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Theoretically, we know that the range and domain of $f(x)=\cot^{-1}x$ is $(-\infty,\infty)$ and $(0,\pi)$, respectively. Its plot is a smooth continuous function.
Strangely, I find that Wolfram Mathematica plots it as an odd
discontinuous (at $x=0$) function in $(-\pi/2,\pi/2)$!
**Question 1:** Have your encoun... |
> Let X be the smallest value obtained when k numbers are randomly chosen from the set 1,...,n. Find E[X] by interpreting X as a negative hypergeometric random variable.
This is Self Test Exercise 7.7 of Sheldon's A First Course in Probability.
The way I approached this is to first consider (for example) P(X = ... |
What is the name of the quantization method where you truncate before adding a half? |
Theoretically, we know that the range and domain of $f(x)=\cot^{-1}x$ is $(-\infty,\infty)$ and $(0,\pi)$, respectively. Its plot is a smooth continuous function.
Strangely, I find that Wolfram Mathematica plots it as an odd
discontinuous (at $x=0$) function in $(-\pi/2,\pi/2)$!
**Question 1:** Have you encount... |
>Consider a dice $D$ with sides $\{1,\ldots,6\}$ and let $p_i$ be the (constant) probability of each side). Let further be $A_n$ the event that any side shows up more than $p_i + h(n)$ for $h(n) = \omega(\sqrt{n})$ times. Show that for $n \rightarrow \infty$ follows $\mathbb{P}[A_n] \rightarrow 0$.
This reminds me o... |
It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that "if A, then B" can be understood as "A *coupled* with B" when it means $\mu(x,y) = T_{norm} (\mu_{A}(x), \mu_{B}(y))$, ins... |
Consider the space $C[0,1]$ equipped with the sup norm. Let $t_0$ be an arbitrary point in $[0,1]$ and define the mapping
$$T: f \to f(t_0)$$
from $X_1 := C[0,1]$ into $X_2 := \mathbb R$. Clearly, $T$ is linear and
$$||Tf||_2=|f(t_0)| \leq \sup_{t \in [0,1]} |f(t)|=||f||_1 $$
with equality holding for co... |
hi i look for transforme of fourier of $xtanh(x)$ in R; for the transform of fourier in $\mathbb{R}-{0}$ the transform of fourier is $2\sum_{n=0}^{\infty}(2n+1)\exp(-(2n+1)\frac{|t|}{2})$ the hint is to compare the transfom of fourier of transforme of fourier of $xtanh(x)$ with transfom of fourier of $|x|$ ,??
than... |
Tldr; can I do the second integral with the method I used for the first integral?
I was reading about some integrals when I saw the following:
$\int_{0}^{\infty}$ $(x^4+2x^2cosh(2t)+1)^{-1}$ = $\frac{\pi}{4cosh(t)}$
and the way to do this is to write the denominator as $(x^2+e^{2t})(x^2+e^{-2t})$ and then thi... |
Why does replacing $\cosh$ by $\cos$ in this biquadratic integral not change anything? |
Consider the function
$$ f(x,y,\lambda)=\frac{1}{8} x \left[\left(\lambda ^2+6 \lambda -4\right) y-(\lambda -1)^2 x\right]$$
over real domain $x>0$, $y>0$ and $\lambda\in[0,1]$.
If we set $y=x$ then we have $f(x,x,\lambda)=\frac{1}{8} (8 \lambda -5) x^2$, meaning that $\lambda=5/8$ is the unique $\lambda$ that s... |
I have some comprehension issues regarding combinatorial mathematics.
More specifically, I am asking about advices on how to approach a given problem in combinatorial theories as I feel that these kind of problems rely on logic "intuition" rather than rigorous method.
