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Does $\int_0^{+\infty}\dfrac{\ln(1+x^n)-\ln(1+x^{n+1})}{(1+x^2)\cdot \ln(x)}\, dx$ converge? How? |
Does $\int_0^{+\infty}\frac{\ln(1+x^n)-\ln(1+x^{n+1})}{(1+x^2)\ln(x)}\, dx$ converge? How? |
Assume $X$ is a compact set (not necessarily convex). If the following equality holds true?
$$\text{Conv}(X)=\text{Conv}(\text{extr(X)})$$
where $\text{extr}$ means the extreme point of $X$. The definition of the extreme point is that a point $a$ is said to be an extreme point of the set $X$ if the point $a$ cannot ... |
In the solution to the following problem (A Walk Through Combinatorics - Chapter 3, problem 21):
>A round robin chess tournament had 2n participants from two countries, n from each country. There were no two players with the same number of points at the end. Prove that there was at least one player who scored at le... |
Suppose I want to cut a regular dodecagon into $n$ congruent simply-connected pieces. For which $n$ is this possible?
I can cut it into 24 right triangles, by cutting from the center to each vertex and the midpoint of each edge; by gluing consecutive such triangles together, I can also get any factor of $24$.
If ... |
My two cents:
The difficulty of rigorously defining "drawing a sample from a distribution $\mathbb{P}$" comes from the fact that, ultimately, probability theory is only an attempt to *describe* or *model* the reality of a "random phenomenon / experiment", and such a "random phenomenon / experiment" is difficult to f... |
1. Regarding what constitutes the metatheory, Kunen defines it as "basic finitistic reasoning" in his *Foundations of Mathematics* (in I.7.2) and elaborates a bit in his section III:
> We conclude with two additional questions: What exactly *is* the metatheory? Why is it beyond reproach, as we said above, for even c... |
I'm reading through Ebbinghaus' *Mathematical Logic* and more specifically chapter 4 where a sequent calculus is constructed. Below is the rule I need clarification on because, according to my definitely-wrong understanding, it leads to a contradiction.
[![enter image description here][1]][1]
Here, $\Gamma$ is ju... |
1. Regarding what constitutes the metatheory, Kunen defines it as "basic finitistic reasoning" in his *Foundations of Mathematics* (in I.7.2) and elaborates in his section III:
> We conclude with two additional questions: What exactly *is* the metatheory? Why is it beyond reproach, as we said above, for even consist... |
1. Regarding what constitutes the metatheory, Kunen defines it as "basic finitistic reasoning" in his *Foundations of Mathematics* (in I.7.2) and elaborates in his section III:
> We conclude with two additional questions: What exactly *is* the metatheory? Why is it beyond reproach, as we said above, for even consist... |
I am trying to understand an example from Thirring's Classical Mathematical Physics, 2nd ed., p. 14.
[![enter image description here][1]][1]
I want to understand how the condition on $M$ satisfies the condition for the implicit function theorem.
Let $f \colon \mathbb{R}^{(n-m)+m} \rightarrow \mathbb{R}^m$ is $... |
Suppose $V$ is a 2 dimensional real vector space. Then, there exists an linear isomorphism $x:\,V\longrightarrow\mathbb R^2$. For any pairs of bases $\mathcal V=(v_1,v_2)$ of $V$ and $\mathcal B=(e_1,e_2)$ of $\mathbb R^2$, we define
\begin{align}
\big[x\big]_{\mathcal V,\mathcal B} = \begin{bmatrix}
\big[x(v_1)\big... |
1. Regarding what constitutes the metatheory, Kunen defines it as "basic finitistic reasoning" in his *Foundations of Mathematics* (in I.7.2) and elaborates in his section III:
> We conclude with two additional questions: What exactly *is* the metatheory? Why is it beyond reproach, as we said above, for even consist... |
[OEIS A169945](https://oeis.org/A169945) introduces the concept of _thickness_ of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(x)$ with coefficients $0$ or $1$ and degree at most $n$, it denotes by $T(n,k)$, the numbe... |
Suppose $V$ is a 2 dimensional real vector space. Then, there exists an linear isomorphism $x:\,V\longrightarrow\mathbb R^2$. For any pairs of bases $\mathcal V=(v_1,v_2)$ of $V$ and $\mathcal B=(e_1,e_2)$ of $\mathbb R^2$, we define
\begin{align}
\big[x\big]_{\mathcal V,\mathcal B} = \begin{bmatrix}
\big[x(v_1)\big... |
Let $0 < c < 2/(n-1)$. Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, \ k=1,...,n-1$.
