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Given vector bundles $E$ and $F$ over a smooth manifold $M$ do we have an isomorphism $$\Gamma(\text{Hom}(E,F)) \cong \text{Hom}(E,F)?$$
That is, if I have a bundle map $E \to F$ do I obtain a section of the bundle $\text{Hom}(E,F)$ and vice versa? I see this being used in multiple places, but I have not found a pro... |
There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted:
$$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg[\underbrace{\ln\left(e^\frac{e^{it}+1}c-b\right)}_v\bigg] \tag{0}$$
as... |
Given vector bundles $E$ and $F$ over a smooth manifold $M$ do we have an isomorphism $$\Gamma(\text{Hom}(E,F)) \cong \text{Hom}_{C^\infty(M)}(E,F)?$$
That is, if I have a smooth bundle map $E \to F$ do I obtain a section of the bundle $\text{Hom}(E,F)$ and vice versa? I see this being used in multiple places, but I... |
It is well-known ghe notion of Joyal's combinatorial species. In his treatment, Joyal assumes finiteness. In fact he works with the groupoid of finite sets. The underlying idea is to capture the notkon of a structure on s finite ground set and to work "combinatorially" with. Ok, finiteness is essential for the Joyal's ... |
There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted as it came across in research:
$$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg[\underbrace{\ln\left(e^\frac{e^{it}+1}c-b\r... |
Let the vertices of the pyramid be $A,B,C,D,V$, where $ABCD$ is the base and $V$ is the apex. Using coordinate geometry, and letting the given sides of the base be $a = AB, b = BC, c = CD, d = DA $, we can assign $A,B,C,D$ as follows
$ A = (0,0,0) , B= (a, 0, 0), C= (x_1, y_1, 0), D = (x_2, y_2, 0) $
with $y_1, ... |
I was solving an apparently simple equation but then it turned out I missed one of the solutions that somebody else find using another method.
$\cos x=\sin x$ was the problem.
I tried the associated angle method, so $\sin(x)=\cos\left(\frac{\pi}{2} - x\right)$.
Performing the substitution I could equal the arg... |
By the multinomial theorem we have that
$$(a + b + c)^n = \sum_{0 \leq i \leq j \leq n} \frac{n!}{i! (j-i)! (n-j)!} a^i b^{j-i} c^{n-j},$$
and so
$$\sum_{0 \leq i \leq j \leq n} \frac{1}{i! (j-i)! (n-j)!} a^i b^{j-i} c^{n-j} = \frac{1}{n!} (a+b+c)^n.$$
I am interested, however, in simplifying the summation given ... |
> Found the subspace/base of the $x-y+z=0$. That is vectors e.g. $(0,1,1), (-1,0,1), (2, -1, -1)$. Checked that they are linearly independent:
$$\begin{bmatrix}0 & 1 & 1\\-1 & 0 & 1\\ 2 & -1 & -1\end{bmatrix} = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}$$ so I guess that is the base (and subspace tha... |
By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$-bimodules and the category of cocontinuous, additive endofunctors on the category of $R$-bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its enveloping ring.
The category of cocontinuous, additive functors carries ... |
Let $\mathbb{F}$ be a field, $V=\mathbb{F}[x]$ the vector space of polynomials. Suppose we have *the first Weyl algebra* $A$, the $\mathbb{F}$-subalgebra of $\operatorname{End}_{\mathbb{F}}V$ generated by $z, d\in \operatorname{End}_{\mathbb{F}}V$ and defined by
\begin{equation}
z(f(x)) = xf(x), d(f(x))=f'(x)
\end{e... |
I want to numerically calculate the low temperature limit of the following [Matsubara sum](https://en.wikipedia.org/wiki/Matsubara_frequency)
$$ S(\Omega) = \pi T \sum_{\omega_n} \frac{4\Delta^2+\Omega^2}{s_1 s_2 (s_1 + s_2)},$$ with $\omega_n = \pi T (2n+1)$, $s_1(\omega_n) = \sqrt{\omega_n^2 + \Delta^2}$ and $s_2... |
Newton's 1676 formulas are
$$
{}_2F_1\left(\genfrac{}{}{0pt}{0}{{1}/{2}+a,{1}/{2}-a}{{1}/{2}};\sin^2x\right)\cos x=\cos(2ax),\tag{1}
$$
$$
{}_2F_1\left(\genfrac{}{}{0pt}{0}{a,-a}{{1}/{2}};\sin^2x\right)=\cos(2ax),\tag{2}
$$
$$
{}_2F_1\left(\genfrac{}{}{0pt}{0}{{1}/{2}+a,{1}/{2}-a}{{3}/{2}};\sin^2x\right)\sin x=... |
Any generalizations of ${}_2F_1\left(\genfrac{}{}{0pt}{0}{a,-a}{{1}/{2}};\sin^2x\right)=\cos(2ax)$? |
What set of points on the 256 by 256 integer grid maximes the number of points in its convex hull? |
It is well known that a Manifold is completely determined by its algebra of smooth functions. How can we describe a submanifold from an algebraic point of view? |
What set of points on the 256 by 256 integer grid maximizes the number of points in its convex hull?