Let's take an example of an exercise :
Suppo... |
$\bigcup \cal{A} := \{x:(\exists a \in \cal{A})x\in a\}$
Theorem:
$\bigcup (A \cup B) = \bigcup A \cup \bigcup B$
Attempt at proof:
$x \in \bigcup (A \cup B) \iff (\exists X \in A \cup B)x \in X $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \iff \exists X[(X \in A$ or $X \in B) \implies x \in X]$
... |
Is this proof of $\bigcup (A \cup B) = \bigcup A \cup \bigcup B$ correct? |
Why is function's root undefined at some point when there is a clear and well-defined solution? |
Tigger and kanga plan to meet at the gym, but will each independently arrive at a uniformly random time between 14:00 and 15:00. Tigger is willing to wait 24 minutes for Kanga before leaving, but Kanga is only willing to wait 6 minutes for tigger before leaving. What is the probability they end up meeting eachother ? |
When $x = y$, $f$ becomes a simpler function of $\lambda$. Specifically:
$$ f(x,x,\lambda) = x^2\lambda - \frac{5 x^2}8. $$
Therefore $f(x,x,\lambda) = 0$ is not a quadratic equation in $\lambda.$
What happens to the quadratic formula when $a = 0$?
Your "general" solution makes an assumption you did not accou... |
Let $\gamma:[0,1]\rightarrow {C}$ be a path. The winding number $\omega(\gamma)$ is defined as $\omega(\gamma)=\frac{\theta(1)-\theta(0)}{2\pi}$. I have read that two paths homotopic to the endpoints have the same winding number also for non closed curves. I would like to proof it but could not find one online. Do you ... |
Two non closed paths homotopic relative to the endpoints have the same winding number? |
This Question asked on [math over flow][1]
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^p+1)^p dy$$
now lets define $a(n,k)$ ... |
Determine extrema of a function $f$, what if the Hessian Matrix is not positive, negative or semi-definite? Do I need to calculate the Eigenvalues? |
Let $X=\mathbb{C}^3$, $Q=V(x^2-yz)$, $H_1=V(y)$, $H_2=V(z)$. The hyperplanes $H_i$ are tangent to $Q$ along a line. Let $U$ be the complement of $Q\cup H_1\cup H_2$. Is the fundamental group $\pi_1(U,\mathrm{pt})$ isomorphic to $\mathbb{Z}^3?$
It helps that all of the varieties involved are conical, so we may consi... |
When you ask what a matrix "really is", you are asking the wrong question. A matrix is, at its heart, just a big block of numbers arranged in a rectangular grid. That's it.
The important question is: what can you *do* with a matrix? That's a much more interesting story! In fact matrices can be used in many dif... |
Looking at [this question][1], I saw user Zhen Lin's comment that when $X$ is a separated scheme over an affine scheme $S$, and $U$ and $V$ are affine open subsets of $X$, we can identify $U \cap V$ with $(U \times_S V) \times_{X \times_S X} X$. I've been able to verify that $U \cap V$ has the universal property of $(U... |
> Let $\{X_n\}_{n \geq 1}$ be a sequence of iid random variables with pdf $$f(x) = \begin{cases} \frac {c} {x^2 \log |x|}, & |x| \gt 2, \\ 0, & \text {otherwise}, \end{cases}$$ where $c$ is a normalizing constant. Let $S_n = \sum\limits_{k=1}^{n} X_k.$ Show that $\frac {S_n} {n} \xrightarrow {p} 0.$
Let us take $\va... |
It's well-known that a $T_1$ topological space is [fully normal][1] if and only if it is $T_2$ and paracompact.
It appears, looking at the proofs from Henno Brandsma's nice exposition [here][2] and [here][3], that we can drop the $T_1$ assumption for the implication that fully normal implies paracompactness (without... |
fully normal implies paracompact without a T1 assumption? |
Consider an asymmetric random walk $(X_n)$ in which the initial point is zero ($X_0 = 1$). It increases by $a$ with a probability of 0.3, remains the same with a probability of 0.4, and decreases by $b$ with a probability of 0.3, where $a<b$ and $a,b \in \mathbb{R}$. Also, suppose this process stops once $X_n \geq \lam... |
I‘m reading [this][1] paper from Keith Conrad and in Example 1.1 he states:
[![enter image description here][2]][2]
[1]: https://kconrad.math.uconn.edu/blurbs/galoistheory/galoisappn.pdf
[2]: https://i.stack.imgur.com/Qdx5R.png
I don‘t the term "$\mathbb Q$-conjugates" - what does that mean? I don‘t get... |
What are $\mathbb Q$-conjugates? |
Let $X$ be a topological space. A **chain of irreducible closed subsets of $X$** is a sequence
$$
Z_0\subsetneq Z_1\subsetneq\cdots \subsetneq Z_n\subset X
$$
where each $Z_i$ is an irreducible closed subset of $X$. For each of the chain, its **length** is either $\infty$ or the integer $n$, and we define the **Kru... |
We say that a topological space is **sober** if every nonempty irreducible closed subset is the closure of a one-point set.
