instruction stringlengths 12 30k |
|---|
I am reading the book introduction to functional analysis by Christian Clason. I need help with the problem 4.2 from the book:
Let $T:C[a, b] \rightarrow C[a, b],\ [Tx](t)=tx(t) \quad \forall t \in [a, b]$
1. Show T is a continuous linear operator
2. Calculate $\|T\|$
3. Show or give a counterexample for the clo... |
I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear polarized Raman measurement. The intensity is given by:
\begin{align}
I_s \propto |e_s\mathfrak{R}(X)e_i|^2+|e_s\mathfra... |
I have a textbook (*Asymptopia* by Joel Spencer, [p.66](https://i.stack.imgur.com/D8FyH.png)) that states that
$$\frac{(n/2)!(n/2)!}{\left(\frac{n+i}2\right)!\left(\frac{n-i}2\right)!} =\prod_{j=1}^{i/2}\frac{\frac n2+1-j}{\frac n2+j}.$$
The equivalence between the LHS and $\left(\prod_{j=1}^{n/2}j^2\right)\lef... |
I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear polarized Raman measurement. The intensity is given by:
\begin{align}
I_s \propto |e_s\mathfrak{R}(X)e_i|^2+|e_s\mathfra... |
My question is a follow up to the famous problem 'Basketball Scores' from the green book. The question can be found [here][1].
I solved the problem using the factorial approach. I also understand how the recurrence is formed and the answer is verified using induction.
> My question is, **how do I solve for P(n, ... |
I am reading the book introduction to functional analysis by Christian Clason. I need help with the problem 4.2 from the book:
Let $T:C[a, b] \rightarrow C[a, b],\ [Tx](t)=tx(t) \quad \forall t \in [a, b]$
1. Show T is a continuous linear operator
2. Calculate $\|T\|$
3. Show or give a counterexample for the clo... |
In the article you linked ["On quadratic variation of martingales"][1], they specifically mention that they will need to use predictable simple process later on to integrate.
Another reason predictability is handy is because it gives uniqueness (see nice lemma 2 here https://almostsuremath.com/2010/01/03/the-stocha... |
How Do I show this Map is a Surjective Submersion? |
Lately I have been playing around and thinking about this function
$$f\left(x\right)=\sum_{n=2}^{1000}\left(\max\left(1,\operatorname{floor}\left(\frac{x}{n}\right)\right)-\max\left(1,\operatorname{floor}\left(\frac{x-1}{n}\right)\right)\right)$$
I am trying to somehow simplify it with some algebraic manipulation... |
Can this summation be converted to an integral? |
Fix a countable transitive model $M$ and consider the collection $\mathbb{P}$ of all forcing extensions of $M$ (i.e., the generic multiverse of $M$), ordered by reverse containment. What happens if we force with the partial order $\mathbb{P}$? Does it collapse $\omega_1$ or $\mathfrak{c}$? Is it equivalent to $\mathrm{... |
There are $\binom{15}6$ ways to choose any $6$ balls from the $15=4+5+6$ balls that we have.
How many ways to choose $6$ balls in such way that at least $1$ is red and at least $2$ are yellow? Let us calculate the opposite thing, it would be easier.
If we pick $0$ red balls, we have to choose all $6$ balls from y... |
I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear polarized Raman measurement. The intensity is given by:
\begin{align}
I_s \propto |e_s\mathfrak{R}(X)e_i|^2+|e_s\mathfra... |
Here I am trying to do the Cobweb diagram for Tent function through MATLAB, and here is the code:
<pre><code>
function cobweb(f,a,b,x0,N)
x0 = 0.8; % initial point
N = 200; % number of iterations
x = linspace(0,1,1000); % space of x
% define the Tent function
for i=1:numel(x)
if x(i) <= 1/2
f(i) = ... |
My question is a follow up to the famous problem 'Basketball Scores' from the green book. The question can be found [here][1].