For some context, I was given an assignment to test algorithms that find convex hulls of a set of points in 2D. One natural thing to test is when we set the input set such that all the points (size *n*) are convex hu... |
I have a Hermitian matrix that is invertible, that is I can write it as:
$H = U^\dagger D U$,
where $U$ is a unitary matrix, and $D$ is a diagonal matrix with the eigenvalues of $H$, which must be real and distinct for it to be invertible.
If I want the resulting $H$ to have a certain number $d$ of non-zero el... |
Let $M$ be a smooth manifold. We will use the following definitions and results.
> Consider a local chart $\,x=\big(x_1,\cdots,x_n\big):\,U\subseteq M\longrightarrow\mathbb R^n\,$ at a point $p\in U$. The **partial derivative operator** at $p$ is
\begin{align*}
\frac{\partial}{\partial x_i}\bigg|_p:\ C^\infty(p)\,... |
I am working on understanding the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. As part of their definition of equations for a 3D surface, they use a function $F(ξ, \bar{ξ})$ where $ξ$ is "the local complex coordinate on the unit 2-sphere in $^3$ obtained by stere... |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
Chapter 20 Problem 23 of Spivak's *Calculus* asks the user to prove the formula for the *Schwarz second derivative* of $f$ at $a$ using a Taylor polynomial. The prompt reads as follows:
>If $f''(a)$ exists, then \begin{align}f''(a)=\lim_{h \to 0}\frac{f(a+h)+f(a-h)-2f(a)}{h^2} \quad \text{Hint: Use a 2nd-degree Tayl... |
What set of points on the 256 by 256 integer grid maximizes the number of vertices in its convex hull?
For some context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in 2D. One natural thing to test is when we set the input set such that all the vertices (... |
### Question
I don't understand why the three axioms of probability (Kolmogorov axioms) are treated as *axioms*, not as definitions. Perhaps it hints at a larger problem: I don't understand why any axiom is treated as an *axiom* instead of a definition in any mathematical system.
Isn't one of the reasons the thre... |
Why are the probability/Kolmogorov axioms called "axioms"? |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial $aX^2 + bX + c$, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. Then, if $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta = b^2 - 4 a c$ of such a polynomial is positiv... |
### Question
I don't understand why the three axioms of probability (Kolmogorov axioms) are treated as *axioms*, not as definitions. Perhaps it hints at a larger problem: I don't understand why any axiom is treated as an *axiom* instead of a definition in any mathematical system.
Isn't one of the reasons the thre... |
I have a really Naive question.
I was reading [this paper][1] on graph theory where it classifies the edge contraction problem as an NP-hard problem where it cannot be solved in a linear time.
On the other hand, I've been thinking about turning graphs into simplicial complexes first and then applying the edge c... |
Let $M$ be a smooth manifold. We will use the following definitions and results.
> Consider a local chart $\,x=\big(x_1,\cdots,x_n\big):\,U\subseteq M\longrightarrow\mathbb R^n\,$ at a point $p\in U$. The **partial derivative operator** at $p$ is
\begin{align*}
\frac{\partial}{\partial x_i}\bigg|_p:\ C^\infty(p)\,... |
### Question
I don't understand why the three axioms of probability (Kolmogorov axioms) are treated as *axioms*, not as definitions. Perhaps it hints at a larger problem: I don't understand why any axiom is treated as an *axiom* instead of a definition in any mathematical system.