Just to clarify, for the definition of sober, does it mean that every nonempty irreducible closed subset is the closure of *some* one-point set; or a subset is nonempty irreducible closed *if a... |
Consider an asymmetric random walk $(X_n)$ in which the initial point is zero ($X_0 = 1$). It increases by $a$ with a probability of 0.3, remains the same with a probability of 0.4, and decreases by $b$ with a probability of 0.3, where $a<b$ and $a,b \in \mathbb{R}$. Also, suppose this process stops once $X_n \geq \lam... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
I know that the Hairy ball theorem plus the following proposition
**Proposition**
For a smooth manifold $M$.
There exists a Lorentz metric on $M$ $\iff$ there is a continuous non-vanishing vector field on $M$
allow one to conclude that there is no Lorenzian metric on even dimensional n-spheres.
**I... |
How do I find a cont. non-vanishing vector field on $\Bbb S^n$, $n=$odd, such that there exists a Lorentz metric? |
Let $X$ be a topological space. It is **irreducible** if it can be written as a union of $U$ and $V$ where $U,V$ are proper closed subsets of $X$.
A **chain of irreducible closed subsets of $X$** is a sequence
$$
Z_0\subsetneq Z_1\subsetneq\cdots \subsetneq Z_n\subset X
$$
where each $Z_i$ is an irreducible clos... |
How many ways can a cube fit into a sphere through its vertices?
I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides, one way is that There is no vertex on the sphere and the whole cube is inside the sphere. Another case is that the cube has only on... |
How many ways can a cube fit into a sphere through its vertices? |
How many ways can a cube fit into a sphere through its vertices? I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides, one way is this. that there are no vertices on the sphere and the entire cube is inside the sphere, the other case is that the cube ... |
Let $X$ be a topological space. It is **irreducible** if it can not be written as a union of $U$ and $V$ where $U,V$ are proper closed subsets of $X$.
A **chain of irreducible closed subsets of $X$** is a sequence
$$
Z_0\subsetneq Z_1\subsetneq\cdots \subsetneq Z_n\subset X
$$
where each $Z_i$ is an irreducible ... |
It's well-known that a $T_1$ topological space is [fully normal][1] if and only if it is $T_2$ and paracompact.
It appears, looking at the proofs from Henno Brandsma's nice exposition [here][2] and [here][3], that we can drop the $T_1$ assumption for the implication that fully normal implies paracompactness (without... |
Let $\mathbf{x}$ be a multivariate Gaussian random vector with zero mean but not necessarily a diagonal covariance matrix(so that its component can be dependent). Let $\mathbf{a}$ be a deterministic vector, and $\mathbf{B}$ be a deterministic symmetric matrix.
Do we have $E[(\mathbf{a}^T\mathbf{x})^i(\mathbf{x}^T\ma... |
How many ways can a cube fit into a sphere through its vertices?
I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides, one way is this. that there are no vertices on the sphere and the entire cube is inside the sphere, the other case is that the cub... |
It makes sense to me that this would be true but I am not sure.
My understanding of (at least one of the definitions of) a continuous function is that
- for every point $p$ in the functions, domain, there exsits a neighborhood $N$ around that point
- There is a neighborhood $f(N)$ in the function's image that... |
Does the exsistance of a continuous function on a topological space imply that that space is Hausdorff? |
Does the existence of a continuous function on a topological space imply that that space is Hausdorff? |
I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides, one way is this. that there are no vertices on the sphere and the entire cube is inside the sphere, the other case is that the cube has only one vertex on the points of the sphere and the other 7... |
I won't be adding anything else to the previous answers but a piece of imagination and loose interpretations to bring a physical intuition about.
**A moving point running along the envelope**
If you look carefully the animation of an sliding bar on the [envelope page](https://en.wikipedia.org/wiki/Envelope_(mathe... |
What does circular inference (benign or vicious) look like symbolically? Is there anything I can do with truth tables or some other means to better understand its structure and what relations among premises and conclusions are circular? |
How can I prove this statement regarding a discrete-time algebraic Riccati inequality? |
I am interested in checking how small $k$ should be with respect to $n$, so that the following inequality holds (asymptotically).
