instruction stringlengths 12 30k |
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**Direct problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we can find the P... |
For most fields $K$ and integers $n$, the general linear group $\mathrm{GL}_n(K)$ has very few normal subgroups - they are precisely the subgroups that either contain the commutator subgroup or are included in the center. The only exceptions to this occur when $K$ either has two or three elements, both with $n=2$. In t... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 15.19 ( or Lemma 14.18 in the second edition ) and stuck at some statement. I've searched for math stack exchnage, but there doesn't seem to be any related questions, so I'm asking this.
**Q.1** Let $X$ be a reduced scheme with finitely many irreducible ... |
Suppose symmetric $A$ and $u$ are given, can we exactly identify the smallest eigenvalue of $A+uu^T$ and its corresponding eigenvector (based on $u$ and given eigenpairs of $A$)? It is okay to add any additional meaningful conditions on $A$ in this question. |
I am given a seperable Banach space and an interval $J$. Pettis' theorem tells that for such a space $Y$, we have: $f : J \rightarrow Y$ is measurable iff for any bounded linear functional $g \in Y^*$, the real function $J \ni t \mapsto g(f(t))$ is measurable.
Now I am supposed to use this theorem to show that any f... |
What does $sin^{-1}x+cos^{-1}y+cos^{-1}xy=π/2$ represent? |
I've been reading The ["Two-State Implicit Filter – Recursive Estimation for Mobile Robots" by Bloesch][1]. The gist of the paper is it tries to find the balance between a Kalman Filter and a Batch optimizer for robotics applications, and introduces a Two State Implicit Filter. Essentially a sliding window estimator of... |
Conjecture :
----------
$$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$
Where $a_n$ is an integer such that :
$$0\leq a_n\leq n$$
>>Some arguments :
I think the best we can say is until we have the (says 20 because of floating point ) first ter... |
$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series . |
In an exact category $(\mathbf{C}, \mathcal{E})$, we define an object $P \in \mathbf{C}$ to be projective if for every admissable epimorphism $g : A \rightarrow B$
$$
\mathbf{C}(P,A) \rightarrow \mathbf{C}(P,B)
$$
is a surjective map of abelian groups. Let $\mathcal{P}$ be the class of projective objects in $(\... |
Does the projective modules in exact category determine the admissable epimorphisms? |
Does the projective modules in an exact category determine the admissable epimorphisms? |
I am having trouble with this exercise in one of my text books
Let $\mathbf{x, y}\in\mathbb{R}^4$, and let $L$ be a set such that
$L = \{\alpha \mathbf{x} +\beta\mathbf{y}: \alpha, \beta\in \mathbb{R}, \alpha+\beta=1\}$
Show that $L$ is an affine subspace of $\mathbb{R}^4$
I am quite stuck on this exercise. I k... |
\begin{equation}
u''+u=\delta(x)
\end{equation}
Using the Fourier integrals we obtain:
$$\mathscr{F}\{u''\}= -(\omega)^2\hat{u}$$
$$\mathscr{F}\{u\}= \hat{u}$$
$$\mathscr{F}\{\delta\}=\frac{1}{\sqrt{2\pi}}$$
We get then:
\begin{equation}
-(\omega)^2\hat{u}+\hat{u}=\frac{1}{\sqrt{2\pi}}
\en... |
It is easy to do with specially induced matrix norms ,which are induced from vector norms.When a matrix norm is induced by a vector norm,what we need to do is to notice the consistence of the matrix norm and the vector norm.
But what's about the others?**What's about the matrix norms which can not be induced by vect... |
Is it true that the spectral radius is always less or equal to any matrix norm? |
In the first answer to the question https://math.stackexchange.com/questions/3608272/prove-that-the-line-xy-goes-through-a-fixed-point-where-x-y-are-on-fixed-con he wrote
>It's well known that there are homographies that take $C$ to a circle.
Let's assume that we can find a specific homography $T$ that maps angles at... |
How to solve this integral over the specific curve and region? |
The definition of a plane in R³ is intuitive. Ax + By + Cz = D, where (A, B, C) is a normal vector to the plane and D is some bias. I can visualize it dividing the space into two halves(because I'm a 3-dimensional creature and it's quite intuitive for me), but how does the generalization of this equation to higher dime... |
How does hyperplane work? |
**Direct problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we can find the P... |
[This answer][1] wrote
>It's well known that there are homographies that take $C$ to a circle.