Isn't one of the reasons the thre... |
### Question
I don't understand why the three axioms of probability (Kolmogorov axioms) are treated as *axioms*, not as definitions. Perhaps it hints at a larger problem: I don't understand why any axiom is treated as an *axiom* instead of a definition in any mathematical system.
Isn't one of the reasons the thre... |
### Question
I don't understand why the three axioms of probability (Kolmogorov axioms) are treated as *axioms*, not as definitions. Perhaps it hints at a larger problem: I don't understand why any axiom is treated as an *axiom* instead of a definition in any mathematical system.
Isn't one of the reasons the thre... |
A concrete category is a pair $(C,U)$ such that $U:C\to \mathbf{Set}$ is a faithful functor. It means that for any if $Y,Z\in \mathrm{ob}(C)$, and a function between the sets $f:U(Y)\to U(Z)$ - if exists a morphism $\phi :Y\to Z$ with $U(\phi)=f$, then it's unique.
Namely, given a function $U(Y)\to U(Z)$, we can say... |
Consider a 2-dimensional rectangular plane sheet of paper $S \subset \mathbb{R}^2$, then push both small sides towards the middle to form a Gaussian-like surface, that we call $D$, see the figure below for only one wave. Unless I am mistaken, this is a developable surface since its Gaussian curvature is everywhere zero... |
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following [$\erf^{-1}(z)$ series](https://www.wolframalpha.com/input?i=series+expansion+inverf%28x%29):
>$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via [Lagrange inversion](https://en.m.wikipedia.org/... |
A concrete category is a pair $(C,U)$ such that $U:C\to \mathbf{Set}$ is a faithful functor. It means that for any $Y,Z\in \mathrm{ob}(C)$, and a function between the sets $f:U(Y)\to U(Z)$ - if exists a morphism $\phi :Y\to Z$ with $U(\phi)=f$, then it's unique.
Namely, given a function $U(Y)\to U(Z)$, we can say th... |
Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits is easy to settle as the following.
> - The last $k$ digits of $a^n$ are given by $a^n\pmod{10^k}$, therefore the sequence of the $k$ last digits is eventually periodic and does not depend on subseque... |
This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $GL(N,\mathbb{C})$. In particular I am dealing with $f:S^1\to GL(N,\mathbb{C})$ and it seems to me that since $\pi_1(GL(N,... |
From @pjs36 's answer to this linked question: https://math.stackexchange.com/questions/1252352/nice-embedding-of-the-permutohedron-of-order-n-in-mathbb-rn-1?rq=1
> [The] $k$-dimensional faces of the permutohedron $P_n$ are in bijection with
> chains of subsets of $[n] = \{1, 2, \ldots, n\}$ of length $n - k$, if
... |
This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $GL(N,\mathbb{C})$. In particular I am dealing with $f:S^1\to GL(N,\mathbb{C})$ and it seems to me that since $\pi_1(GL(N,... |
Is the the reverse implication true for the Root of a simple Lie algebra? |
In book called *Methods of theoretical physics* from Morse and Feshbach, there is identity, which I wanted to prove ($P_n(x)$ are Legendre polynomials):
$$\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; P_n(x) =(2\cosh u - 2x)^{-1/2} \;$$
I did the following:
$$\left( 2\cosh u - 2x \right)^{-1/2} = \left( 2\frac{e^... |
From @pjs36 's answer to this linked question: https://math.stackexchange.com/questions/1252352/nice-embedding-of-the-permutohedron-of-order-n-in-mathbb-rn-1?rq=1
> [The] $k$-dimensional faces of the permutohedron $P_n$ are in bijection with
> chains of subsets of $[n] = \{1, 2, \ldots, n\}$ of length $n - k$, if
... |
I'm currently reading Spivak's Calculus and I'm having trouble understanding the example with the limit of Thomae's function. The function
\begin{align}
f(x) = \begin{cases}
0 & \text{$x$ irrational, $0 < x < 1$}\\
\frac{1}{q} & \text{$x = p/q$ in lowest terms, $0 < x < 1$}
\end{cases}
\end{align}
approaches $0... |
It is known that the pdf $f_{ac}$of the product of two independent random variables $a$ and $c$ with standard normal distribution is (see [here][1]):
$$f_{ac}(t)=\frac{1}{\pi}K_0(|t|), \quad t\in \mathbb R$$
where $K_0(t)$ denotes [the modified Bessel function of the second kind][2].