$\textbf{Question}$: For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I've been trying to ... |
Let $0 < c < 2/(n-1)$. Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, \ k=1,...,n-1$.
$\textbf{Question}$: For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I've been trying to ... |
This is getting almost unbearably tedious. Let's go step by step.
Do you know how to show the algebraic quadratic equation relating $X$ an $Y$? Just get $x'$ and $y'$ from the two equations you showed in the post using $X$ and $Y$, and plug them into the condition
$$
x'^2 + 2\gamma x' y' + y'^2=1.
$$
Now the... |
Let $0 < c < 2/(n-1)$. Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, \ k=1,...,n-1$.
$\textbf{Question}$: For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I haven't solved the... |
Consider the following ODE system:
> $\frac{dx}{dt}=x^2+y^2-y\\ \frac{dy}{dt}=-2xy-x$
I want to prove that there exists a solution $(x(t),y(t))$ such that \
$\lim_{t\to +\infty}(x(t),y(t))=\lim_{t\to -\infty}(x(t),y(t))=(0,0)$ and $(x(t),y(t))\neq (0,0)$ for any $t$.\
I can prove that when $F(x,y)=6x^2y+3x^2+2y... |
The smallest number which is divisible by first $10$ natural numbers is $2520$, but why $0$ is not the smallest divisible number? |
I'm writing solutions for students who are taking a competition exam and I took problems from old [purple comet][1] competition problems. This problem is the last one from the 2004 middle school contest and says:
> In the addition problem WHITE+WATER=PICNIC each distinct letter represents a different digit. Find the... |
**QUESTION:**
Let $H$ be Hilbert space, $T$ is a bounded operator on $H$, $TT^{\ast }\geqslant T^{\ast }T$, proof $T$ is normal operator($TT^{\ast } =T^{\ast }T$).
I guess this question is missing condition (maybe $T$ is a compact operator?).
I don't know what "$TT^{\ast }\geqslant T^{\ast }T$" means, here's m... |
Solving the Pell equation $ x^2 - (k^2-1)y^2 = 1 $, the general solutions for $y$ are generated by the recurrence relation
$y_{n+2} = 2k\cdot y_{n+1} - y_n, y_0 = 0, y_1 = 1$
which is the same of the [Chebyshev Polynomial of the Second Kind][1] and so the $n$th-solution for $y$ is $y= U_{n}(k) $
Now if if we ... |
Suppose you roll a standard six sided dice twice. What is the probability that you get a composite number both times?
Considering that 4 and 6 are the only composite numbers that we have here, the probability of getting a composite number both times should be 2/6 x 2/6 = 1/9. Right? Well, the correct answer is 1/4. ... |
Solving the Pell equation $ x^2 - (k^2-1)y^2 = 1 $, the general solutions for $y$ are generated by the recurrence relation
$y_{n+2} = 2k\cdot y_{n+1} - y_n, y_0 = 0, y_1 = 1$
which is the same of the [Chebyshev Polynomial of the Second Kind][1] and so the $n$th-solution for $y$ is $y= U_{n}(k) $
Now if if we ... |
Suppose you roll a standard six sided dice twice. What is the probability that you get a composite number both times?
Considering that $4$ and $6$ are the only composite numbers that we have here, the probability of getting a composite number both times should be $2/6 \times 2/6 = 1/9$. Right? Well, the correct answ... |
Since $1.6\color{red}{1}81>\phi$, to show $e^{e^2}>1000\phi$ it is enough to show
$$e^2>3\ln10+\ln2+\ln(1-0.19095)$$
We have
$$e^2>(2.71828)^2>7.38904$$
$$6.90776>3\ln10$$
$$0.69315>\ln2.$$
And utilising the fifth degree Maclaurin polynomial of $\ln(1-x)$
$$-0.21188>\ln(1-0.19095).$$
Check: $$7.38904\stackrel{?... |
Let $X = (X_1, X_2)$ be a non-negative bivariate random vector. Let $Y_i = \phi _i(X_i), i = 1, 2,$ be one-to-one
transformations with $\phi_i(X_i)$ differentiable functions, and let $Y = (Y_1, Y_2)$ be a non-negative bivariate random
vector connected with $X$ by the relation $Y_i = \phi_i(X_i)$.
