instruction stringlengths 12 30k |
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$\require{AMScd}$I make reference to [this](https://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf) paper. I've recently become aware that there's an annoying variety of seemingly distinct definitions of triangulated categories, partly because allegedly no one knows an example of a weakly but not "strongly" triangulate... |
Is the octahedral axiom really equivalent to the 4 x 4 lemma? |
I assume you meant $U = \{f \in X | \sup_{a \in A} |f(a)| \geq 1\}$ ($f(a)$ in the absolute value instead of $f(x)$)? In that case no, assuming $A$ is $T_1$ and infinite. Indeed, just choose a sequence of distinct points $\{a_n\}_{n \in \mathbb{N}} \subset A$. Then $f_n = 1_{\{a_n\}} \in U$, but $f_n \to 0$ pointwise, ... |
I was trying to come up with a problem where the solutions would be the numbers 5, 6, 10 and 11. It seems like "finding all integers n, where $\frac{n(n+1)}{10}$ is a prime power", is a good attempt, as there are at least no other solutions for n < 1.000.000 (tested with python).
However, I didn't manage to prove th... |
Let $(V,+,\cdot)$ be a vector space over a field $\mathbb K$. For a $v\in V$ Let $-v$ be its inverse in $V$ and for $a\in \mathbb K$ let $a’$ be its additive inverse in $\mathbb K$
Now is $-v$ the same as $(1)’\cdot v$? Where the RHS is the vector $v$ scaled by the additive inverse of $1$ in $\mathbb K$. I don’t kno... |
Your equation for $C(t)$ can be broken down into equations for $x$ and $y$ separately:
$$
x(t) = \frac{1-t^2}{1+t^2} \quad ; \quad
y(t) = \frac{2t}{1+t^2}
$$
It’s easy to check that $[x(t)]^2 + [y(t)]^2 = 1$ for all $t$. This means that every point $C(t)= (x(t),y(t))$ lies on the unit circle. Also it’s clear that ... |
I was trying to come up with a problem where the solutions would be the numbers 5, 6, 10 and 11. It seems like "finding all integers n, where $\frac{n(n-1)}{10}$ is a prime power", is a good attempt, as there are at least no other solutions for n < 1.000.000 (tested with python).
However, I didn't manage to prove th... |
Consider the stereographic projection chart on $S^2$ which doesn't include the north pole $$(X,Y)=\varphi(x,y,z)=\left(\frac{x}{1-z}, \frac{y}{1-z}\right).$$
I want to pull back the 1-form $\omega = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}$ to $S^2$ from $\mathbb{R}^2$ to $S^2$ but I am not sure about a step in the calculati... |
Let $A$ be a compact topological space equipped with the Borel $\sigma$-algebra, and $X=B_b(A)$ be the vector space of bounded measurable functions. Let $Y=\mathcal M(A)$ be the vector space of finite signed measure on $A$. Define the dual pair $<\cdot, \cdot>$ between $(X,Y)$ such that $
<f, \mu >=\int_A f(a)\mu(da... |
I’m trying to proof that if $A$ and $B$ are two convex sets in a **finite dimensional** vector space $V$ than there exists an hyperplane that separates $A$ and $B$, but I’m not understanding how to proceed. Does someone have any ideas? |
Consider the following logical statement.