Then, the probability can... |
Let $M$ be a smooth manifold. We will use the following definitions and results.
> Consider a local chart $\,x=\big(x_1,\cdots,x_n\big):\,U\subseteq M\longrightarrow\mathbb R^n\,$ at a point $p\in U$. The **partial derivative operator** at $p$ is
\begin{align*}
\frac{\partial}{\partial x_i}\bigg|_p:\ C^\infty(p)\,... |
I am trying to understand a statement in Andrew Snowden's [Notes](https://www.mit.edu/~NivenT/assets/pdf//UMich_679_Notes.pdf) regarding the representability of the functor $F_{\Gamma(3)}$ (Section 14.1).
Here, $F_{\Gamma(N)}(S)= \{isom \; classes \; of \; (E,(P,Q)) \; over \; S\}.$
In Section 14.1, the notes ... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
>Let $(a_n)$ and $(b_n)$ be two real sequences. Suppose that $(a_n)$ is a subsequence of $(b_n)$ and $(b_n)$ is a subsequence of $(a_n)$. Does it follow that they are the same sequence?
I am not sure, I think if $ (a_n) $ and $ (b_n) $ are real sequences such that $ (a_n) $ is a subsequence of $ (b_n) $ and $ (b_n) ... |
Does it follow that $(a_n)$ and $(b_n)$ are the same sequence? |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
A concrete category is a pair $(C,U)$ such that $U:C\to \mathbf{Set}$ is a faithful functor. It means that for any $Y,Z\in \mathrm{ob}(C)$, and a function between the sets $f:U(Y)\to U(Z)$ - if exists a morphism $\phi :Y\to Z$ with $U(\phi)=f$, then it's unique.
Namely, given a function $U(Y)\to U(Z)$, we can say th... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
Recently, I am exposed to SDE, I still did not fully get the concepts. I want to solve nonlinear undamped oscillator
$$\frac{d^2x}{dt^2}=-\alpha^2 g(x) + f(t)$$
where $g(x)= sin(x),\ cos(x),\ exp(ix),....$ and f(t) is force due to diffusion where $ <f(t)>=0$ and $<f(t)f(\tau)>=2D\delta(t-\tau)$.
I have the following... |
> Can we expect that certain inequalities hold in tangent spaces of any
> Riemannian $N$-manifold if it holds in $\mathbb{R}^N$?
For example, take the Cauchy-Schwarz inequality, for $x,y\in\mathbb{R}^N$ we have $$\langle x,y\rangle\leq|x||y|$$ Now if we consider some Riemannian $N$-manifold, say $(M,g)$, we can def... |
On page $30$ in Hirsch's Differential Topology one can find the sentence
> These are images of embeddings, and should be "submanifolds".
This would imply that the definition of $C^r$-submanifold of a $C^r$-manifold $M$ (possibly with boundary) should be that it is a subset $N \subseteq M$, such that there is a $C^... |
Contradicting definitions for submanifold in Hirsch Differential Topology? |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
On page $30$ in Hirsch's Differential Topology one can find the sentence
> These are images of embeddings, and should be "submanifolds".
This would imply that the definition of $C^r$-submanifold of a $C^r$-manifold $M$ (possibly with boundary) should be that it is a subset $N \subseteq M$, such that there is a $C^... |
The original title of my post was
<h3>Why is it so hard for mathematicians to say <i>exactly one</i>?</h3>
The proof under critique seems to be an elaborate way of saying: given a function $s$ let $a=s\left(x\right)$ and $b=s\left(y\right)$ then $x=y\implies a=b.$ Which is the definition of a function.
The w... |
I attached the figure I am coloring below.
