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**Expression:**
$$(xy)' + (yz)$$
**My attempt:**
1. $(xy)' + (yz)$
2. $(x'+y') + (yz) \quad \textit{After applying de Morgan's Axiom}$
3. $x' + (y' + yz) \quad \textit{After applying 1st Distributive Axiom}$
4. $x'+ (y' + y) + (y'z) \quad \textit{Rewriting}$
5. $x' + 1 + (y'z) \quad \textit{After a... |
Two functors $F: →$ and $G: →$ are _adjoint_ iff there is a family of isomorphisms
\begin{align}
η_{X, A}: (FX, A)\tilde{\to} (X, GA)
\end{align}
which is natural in $X$ and $A$ (i.e., $(F\_, \_)≅(\_, G\_)$ as bifunctors).
In my mind, this definition is “clearly useful”, because I've known it for years and I... |
Suppose we have an underwater robot that can travel at a fixed velocity, v, if there is no current. This robot needs to travel a fixed distance, d. While in the water, the current acts on the robot affecting its velocity. The current varies over time. How do we determine how long it takes for the robot to travel d? Tha... |
So I have come across the following claim in [this paper by B. Poonen](https://math.mit.edu/~poonen/papers/bertini.pdf) (Lemma 2.2): Let $X$ is a smooth quasi-projective scheme over $\mathbb{F}_q$, $P$ a closed point, $\mathfrak{m}$ the ideal sheaf corresponding to $P$ and $Y$ the closed subscheme corresponding to $\ma... |
> Solve the equation:
> $$4x^4 - 36x^3 + 61x^2 + 90x + 25 = 0.$$
This problem was lifted from
[this Youtube Video](https://www.youtube.com/watch?v=M4bKGwrKGxs). I didn't
particularly like the video's approach, so I used my own approach, which I have never
seen discussed anywhere. To understand the approach... |
I am aware that related topics have been discussed in other threads. However, not being an expert, I would like to know the answers to the following questions:\
Let $E$ be a set having n elements ; $ |E|=n.$\
$A_p$ is a subset of $E$ with $p$ elements $(1 \le p \le n)$, chosen uniformly at random from the set of $\b... |
I am new to multivariable calculus and I am trying to learn how surface area double integrals work. I am stuck on how to parameterize this function:
$\int \int_S x dS$ where S is the part of the cylinder $x^2 + 2y^2 = 2$ bounded between the planes $ z = -4$ and $z = y + 4x + 12$.
I first tried to convert to pol... |
How do I parameterize this double integral? |
In
> <cite authors="Cichon, E. A.">_Cichon, E. A._, [**A short proof of two recently discovered independence results using recursion theoretic methods**](https://doi.org/10.2307/2043364), Proc. Am. Math. Soc. 87, 704-706 (1983). [ZBL0512.03028](https://zbmath.org/?q=an:0512.03028).</cite>
the following process is... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> **If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integer... |
For some reason, you have stopped reading Cunningham's "Theoretical Properties of the Network Simplex Method" *just* as it was about to answer your question. We read on:
> **Lemma 1.** Let $T, T'$ be strongly feasible trees, such that $T'$ is a successor of $T$ with entering edge $e$ and leaving edge $f$, and suppos... |
Of course this is a terribly late answer, but I believe that note is an early draft of the book https://bookstore.ams.org/view?ProductCode=GSM/235, Ricci Solitons in low dimensions, by Professor Chow, which is published last year (2023) |
Let $\mathcal{C}$ be a family of subsets of some finite set $X$, and let $\Omega$ be the $\sigma$-algebra generated from $\mathcal{C}$. Let $\mathcal{A}$ be the set of subsets of $S \subseteq X$ such that
1) For any set $C \in \Omega$, either $S \cap C = \emptyset$ or $S \cap C = C$.
2) There is no proper subset of... |
Complementary slackness states that if $x$ and $y$ are solutions to primal and dual respectively, then they satisfy: $x$ and $y$ are optimal solutions to primal and dual respectively if and only if they are feasible to primal and dual respectively and they satisfy the complementary slackness condition.
My professor ... |
**Question**
I am **unsure of the validity or completeness of the logic** in my solution proof to Exercise 3.4.6 of Tao's Analysis I 4th Ed. I am self-teaching and the skills involved in logic, proof, quantifiers, etc are new to me.
> Prove Lemma 3.4.10. (Hint: start with the set $\{0, 1\}^X$ and apply
> the rep... |
There is one part I don't get from the proof of Baby Rudin's theorem 7.32:
[![][1]][1]
The reason we require that the polynomial $\sum_{i=1}^n c_i y^i$ to vanish when $y=0$ because we want this property to hold: if $f(x)=0$ for some $x\in K$, then $g(x)=0$ as well. Is that correct?