$$ (A \land B) \implies (C \land D)$$
**Question:** Under what conditions does the following statement follow from the above?
$$ (A\implies C) \land (B \implies D)$$
---
**Context:** I realised the heart of my previous [question][1] is this simpler to state, more... |
When does $ (A \land B) \implies (C \land D)$ imply $ (A\implies C) \land (B \implies D)$? |
It is said that category theory serves as an alternative foundation of mathematics, as such it must define natural numbers in terms of categories as it is done in set theory, when we consider set theory as a foundation for mathematics and where we define recursively natural numbers as
$0:=\emptyset$ and $n+1:=n\cup ... |
I’m trying to proof that if $A$ and $B$ are two convex sets in a **finite dimensional** vector space $V$, with $A\cap B= \emptyset $ , than there exists an hyperplane that separates $A$ and $B$, but I’m not understanding how to proceed. Does someone have any ideas? |
I read in some lecture notes the following definition of contraction rate:
Definition (Posterior rate of contraction) The posterior distribution $\Pi_n\left(\cdot \mid X^{(n)}\right)$ is said to contract at rate $\epsilon_n \rightarrow 0$ at $\theta_0 \in \Theta$ if $\Pi_n\left(\theta: d\left(\theta, \theta_0\right)... |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
Investigate at which points on $\partial D$ the function can be continuously extended, and in this case, provide the continuous extension.
$f : D \rightarrow \mathbb{R}$, $x \mapsto \frac{{x_1 \sin(x_2) + x_2 \sin(x_1)}}{{\sqrt{x_1^2 + x_2^2}}}$
I have tried to disprove that $f(x_1, x_2)$ has a continuous extensi... |
We know that
$$\sin x = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \cdots = \sum_{n\ge 0} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$
But how could we derive this without calculus?
There were some approach using $e^{ix} = \cos x + i \sin x$, pls notice I also would like to avoid such definition as to prove $e^{ix} = \cos x ... |
Consider the stereographic projection chart on $S^2$ which doesn't include the north pole $$(X,Y)=\varphi(x,y,z)=\left(\frac{x}{1-z}, \frac{y}{1-z}\right).$$
I want to pull back the 1-form $\omega = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}$ to $S^2$ from $\mathbb{R}^2$ to $S^2$ but I am not sure about a step in the calculati... |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
How can you show that the absolute value of $\sin{x}$ diverges? |
$\require{AMScd}$I make reference to [this](https://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf) paper. I've recently become aware that there's an annoying variety of a priori distinct (but "known" to be equivalent up to adding other hypotheses) definitions of triangulated categories, partly because allegedly no one... |
How can I convert a number into a seemingly random number in a process that is actually *deterministic*?
For example I'd like to transform: `123456` into `51`.
Or `28`. Doesn't matter as long as it's a number between 1 - 100.
The value should be significantly different for similar inputs.
Transforming `123456... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain.
... |
If X has Poisson (lambda) distribution then what would be the coverage probability of its confidence interval? |
How can I convert a number into a seemingly random number in a process that is actually *deterministic*?
For example I'd like to transform: `123456` into `51`.
Or `28`. Doesn't matter as long as it's a number between 1 - 1000.
The value should be significantly different for similar inputs.
Transforming `1234... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain but t... |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain but t... |
Are elliptic differential operators between vector bundles epimorphisms of sheaves? |
How can I convert a number into a seemingly random number in a process that is actually *deterministic*?
For example I'd like to transform: `123456` into `51`.
Or `28`. Doesn't matter as long as it's a number between 1 - 1000.
The value should be significantly different for similar inputs.
Transforming `1234... |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
Let $n\ge 2$ and $F:\mathbb R\to\mathbb R^n$ be a positive function, meaning that
$$ F(x)=(F_1(x),\dots, F_n(x))$$
and $F_i(x)\ge 0$ for any $i\in\{1,\dots, n\}$.
My question is: is always possible to find the minimum (or the maximum) between the $n$ components of $F$? More precisely, is it possible to find $i\in\... |
If $F(x)=(F_1(x),\dots, F_n(x))$, is always possible to find $F_i(x)=\min\{F_1(x),\dots, F_n(x)\}$? |
In the paper *On The Closure of Characters and the Zeros of
Entire Functions* by Beurling and Malliavin they make the following claim in the introduction.
The closure radius $\rho = \rho(\Lambda)$ defined as the upper bound of the numbers $r$ such that set $\{e^{i \lambda x}\}_{\lambda \in \Lambda}$ span the space... |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
Let $n\ge 2$, ($n$ finite) and $F:\mathbb R\to\mathbb R^n$ be a positive function, meaning that
$$ F(x)=(F_1(x),\dots, F_n(x))$$
and $F_i(x)\ge 0$ for any $i\in\{1,\dots, n\}$.