The long sides of the figure are one side and not two short sides. We do not count reflections as indistinguishable
I was trying to solve it for using casework: When the cross in the middle is all one color, and the short sides are both the same color, there are 3*3=9 ... |
I attached the figure I am coloring below.
The long sides of the figure are one side and not two short sides. We do not count reflections as indistinguishable
I was trying to solve it for using casework: When the cross in the middle is all one color, and the short sides are both the same color, there are 3*3=9 ... |
This doesn't answer the question that you asked, but if we let ${\rm Perf}(\le n)$ be the number of isomorphism classes of perfect groups of order up to $n$, then we have the bounds
$$n^{l(n)^2/108-cl(n)} \le {\rm Perf}(\le n) \le n^{l(n)^2/48 + l(n)},$$
where $c$ is a constant and $l(n)= \log_2(n)$. This is
proved ... |
Consider $n$ sets with respective cardinalities $c_1$, $\ldots$, $c_n$. How many possible Venn diagrams of these set are there? I didn't find with Google. |
The Pearson mean is a weighted mean defined as
$$m_t = \frac {\sum_{i=0}^t w_ix_i} {\sum_{i=0}^t w_i}$$
The exponential mean weight average (EMWA) is the following recurrence where $0 < \alpha <1$.
$$m_t = (1-\alpha)m_{t-1} + \alpha x_t = m_{t-1} + \alpha(x_t - m_{t-1})$$
Is this recurrence a Pearson mean w... |
Is the exponential mean weight average (EMWA) a Pearson mean? |
I teach a 5th grade class geometry, and I came up with the following alternative proof (?) to show that the internal angle sum of a triangle is $180^\circ$. I remember reading that this result is equivalent to the [parallel postulate](https://en.wikipedia.org/wiki/Parallel_postulate), however I can't see where I have u... |
Every extension of R that is a subfield of C is finite. This statement is false and I have a couple of counter examples, are they correct?
1) Consider the field extension $\mathbb{R}(x)$, the field of rational functions over $\mathbb{R}$. This field contains infinitely many transcendental elements, including the var... |
I have short question regarding the universal property of the Clifford algebra. Suppose $(V,q)$ is a quadratic $\mathbb{F}$-vector space and $(\mathrm{Cl}(V,q),j)$ is the Clifford algebra. Recall that $\mathrm{Cl}(V,q)$ is an associative and unital algebra and $j:V\to\mathbb{F}$ is a linear map such that $j(v)^{2}=-q(v... |
Gerry has given a nice hint in the comments. First, note that apart from $(0,0)$, there is no solution where one of the variables is 0. Write the first equation as
$$3x^2y+3y^3+6x^3+6xy^2 -(3y+6x) = x$$
$$\iff (x^2+y^2)(3y+6x) - (3y+6x) = x$$
$$\iff (x^2+y^2-1)(3y+6x)=x \tag{1}$$
Similarly for the second equation
... |
I teach a 5th grade class geometry, and I came up with the following alternative proof (?) to show that the internal angle sum of a triangle is $180^\circ$. I remember reading that this result is equivalent to the [parallel postulate](https://en.wikipedia.org/wiki/Parallel_postulate), however I can't see where I have u... |
Let the grid consist of r rows and k columns. Two players take turns moving a piece to an adjacent square (no diagonal moves). Once a square has been visited it cannot be visited again. The piece starts in the top left square, this square can therefore not be visited again. The loser is the player that does not have a ... |
Consider two functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $g: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$. Let $g$ be continuously differentiable with respect to $y$.
I would like to define $f$ as follows: $f(x)=y$, where $y\in\mathbb{R}^m$ is such that
$$
\nabla_y g(x,y) = \frac{\partia... |
> How many compositions of $n$ are there where each part is at least $m$?
**My attempt:**
So far I have found the generating function and have that the desired number of compositions is
$$
[x^{n}]\frac{1-x}{1-x-x^{m}}.
$$
But I am unsure how to proceed, I have made a few attempts to transform this (for exa... |
In the book Introduction to Partial Differential Equations by Folland, he states the claim
>> Let $ \Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $ S$. The Green's function $G$ for $\Omega$ exists and for each $x \in \Omega , G(x, \cdot) \in C^\infty(\bar{\Omega} \backslash \{x\})$.