I want to tran... |
>Let $X$ be a continuous random variable with CDF $F_X$, and $Y=X^2$. Find the CDF of $Y$.
My solution:
$$F_Y(t)=P(Y \leq t) = P(X^2 \leq t) =
P( -\sqrt{t} \leq X \leq \sqrt{t}) = $$
$$P(X \leq \sqrt{t}) - P(X \leq -\sqrt{t}) = F_X(\sqrt{t}) - F_X(-\sqrt{t})$$ (assuming $t \geq 0$, otherwise $F_Y(t)=0$).
Is... |
This is exercise I.2.14 in Hartshorne's *Algebraic Geometry*:
> Define $\psi : \mathbb{P}^{n}\times \mathbb{P}^{m}\longrightarrow \mathbb{P}^{N}$ where $N=rs+r+s$ by $(a_0,\dots,a_r)\times (b_0,\dots,b_s)=(\dots,a_ib_j,\dots)$
>Show that $Im\psi$ is a subvariety of $\mathbb{P}^N$.
Hartshone gave a hint as follow... |
I have two questions related to an extension of the [Stone - von Neumann Theorem][1]:
(1) Are there unitary groups with uncountably many elements indexed over the complex plane?
(2) Can the Stone - von Neumann Theorem be formulated over $\mathbb {C}$ instead of over $\mathbb {R}$, to establish a one-to-one corres... |
Complex extension of the Stone - von Neumann Theorem? |
Let $a<b$ be real numbers. I'm asked to prove that for every $f\in L^2(a,b)$ there exists a unique polynomial $p_n$ of degree less than $n$ such that
$$||f-p||_2\geq ||f-p_n||_2,$$
for every polynomial $p$, $\text{deg}\,p\leq n$. My idea is to use that the set
$$S:=\{p:p\text{ polynomial, }\text{deg}\,p\leq n\}$$
i... |
Is the set of polynomials of degree less than $n$ is closed in $L^2(a,b)$? |
We can also derive by power series / Fourier series in $\mathbb{C}$.
$$
\ln(1+z) = z - \frac{z^2}{2} + \frac{z^3}{3} \pm \cdots, \quad |z| \leq 1 \land z \neq -1.
$$
Substitute $z = e^{2i \theta}$ and take the real part, we get
$$
\ln 2 + \ln\cos\theta = \cos(2\theta) - \frac{\cos(4\theta)}{2} + \frac{\co... |
Is the set of polynomials of degree $\le n$ closed in $L^2(a,b)$? |
Let $a<b$ be real numbers. I'm asked to prove that for every $f\in L^2(a,b)$ there exists a unique polynomial $p_n$ of degree less than or equal to $n$ such that
$$\|f-p\|_2\ge\|f-p_n\|_2,$$
for every polynomial $p$, $\text{deg}\,p\leq n$. My idea is to use that the set
$$S:=\{p:p\text{ polynomial, }\deg p\leq n\}$$... |
Does this epsilon proof work?
$x>K ⟹|()−|<$
$f(x)=\dfrac{1}{x}+\log \left(\dfrac{x}{1+x}\right)\rightarrow 0$ with $x \rightarrow \infty$.
Using this inequality:
$\frac{x-1}{x}\leq \log(x) \leq x-1$
$\dfrac{\dfrac{x}{1+x}-1}{\dfrac{x}{1+x}} \leq \log \left(\frac{x}{1+x}\right) \leq \frac{x}{1+x}-1$
$-\f... |
My question is about what I think is a mistake in Roman Kossak's book *Model theory for beginners* (which is in my opinion a good book; I at least enjoyed a lot what I've read so far).
In chapter 5, Kossak defines the **type** $\operatorname{tp}^{\mathfrak M}(\bar a)$ of a tuple $\bar a \in M^{< \omega}$ in the doma... |
Define an equivalence relation on Cauchy sequences in $\mathbb{Q}$ by $(x_n)\equiv (y_n)$ if $\lim (x_n-y_n)=0$.
The question I am trying to prove is that
>If $(x_n)$ and $(y_n)$ are Cauchy sequences in $\mathbb{Q}$ such that $(x_n)(y_n)\equiv (0)$ then $(x_n)\equiv (0)$ or $(y_n)\equiv (0)$.