[1]: https://i.stack.imgur... |
so recently, i have been thinking about the following question:
if there exist $\frac{{d^n}}{{dx^n}} g(x) = n \frac{{d^n}}{{dx^n}} f(x)$
can we express it as some $h(f(x))$?
so the answer is yes. We can express it as $(x-a)\frac{d}{dx}f(x)$
by transforming the function $f(x)$ into a Taylor series, and then ... |
Let $A \in \mathcal{M}_{n\times n} (\mathbb{R})$ such that $A^2=\alpha A$ for some $\alpha \neq 0$.
Under this assumption, we have by induction that $A^{n}=\alpha^{n-1}A$; $n \geq2$; then:
$$\exp(tA)=I+ tA+\frac{t^2A^2}{2!}+ \dots \ =I+At+\frac{A\alpha t^2}{2!} + \frac{A\alpha^2 t^3}{3!} + \dots = \\ = I+ \frac{1... |
I am studying the following semi-definite problem:
$$\begin{array}{rl} \textrm{Given:} & W \in \mathbb{R}^{n \times n} \\
\textrm{Minimize:} & u_1 + u_2 + \ldots + u_n \\
\textrm{Subject to:} & \forall i \in [n]:\\
& \begin{bmatrix} X & e_i \\ e_i^\intercal & u_i \end{bmatrix} \succeq 0 \\
& (W^\intercal X W)_{i... |
Your (implicit) claim that $$e_1:=(1,1,1,1),\quad e_2:=(0,1,1,1),\quad e_3:=(0,0,0,1),\quad e_4:=(0,1,0,0)$$ make a basis of $\Bbb R^4$ is correct (how would you prove it?), hence for any choice of four vectors $f_1,f_2,f_3,f_4\in\Bbb R^4$, there is a unique linear transformation $T:\Bbb R^4\to\Bbb R^4$ that maps $e_i$... |
Your (implicit) claim that $$e_1:=(1,1,1,1),\quad e_2:=(0,1,1,1),\quad e_3:=(0,0,0,1),\quad e_4:=(0,1,0,0)$$ make a basis of $\Bbb R^4$ is correct (how would you prove it?), hence for any choice of four vectors $f_1,f_2,f_3,f_4\in\Bbb R^4$, there is a unique linear transformation $T:\Bbb R^4\to\Bbb R^4$ that maps $e_i$... |
I am new to multivariable calculus and I am trying to learn how surface area double integrals work. I am stuck on how to parameterize this function:
$\iint_S x dS$ where $S$ is the part of the cylinder $x^2 + 2y^2 = 2$ bounded between the planes $ z = -4$ and $z = y + 4x + 12$.
I first tried to convert to polar... |
I am new to multivariable calculus and I am trying to learn how surface area double integrals work. I am stuck on how to parameterize this function:
$\iint_S x dS$ where $S$ is the part of the cylinder $x^2 + 2y^2 = 2$ bounded between the planes $ z = -4$ and $z = y + 4x + 12$.
I first tried to convert to polar... |
As part of a physics project studying renormalization group flows of scalar field theories, I've come across the following quartic plane curve in the variables $m$ and $n$:
$$
36 + 16 m^4 - 108 n + 105 n^2 - 18 n^3 + n^4 - 8 m^3 (-18 + 17 n) +
m^2 (420 - 468 n + 81 n^2) + m (216 - 408 n + 234 n^2 - 34 n^3)=0.
$$... |
Currently, I am writing my thesis (in molecular biology - not mathematics), and I am puzzled over the results.
I measured an increase in a signal and did a one-phase association fit in GraphPad. Now, I have four curves, and they look as we would expect (attached image).
The bit that puzzles me is: why does the ra... |
Currently, I am writing my thesis (in molecular biology - not mathematics), and I am puzzled over the results.
I measured an increase in a signal and did a one-phase association fit in GraphPad. Now, I have four curves, and they look as we would expect (attached image: https://i.stack.imgur.com/b4q3T.png).
The bi... |
> If $f:[a,b]\to\mathbb{C}$ is continuous,
> $M=\sup\limits_{z\in[a,b]}|f(z)|$ then
> $$\left\vert\int\limits_a^bf(z)dz\right\vert\leq M(b-a).$$
It's easy to prove the inequality. But I don't know how to show the above inequality becomes equality if and only if $f$ is constant.
I just can deduce that if $\leq... |
How to show when the inequality holds? |
Currently, I am writing my thesis (in molecular biology - not mathematics), and I am puzzled over the results.