I solved the problem using the factorial approach. I also understand how the recurrence is formed and the answer is verified using induction.
> My question is, **how do I solve for P(x, ... |
Let's say we have a vector space $X$ and a function $p:X\to \mathbb{R}^+$ such that $p(ax)=|a|p(x)$ for all $a\in \mathbb{R}$ and $x\in X$, and such that $p(x)=0$ iff $x=0$.
Then $p$ satisfies the triangle inequality (and is therefore a norm) iff it's a convex function. This is where our connection between the tria... |
I found a question in Modern Real Analysis by William P. Ziemer. The question can be found in Section 2.1 Question 1. It goes:
Use the fact that $\mathbb N =$ {$ n: n = 2k$ for some $k \in \mathbb N$} $\cup$ {$ n: n = 2k+1$ for some $k \in \mathbb N$} to prove $c \cdot c = c$.
**What could they be referring to by c?... |
In the article you linked ["On quadratic variation of martingales"][1], they specifically mention that they will need to use predictable simple process later on to integrate.
Another reason predictability is handy is because it gives uniqueness (see nice lemma 2 here https://almostsuremath.com/2010/01/03/the-stocha... |
Alternatively: since $(3\pm\sqrt{-55})/2$ has norm $2^4$, the smallest power of $2$ that can occur as a norm of an imaginary element in the ring, we have $(2)=P^n(\overline{P})^{4-n}$ for $n\in\{0,1,2,3,4\}$; but if both exponents were greater than zero then $2$ would have a proper divisor in the ring which is false. S... |
If I want to turn the "p or q" part of an "if p or q then r" statement into an if then statement using the conditional identity then how do I resolve the nested if then statement that results from that. It seems like it would literally become if if p then q then r. Does that become if p then q and r? |
Prove that the division rings $\mathbb{C}((z;\sigma))$ and $\mathbb{H}((x))$ are not isomorphic. The following statement is needed:
\begin{equation}
\text{any ring automorphism $\phi: \mathbb{R}((x)) \rightarrow \mathbb{R}((x))$ has the form:}
\end{equation}
\begin{equation}
\forall a \in \mathbb{R}, \phi(a)=a;
\... |
I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear polarized Raman measurement. The intensity is given by:
\begin{align}
I_s \propto |e_s\mathfrak{R}(X)e_i|^2+|e_s\mathfra... |
Let be a unital C*-algebra. Is there a unital positive self-map :→ which is invertible (i.e. injective and surjective) but not a ∗-automorphism? If yes, how does appear its Gelfand-Naimark-Segal covariant representation associated to a state on which is invariant under ? Are there examples of such maps which are inv... |
I'm writing my physics bachelor on the Raman scattering effect in solids. I'm trying to evaluate the scattering intensity response to varying polarization angle. This is the well known linear polarized Raman measurement. The intensity is given by:
\begin{align}
I_s \propto |e_s\mathfrak{R}(X)e_i|^2+|e_s\mathfra... |
I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one sinusoid and mix this signal with two sinusoids. This is shown in the following figure:
[![Signal filtering and mixing][1]... |
What is the intuition behind the single-pass algorithm for the corrected sum of squares? |
Suppose you have a deck of 52 cards with 13 cards in each of the 4 suits. What is the probability of drawing 2 cards and getting a pair? If you replace the first card and shuffle before drawing the second card. |
>Show that the number of reduced residues $a \bmod m$ such that $a^{m-1} \equiv 1 \mod m$ is exactly $$\displaystyle \prod_{p \mid m} \gcd(p-1,m-1).$$
Suppose $f(x) = x^{m−1}−1$ and let $m = p^\alpha_1 \cdots p^\alpha_n$ denote the prime factorization of $m.$ If $p$ is prime, $p \mid m$ and $\alpha \geqslant 1,$ the... |
I'm working on a problem where I need to determine the conditions under which $2^n - 46$ is divisible by 4, where $n$ is a non-negative integer. I understand that for any power of 2 greater than $2^2$, the result is always divisible by 4. However, when subtracting 46 from $2^n$, I'm unsure how to systematically approac... |
A simple question, hopefully - consider, for example, the standard BVP defined as follows
$$y^{\prime\prime} + \lambda y = 0\qquad y\left(0\right) = 0 \qquad y\left(2\pi\right) = 0$$
The traditional approach here is to find the eigenvalues and associated eigenfunctions, which would be something like
$$\lambda_... |
So the given subspaces are
$ S = \mathbf{span} \{ (1,0,1,-1)^{T}, (0,1,0,-1)^{T} \} $
and
$ T = \mathbf{span}\{(-1,1,3,1)^{T}\} $
Using identities
$ (S + T)^{0} = S^{0} \cap T^{0}$ and
$ (S \cap T)^{0} = S^{0} + T^{0}$
I can determine basis for $S^{0}$ and $T^{0}$ first.