> $\binom{n-1}{k-1} \Big(\binom{n-1}{k-1}(1-k) + \binom{n-k-1}{k-1}k +1 - 2k\Big) + \binom{n-k-1}{k-1}\cdot (k-2) - k \geqslant 0$.
I tried to express $\binom{n-k-1}{k-1}$ as $\binom{n... |
Let $LC = \{\lambda v \mid \lambda \in L, v \in C\},$ where $C$ is some convex set of the vector space $V.$ If $L$ is an interval, is $LC$ convex?
I thought and assumed so for a while, and some drawings suggested this to be true, however, today I tried to prove it analytically and the proof escaped me.
We start w... |
If $L$ is an interval and $C$ is convex, is $LC$ convex too? |
I read theorem 6.13 out of Folland’s book. I was wondering what goes wrong if you take q = infty. Is there a nice counterexample?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/yq3p4.jpg |
Let $F_2$ be the free group of rank 2. What is the abelianization of $Aut(F_2)$?
There is a surjection $Aut(F_2) \rightarrow GL_2(\mathbb{Z})$, so we at least have $\mathbb{Z}/2$ as a quotient. Is this the full abelianization? |
What is the abelianization of $Aut(F_2)$? |
I am studying rotations matrixes and I'm confused about the order of rotations.
Let's say I would like to rotate vector $\vec a$ into $\vec b$. They are defined in the cartesian coordinate system. I would like to do the rotation first around the $x$ axis, then the $y$ axis, and then the $z$ axis. Does that mean I ne... |
Let $F_2$ be the free group of rank 2. What is the abelianization of $\operatorname{Aut}(F_2)$?
There is a surjection $\operatorname{Aut}(F_2) \rightarrow\operatorname{ GL}_2(\mathbb{Z})$, so we at least have $\mathbb{Z}/2$ as a quotient. Is this the full abelianization? |
What is the abelianization of $\operatorname{Aut}(F_2)$? |
[Kaplansky (1951)][1]'s original definition of an AW$^*$-algebra is a C$^*$-algebra $A$ such that (i) each family of orthogonal projections has a least upper bound (LUB) in $P(A)$ (the set of projections in $A$) and (ii) each maximal abelian $*$-subalgebra (masa) of $A$ is generated by its projections.
Moreover, a ... |
If I sample $N+1$ uniform $k$-bit integers $x, x_1, \ldots, x_N$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{x_i\}$? Upper/lower bounds and/or asymptotic approximations would also be of interest.
For prime $p$ and $x, x_1, \ldots, x_N$ as above, say $x$ is $p$*-uniqu... |
> Let $G$ be a group with the following property: for every $H, K \leq G$, then either $H \subseteq K$ or $K \subseteq H$. Show that there exists a prime number $p$ such that the order of every element of $G$ is a power of $p$.
It is straightforward to show the result above when $G$ is finite. Indeed, fix $g \in G... |
I have no idea how to solve this. It's easy to show that the open ball or the open cube is convex, but how do we show that the open rectangle is? (The same holds for the closed rectangle).
Thanks |
Fully normal implies paracompact without a $T_1$ assumption? |
**Problem:**
>Show by hand without any computer assistance:
$$e^{e^2}>1000\phi,$$
where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$.
I come across this limit showing:
$$\lim_{x\to 0}x!^{\frac{x!!^{\frac{2}{x!!-1}}}{x!-1}}=e^{e^2}.$$
I cannot show it without knowing some decimals and ... |
My textbook states that `$x = x+1$` is not a proposition.
Why? I thought about domains but no matter the domain this statement is always false because 0 is not equal to 1.
My textbook defines a proposition in the following way:
> *A boolean **proposition** (or simply proposition) is a statement which is eith... |
Propositional Logic for variable tautologies? |
My textbook states that `x = x+1` is not a proposition.
Why? I thought about domains but no matter the domain this statement is always false because 0 is not equal to 1.
My textbook defines a proposition in the following way:
> *A boolean **proposition** (or simply proposition) is a statement which is either... |
Let $T$ be a regular tetrahedron, and $\ell$ a line from the midpoint of one edge $E_1$, through the center $C$, to the midpoint of another edge $E_2$.