Then he... |
Let $(X_i)_{i \in \mathbb{N}} \sim \mathcal{N}(0,1)$, where all variables are i.i.d. For the random vector
$$\mathbf{X} = \left [ X_1, \ \ \ \ X_1 + X_2, \ \ \ \ X_2 + X_3, \ \ \ \ \cdots \ \ \ \ X_{n-2} + X_{n-1}, \ \ \ \ X_{n-1} \right ]^T,$$
how can I find $\mathbb{P}(\mathbf{X} \geq 0)?$ In other words, th... |
I am reading the book *Measure and Integration: An introduction to Real Analysis* by Richard L.Wheeden, Antoni Zygmund, second edition. In chapter 5, the authors give a definition of a distribution function (p. 97):
>It turns out that there is a remarkably simple and useful representation of Lebesgue integrals over su... |
In the book Introduction to Partial Differential Equations by Folland, he states the claim
>> Let $ \Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $ S$. The Green's function $G$ for $\Omega$ exists and for each $x \in \Omega , G(x, \cdot) \in C^\infty(\bar{\Omega} \backslash \{x\})$.
Then he... |
Calculate a integral of :
$$I = \int \int_{V} y^{-2} d \lambda_2(x,y)$$
where: $V = \{ x^2 + y^2 > 1, |x| < \frac{1}{2}, y > 0 \}$
-----------------
Using help provided by @geetha290krm I was able to write the solution.
[![][1]][1]
From the drawing I see that it would be appropriate to integrate $ \f... |
Given a matrix space $\mathbb{R}^{m\times n}$, a vector norm $\|\cdot\|_p$ in $\mathbb{R}^{n}$ and a vector norm $\|\cdot\|_q$ in $\mathbb{R}^{m}$ with $p,q \ge 1$. The matrix norm induced by the two vector norms is defined as $$\| A \|_{p,q} = \sup{\frac{\|Ax\|_q}{\|x\|_p}}$$
The Frobenius norm is defined as $$\| A \... |
I have this problem as follows:
If $f: X \rightarrow Y$ and $g: Y \rightarrow X$ are homotopy inverses, the homotopy $gf \sim id_{X}$ gives a path $\gamma$ in X from $gf(x_0)$ to $x_0$. Prove that the induced homomorphism $g_{\ast}f_{\ast}$ is equal to the change-of-basepoint isomorphism $\beta_{\gamma}$.
So I k... |
I am trying to understand a statement in Andrew Snowden's [Notes](https://www.mit.edu/~NivenT/assets/pdf//UMich_679_Notes.pdf) regarding the representability of the functor $F_{\Gamma(3)}$ (Section 14.1).
Here, $F_{\Gamma(N)}(S)= \{isom \; classes \; of \; (E,(P,Q)) \; over \; S\}.$
In Section 14.1, the notes ... |
The stability of discrete systems can be understood by partial-fraction expansion of the discrete transfer function and finding the region of conversion for each divided fraction. This derives us into this conclusion that in order to be stable, the system poles must remain inside the unit circle within the $z$ plane in... |
Consider the set $V$ consisting of all infinite sequences $a = (a_0, a_1, \ldots)$ where each $a_i \in \mathbb{R}$ and $a_0 \neq 0$. How can we demonstrate that the operation $(a + b) + c = a + (b + c)$ is associative for all $a, b, c \in V$?
$a + b = (a_0b_0, a_0b_1 + a_1b_0, \ldots)$ and it is equal to: $(a + b)_... |
What set of points on the 256 by 256 integer grid maximizes the number of vertices in its convex hull?
For some context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in 2D. One natural thing to test is when we set the input set such that all the vertices (... |
I have short question regarding the universal property of the Clifford algebra. Suppose $(V,q)$ is a quadratic $\mathbb{F}$-vector space and $(\mathrm{Cl}(V,q),j)$ is the Clifford algebra. Recall that $\mathrm{Cl}(V,q)$ is an associative and unital algebra and $j:V\to \mathrm{Cl}(V,q)$ is a linear map such that $j(v)^{... |
Determine the number of sequences $(a_1, a_2, . . . , a_n)$, with elements from the set $\{ 0, 1, 2 \}$, satisfying the condition $|a_k - a_{k-1}| \leq 1$ for $k = 2, 3, . . ., n$.