Let's assume that we can find a specific homography $T$ that maps angles at $P$ to equal angles at $P'$. I.e. $C'$ is a circle, and for all points $A,B$
$$∠APB=∠A'P'B'.$$
so I try to find all projective transformations ... |
The definition of a plane in $R^3$ is intuitive : $Ax + By + Cz = D$ , where $(A, B, C)$ is a normal vector to the plane and $D$ is some bias.
I can visualize it dividing the space into two halves (because I'm a 3-dimensional creature and it's quite intuitive for me) , but how does the generalization of this equatio... |
Recently, My Professor taught us about group extension. It is the following: A group $G$ is an extension of $Q$ by $N$ if we have the following short exact sequence:
$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$.
After learning this definition I observed that for $n \geq 3$, the symmetric group on $n$... |
[This answer][1] wrote
>It's well known that there are homographies that take $C$ to a circle.
Let's assume that we can find a specific homography $T$ that maps angles at $P$ to equal angles at $P'$. I.e. $C'$ is a circle, and for all points $A,B$
$$∠APB=∠A'P'B'.$$
A construction of the projective transformation ... |
To calculate powers modulo n you can use [Exponentiation by squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring) together with [Modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic); the Python built-in function pow does that (not math.pow, which accepts only 2 parameters).
pow (x, y, ... |
I am just beginning my study for field theory and I am trying to solve a problem:
>Let $p$ be a prime and $\zeta$ be the primitive $p$th root of unity. Then calculate $[\mathbb{Q}(\sqrt[p]{2},\zeta) : \mathbb{Q}]$.
Here is my attempt:
Since we have $\mathbb{Q} \subseteq \mathbb{Q}(\sqrt[p]{2}) \subseteq \mathb... |
Assume $f:[0,1]\to [0,1]$ is an orientation-preserving homeomorphism, and $f$ is absolutely continuous on $[0,1]$, $\log f'\in H^{1/2}([0,1])$, that is to say,
$$\int_{0}^{1}\int_{0}^{1}\frac{|\log f'(x)-\log f'(y)|^2}{|x-y|^2}dx dy<\infty,$$ my question is if there exists $M>0$ independent of $u,v$ such that
$$\f... |
[This answer][1] wrote
>It's well known that there are homographies that take $C$ to a circle.
Let's assume that we can find a specific homography $T$ that maps angles at $P$ to equal angles at $P'$. I.e. $C'$ is a circle, and for all points $A,B$
$$∠APB=∠A'P'B'.$$
A construction of the projective transformation ... |
In the process of dealing with a certain mathematical question I have found myself dealing with pairs of (square) matrices $(A,B)$ such that $\ker(A) = \operatorname{im(B)}$ and $\ker(B) = \operatorname{im}(A)$. I am curious if there is any established theory on such classes of matrices; and in particular if given a ma... |
I am reading Serre's book *Complex Semisimple Lie Algebras*, and I have a problem about how to do a computation involving matrix exponential.
Let $V$ be a representation of $\mathfrak{sl}_2$. Let $X = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$ and $H = \begin{b... |
We know that $\begin{bmatrix} B_T \\ B_t \end{bmatrix} $ is a normal Gaussian vector and $\text{Cov}(B_T, B_t) = \min\{T,t\} = t,$ i.e,
$$
\begin{bmatrix} B_T \\ B_t \end{bmatrix} \sim \mathcal{N}(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} T & t \\ t & t \end{bmatrix}).
$$
When deriving that $E[B_t... |
[This answer][2] wrote
>It's well known that there are homographies that take $C$ to a circle.
Let's assume that we can find a specific homography $T$ that maps angles at $P$ to equal angles at $P'$. I.e. $C'$ is a circle, and for all points $A,B$
$$∠APB=∠A'P'B'.$$
A construction of the projective transformation ... |
About a month ago, I asked a [similar question][1] about a certain class of polynomials that seem to defy the odds of "eventual divisibility", i.e. iterating an "eventually divisible" polynomial we can guarantee that any input will yield a number that is divisible by a certain prime $p$ at least once (and thus infinite... |
[This answer][2] wrote
>It's well known that there are homographies that take $C$ to a circle.