My question is: is always possible to find the minimum (or the maximum) between the $n$ components of $F$? More precisely, is it possible ... |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
I want to derive $\sin(A-B) = \sin A \cos B - \cos A \sin B$ from $\cos(A-B)=\cos A \cos B +\sin A \sin B$
$$\cos(A-B)=\cos A \cos B +\sin A \sin B$$ substitute $A$ for $A + \pi$
$$\cos(A+\pi-B)=\cos( A+\pi) \cos B +\sin( A+\pi) \sin B$$
$$\sin(A-B)=\sin(A) \cos B +\cos(A) \sin B$$
but this is wrong. However yo... |
I confirm your guess for the case $N=3$.
By your hint in the comments,
> $K$ is a constant in $[-\pi, \pi]$
follows, $k_{1..3}$ are almost free. That is why I disbelieved your minimum if $k_{1..3}=K/N$ assumption.
My reasoning, with $k_1=k_2=0$ the "terms" $\cos(k_1)$ and $\cos(k_2)$ have a maximum impact ... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain but t... |
Let $A$ be a compact metric space,
and $E\subset A$ be a Borel measurable set. Does there exist a finite positive measure $\mu$ on the Borel sets of $A$ such that $\mu(E)>0$ and $\mu(A\setminus E)=0$?
-----
The question is related to [this question](https://math.stackexchange.com/questions/208305/probability-... |
[enter image description here][1]
[1]: https://i.stack.imgur.com/cpS8K.png
How do I label the position vectors & what would be an efficacious way to continue the proof? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
>1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
>2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entr... |
I wouldn't say this isomorphism is "really simple" when learning these things for the first time. It involves an isomorphism in infinite Galois theory and an isomorphism between a certain subgroup of $\mathbf Z_p^\times$ with $\mathbf Z_p$. In particular, the group
${\rm Gal}(F(\mu_{p^\infty})/F(\mu_{p}))$ is not r... |
There is a similar question here: https://math.stackexchange.com/questions/3836805/proving-a-certain-implication-in-the-equivalent-formulations-of-hausdorff-spaces/4890244#4890244 but it does not ask for proof of this exact equivalence that I am asking.
My book has the following equivalences for a Hausdorff topolog... |
Here is an image of the proposition:
[![enter image description here][1]][1]
An highlighted below is the part of the proof I'm having trouble with.
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/DDSYD.png
[2]: https://i.stack.imgur.com/3agfb.png
Why is the equation true ... |
Pederson's Analysis now Proposition 4.5.10, Why is this operator equal to $0$? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
>1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
>2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entr... |
I'm reading [the paper](https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8715594)(I can't find one arXiv version of this paper...) and suspect the correctness of one theorem inside. A [hankel matrix](https://en.wikipedia.org/wiki/Hankel_matrix) $H$ is a square matrix in which each ascending skew-diagonal from l... |
Does every positive semidefinite hankel matrix obeys one Vandermonde decomposition? |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
There is a similar question here: https://math.stackexchange.com/questions/3836805/proving-a-certain-implication-in-the-equivalent-formulations-of-hausdorff-spaces/4890244#4890244 but it does not ask for proof of this exact equivalence that I am asking.
My book has the following equivalences for a Hausdorff topolog... |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
I am studying stochastic processes using *An Introduction to Stochastic Modeling* by Pinsky and Karlin. I stuck on this question 3.4.18 in Chapter 3. I would really appreciate if someone could help me with it! **Here is the question:**
> A well-disciplined man, who smokes exactly one half of a cigar each day, buys a... |
In [this][1] question it is proved in the answers that If $f :(a , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+1) - f(x) )=l$ implies $\lim\limits_{x \to \infty }\frac{f(x)}{x}=l$.