I measured an increase in a signal and did a one-phase association fit in GraphPad. Now, I have four curves, and they look as we would expect.
[![enter image description here][1]][1]
The bit that puzz... |
In Umemoto's thesis [1] on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind these angles is unclear. Could someone explain why these angles are chosen and their role in the proof?
### Fig.18
... |
Given $$ X_{1}(t) = (\exp(t),0,\exp(t)) ;X_{2}(t)=(2,1,0); X_{3}(t)=(0,2\exp(-t),-\exp(-t))$$
if I'm interested in a LHCC system that has these three functions as its solutions, then there would be only one system satisfying these conditions as there is only one linear transformation with the eigenvalues and associat... |
Instead of looking at the collection $X_1,...,X_n$ we can look at the vector $Y = (X_1,...,X_n)$ for which $\sigma(Y) = \sigma(X_1,...,X_n)$, so it is enough to consider case of single random variable $Y$ (but taking values in some product space $\mathbb R^n$ or any different).
Thus, let us assume that $(\Omega,\mat... |
I have proved that $\phi \in \operatorname{Aut}(\mathbb{D})$ has a fixed point in $\mathbb{D}$ iff its correspondent from $\operatorname{Aut}(\mathbb{H})$ has a fixed point in $\mathbb{H}$, where $\mathbb{D}$ is open unit disk, and $\mathbb{H}$ is upper half-plane.
But I don't know how to show that $\phi \in \opera... |
In Umemoto's thesis [1] on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind these angles is unclear.
For instance, I thought, as its reflection $/theta6$ of bottom left corner of the top tria... |
I have proved that $\phi \in \operatorname{Aut}(\mathbb{D})$ has a fixed point in $\mathbb{D}$ iff its correspondent from $\operatorname{Aut}(\mathbb{H})$ has a fixed point in $\mathbb{H}$, where $\mathbb{D}$ is open unit disk, and $\mathbb{H}$ is upper half-plane.
But I don't know how to show that $\phi \in \opera... |
Currently, I am writing my thesis (in molecular biology - not mathematics), and I am puzzled over the results.
I measured an increase in a signal and did a one-phase association fit in GraphPad. Now, I have four curves, and they look as we would expect.
[![enter image description here][1]][1]
The bit that puzz... |
I have proved that $\phi \in \operatorname{Aut}(\mathbb{D})$ has a fixed point in $\mathbb{D}$ iff its correspondent from $\operatorname{Aut}(\mathbb{H})$ has a fixed point in $\mathbb{H}$, where $\mathbb{D}$ is open unit disk, and $\mathbb{H}$ is upper half-plane.
But I don't know how to show that $\phi \in \opera... |
COMMENT.- It is known that the function $x\to e^x$ **operates** over the algebra $\mathcal{M}_{n\times n} (\mathbb{R})$ (*I don't know how the french word **opère** is translated in the English of Functional Analysis World*). In simplest English, if $A\in\mathcal{M}_{n\times n} (\mathbb{R})$, the matrix $e^A\in\mathcal... |
When one is first introduced to a differential of a real-valued function, it is defined as a linear function that takes tangent vectors to real numbers. That is, differential is a linear functional on the tangent space at a point, where tangent space is thought of as the space of all arrows emanating from the point (se... |
I'm reading about Cuntz-Pimsner algebras at the moment, and something simple albeit annoying has been bothering me.
Given a $C^{*}$-Correspondence $(\sf{X},\mathfrak{A})$, Pimsner defines an 'augmented Cuntz-Pimsner algebra' as 'the $C^{*}$-algebra generated by $\mathcal{O}_{\sf{X}}$ and $\mathfrak{A}$'.
Preseuma... |
$C^{*}$-Algebras Generated by Sets & other C*-Algebras? |
On page $16$ of Hatcher's, following proof is given:[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/MdN6U.png
but I can't see the induced deformation retract's existence.
My attempt is to use "passing to the quotient", but somehow at the end I can't guarantee the final step of my arg... |
>Let $U \subset \mathbb{R}^2$ be a bounded open set with Lipschitz boundary. Suppose $f^n \rightharpoonup f$ in $W^{1,p}(U)$. Then $f_n \to f$ in $L^{\infty}(U)$ for $2<p<\infty$.
Attempt: Since $p>n=2$, I know we have to use Morrey's inequality. As $f_n$ is weakly convergent, it is uniformly bounded in $W^{1,p}$. T... |
I hope this is helpful. Joriki's brute force calculations of minimal $m$ for small $n$ are quite interesting and indeed come close to your estimate of $\log_2(n!)$. If you're willing to sacrifice such efficiency in exchange for a simpler, algorithmic design, here's an alternative approach, following a well-trodden path... |
In Umemoto's thesis [1] on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind these angles is unclear.
