequality: for all $y\in B$ such that $\|y\|=\|x\|,$ whenever $f(y)... |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denote the space of hamiltonian symplectomorphisms of $(M,\omega)$
I have the following statement in my lecture notes:
**Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
I was studying some measure theory and upon searching for additional exercises on the internet I came upon one that said:
> Given $H$ a Borel set subset of the real numbers, prove that if
> $\lambda(H)=0$ (its Lebesgue measure) then there exists a number
> $\alpha$ such that $\alpha + H$ , the set of all the re... |
I was studying some measure theory and upon searching for additional exercises on the internet I came upon one that said:
> Given $H$ a Borel set subset of the real numbers, prove that if
> $\lambda(H)=0$ (its Lebesgue measure) then there exists a number
> $\alpha$ such that $\alpha + H$ , the set of all the re... |
A is square matrix that applies |A + I| = |A − 3I| = 0 and also rank(A)= 2
I need to find characteristic polynomial of A and to prove that A is diagonal.
How to start with this question and is there any tips I can get to this type of questions?
Thank you :) |
Introduction:
Find a such family of polynomial then we can always use Newton's method provided we have a procedure to start.
We introduce a rational function:
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots x}}}}\right)-C=0... |
$A$ is square matrix that applies $|A + I| = |A − 3I| = 0$ and also $\text{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is diagonal.
How to start with this question and is there any tips I can get to this type of questions? |
How to find the characteristic polynomial of a square matrix and how to proof if it's diagonal? |
Let $A,B,C$ be objects in a category $\mathrm{C}$ and we have the pushout diagram of monomorphisms$\require{AMScd}$
\begin{CD}
C @>>> A\\
@VVV @VVV\\
B @>>> A\bigsqcup\limits_C B\end{CD} Now consider another object $X$ in $\mathrm{C}$ such that the below diagram is also a commutative square of monomorphisms. \begin... |
$A$ is square matrix that applies $|A + I| = |A − 3I| = 0$ and also $\text{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is diagonal.
I thought of using the characteristic polynomial equation |λI - A| = 0 , and for prove that $A$ is diagonal using the $P$^-1$A$$P$ and proving ... |
$A$ is square matrix that applies $|A + I| = |A − 3I| = 0$ and also $\text{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is diagonalizable.
I thought of using the characteristic polynomial equation |λI - A| = 0 , and for prove that $A$ is diagonal using the $P$^-1$A$$P$ and pr... |
**Q.** Let $R$ be a normal domain with finite extension of fields $K' \subseteq K'' \subseteq K(R) $ ( $K(R)$ is the quotient field ), possibly simple extension. Consider $R':=K'\cap R$ and $R'':=K''\cap R$.
Then, 1) $R' \hookrightarrow R''$ is an integral extension?
2) The field extension $K(R'')/ K(R'... |
Let $C_n$ denote the cyclic group of order $n$ and let $R=\mathbb{Z}[C_n]$ be its group ring.
Any automorphism on $C_n$ defines a ring automorphism of $R$.
Are these all ring automorphisms of $R$?
Equivalently asked, for a given ring automorphism $\phi:R\to R$ is it true that $\phi(C_n)\subset C_n$? |
$A$ is square matrix of order n , that applies $|A + I| = |A − 3I| = 0$ and also $\text{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is diagonalizable.
I thought of using the characteristic polynomial equation |λI - A| = 0 , and for prove that $A$ is diagonal using the $P$^-1... |
$A$ is square matrix of order n , that applies $|A + I| = |A − 3I| = 0$ and also $\text{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is diagonalizable.
I thought of using the characteristic polynomial equation |λI - A| = 0 , and for prove that $A$ is diagonalizable using the ... |
Let a set of $N$ pairs of matched measured points $p_i, c_i$ with some noise. Using an optimization method, we optimize a model $m$ that estimates $\hat{c_i}$ from $p_i$:
$$\hat{c_i} = m(p_i)$$
Usually, the real-mean-square-error (rmse), is of an $O(n)$ computational complexity. However, due to the nature of my s... |
Let $f$ be a differentiable function in $\mathbb{R}$ such that $f'(x) = \alpha f(x)$ for all $x \in \mathbb{R}$, where $\alpha$ is a constant. Prove that $f(x) = f(0)e^{\alpha x}$ for all $x \in \mathbb{R}$.
My attempt:
Consider the map $g:\mathbb{R} \to \mathbb{R}$ defined by $g(x) = f(x)e^{-\alpha x}$. Then $g... |
$A$ is square matrix of order n , that applies $|A + I| = |A − 3I| = 0$ and also $\text{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is diagonalizable.
I thought of using the characteristic polynomial equation |λI - A| = 0 , and for prove that $A$ is diagonalizable using the ... |
Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power?
While I playing my calculator, I noticed the following:
$(\lfloor\sqrt{1!+2!+3!+4!+5!}\rfloor+1)^{2} - (1!+2!+3!+4!+5!)=16=2^{4}$
$(\lfloor\sqrt{1!+2!+3!+\cdots+6!}\rfloor+1... |
Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power? |
I encounter some difficulties to understand some notations in the proof of **Theorem 1.2.17** of the book "*Positivity in Algebraic Geometry I*" by Robert Lazarsfeld. To make things clear, a scheme is a seperated algebraic scheme of finite type over $\mathbb C$.