This condition looks suspiciously similar to ... |
If $\lim\limits_{x\to\infty}\frac{f(x+1)-f(x)}{g(x+1)-g(x)}=l$ what are sufficient conditions to make $\lim\limits_{x\to\infty}\frac{f(x)}{g(x)}=l$? |
In [this][1] question it is proved in the answers that If $f :(a , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+1) - f(x) )=l$ implies $\lim\limits_{x \to \infty }\frac{f(x)}{x}=l$.
This condition looks suspiciously similar to ... |
I'm currently working on an assignment in my calculus course where we have to prove the formula to calculate area D given in exercise 1.2.14 of the book Introduction to Mathematics of Medical Imaging by Charles L. Epstein. However hard I try to find possible proof of this formula, either it's doing it myself, asking pe... |
Here is the binary operation $ *: \mathbb{R}\times \mathbb{R} \backslash (0,0) $ defined by $ (a,b)(c,d)=(ac-bd,ad+bc) $. My idea is that to show this is a group ($\mathbb{R}\times \mathbb{R} \backslash (0,0), * $), I need to show that $ * $ is well-defined and associative and then show it has an identity and inverse. ... |
How did Artin discover this function? |
In Artin's "Galois Theory" P38, he said the function
$$f(x) = \frac{(x^2 - x + 1)^3}{x^2(x-1)^2}$$ satisfies the properties of $f(x)=f(1-x)=f(\frac{1}{x})$.
> Is the function given by some rational step or just by a flash of insight？
If $f(0)$ is a number, then $f(0) = f(\frac{1}{0})$. So that the domain of de... |
How did Artin discover the function $f(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}$ with the properties $f(x)=f(1-x)=f(\frac{1}{x})$? |
Let's state this question in cokernels first, and hopefully duality will take care of the kernels.
Let $0 \to P \to N \to M \to 0$ be a short exact sequence of modules over a ring $R$. Given free resolutions $0 \to L_n \to L_{n-1} \to \cdots \to L_0 \to N \to 0$ and $0 \to K_m \to K_{m-1} \to \cdots \to K_0 \to P \t... |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
Definition 3.1.1 in page 25 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3:
Now... |
Given a set of Pauli matrices ${X, Y, Z, I}$ and $n$ is the number representing the length of tensor products I want to produce from Paulie matrices. I construct a set of $2n-2$ such matrices, for example, the set $S=\{Z_1Z_2Y_1,Z_3Y_2,Y_3,Y_2\}$ with n=3 and $Y_2 = I\otimes Y \otimes I$ (the same notation with other e... |
How to check if a group of matrices generate a set of matrices? |
In [this][1] question it is proved in the answers that If $f :(a , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+1) - f(x) )=l$ implies $\lim\limits_{x \to \infty }\frac{f(x)}{x}=l$.
This condition looks suspiciously similar to ... |
How to check if a group of matrices generates a set of matrices? |
If $A$ and $B$ aren't disjoint and $A \cup B \neq \Omega$, then is $P(A \cup B) \geq P(A)P(B)$?
My only idea is to use $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ but there's a minus in front of the intersection and the events don't have to be independent. |
To derive the determinant of a square matrix, we can always use elimination to convert it into an upper triangular matrix. For example, in the $M_{2 \times 2}$ case, we can always use elimination to convert the $M_{2 \times 2}$ to an upper triangular matrix. There are two cases to consider: when $a \neq 0$ and when $a=... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain but t... |
I have to show the equivalence of this
Let $f(x) \in F[x]$, and $K / F$ an extension which contains $R_f$, the set of all root of $f(x)$. show the equivalence for a subfield $D \leq K$ :
(a) $D$ is the least element of the set $\left\{E \leq K \mid F \leq E, R_f \subseteq E\right\}$.
(b) $D$ is the min... |
When the boat's speed was 45 km/h, its engine suddenly stopped.
7 minutes after the engine stopped, the boat's speed was 12 km/h.
It is known that on the boat, in the opposite direction to its movement, the frictional force of the water acts, which is proportional to the speed of the boat.