For instance, I thought, as its reflection $\theta_6$ of bottom left corner of the top tri... |
About a month ago, I asked a [similar question][1] about a certain class of polynomials that seem to defy the odds of "eventual divisibility", i.e. iterating an "eventually divisible" polynomial we can guarantee that any input will yield a number that is divisible by a certain prime $p$ at least once (and thus infinite... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
OK, let's try to make sense of your question. First of all, what is ${\mathbb R}^{p,q}$? Let $n=p+q$. Then ${\mathbb R}^{p,q}$ is the vector space ${\mathbb R}^{n}$ equipped with the quadratic form
$$
Q({\mathbf x})= x_1^2+...+x_p^2 - x_{p+1}^2- ... - x_n^2.
$$
In particular, as a **topological space,** ${\mathbb ... |
Instead of looking at the collection $X_1,...,X_n$ we can look at the vector $Y = (X_1,...,X_n)$ for which $\sigma(Y) = \sigma(X_1,...,X_n)$, so it is enough to consider case of single random variable $Y$ (but taking values in some product space $\mathbb R^n$ or any different).
Thus, let us assume that $(\Omega,\mat... |
I'm reading about Cuntz-Pimsner algebras at the moment, and something simple albeit annoying has been bothering me.
Given a $C^{*}$-Correspondence $(\sf{X},\mathfrak{A})$, Pimsner defines an 'augmented Cuntz-Pimsner algebra' as 'the $C^{*}$-algebra generated by $\mathcal{O}_{\sf{X}}$ and $\mathfrak{A}$'.
Preseuma... |
OK, let's try to make sense of your question. First of all, what is ${\mathbb R}^{p,q}$? Let $n=p+q$. Then ${\mathbb R}^{p,q}$ is the vector space ${\mathbb R}^{n}$ equipped with the quadratic form
$$
Q({\mathbf x})= x_1^2+...+x_p^2 - x_{p+1}^2- ... - x_n^2.
$$
In particular, as a **topological space,** ${\mathbb ... |
Instead of looking at the collection $X_1,...,X_n$ we can look at the vector $Y = (X_1,...,X_n)$ for which $\sigma(Y) = \sigma(X_1,...,X_n)$, so it is enough to consider case of single random variable $Y$ (but taking values in some product space $\mathbb R^n$ or any different).
Thus, let us assume that $(\Omega,\mat... |
I want to find the ensemble mean of an equation $\frac{dA}{dt}=-\beta A+c$. where $A$ is a r.v. and can be written as sum of mean and fluctuation, i.e., $A=E[A]+A'$. $\beta$ and $c$ are some constant parameters. So with this, I can write the ensemble mean of the equation as:
$E \left[ \frac{d\left(E[A]+A'\right)}{d... |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
$\newcommand{\C}{\mathscr{C}}$
$\newcommand{\R}{\mathbb R}$
Let $\C$ be $C^\infty$, that is a set with operations:
$$\Phi_f:\C^n\longrightarrow \C$$
for every smooth function $f$ such that if $g:\mathbb R^n\rightarrow \mathbb R$, and $f_i:\R^m\rightarrow \R$ are smooth functions for $1\leq i\leq n$ then:
\begin{al... |
How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix}
0&I_6&0\\I_6&0&0\\0&0& -I_{16}\end{bmatrix} \tag{2}$$
I know that a symmetric matrix has $28 \times 29/2=406$ free parameters and I know that the final a... |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
FullofDill's answer is perfectly acceptable, but I just wanted to challenge myself to create a non-piecewise PMF for these, because it seemed like a neat challenge!
For Part a, Obviously we start knowing that our denominator for our probability will be 10, and we can see that as long as $y$ is in range, $y\geq2$ - S... |
can anyone help me?
Be $(X,\|.\|)$ a normed space that satisfies the parallelogram law. Let's define $\Phi : X\times X \rightarrow \mathbb{C}$ by:
$\Phi (x,y)=\dfrac{1}{4}[ \|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2]$
Prove
$\Phi(x,y)=\overline{\Phi(y,x)}$
I resolved
$\overline{\Phi(x,y)}=\dfrac{1... |
Is $\langle x,y \rangle=\overline{\langle y,x \rangle}$? |
I have proved that $\phi \in \operatorname{Aut}(\mathbb{D})$ has a fixed point in $\mathbb{D}$ iff its correspondent from $\operatorname{Aut}(\mathbb{H})$ has a fixed point in $\mathbb{H}$, where $\mathbb{D}$ is open unit disk, and $\mathbb{H}$ is upper half-plane.