How do I prove that the reflection of each point on the tetrahedron across this line $\ell$ is also on the tetrahedron? I can see this visually (especially when I ro... |
From the general definition of orthogonality for functions set of trig functions $\phi_k=\cos(kx)$ is orthogonal on $[a,b]$ if
\begin{equation}
\int_a^b w(x) \cos(jx)\cos(kx) dx =\alpha_j \delta_{jk},
\end{equation}
where $w(x)$ is the weight function and $\delta_{jk}$ is the Dirac delta. On the interval $a=-\... |
Ways to determine (or estimate) the weight function that makes trig functions orthogonal on a given interval? |
Let $H_1,H_2$ be Hilbert spaces with inner products $\langle \cdot, \cdot \rangle_1$ and $\langle \cdot, \cdot \rangle_2$. Corresponding to every $T \in \mathcal B(H_1,H_2)$, there is a unique element $T^*$ of $B(H_2,H_1)$ determined by the relation
$$\langle T x_1, x_2 \rangle_2=\langle x_1, T^* x_2 \rangle_1$$
... |
Ask a question on adjoint operator? |
Question from an old exam:
>The King of Squares sets out to patrol the City of Squares. The City of Squares is an infinite ladder, i.e., a graph $(V, E)$ where $V = \mathbb{N} \times \{0,1\}$ and the vertex $(x, y)$ is connected to $(x-1, y)$ for $x>0,(x+1, y)$ and $(x, 1-y)$. Unfortunately, due to the revolt, some pa... |
>**Question:** Why $\| T^* x_2 \|^2_1= | \langle x_2，T T^* x_2 \rangle_2 |$?
Because you just proved that $T^*$ exists and it is the adjoint of $T$. Then
$$
\|T^*x_2\|_1^2=\langle T^*x_2,T^*x_2\rangle_1=\langle T(T^*x_2),x_2\rangle_2
=\overline{\langle x_2,TT^*x_2\rangle_2}=\langle x_2,TT^*x_2\... |
Question from an old exam:
>The King of Squares sets out to patrol the City of Squares. The City of Squares is an infinite ladder, i.e., a graph $(V, E)$ where $V = \mathbb{N} \times \{0,1\}$ and the vertex $(x, y)$ is connected to $(x-1, y)$ for $x>0,(x+1, y)$ and $(x, 1-y)$. Unfortunately, due to the revolt, some pa... |
Prove that if integer vector system is linear independent over $Z_p$ ($p$ is prime) then this system is linear independent over $Q$.
I understand how to solve this problem (saw this solution: https://math.stackexchange.com/questions/3997526/linearly-independent-over-f-p-to-q)
but there's one point that seems uncle... |
$
\def\l{\lambda} \def\o{{\tt1}}
\def\BR#1{\Big[#1\Big]}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\frob#1{\left\| #1 \right\|_F}
\def\qiq{\quad\implies\quad}
\def\qif{\quad\iff\quad}
\def\p{\partial} \def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red}{#1}}
... |
$
\def\l{\lambda} \def\o{{\tt1}}
\def\BR#1{\Big[#1\Big]}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\frob#1{\left\| #1 \right\|_F}
\def\qiq{\quad\implies\quad}
\def\qif{\quad\iff\quad}
\def\p{\partial} \def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red}{#1}}
... |
Prove that if integer vector system is linear independent over $Z_p$ ($p$ is prime) then this system is linear independent over $\mathbb{Q}$.
I understand how to solve this problem (saw this solution: https://math.stackexchange.com/questions/3997526/linearly-independent-over-f-p-to-q)
but there's one point that se... |
$
\def\l{\lambda} \def\o{{\tt1}}
\def\BR#1{\Big[#1\Big]}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\frob#1{\left\| #1 \right\|_F}
\def\qiq{\quad\implies\quad}
\def\qif{\quad\iff\quad}
\def\p{\partial} \def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red}{#1}}
... |
>There is a coin that is unbalanced in a way so that the probability of outputting a digit is $\frac{1}{4}$. If it is tossed $n$ times, the probability that there will be exactly $2$ digits is equal to the probability that there will be exactly $3$ digits. The value of $n$ is...