So for S^{0} I find... |
Prove that the division rings $\mathbb{C}((z;\sigma))$ and $\mathbb{H}((x))$ are not isomorphic. The following statement is needed:
\begin{equation}
\text{any ring automorphism $\phi: \mathbb{R}((x)) \rightarrow \mathbb{R}((x))$ has the form:}
\end{equation}
\begin{equation}
\forall a \in \mathbb{R}, \phi(a)=a;
\... |
I am working on a proof for the statement:
Suppose $x, y \in \mathbb{R}$. If $5|x$ and $5|y$, then $5|xy$
Let's denote the propositions as follows:
- $P: 5|x$
- $R: 5|y$
- $Q: 5|xy$
Logically, this is represented as $(P \land R) \rightarrow Q$.
Now, I am aiming to conduct a contrapositive proof, transfor... |
How to prove the implication $(\neg q \rightarrow (\neg p \lor \neg q))$? |
Let $f:\mathbb{R}^2\to\mathbb{R}$ be nonnegative and convex in the second argument everywhere. That is, $f(p,\cdot)\ge 0$ is convex for all $p\in\mathbb{R}$. Moreover, assume $f(p,p)=0$ for all $p$. A primary example of such function is $f(p,q)=(q-p)^2$. If necessary $f$ can be twice-continuously differentiable.
Now... |
Does a tringle inequality hold for any nonnegative convex function? |
### Question
Prove that $\int f \ln(f) d \mu = \sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$
With $f$ verifying $ \int f d \mu = 1 $ and $ f \cdot \ln(f) $ is integrable, with $ \phi $ a real measurable function s.t. $ f \phi $ is $\mu$-integrable
### Answer
1- $\int f \cdot... |
If $f(x) =\sin^{1200}(x) \ln(1+x)^{500}\arctan^{300}(x)$ how to find $f^{(2000)}(0)$? |
Does a triangle inequality hold for any nonnegative convex function? |
Say we have only defined the rational numbers. If we have two rational numbers, s and r, and we can show that $|s-r|<\epsilon$, for all positive rational $\epsilon$, is that enough to show that $s = r$? Is this a trivial question? or is there a definition of equality or an equivalence relation that would satisfy this p... |
Is the following proof of equality complete? |
I am not even sure I have a question anymore (I will just give up)... in the past month or so I have been researching cubic Bézier curves. The idea was to find a fit through data points, using piecewise Bézier curves and the Euler-Lagrange equation to minimize an "energy" proportional with the square of the length of t... |
Say we have only defined the rational numbers. If we have two rational numbers, s and r, and we can show that $|s-r|<\epsilon$, for all positive rational $\epsilon$, is that enough to show that $s = r$? Is this a trivial question? or is there a definition of equality or an equivalence relation that would satisfy this p... |
Given a finite semigroup $S$ containing a unique idempotent $e$, I can show that every element $s\in S$ has an ''inverse" $s^{-1}\in S$ in the sense that $s s^{-1} = s^{-1}s = e$:
Since $S$ is finite, every element $s$ has an idempotent power $s^{p_s}$ with $p_s\geq1$. It follows that $s^{2p_s}$ is also idempotent. ... |
Is a finite semigroup with a unique idempotent $e$ always a group with unit $e$? |
You will not get better by having others give you the answers. So I will just give you some ideas on what to try.