We know that if I define the number of sequences for a given $n$ as $a_n$, then:
$$a_n = 3a_{n-1} + 2a_{n-1} + 2a_{n-1} = a_n = 7a_{n-1... |
**the question**
We are asking for the perfect squares of four digits whose square root is the sum of the numbers obtained if we separate the first two digits from the last two
**the idea**
let the number be $\overline{abcd}$, where a,b,c,d are digits
$32^2\leq \overline{abcd} \leq 99^2$, which means $32\le... |
In the book Introduction to Partial Differential Equations by Folland, he states the claim
>> Let $ \Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $ S$. The Green's function $G$ for $\Omega$ exists and for each $x \in \Omega , G(x, \cdot) \in C^\infty(\bar{\Omega} \backslash \{x\})$.
Then he... |
Let's assume that I have a sphere of radius $R$ with the spherical coordinates $(r, \theta ,\phi)$
The surface element can be expressed as $dA = R^2 \sin\theta d\theta d\phi$.
By integrating over the whole surface, $A = 4\pi R^2$
Now, let's say I consider a point not at the origin but at a distance $d$ from the c... |
How do I solve this differential-integral equation? |
This is a question that my professor asked to the class.
There are 52 cards with numbers ranging from 1 to 52 on them. There will be 2 players, and their aim is to find a card on the card list in at most 26 tries. They will have time to speak before game starts. After game starts, Player 2 will leave the room and ... |
**What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?**
For some context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in 2D. One natural thing to test is when we set the input set such that all the ... |
Let's assume that I have a sphere of radius $R$ with the spherical coordinates $(r, \theta ,\phi)$
The surface element can be expressed as $dA = R^2 \sin\theta d\theta d\phi$.
By integrating over the whole surface, $A = 4\pi R^2$
Now, let's say I consider a point not at the origin but at a distance $d$ from the c... |
I want to compute the following integral
$$
\int_{\mathbb{R}^3}e^{-(x^2+y^2+z^2)/a^2}\,dxdydz
$$
and I thought that using spherical coordinates could make it easier, since $r^2=x^2+y^2+z^2$. With this in mind, I tried
$$
\int_{\mathbb{R}^3}e^{-(x^2+y^2+z^2)/a^2}\,dxdydz=\int_0^\infty\int_0^{2\pi}\int_0^\pi r^2\si... |
Define $$g=\gcd(2^{k+3}-1,-2×(2^{k+1}-1)).$$
Then we know that there exist some integers $r$ and $s$ such that $$g=r×(2^{k+3}-1)-(2×s)(2^{k+1}-1).$$
Then my question is: Can we find an explict formula for $r$ and $s$ as a function of $k$?
I remark that if $k$ is odd then $g=3$ and if $k$ is even, then $g=1$. S... |
Define $$g=\gcd(2^{k+3}-1,-2×(2^{k+1}-1)).$$
Then we know that there exist some integers $r$ and $s$ such that $$g=r×(2^{k+3}-1)-(2×s)(2^{k+1}-1).$$
Then my question is: Can we find an explicit formula for $r$ and $s$ as a function of $k$?
I remark that if $k$ is odd then $g=3$ and if $k$ is even, then $g=1$. ... |
Random variables on $\mathbb{R}$ with finite expectation form an $\mathbb{R}$-linear space with respect to addition of random variables and multiplication by elements of $\mathbb{R}$. Is there an understandable way to characterize all linear functionals on random variables, i.e. $\mathbb{R}$-linear functions that map r... |
How can one characterize all linear functionals on random variables on $\mathbb{R}$ with finite expectation? |
Let $A,B,C$ be rings. I know if $A \to B \to C$ where $A \to B$ and $B \to C$ are finite, then $A \to C$ is finite (i.e. as modules). My question is, if we know $A \to C$ is finite, does it follow that $A \to B$ or $B \to C$ is finite. Specifically, what if in this case $A$ is Noetherian and $B = A[x]$ is also Noetheri... |
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