The following is a solution that I fo... |
$
\def\l{\lambda} \def\o{{\tt1}}
\def\BR#1{\Big[#1\Big]}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\frob#1{\left\| #1 \right\|_F}
\def\qiq{\quad\implies\quad}
\def\qif{\quad\iff\quad}
\def\p{\partial} \def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red}{#1}}
... |
Given a partial differential equation, after computing the Lie point symmetries of this PDE, means a one-parameter group (Lie group) of transformation that leaves the set of solution of the PDE invariant using Lie's theorem (infinitesimal criterion of symmetry), this one parameter group is generated by a vector field (... |
why or how to prove that the infinitesimal symmetries of a differential equation forms a Lie algebra? |
Let $X,Y,Z$ be topological spaces, and let $f:X\to Y$ and $g:X\to Z$ be continuous functions.
Suppose that $f$ is constant on each fiber of $g$, i.e. for every $z\in Z$, the fiber $g^{-1}(z) \subseteq X$ is sent to a single point $y_z\in Y$. Then clearly there exists a mapping $Z \to Y$ sending $z$ to $y_z$.
Is t... |
Is this induced mapping continuous? |
Let $T$ be a regular tetrahedron, and $\ell$ a line from the midpoint of one edge $E_1$, through the center $C$, to the midpoint of another edge $E_2$.
How do I prove that the reflection of each point on the tetrahedron across this line $\ell$ is also on the tetrahedron? I can see this visually (especially when I ro... |
I found the following analyses in the paper *Bottlenecks: sectoral imbalances and the US productivity slowdown* by Acemoglu, Autor and Patterson (2023) - working paper version [here][1]. This is a paper whose core analysis is a derivation from a simple theoretical economic growth model with linkages between industries ... |
Making this a community wiki answer.
As OP indicated, this problem is translated from a different language, with attendant issues. (The [OSN](https://en.wikipedia.org/wiki/Indonesia_National_Science_Olympiad) is the Indonesian National Science Olympiad.) We'll assume the question is something along the lines of:
... |
Only the rotations through the center and a vertex or edge midpoint send a tetrahedron to itself: geometric proof? |
> Let $G$ be a group with the following property: for every $H, K \leq G$, either $H \subseteq K$ or $K \subseteq H$. Show that there exists a prime number $p$ such that the order of every element of $G$ is a power of $p$.
It is straightforward to show the result above when $G$ is finite. Indeed, fix $g \in G$ and... |
Why or how to prove that the infinitesimal symmetries of a differential equation form a Lie algebra? |
I know (see for example page 47 in Brown) that a short exact sequence of groups
$$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$$
gives rise to a 5 term exact sequence in their homology groups
$$H_2(G) \rightarrow H_2(Q) \rightarrow H_1(N)_Q \rightarrow H_1(G) \rightarrow H_1(Q) \rightarrow 0$$
... |
How can I find a route through a partitioned weighted graph that visits as many of the partitions as possible, for a given maximum distance? Start and end anywhere.
A real-world example could be to find a route that visits as many European countries as possible in $2000$ miles.
EDIT: by partitioned graph here, I ... |
If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?
$\cup_i\mathcal{F}_i$ seems to be closed under complement since for all $x$ in the union, $x$ has to belong to some $\mathcal{F}_i$, and so must its complement.
It... |
So, we can make this a little less complex by reframing the question: What we're asking is "given a target number, what is the probability of more than 0 dice equalling or exceeding that target?". Ultimately, it doesn't really matter which die has the highest number in this scenario, we really just care if any of the d... |
Let $T(x)=\frac{x}{\sqrt{1+x^2}}$ be a function from $R$ to $R$. Prove that the function is firmly non-expansive.
Definition:
$$||Tx-Ty||^2 + ||(Id-T)x-(Id-T)y||^2 \leq ||x-y||^2$$
If the space is Hilbert, this simplifies to,
$$||Tx-Ty||^2 \leq \hspace{3mm} <Tx-Ty, x-y> $$
My attempt:
Since $R$ is... |
I would like to ask a fairly general question: how much of naïve linear algebra remains true for vector bundles? For example,
1. If $F$ and $F’$ are two subbundles of $E$ such that the fiber $E_x$ is the internal direct sum $F_x$ and $F’_x$ at each point $x$, is it true that $E$ is isomorphic to the direct sum of $... |
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