Let's assume that we can find a specific homography $T$ that maps angles at $P$ to equal angles at $P'$. I.e. $C'$ is a circle, and for all points $A,B$
$$∠APB=∠A'P'B'.$$
A construction of the projective transformation ... |
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $\overline{\mathbb{Q}}_p$ a fixed algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $R$ be a complete Noetherian local $\mathbb{Z}_p$-algebra. It is not hard to see that if $R$ is a finite $\mathbb{Z}_p$-algebra, then the set $\text{Hom}_{\mat... |
A complete local ring is finite over $\mathbb{Z}_p$ if and only if it has finitely many $\overline{\mathbb{Q}}_p$-valued points? |
$$
{\displaystyle dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}}
$$
I was reading on wikipedia about geometric brownian motion (https://en.wikipedia.org/wiki/Geometric_Brownian_motion) and this question popped into my mind. To be clear, I understand the interpretation of the above equation, but is it technically correct... |
In calculus, if dy/dx is not in fact a fraction, is the equation below for geometric brownian motion technically incorrect? |
I have an intriguing question. There's a proposition or exercise stating that if $X$ is a simply connected space, then the map $f: S^1 \longrightarrow X$ can be continuously extended to a disk $D^2$.
On the other hand, we know that if $f$ is null-homotopic, it can also be extended to $D^2$. Utilizing this insight,... |
Q) **How to Integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$ ?**
First of all let me tell what I think about this question.
In my Coaching Institute, the chapter **'Integration'** is over. This question came in my mind while I was solving the questions of **'Integration By Partial Fraction Decomposi... |
How to integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$? |
By saying "only", I mean that no words are involved except "commutative". Just draw a commutative diagram, and make some arrows dotted lines to mean "the other arrows already exist, and this is what the theorem demonstrates to exist".
To prove a theorem like this, we first draw all the conditions as solid arrows, a... |
Is it possible to express and prove a theorem only with commutative diagrams? |
On page 22 of Multidimensional Stochastic Processes as Rough Paths by Friz and Victoir, they state that a map $\omega \colon \Delta_T \to [0,\infty)$, where $\Delta_T = \{(s,t) : 0 \leq s \leq t \leq T\}$, is defined as superadditive if for all $s \leq t \leq u$ in $[0,T]$, $\omega(s,t) + \omega(t,u) \leq \omega(s,u)$.... |
Why is $(s,t) \mapsto \lvert t-s \rvert$ a control? |
I was [reading][1] the other day (Chapter 3 Introduction), that
> sequential calculus is also called weak second order monadic logic with one successor or WS1S.
I understand the difference between first-order and second-order logic and also the one successor parts. What is a "weak" MSO? SO, what is it weaker tha... |
Is it *problematic* to allow for expressions such as
$\forall x\exists x (x < x)$
to count as well formed sentences? No (as others have explained). You can set up your formal first-order logical languages that way, while keeping everything working systematically in an acceptable way.
Is is *necessary* to allow fo... |
To calculate powers modulo n you can use [Exponentiation by squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring) together with [Modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic); the Python built-in function pow does that (not math.pow, which accepts only 2 parameters).
pow (x, y, ... |
Let $f:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ be a proper convex lower semicontinuous function. Suppose that $f$ is bounded by below, and for simplicity that $\inf f = 0$. Set $\varphi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ as $\varphi(x):=\sqrt{f(x)}$. It is known that
\begin{align*}
\partial\varphi(x) = \frac{1... |
Is there a name for this property of a binary operation? |
I encountered the following problem but I have no idea how to proceed.. Basically, I have a class of periodic function satisfying certain properties, and since I do not have explicit form of them I use their Fourier expansions, and hope to arrive the minimum value of $g(x)+g(\alpha-x)$ over interval $x\in(0,\alpha]$, w... |
In the book I am currently reading, it says
> Gradient-based methods are useful for one global optimum and no
> additional local optima: (quasi-)concave for maximums, (quasi-)convex for
> minimums.
It is clear to me why concave and convex functions work, as it implies that there is only one maximum/minimum and ... |
Gradient-based methods on quasi-convex problems? |
What does it mean $\frac{n-1}{2}K_{2}$?