First of all, when dealing with "prove or disprove" questions, don't just decide on some proof-strategy and get stuck immediately. First get some feeling for the question by fiddling around with examples... |
I saw this challenging problem: If $f(x) =\sin^{1200}(x) \ln(1+x)^{500}\arctan^{300}(x)$ how to find $f^{(2000)}(0)$.
Applying Leibniz rule seems to be the only way to solve this must it quickly became a nightmare finding a pattern:
$$f'(x)=1200 \sin^{1199}(x) \ln(1+x)^{500}\arctan^{300}(x)\cos(x)+\sin^{1200}(x... |
Suppose $(X,Y)$ is a 2-dimensional multivariate Gaussian with respective means $\mu_X,\mu_Y$, variances $\sigma^2_X,\sigma^2_Y$ and covariance $\rho_{XY}$. Note that I'm using the covariance rather than correlation just because it is simpler. Then $(X,Y) \sim \mathcal{N}(\mathbf{\mu},\mathbf{\Sigma})$, where
$$\mathb... |
In my lecture notes, I came across the statement: 'Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix.'
Can someone explain why this is the case?
The context behind this question is finite element method. |
Suppose we have $N$ independently identically distributed (i.i.d.) samples $X_1,\cdots,X_N$ generated from a sub-Gaussian random variable $X \sim \mathbb{P}$. Then by definition there exists the smallest $\sigma >0$ such that
$$\mathbb{E}[\exp(\theta (X - \mathbb{E}[X]))] \le \exp(\theta^2 \sigma^2/2) ~~ \forall \the... |
How to estimate the best variance-proxy of a sub-Gaussian distribution from data? |
Let's say we have a vector space $X$ and a function $p:X\to \mathbb{R}^+$ such that $p(ax)=|a|p(x)$ for all $a\in \mathbb{R}$ and $x\in X$, and such that $p(x)=0$ iff $x=0$.
Then $p$ satisfies the triangle inequality (and is therefore a norm) iff it's a convex function. This is where our connection between the tria... |
The statement of the exercise from the textbook goes:
Let $a \in E$, where $E$ is an algebraic extension of a field $F $ of prime characteristic $p$. Let $m(X)$ be the minimal polynomial of $a$ over the field $F(a^p)$. Show that $m(X)$ splits over $E$, and in fact $a$ is the only root, so that $m(X)$ is a power of ... |
Is there another answer or approach to this question? |
On my lecture notes, I came across the statement: 'Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix.'
Can someone explain why this is the case?
The context behind this question is finite element method. |
Let $f:\mathbb{R}^2\to\mathbb{R}$ be nonnegative and strictly convex in the second argument everywhere. That is, $f(p,\cdot)\ge 0$ is convex for all $p\in\mathbb{R}$. Moreover, assume $f(p,p)=0$ for all $p$. A primary example of such function is $f(p,q)=(q-p)^2$. If necessary $f$ can be twice-continuously differentiabl... |
This is an elementary question asked by the beginner at the frontdoor of higher mathematics.
In Burton's *Elementary Number Theory* ( seventh edition, page 9) I find the following equality written under the (induction) hypothesis that the binomial formula works for $( a+b)^m$:
$$\Large b \space (a + b)^m = b \... |
Let $f:\mathbb{R}^2\to\mathbb{R}$ be nonnegative and strictly convex in the second argument everywhere. That is, $f(p,\cdot)\ge 0$ is convex for all $p\in\mathbb{R}$. Moreover, assume $f(p,p)=0$ for all $p$. A primary example of such function is $f(p,q)=(q-p)^2$. If necessary $f$ can be twice-continuously differentiabl... |
Does a reverse triangle inequality hold for any nonnegative convex function? |
Can the Residue Integration method be used if the interval of integration does not result in a closed contour, or cannot be leveraged due to symmetry?