*Here, I am cons... |
After finding the exact value of the [integral](https://www.quora.com/How-do-I-evaluate-the-double-integral-displaystyle-int_-0-1-int_-0-1-left-frac-ln-1-xy-1-xy-right-2-dx-dy/answer/Lai-Johnny?ch=17&oid=1477743739641358&share=6743e386&srid=CCOmo&target_type=answer), $$\displaystyle \int_0^1 \int_0^1\left(\frac{\ln (1... |
Any single variable polynomial in gf2 is reduced to first degree x**n=x So all polynomials of n variables will be like x+y+x*y We can transform any 3_sat as a system of polynomial equations in gf2
x or y is equivalent to x+y+XY in gf2
x and y is equivalent to XY in gf2 and by adding some rules of solving for a va... |
I want to know the behavior of the solution without solving the ordinary differential equation for the following enzyme-substrate reactions.
[![enter image description here][1]][1]
**My question**
>Perhaps from the first equation, we expect to be able to say "that the concentration of substrate S does not become z... |
Let $k$ be a field that is finitely-generated over its prime field and $\bar k$ a separable closure. I would like to compute
$$
H^1_{cts}(\mathrm{Gal}_k, \hat{\mathbb{Z}}(1))
$$
Since $\mathbb{Z}(1) = \varprojlim \mu_{n, \bar k}$, we have
$$
H^1_{cts}(\mathrm{Gal}_k, \hat{\mathbb{Z}}(1))
= \varprojlim_n H^... |
After finding the exact value of the [integral](https://www.quora.com/How-do-I-evaluate-the-double-integral-displaystyle-int_-0-1-int_-0-1-left-frac-ln-1-xy-1-xy-right-2-dx-dy/answer/Lai-Johnny?ch=17&oid=1477743739641358&share=6743e386&srid=CCOmo&target_type=answer), $$\displaystyle \int_0^1 \int_0^1\left(\frac{\ln (1... |
Russel's Paradox involves a test whether a set x is an element of itself.
But from a naive set theory perspective, sets have a hiearchy, per definitionem a set cannot be an element of itself. This seems to be a formalisation problem.
What axioms are meant to preclude this? |
My attempt: take the interval (0,1) . Assume (0,1) is countable then there exist a bijection from N to H where $H=(0,1)$. Define elements of H as a monotonically increasing sequence $\{a_i\}_{i=1}^{\infty}$ , $H=\{a_i | i \in N\}$. However $a_1^{2} \in (0,1)$ and is not an element of H since $a_1^{2}<a_1$. Thus $H \neq... |
Is it true that
$$
\tan^{-1}\left(\frac{x}{y}\right)+\tan^{-1}\left(\frac{y}{x}\right)=\text{sgn}(xy)\frac{\pi}{2}
$$
If so, I do I show it? Any hint would be useful. |
Does $\tan^{-1}\left(\frac{x}{y}\right)+\tan^{-1}\left(\frac{y}{x}\right)=\text{sgn}(xy)\frac{\pi}{2}$? |
I am working on a computer vision project with a sphere of calibrated images.
And I would like to (relatively) quickly estimate the center of the sphere by simply computing the intersection of all the rays of the camera direction.
Here is an example of an [image sphere][1].
For each ray, I have a origin $o$ and ... |
Is it true that
$$
\tan^{-1}\left(\frac{x}{y}\right)+\tan^{-1}\left(\frac{y}{x}\right)=\text{sgn}(xy)\frac{\pi}{2}
$$
for $x,y\in\mathbb{R}\backslash \{0\}$? If so, I do I show it? Any hint would be useful. |
First, some generalities from algebraic topology (you can find these in any algebraic topology textbook). Suppose that $M$ is, say, a connected manifold (this works for a much larger class of topological spaces but we will not need this). Then the covering spaces of $M$ are classified (up to an isomorphism) by subgroup... |
>Let $T:\mathcal{H}\rightarrow \mathcal{H}$ be a linear continuous operator between Hilbert spaces and $\{b_i\: |\: i \in I\}$ an orthonormal basis. Prove that if:
>$$\sum_{i\in I}\lVert T b_i\rVert^2<\infty$$ Then, $T$ is compact.
Hilbert spaces are reflexive, as is easily seen from applying Riesz Representation T... |
Kissing number in 8 dimensions is supposedly [240](https://en.wikipedia.org/wiki/Kissing_number), for spheres that are centered around points of E8 lattice.