In general which graph am I referring to if I write $l\times K_{2}$? |
Let $X$ be a topological space. Let's say $X$ is Hausdorff, compact. Let $A \subset X$ be a subspace of $X$, let's say $A$ is open (though I'm not sure it's relevant).
The topological cone $CX$ of $X$ is defined as the quotient of $(X \times [0,1])/ (x,1)\sim(x',1)$. We define $CA$ similarly. In fact, the cone oper... |
When does the (topological) cone $CA$ of an (open) subspace $A \subset X$ embed into the cone $CX$? |
The definition of the outer measure of a set $A\subseteq\mathbb{R}$ is as follows:
$$
|A| = \inf \left\{ \Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k): I_1, I_2,\dots\text{ are open intervals such that }A\subseteq\bigcup_{k=1}^{\infty} I_k \right\}.
$$
My quick observation is that if two open intervals intersect, we ... |
Let $(\Omega,M,P)$ be a probability space, $ X_{1} $ and $ X_2 $ be two random variavles, is it ture that $\sigma(X_1+X_2) \subseteq \sigma (X_1,X_2)$ ? How can we prove this?
Someone told me that this is a conclusion of Doob-dynkin's theorem.
**Doob-Dynkin's Theorem**: Lex $X=(X_1, X_2,...,X_n)$ be a $R^n$ rand... |
Is $\sigma(X+Y) \subseteq \sigma (X,Y)$? |
A question I found and that I could not answer so far. Assuming that A is a nxn Matrix with non-negative entries, that fulfills the equation A*A = 0, where 0 is the zero matrix. What is the maximal number of positive entries for A? |
I encountered the following problem but I have no idea how to proceed.. Basically, I have a class of periodic function satisfying certain properties, and since I do not have explicit form of them I use their Fourier expansions, and hope to arrive the minimum value of $g(x)+g(\alpha-x)$ over interval $x\in(0,\alpha]$, w... |
Since I’m researching and experimenting with special kinds of primes (I asked [this question][1] about them, you can see their definition there), which I just realized are called “additive primes” in mathematical terms (didn’t know), I have a question about them:
**Is the number of additive prime numbers between $1$... |
Is the number of additive primes between $1$ and $1\text{,}000 \text{,} 000 \text{,} 000$ known? |
I want to test the primality of some numbers.
To do this I have tried PARI/GP and from the command line:
isprime(1797985968535216664944496004358374343509215681442061471412191560717902567100922783048272325884204171542595289833292525444657649015706240748603258905587228070218364321463272748372578381297752753873677... |
In the book I am currently reading, it says
> Gradient-based methods are useful for one global optimum and no
> additional local optima: (quasi-)concave for maximums, (quasi-)convex for
> minimums.
It is clear to me why concave and convex functions work, as it implies that there is only one maximum/minimum and ... |
Studying Gabriel Navarro's book "Character Theory and the McKay Conjecture", I've come across the following problem. First, let's fix some notation: $G$ will be a finite group, $R$ will denote the algebraic integers of $\mathbb{C}$ over $\mathbb{Q}$, $p$ will be a prime, $M$ will be a maximal ideal of $R$ containing $p... |
From https://brilliant.org/wiki/linearity-of-expectation/
[![enter image description here][1]][1]
I have two questions:
### First question
The "obvious" solution is by using linearity of expectations. My idea was to write the states as $E(n - 1|n)$, where $n$ is the number of steps she needs to get her acorn. So th... |
Suppose a countable group $G$ acts transitively on a countable set $X$. Let $G = G_0\supset G_1\supset\cdots$ be a descending chain of subgroups. Suppose $x,y\in X$ lie in the same orbit of $G_i$ for all $i$. Must they lie in the same orbit of $\cap_i G_i$?
What if we assume that the stabilizer of $G_i$ is cyclic? T... |
A question I found and that I could not answer so far. Assuming that A is a nxn Matrix with non-negative entries, that fulfills the equation $A*A = A^2= 0$, where $0$ is the zero matrix. What is the maximal number of positive entries for A?