For example:
<p><span class="math-container">$$\int_{a}^{b}\frac{1}{k+cosx}\,dx \\$$</span></p>
Edit: k>1
Letting a=0, b=2$\pi$, then substituting z=$e^{ix... |
I need to find the number of numbers from 1 to n that will have an odd number of divisors. For example, 4 has 3 divisors - 1,2,4. Knowing that such numbers are formed by raising prime numbers to an even power in their product (2^2 = 4, 3^2 = 9, 2^2 * 3^2 = 36, etc.), I I want to generate these numbers until I reach n a... |
Given a finite semigroup $S$ containing a unique idempotent $e$, I can show that every element $s\in S$ has an ''inverse" $s^{-1}\in S$ in the sense that $s s^{-1} = s^{-1}s = e$:
Since $S$ is finite, every element $s$ has an idempotent power $s^{p_s}$ with $p_s\geq1$. It follows that $s^{2p_s}$ is also idempotent. ... |
Suppose $(X,Y)$ is a 2-dimensional multivariate Gaussian with respective means $\mu_X,\mu_Y$, variances $\sigma^2_X,\sigma^2_Y$ and covariance $\rho_{XY}$. Note that I'm using the covariance rather than correlation just because it is simpler. Then $(X,Y) \sim \mathcal{N}(\mathbf{\mu},\mathbf{\Sigma})$, where
$$\mathb... |
How do I find constants so that the curvature of an oscillating funtion is between them? |
Assume $E$ is an infinite dimensional vector space. Assume $\tau_1, \tau_2$ are two topologies that make $E$ into topological vector spaces. Assume $\tau_1$ is strictly finer than $\tau_2$.
Is there any examples of $E, \tau_1, \tau_2$, such that the dual of $(E, \tau_1)$ is the same (as set of linear functions $E\t... |
Suppose you have a deck of 52 cards with 13 cards in each of the 4 suits (standard poker deck). What is the probability of drawing 2 cards and getting a pair if you replace the first card and shuffle before drawing the second card? |
Suppose you have a deck of 52 cards with 13 cards in each of the 4 suits. What is the probability of drawing 2 cards and getting a pair if you replace the first card and shuffle before drawing the second card? |
Let $f:\mathbb{R}^2\to\mathbb{R}$ be nonnegative and strictly convex in the second argument everywhere (that is, $f(p,\cdot)$ is strictly convex for all $p\in\mathbb{R}$). Moreover, assume $f(p,p)=0$ for all $p$. A primary example of such function is $f(p,q)=(q-p)^2$. If necessary $f$ can be twice-continuously differen... |
Intuitively, a point is not a limit point of a sequence if its smallest distance to an element of the sequence (not including itself) is a positive number. What $n$ minimizes $d=|\frac{4n-3}{2n-1}-5/3|?$
$\frac{4n-3}{2n-1}-5/3=\frac{12n-9-5(2n-1)}{3(2n-1)}=\frac{2(n-2 )}{3(2n-1)}$
$\frac{2(n-2 )}{3(2n-1)}=\fra... |
All the divisors of a natural number $n$ are split into pairs in the following way: $a$ and $b$ go in pair if $ab=n$. Since all the divisors are in pairs, there is an even number of them.