However, if we simply take an 8-cube, it has 256 vertices. Using each vertex as a center of a sphere, we can put another sphere in the middle of this cube, touc... |
In my scenario, I have a map of 2d points and a set of disjoint circles of radius 5 that are on the map. I'm shooting a laser beam from position (0,0) at some angle until I shoot a circle. The probability of observing the circle is 1 along its whole area.
The goal is to find the probability distribution of the circle ... |
Darboux Theorem: If $f$ is differentiable on $[a,b]$ and $\eta$ between $f'(a)$ and $f'(b)$, then exists $c\in[a,b]$ such that $f'(c)=\eta$.
The common counterexample about the function at $c$ not necessarily continuous is $f(x)=x^2\sin\frac{1}{x}(\mathbb{R}-\{0\}); 0(0)$, $\eta=0$ and $f'(0)=0$ where it's not cont... |
Can Darboux Theorem be stronger by making the function at c continuous? |
In my scenario, I have a map of 2d points and a set of disjoint circles of radius 5 that are on the map. I'm shooting a laser beam from position (0,0) at some angle until I shoot a circle. The probability of observing the circle is 1 along its whole area.
The goal is to find the probability distribution of the circle ... |
Very much late to the party. The Jordan volume/measure/content is defined on the Jordan measurable sets (subsets of $\mathbb{R}^{n}$ whose indicator functions are Riemann integrable). These sets do not form a $\sigma$-algebra because they're not closed under countable unions, as we can see for the classical example $\m... |
Is there a theory of "quadratic" Hamiltonian evolutions on Poisson manifolds? |
I have an expression of the form
\begin{equation}
i\operatorname{tr}(ABC - B^\dagger AC),
\end{equation}
where $A$ and $C$ are Hermitian matrices, but $B$ is not. $B-B^\dagger := i D$, where $D$ is also a Hermitian matrix. It can be seen that the above is real by cyclic permutation of the second term inside the tra... |
I am working on a computer vision project with a sphere of calibrated images.
And I would like to (relatively) quickly estimate the center of the sphere by simply computing the intersection of all the rays of the camera direction.
Here is an example of an image sphere:
[![enter image description here][1]][1]
Fo... |
The calculation is all fine, but it falls short at the end. Setting $\varepsilon$ to be a function of $K$ is a misstep; you don't get to determine what form $\varepsilon$ takes, as it is an arbitrary positive number.
The next step, where you give $K$ as a function of $\varepsilon$ is better, however, there is a smal... |
Can Darboux Theorem be stronger by making the derivative at c continuous? |
Let's look at the problem from another angle. Assume $b_1, b_2, b_3, ...$ is an increasing sequence such that $b_1=1$ and for every $i \in \mathbb N,$ $2^{b_{i+1}}$ has $b_i+1$ digits. **At first glance, this definition may seem strange, but what does that mean? In fact, what it means is the reasoning behind its defini... |
Darboux Theorem: If $f$ is differentiable on $[a,b]$ and $\eta$ between $f'(a)$ and $f'(b)$, then exists $c\in[a,b]$ such that $f'(c)=\eta$.
The common counterexample about $f'$ at $c$ not necessarily continuous is $f(x)=x^2\sin\frac{1}{x}(\mathbb{R}-\{0\}); 0(0)$, $\eta=0$ and $f'(0)=0$ where $f'$ is not continuou... |
I am reading the paper [A Zvonkin's transformation for stochastic differential equations with singular drift and applications](https://www.sciencedirect.com/science/article/pii/S0022039621004174) where the authors introduce the following function spaces:
---
We fix $\theta \in [0,2]$ and $p, q \in [1,\infty]$.
-... |
Let's look at the problem from another angle. Assume $b_1, b_2, b_3, ...$ is an increasing sequence such that $b_1=1$ and for every $i \in \mathbb N,$ $2^{b_{i+1}}$ has $b_i+1$ digits. **At first glance, this definition may seem strange, but what does that mean? In fact, what it means is the reasoning behind its defini... |
Evaluate $ \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi}{2})} \frac{\cos(y)-\sin(2y)}{\cos(x)\cos(y)}$
So far my steps have included: $ = \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi}{2})} \frac{\cos(y)-2\sin(y)\cos(y)}{\cos(x)\cos(y)}$ using the double angle identity
$ = \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi... |
Let $M$ be a type III$_1$ factor and $N$ be the semi-finite von Neumann subalgebra of $M$.