Find the law of th... |
As I learned, for a differential manifold with an affine connection $(M,\nabla)$, its torsion is defined as
$$\begin{aligned}
T:\mathscr{X}(M)\times \mathscr{X}(M) &\rightarrow \mathscr{X}(M)\\
(X,Y) &\mapsto T(X,Y):= \nabla_XY-\nabla_Y X-[X,Y]
\end{aligned} $$
And I can check the multi-linearity.
My ques... |
Why is torsion (curvature) a tensor field? |
2Orthogonal Matrices and Change of Basis
Let be an ordered basis for the vector space . Recall that the coordinate matrix of a vector in is the column vector
If is another basis for , then the transition matrix from to changes a coordinate matrix relative to into a coordinate matrix relative to ,
The que... |
When the boat's speed was 45 km/h, its engine suddenly stopped.
7 minutes after the engine stopped, the boat's speed was 12 km/h.
It is known that on the boat, in the opposite direction to its movement, the frictional force of the water acts, which is proportional to the speed of the boat.
Find the law of th... |
2 Players playing a game to guess a number between 1 and 100 such that the higher of the two guesses wins 100 - max(guess1, guess2). What should be the ideal guess in this game?
My thought is first to assume uniform distribution for the opponent as we dont know anything how they might guess and in turn have a expect... |
The question is:
> Prove that the following statement is true.
>
> For any rectangular $m\times n$ matrix A, $(\mathrm{Nul}( A ))^\perp = \mathrm{Row} (A^T A)$
Now, my understanding of the Row space is that it is simply $\mathrm{Col}\ A^T$ (since it is just the Span of all linear combinations of row vectors of A... |
>Let $(H, \ast)$ be a group, where $H \subseteq (0, \infty)$, which has these properties:
>- $x \in H \Rightarrow \frac{1}{x} \in H$
>- $2023 \in H$, and
>- $x \ast y = \frac{1}{x} \ast \frac{1}{y}$ for any $x, y$.
>
>Prove that $H$ is not an interval.
As far as my attempts go, I haven't been able to do much, a... |
I tried it using the double angle identity
$$\sin{2x}=2\sin x\cos x$$
The answer that I got is
$$\frac{-\cos 2x}{4} +c$$
However I've also tried it using $u$-substitution.
I let $u=\sin x$. Thus obtaining $\cos x$ when differentiating. And cutting the $\cos x$ in $2\sin x\cos x$ out with the $\cos x$ in the ... |
I tried it using the double angle identity
$$\sin{2x}=2\sin x\cos x$$
The answer that I got is
$$\frac{-\cos 2x}{4} +c$$
However I've also tried it using $u$-substitution.
I let $u=\sin x$. Thus obtaining $\cos x$ when differentiating. And cutting the $\cos x$ in $2\sin x\cos x$ out with the $\cos x$ in the ... |
Let's state this question in cokernels first. The case of kernels are easily solved since free modules are projective.
Let $0 \to P \to N \to M \to 0$ be a short exact sequence of modules over a ring $R$. Given free resolutions $0 \to L_n \to L_{n-1} \to \cdots \to L_0 \to N \to 0$ and $0 \to K_m \to K_{m-1} \to \cd... |
I was recently reading about parallel lines which by definition are lines on the same plane which never meet and are equidistant. I came across a piece of text where the concept of parallel lines was also extended to curves which is all right.
My question is that can two concentric circles also be considered as para... |
2 Players playing a game to guess a number between 1 and 100 such that the higher of the two guesses wins $100 - \max(guess_{1}, guess_{2})$. What should be the ideal guess in this game?
My thought is first to assume uniform distribution for the opponent as we don't know anything how they might guess and in turn hav... |
Do you know modular arithmetic? This problem is much clearer to understand in that setting (it becomes obvious why the exponent $m$ is being taken as odd), while it seems totally opaque and strange as a plain induction problem. |
Consider $\displaystyle(\ell_2, \lVert\cdot\rVert_\star), \lVert x\rVert_\star = \sum\limits_{k=1}^{\infty}\frac{|x(k)|}{k}$. What is its dual space? Is this space reflexive?