When would this reasoning not work? When $a$ and $b$ are the same number. So $n=a^2$. Hence squares have always odd number of div... |
Well, the title says it all ... I read that given a family of boolean subalgebras, their union is not in general a boolean algebra except if the family is directed ... What is an example of a family of boolean subalgebras whereby their union is not a boolean algebra ? |
Given a finite semigroup $S$ containing a unique idempotent $e$, I can show that every element $s\in S$ has an ''inverse" $s^{-1}\in S$ in the sense that $s s^{-1} = s^{-1}s = e$:
Since $S$ is finite, every element $s$ has an idempotent power $s^{p_s}$ with $p_s\geq1$. It follows that $s^{2p_s}$ is also idempotent. ... |
Given a finite semigroup $S$ containing a unique idempotent $e$, I can show that every element $s\in S$ has an ''inverse" $s^{-1}\in S$ in the sense that $s s^{-1} = s^{-1}s = e$:
Since $S$ is finite, every element $s$ has an idempotent power $s^{p_s}$ with $p_s\geq1$. It follows that $s^{2p_s}$ is also idempotent. ... |
Is a finite simple semigroup with a unique idempotent $e$ always a group with unit $e$? |
All of the questions I found are about literally covering "What is the most effective way to cover a circular pot with a square lid etc..."
Here's an example to show what I mean: I have a circular farm of 10,000 sq m in area. I have a crop duster that can cover a ten meter wide area (10 meter wingspan) while travelin... |
All of the questions I found are about literally covering "What is the most effective way to cover a circular pot with a square lid etc..."
Here's an example to show what I mean: I have a circular farm of 10,000 sq m in area. I have a crop duster that can cover a ten meter wide area (10 meter wingspan) while travelin... |
Given a finite semigroup $S$ containing a unique idempotent $e$, I can show that every element $s\in S$ has an ''inverse" $s^{-1}\in S$ in the sense that $s s^{-1} = s^{-1}s = e$:
Since $S$ is finite, every element $s$ has an idempotent power $s^{p_s}$ with $p_s\geq1$. It follows that $s^{2p_s}$ is also idempotent. ... |
Is a finite semigroup with a unique idempotent $e$ always a group with unit $e$? |
Suppose I have a sequence of positive random vectors $\vec{X}_N$ of fixed length $l$.
That is, $\vec{X}_N = (x_N^{(1)}, x_N^{(2)},\cdots, x_N^{(l)})$ where each entry $x_N^{(i)} > 0$.
Suppose I further have that $\mathbb{E}\left[ \sum_{i=1}^N \left( \frac{(x_N^{(i)}-1)^2}{x_N^{(i)}}\right) \right] = O(1/N)$.
It... |
Suppose I have a sequence of positive random vectors $\vec{X}_N$ of fixed length $l$.
That is, $\vec{X}_N = (x_N^{(1)}, x_N^{(2)},\cdots, x_N^{(l)})$ where each entry $x_N^{(i)} > 0$.
Suppose I further have that $\mathbb{E}\left[ \sum_{i=1}^l \left( \frac{(x_N^{(i)}-1)^2}{x_N^{(i)}}\right) \right] = O(1/N)$.
It... |
Suppose I have a sequence of positive random vectors $\vec{X}_N$ of fixed length $l$.
That is, $\vec{X}_N = (x_N^{(1)}, x_N^{(2)},\cdots, x_N^{(l)})$ where each entry $x_N^{(i)} > 0$.
Suppose I further have that $\mathbb{E}\left[ \sum_{i=1}^l \left( \frac{(x_N^{(i)}-1)^2}{x_N^{(i)}}\right) \right] = O(1/N)$ so the... |
I wish to compute the integral of
$$\displaystyle \int_0^{\infty} \dfrac{1}{x^2} \exp{\left(-\dfrac{a}{x}-bx\right)} dx$$
I found something similar in Gradshteyn and Ryzhik 7th Ed. p336 (Sec 3.324) in terms of the modified Bessel function of the second kind $K_\nu (x)$ but it didn't include the factor of $\dfrac{... |
I'd like to ask a general question and then a more specific question about extending a specific property of the reals to the complex numbers.