What is the type of relative commutant $N'\cap M$? Is it semi-finite? |
I'm going through the first openstax calculus book and i'm struggling to understand something.
Let me process this limit as I think it goes and then ask the question about it.
Evalulate $$\require{cancel}\lim_{\Theta \to 0}\frac{1-\cos \Theta}{\sin \Theta}$$
>
\begin{align}
\lim_{\Theta \to 0}\frac{1-\cos \The... |
Is there a closed-form solution for the recurrence relation $x_{t+1} = x_{t}^\alpha + 1$, where $\alpha \in (0,1)$? If it were $x_{t+1} = x_{t}^\alpha$, then taking the Log of both sides would make it linear and easy to solve. However, the same thing does not work in the example. Taking the exponent of both sides does ... |
Let's look at the problem from another angle. Assume $b_1, b_2, b_3, ...$ is an increasing sequence such that $b_1=1$ and for every $i \in \mathbb N,$ $2^{b_{i+1}}$ has $b_i+1$ digits. **At first glance, this definition may seem strange, but what does that mean? In fact, what it means is the reasoning behind its defini... |
I am working on Weibel's K-book and have, while meditating phantom maps and weak homotopies, asked myself if in the same way as their is a category $Ho(Top)$ identifying homotopic maps, is their a category $Ho_w(Top)$ identifying weakly homotopic maps? <br>
To recall, two maps are weakly homotopic if they induces th... |
Let $F: I \to \mathscr{C}$ be a connected functor into an $\infty$-category which factors over $\mathscr{C}^{\mathrm{core}} \to \mathscr{C}$. Why is $\operatorname{lim}{F} \simeq F(i)$ for any $i \in I$?
Orally: Take a limit over a diagram of equivalences, then the limit is any of the equivalent objects. It seems ev... |
I figured it out by myself. My question is basically an variation on the question whether a single point is inside a convex hull. This is just an adapted version of [this answer](https://stackoverflow.com/questions/4901959/find-if-a-point-is-inside-a-convex-hull-for-a-set-of-points-without-computing-th/11731437#1173143... |
Let $f,g \in L^{\infty}([0,1],\mathbb{R}^d$ and $F:\mathbb{R}^n \times \mathbb{R}^d \to \mathbb{R}^n$ be a $C^1$ function. Let us consider the two differential equations:
$$
\dot{x}(t)=F(x(t),f(t)), \quad \dot{x}(t)=F(x(t),g(t)), \quad x(0)= a \in \mathbb{R}^n
,$$
for almost every $t\in [0,1]$. Can I deduce necessa... |
While studying complex Lie groups theory, and more generally complex geometry, I've found two different objects which are called "complex tori".
1. Consider the multiplicative group $\mathbb C^*$, of course this is a non-compact complex Lie group. The direct product of $n$ copies of this is of course a complex Lie ... |
I am working on Weibel's K-book and have, while meditating phantom maps and weak homotopies, asked myself if in the same way as their is a category $Ho(Top)$ identifying homotopic maps, is their a category $Ho_w(Top)$ identifying weakly homotopic maps? <br>
To recall, two maps are weakly homotopic if they induces ho... |
While playing around with repeated integrals with my friend, I came across the following integral.
$$\int_{0}^{1} \dfrac{\ln (1-x)}{1+s^2x^2} dx=\dfrac{\arctan s \ln(1+s^2)}{2s}-\sum_{n≧0} \dfrac{(-1)^n}{(2n+1)^2} s^{2n}$$
I solved the above integral in the following way.