So far I have tried going through the specific entries, namely to solve $0 =... |
> Let $X$ be an RV, $\mathcal{F}$ a $\sigma$-algebra, $\phi$ the characteristic function of a distribution $\nu$. Assume that for the conditional characteristic function
>
> $$\phi_\mathcal{F}(t) := \mathbb{E}[\exp(itX) \mid \mathcal{F}]$$
>
> holds $$\phi_\mathcal{F}(t) = \phi(t) \quad a.s. \qquad (2)$$
>
> an... |
Consider the group $G = \mathrm{SO}(3)$. It is formed by real $3 \times 3$ matrices that are orthogonal, i.e., $R^\intercal R = \mathbb{1}$. Consider now the subspace $H$ of matrices with the form
$$R = \begin{pmatrix}0 & 0 & 0 \\ 0 & a & b \\ 0 & c & d\end{pmatrix}$$
where
$$S = \begin{pmatrix} a & b \\ c & d\end{p... |
Does a subgroup need to have the same identity as the original group? |
The Besov spaces can be defined using the dyadic decomposition. For this purpose, let $\varphi \in C^\infty_c(\mathbb R^n)$ be such
$ {\rm supp}(\varphi) \subseteq \left\{x; \, \frac 12< |x| < 2\right\}$ and denote
$$\varphi_j(x) = \varphi\left(2^{-j}x\right)\quad\text{for} \quad j\in \mathbb N^\ast\quad \text{a... |
In QR decomposition of a matrix, Q represents the orthogonal matrix and R represent upper triangular. I was wondering if a matrix could be decomposed into orthogonal and lower triangular and found LQ decomposition. Why is it not QL ? Why is the rotation first? |
Let $(\Omega,M,P)$ be a probability space, $ X_{1} $ and $ X_2 $ be two random variavles, is it ture that $\sigma(X_1+X_2) \subseteq \sigma (X_1,X_2)$ ? How can we prove this?
Someone told me that this is a conclusion of Doob-dynkin's theorem.
**Doob-Dynkin's Theorem**: Lex $X=(X_1, X_2,...,X_n)$ be a $R^n$ rand... |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. The latter example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already ... |
In QR decomposition of a matrix, Q represents the orthogonal matrix and R represent upper triangular. I was wondering if a matrix could be decomposed into orthogonal and lower triangular and found LQ decomposition. Why is it not QL ? Why is the rotation first?
In general why is QR decomposition not RQ ? Is there an... |
Suppose a countable group $G$ acts transitively on a countable set $X$. Let $G = G_0\supset G_1\supset\cdots$ be a descending chain of finite index subgroups. Suppose $x,y\in X$ lie in the same orbit of $G_i$ for all $i$. Must they lie in the same orbit of $\cap_i G_i$?
What if we assume that the stabilizer of $G_i$... |
Let $bp(G)$ be the number of complete bipartite graphs needed to partition the edge set of G. This means, $ \forall e \in E(G) $, e is in exactly 1 complete bipartite graph. Now, how to prove bp(G) $ \geq \log_{2}(\chi(G)) $ ?
Some things I thought of is that we can consider $ V_{i} $ the independence sets of G that... |
can anyone help me?
Be $(X,\|.\|)$ a normed space that satisfies the parallelogram law. Let's define $\Phi : X\times X \rightarrow \mathbb{C}$ by:
$\Phi (x,y)=\dfrac{1}{4}[ \|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2]$
Prove
$\Phi(x,y)=\overline{\Phi(x,y)}$
I resolved
$\overline{\Phi(x,y)}=\dfrac{1... |
Is $\langle x,y \rangle=\overline{\langle x,y \rangle}$? |
Consider the cubic equation
$$ ax^3 + bx^2 + cx + d = 0\,.$$
My question is: do the zeros of this equation change on the same order as the change in its coefficients? I'm interested in $d$ specifically, so in this case, suppose $|d -d'| < \epsilon$. Then, are the zeros of
$$ax^3 + bx^2 + cx + d' = 0$$
within $\... |
Are zeros of a cubic polynomial lipschitz wrt the constant term? |
Are these 2 equations equivalent? If so how can I prove it disprove it? |
Suppose a countable group $G$ acts transitively on a countable set $X$. Let $G = G_0\supset G_1\supset\cdots$ be a descending chain of finite index subgroups. Suppose $x,y\in X$ lie in the same orbit of $G_i$ for all $i$. Must they lie in the same orbit of $\cap_i G_i$?