What is the general structure of extending a proof to the complex numbers? Logically, in my opinion, the complex numbers should have some holes that just simply don't follow t... |
The following clip is taken from *Cantor: Detour through Infinity* (Davis, 2018, p. 56). When putting odd numbers after even numbers, what should the index for the first odd number ($1$ in the sequence) be, $\omega$-th or $(\omega + 1)$-th? Davis put the former, but I found the latter elsewhere on the internet. I am co... |
$\omega$-th or $(\omega + 1)$-th when putting odd numbers after even numbers? |
The following clip is taken from Chapter 4 - *Cantor: Detour through Infinity* (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ in the sequence) be, $\omega$-th or $(\omega + 1)$-th? Davis put the former, but I found the latter elsewhere on the in... |
What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares? |
I read that given a family of Boolean subalgebras, their union is not in general a Boolean algebra except if the family is directed. What is an example of a family of Boolean subalgebras whose union is not a Boolean algebra? |
The following clip is taken from Chapter 4 - *Cantor: Detour through Infinity* (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ in the sequence) be, $\omega$-th or $(\omega + 1)$-th? Davis put the former, but I found the latter elsewhere on the In... |
Unit circle trigonometric functions are defined on the graph $x^2+y^2=1$. Hyperbolic trigonometric functions are defined on the hyperbola $x^2-y^2=1$.
If I would want to define other trigonometric functions on a completely different graph,
a) What properties would be useful to preserve? I'm not an expert on t... |
This might seem like a very vague question, but the details are really confusing me. So, for example, say we are studying the category of $A$-modules $\mathsf{Mod}_A$ where $A$ is a commutative unital ring.
At first, I was confused how $\mathsf{Mod}_A$ is an abelian category, as in the hom-sets having an abelian gro... |
How much data does a category contain? |
I'm having trouble understanding how to calculate the first order derivative of matrix, and was wondering if anyone could help me.
The function with respect to P is
f(P)=*Tr*($XPAP^TPA^TP^TX^T$)=$||XPAP^T||^2_F$
where *Tr* is the trace.
The derivative of *Tr(AP)* is $A^T$, but how to calculate the derivativ... |
> A cone is cut by a plane, making a closed curve. Two spheres, each inscribed into the cone, are tangent to the plane, at points $A$ and $B$ respectively.
>
> Find the point $C$ on the cut line (i.e. the curve formed by the intersection of the cone and plane) such that $CA + CB$ is at an extrema.
>
> --Vladimir ... |
At what points will the cut line of a plane intersecting a cone maximize the distance to tangent spheres? (V. Arnold) |
I'm having trouble understanding how to calculate the first order derivative of matrix, and was wondering if anyone could help me.
The function with respect to P is
f(P)=*Tr*($XPAP^TPA^TP^TX^T$)=$||XPAP^T||^2_F$
where *Tr* is the trace, $A$ is a matrix.
The derivative of *Tr(AP)* is $A^T$, but how to calcul... |
The Axiom of Completeness can be formulated as:
There exists a set $R$ such that:
1. $R$ is an ordered Archimedean field
2. Any nonempty subset of $R$ with an upper bound has a least upper bound.
Recently, I read something that suggested this is logically equivalent to the following:
There exists a set $R$ s... |
Is the classical interpretation of relative entropy wrong? |
First of all, let's split the problem into two parts. (1) What is the correspondence between _position in the sequence_ and _letter_? (2) What is the correspondence between _position in your rectangular table_ and _position in the sequence_?
Of these, #1 is the trickier one. Let's number our sequence starting at 0; ... |
I saw this challenging problem: If $f(x) =\sin^{1200}(x) \ln(1+x)^{500}\arctan^{300}(x)$ how to find $f^{(2000)}(0)$.
Applying Leibniz rule seems to be the only way to solve this must it quickly became a nightmare finding a pattern:
$$f'(x)=1200 \sin^{1199}(x) \ln(1+x)^{500}\arctan^{300}(x)\cos(x)+\sin^{1200}(x... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.