$$proof:$$
$$\int_{0}^{1} \dfrac{\ln (1-x)}... |
I figured it out by myself. My question is basically an variation on the question whether a single point is inside a convex hull. This is just an adapted version of [this answer](https://stackoverflow.com/questions/4901959/find-if-a-point-is-inside-a-convex-hull-for-a-set-of-points-without-computing-th/11731437#1173143... |
Let's look at the problem from another angle. Assume $b_1, b_2, b_3, ...$ is an increasing sequence such that $b_1=1$ and for every $i \in \mathbb N,$ $2^{b_{i+1}}$ has $b_i+1$ digits. **At first glance, this definition may seem strange, but what does that mean? In fact, what it means is the reasoning behind its defini... |
Let $X$ be an $n \times n$ random matrix with *i.i.d. standard Gaussian entries*, i.e., the entries $X_{ij}$ are independent standard normal random variables. Given any deterministic unit vectors $u, v \in \mathbb R^n$, calculate the *mean and variance* of $Z_n := u^\top X^\top Xu + \sqrt n v^\top Xv$, and prove the ce... |
I have been reading the solution of the following question:
We can regard $\pi_1(X, x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S^1,s_0) \to (X, x_0).$ Let $[S^1, X]$ be the set of homotopy classes of maps $S^1\to X,$ with no conditions on basepoints. Thus there is a natural map $\Phi: \pi_... |
What exactly is the proof of onto here? |
I assume $N_i \subset B_{\mathfrak{q}}$, rather than $N_i \subset B$?
Any simple tensor of $N \otimes_{A_{\mathfrak{p}}} M_{\mathfrak{p}}$ is of the form $(b/t) \otimes (m/s)$ for $b \in B$, $t \in B-\mathfrak{q}$, $m \in M$ and $s \in A-\mathfrak{p}$, which is just $(b/f(s)t) \cdot (1/1)\otimes (m/1)$, considering ... |
I wonder, if given a graph A represented by an adjacency matrix C and one would invert C, is there a proof that shows what the structure of A^-1 would look like? |
What does the inverse of a graph look like? |
I wonder, if given a graph A represented by an adjacency matrix C and one would invert C, is there a proof that shows what the structure of A^-1 would look like? For example looking at the structure of Molecules using the Hückel matrix. |
Can anyone tell me exactly what is the question that is being asked in English for this equation? I am confused.
Please check the link for the video:
https://youtube.com/shorts/mGDnsbKr7IM?si=3eLrjOzLffTp3SBb
A book costs $1 plus half its price. How much does it cost?
A. $1
B. $1.50
C. $2
Is the answ... |
Can anyone tell me exactly what is the question that is being asked in English for this equation? I am confused.
Please check the link for the video:
https://youtube.com/shorts/mGDnsbKr7IM?si=3eLrjOzLffTp3SBb
A book costs $1 plus half its price. How much does it cost?
A. $1
B. $1.50
C. $2
Is the answ... |
I was playing around with the following series
$$S(x) = \frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+...=\sum_{p\in primes}\frac{1}{x^p}$$
for $x\in\mathbb{R}$ and $|x|>1$.
And it seems, that for all $x\ge2$ the value of this sum is pretty close numerically to the following expression:
... |
Question from an old test:
>The bakery produces doughnuts with weights $W_1 \text{g}, W_2 \text{g}, W_3 \text{g}, \ldots$, where $W_i$ for each doughnut is drawn from a distribution with a finite expected value $\mu$ and variance $\sigma^2$. Let $p_n$ denote the probability that the weight of $n$ doughnuts exceeds $n ... |
Problem:
Solve the following differential equation:
$$ 6x^2y \, dx - (x^3+1) \, dy = 0 $$
Answer:
\begin{align*}
6x^2 \, dx - \dfrac{ (x^3+1)}{y} \, dy &= 0 \\
\dfrac{ 6x^2}{x^3+1} \, dx - \dfrac{ dy}{y} &= 0 \\
2 \ln{\left( x^3+1\right) } - \ln{|y|} &= C_1 \\
\ln{ \left( \dfrac{ (x^3+1)^2}{|y|} \right)... |
Solving a first order differential equation - does it matter where you put the absolute value sign? |
How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?
This equation is called “Gauss's Law” in physics. I seek for a rigorous mathematical proof for it (my original physics textbook doesn't provide a rigorous proof).
Now, I want to describe my question in detail... |
How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously? |
Okay, the information we have:
$Z$~$N(0,1)$
$Z$ is statistic independent of $X$
$Y=\begin{cases}a\left|X\right|&Z\ge 0\\ b\left|X\right|&Z<0\end{cases}$
We know that $X$ is a continuous random variable that $P\left(-1\le X\le 1\right)=1$ and $f_X\left(x\right)<\infty$ and $E[X]=\mu$ and $Var[X]=\sigma^2$
I ... |
Let $M$ be a type III$_1$ factor and $N$ be the non-trivial ($N\neq \Bbb C 1$)semi-finite von Neumann subalgebra of $M$.
What is the type of relative commutant $N'\cap M$? Is it semi-finite? |
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