My idea is to consider $\displaystyle\varphi: (\ell_2, \lVert\cdot\rVert_\star) \rightarrow (\ell_1, \lVert\cdot\rVert_1), \varphi(\{x(k)\}_{k... |
A corollary on page 287 of Hungerford's Algebra is
> Corollary 6.13. If $F$ is an extension field of $E$ and $E$ is an extension field of $K$, then $$[F:E]_s[E:K]_s=[F:K]_s\mbox{ and } [F:E]_i[E:K]_i=[F:K]_i.$$
Here $[F:K]_s$ means the separable degree of $F$ over $K$ and $[F:K]_i$ the inseparable degree of $F$ o... |
For this question I am asked to find the *radius and centrepoint* of the
circle of curvature for the following function:
> −7.87e^(2.65x)
I calculate the radius correctly with the formula: R =1/ρ
which is:
R = 0.9838228573
[working out for radius][1]
However, when it comes to finding *the centrepoint*... |
For this question I am asked to find the *radius and centrepoint* of the
circle of curvature for the following function:
> −7.87e^(2.65x) at the point when x = -1.25
I calculate the radius correctly with the formula: R =1/ρ
which is:
R = 0.9838228573
[working out for radius][1]
However, when it comes ... |
In a paper by Frink (1987), it was shown that if $(x,y,z)$ are solutions to the equation $x^2+y^2 = z^2 +1$ and $(a,b,c)$ are Pythagorean triples i.e, solutions to $a^2 + b^2 = c^2$ then:
$$(x,y,z) = (x+a, y+ b, z+c) \\
(x', y',z') = (x'+ a, y'+b, z'+c) $$
where $x+ x' = a, y+y' = b, \mbox{and } z+z' =c$ are ... |
In [this][1] question it is proved in the answers that If $f :[0 , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+1) - f(x) )=l$ implies $\lim\limits_{x \to \infty }\frac{f(x)}{x}=l$.
This condition looks suspiciously similar to ... |
Suppose $n \geq p \geq m$. Let $A \in \mathbb{R}^{m \times n}$ be of full row rank and $B \in \mathbb{R}^{n \times p}$ be of full column rank. If $A B$ is an $m$-by-$p$ matrix of full row rank. Can we have an estimation of $\|(A B)^{\dagger}\|$ with respect to $\|A^{\dagger}\|$ and $\|B^{\dagger}\|$ (and maybe $\|A\|$ ... |
How to solve this limit with e power in both nominator and denominator? |
That means if a one parameter differentiable group $\Phi(x,t):\mathbb{R}^d\times \mathbb{R}\to\mathbb{R}^d$ satisfies $$\Phi(\Phi(x_0,t_1),t_2)=\Phi(x_0,t_1+t_2),\Phi(x_0,0)=x_0$$
holds for $\forall x_0\in\mathbb{R}^d,t_1,t_2\in\mathbb{R}.$ Then is there exist a suitable function $F$ on $\mathbb{R}^d$ s.t. $\dot{\Phi(... |
Is solution's group property is a characteristic property of autonomous differential dynamic systems? |
I'm working through a paper on Proximal Policy Optimization (PPO) and am trying to understand the derivation of the optimal policy probabilities for the off-policy case as expressed in Equation 16. "Off-Policy Proximal Policy Optimization" is the name of this paper, and it can be found easily online. The derivation u... |
In a paper by Frink (1987), it was shown that if $(x,y,z)$ are solutions to the equation $x^2+y^2 = z^2 +1$ and $(a,b,c)$ are primitive Pythagorean triples i.e, solutions to $a^2 + b^2 = c^2$ and $\gcd(a,b,c) = 1$ (pairwise relatively prime), then:
$$(x,y,z) = (x+a, y+ b, z+c) \\
(x', y',z') = (x'+ a, y'+b, z'+c)... |
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