> **Theorem 1.2.17. (Amplitude in families) :** Let $f... |
Let $A,B,C$ be objects in a category $\mathrm{C}$ and we have the pushout diagram of monomorphisms$\require{AMScd}$
\begin{CD}
C @>>> A\\
@VVV @VVV\\
B @>>> A\bigsqcup\limits_C B\end{CD} Now consider another object $X$ in $\mathrm{C}$ such that the below diagram is a pullback square of monomorphisms. \begin{CD}
C ... |
Does the definition of $R^3$ requires orthonormal basis?
If I have a vector space V that consists of 3 linear independent
basis {$v_1,v_2,v_3$} which are not orthonormal, can I
still call it $R^3$?
It seemed people generall use $R^3$ to mean orthonormal basis,
What's the definition of $R^3$(or even, $R^n$)? do... |
Does the definition of $R^3$ requires orthonormal basis? |
Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power?
While I playing on my calculator, I observed the following:
$(\lfloor\sqrt{1!+2!+3!+4!+5!}\rfloor+1)^{2} - (1!+2!+3!+4!+5!)=16=2^{4}$
$(\lfloor\sqrt{1!+2!+3!+\cdots+6!}\rflo... |
I am trying to understand how a joint distribution is formed when two regular conditional distributions are involved that are conditional with respect to different random variables.
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let there be the three random variables $X:(\Omega, \mathcal{A})\ri... |
I am working on visualising a polygon in a hyperbolic space using a `HyperbolicRegularPolygon` class in SageMath, which constructs regular polygons in the hyperbolic plane. The construction algorithm involves several steps:
### Algorithm: [URL](https://github.com/sagemath/sage/blob/develop/src/sage/plot/hyperbolic_r... |
I have begun reading about quaternions after a long time. However I should answer this soon to other members of my work group.
My understanding of quaternions is composed of four values w,x,y and z and that they can be used to represent rotations.
I have been told one condition of the quaternions is they have to ... |
quaternions with negative w? |
A common assumptions in probability theory is that the probability space is complete, i.e. if $\left(\Omega, \mathcal{A}, \mathbb{P}\right)$ is a probability space and $A \in \mathcal{A}$ satisfies $\mathbb{P}\left(A \right) = 0$ and $B \subset A$, then $B \in \mathcal{A}$. Another common assumption is that the $\left... |
Complete, standard Borel spaces? (Complete Borel isomorphic probability spaces) |
Does the definition of $R^3$ requires orthonormal basis?
If I have a vector space V that consists of 3 linear independent
basis {$v_1,v_2,v_3$} which are not orthonormal, can I
still call it $R^3$?
It seemed people generally use $R^3$ to mean orthonormal basis,
What's the definition of $R^3$(or even, $R^n$)? d... |
Let $B$ be a Banach space. Let $x\in B.$ Then by Hahn-Banach there exists $f\in B^\ast$ such that $\|f\| =1$ and $f(x) = \|x\|.$
Question: can we strengthen this result so that now, for each $\epsilon>0$ we can find $f$ satisfying the following inequality: for all $y\in B$ such that $\|y\|=\|x\|,$ whenever $f(y)=\|x... |
I have begun reading about quaternions after a long time. However I should answer this soon to other members of my work group.
My understanding of quaternions is composed of four values w,x,y and z and that they can be used to represent rotations.
I have received data that supposedly represent quaternions
What... |
I know that a vector space equipped with a seminorm can be transformed into a normed space by taking the equivalence classes of the equivalence relation $f \sim g$ iff $\|f-g\| = 0$.
Can a similar procedure be applied to make a vector space equipped with a positive [sesquilinear form][1] into an inner product space... |
Transforming a sesquilinear form into an inner product? |
Let $P_{m}^{\left(a,b\right)}\left(s\right)$ the $m$-th [Jacobi polinomial][1]. An old result of Bateman shows that $$
\left(1-x\right)^{\alpha+\mu}\frac{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(x\right)}{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(1\right)}=\frac{\Gamma\left(\alpha+\mu+1\right)}{\Gamma\left(... |
Let $\mathbb{Q}_{\mathrm{Br}}$ be the field of bring-jerrard rational numbers, this is, the field defined by the infinite algebraic extension $\mathbb{Q}_{\mathrm{Br}}=\mathbb{Q}(S)$, were
$$S=\{\tau\in\mathbb{C}|\tau^5+p\tau+q=0,\{p,q\}\subseteq\mathbb{Q}\}$$
That $\mathbb{Q}_{\mathrm{Br}}$ is a field comes from the... |
What is the least degree polynomial whose Galois group is not solvable in the field of Bring rational numbers? |
Is this inequality true?
$\int_{0}^{1} \int_{a}^{x} f^2(t) \, dt \, dx \leq \int_{0}^{1} f^2(t) \, dt$
I've seen something like this used in a solution, where instead of $f^2$ I had $f^{2p-2}+(f')^2$ and I didn't understand how they got there. If it helps, all i know about f is that it reaches 0 in at least 1 poin... |
Double integral smaller than single integral? |
The following question is from a worksheet on polynomials.
The polynomial $x^3 + cx + d=0$ has a solution when $x=d$. Prove that $d$ is the only solution provided $-2 < d < 2 $, $d ≠ 0$
My initial approach to this was through long substitution by factor $(x-d)$, solve for the remainder which would equal zero and ... |
How can I show that the function $h(x):= \frac{\log(x)}{1-x}$ is increasing on $\mathbb{R}_{>0}$? The derivative of $h$ is $h'(x)= \frac{x\log(x)-x+1}{x(1-x)^2}$ but I don't see why $h'(x)$ is always non-negative. |
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