What if we assume that the stabilizer of $x$ i... |
I know that this is a very general fact about limits and colimits, but I would like to prove it directly for ends and coends. If $\mathcal V$ is a closed braided monoidal category, $V$ an object in $\mathcal V$ and $P\colon\mathcal M^\mathsf{op}\otimes\mathcal M\to \mathcal V$ is a $\mathcal V$-functor, I can define it... |
A construction of the projective transformation is as follows:
>Given a fixed line $L$ and two fixed points $S,S'$.\
>For any point $P$ on the plane, let the line $PS'$ intersect $L$ at $Z$.\
>Draw the line through $S'$ parallel to $SP$ and this line intersect the line $SZ$ at $P'$.\
>The map $P\mapsto P'$ maps ang... |
**Question:**
What is the sum of all the roots of the equation $x|x|-5|x+2|+6=0$?
----------
**My Method of Solving:**
3 cases arise here.
**Case 1:**
Considering that $x\geq0$, the equation becomes $x^2-5(x+2)+6=x^2-5x-4=0$
Using the fact that sum of roots = $-\frac{b}{a}$, I get $\mathbf{5}$... |
What is the sum of all the roots of the equation $x|x|-5|x+2|+6=0$? |
Just try it:
$$ab \stackrel{?}{=} 1 - (1-a) \cdot (1-b)$$
Well:
$1 - (1-a) \cdot (1-b) = 1 - (1 - a - b + ab) = a + b - ab \ne ab$
So, no. Those equations are not equal. |
A construction is as follows:
>Given a fixed line $L$ and two fixed points $S,S'$.\
>For any point $P$ on the plane, let the line $PS'$ intersect $L$ at $Z$.\
>Draw the line through $S'$ parallel to $SP$ and this line intersect the line $SZ$ at $P'$.\
>The map $P\mapsto P'$ maps lines through $S$ to parallel lines ... |
Suppose we have
$F(z)=f(\phi(z))$ where $\phi$ is a Möbius transformation which maps points of the unit circle to points of the unit circle. Suppose also that there is an interval of length $\pi$ such that on that interval $f(e^{it})-f(e^{-it})>0$. How to prove then this is also true for $F$?
Here, I defined:... |
Suppose $b > a > 0$, $\delta > 0$.
Let $M_k$ be defined as follows:
For $k < \frac{m}{2}$,
$M_k = \frac{{(a + \delta)_k(b)_{m-k} + (a + \delta)_{m-k}(b)_{k} - (a)_{k}(b + \delta)_{m-k} - (a)_{m-k}(b + \delta)_{k}}}{{k!(m - k)!}}$
For $k = \frac{m}{2}$,
$M_k = \frac{{(a + \delta)_{k}(b)_{m-k} - (a)_{k}(b + ... |
I encountered the following problem but I have no idea how to proceed.. Basically, I have a class of periodic function satisfying certain properties, and since I do not have explicit form of them I use their Fourier expansions, and hope to arrive the minimum value of $g(x)+g(\alpha-x)$ over interval $x\in(0,\alpha]$, w... |
There is an engineering question that occurred to me several years ago, but I could not solve it
Assuming $H$ is the point of intersection of the heights, $I$ is the point of intersection of the bisectors, and $G$ is the point of intersection of the averages in triangle $∆ABC$, and the points $H$, $I$, and $G$ are ... |
Is it possible to create triangle ∆ABC Depending on the points H,I,G? |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. The latter example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already ... |
We use induction on $\chi(G)$. I'll let you handle the base cases yourself.
Let $\chi(G) = k\geq 3$ and let
\begin{equation*}
C_i:=\{v \in V(G) \mid \text{ the colour used for $v$ is $i$ }\},
\end{equation*}
and $\{C_1, C_2, \dots, C_{k} \}$ be the disjoint collection of colour classes of $G$. Let $I = \{1, ... |
How do I calculate an integral $\int_{-\infty}^{\infty} \frac{z-a}{\sinh(z-a)}\frac{z+a}{\sinh(z+a)} dz$ for $a>0$?
Expanding the integrand and integrating term-by-term (a-la polylog) gives rise to an ugly looking **double** series.
Edit: it surely can be done this way, but what is the most straightforward way to cal... |
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