But I don't know how to show that $\phi \in \opera... |
One of the corollaries of soundness says that if $\Gamma$ is satisfiable, then $\Gamma$ is consistent. I am wondering whether we can conclude that Peano's axioms $\mathsf{PA}$ is consistent from the fact that the standard model of arithmetic $\mathcal{M}_A=(\mathbb N, 0^{\mathcal{M}_A}, s^{\mathcal{M}_A}, +^{\mathcal{M... |
Can we conclude that Peano's axioms consistent from soundness? |
When one is first introduced to a differential of a real-valued function, it is defined as a linear function that takes tangent vectors to real numbers. That is, differential is a linear functional on the tangent space at a point, where tangent space is thought of as the space of all arrows emanating from the point (se... |
Have you tried working through an algebraic derivation of the primitive Pythagorean triple parametric formula in $\mathbf Z$ to get an analogue in $\mathbf Z[i]$? Because $-1$ is a square, the terms in $\alpha^2 + \beta^2 = \gamma^2$ are essentially symmetric in $\alpha$, $\beta$, and $\gamma$, so we can assume withou... |
In the build-up to the enunciation of the Birch, Swinnerton Dyer conjecture (BSD), the following back-of-the-envelop idea comes up:
Because half of the $1$ to $n-1$ elements mod $p$ are quadratic residues (squares), for the expression $y^2= x^3 +\cdots$ mod a certain prime $p,$ **on average** $1/2$ of the input $x$ ... |
Why is $Np / p = 1$ the expectation and not the upper bound in the limit for elliptic curves over a finite field? |
>Page 113 - Dummit and Foote - Group actions
>
>Two group elements induce the same permutation on $A$ if and only if they are in the same coset of the kernel.
What does this mean? Two permutations induce the same permutation on $A$ looks like it means $\sigma_1 A = A' = \sigma_2 A$ and then I don't really understa... |
In this paper: https://www.tandfonline.com/doi/epdf/10.1080/01621459.2017.1285773?needAccess=true I do not understand why there are K^n terms in the integral in Equation (9) instead of n^K. If we have n observations, and each observation can belong to one of K clusters, then we should have n^K possible assignments. Can... |
Let $G=(V, E)$ be an undirected graph where $V$ is the set of vertices and $E = \emptyset$. $G$ is clearly chordal since it has no cycles. We are given a function $f(u,v,w,G)$ who takes as input $G$ and $u,v,w \in V$($u$, $v$ and $w$ are distinct vertices). It returns a new graph $G'=(V,E \cup U)$ where $U = \{ pq| pq ... |
This is an easy question about taking expectations that I'm confusing myself with. To describe the simplest setting, suppose that I have $N$ machines, which when chosen outputs a value $1$. I'll choose $K$ machines simultaneously, as a batch. I'll denote the probability of each machine $i$ being chosen in this ($K$-bat... |
Here's the problem, as far as I understand it. For given measure spaces:
$$(\mathcal{X}_0, B(\mathcal{X}_0), \mathbb{P}_0)$$
$$(\mathcal{X}_1, B(\mathcal{X}_1), \text{don't-care-about-this-measure-yet?})$$
Chopin and Papaspiliopoulos, in their 2021 Sequential Monte Carlo book, introduce the idea of a probabi... |
a) Suppose $S$ and $T$ are injective. Is $S+T$?
b) Suppose $S$ and $T$ are surjective. Is $S+T$?
My answer for a) was:
Let $v_1,v_2 \in V$. Consider:
$(S+T)(v_1) = (S+T)(v_2)$
$(S+T)(v_1) - (S+T)(v_2) = 0$
$S(v_1) + T(v_1) - S(v_2) - T(v_2) = 0$
$S(v_1 - v_2) + T(v_1 - v_2) = 0$
$S(v_1 - v_2) = -T... |
The following is a probability question I got during an interview:
I had two travels in the last year. At least one of them is an international travel in December, what is the probability that both travels are international? Given that the probability of traveling in one specific month is $\frac {1}{12}$.
The c... |
Conventionally, we approximate a binomially-distributed variable $X\sim B(n, p)$ with the Poisson-distributed variable $Y\sim Po(np)$, with the mean of $X$ and $Y$ being identical.
However, we could also approximate $X$ with $Z\sim Po(np(1-p))$. While the mean of $X$ and $Z$ are different, the variances are identica... |
What's the optimal way to approximate a binomial distribution with a Poisson distribution? |
This may be best answered by looking for an improvement to that formula, and explaining how to find it.
The basic idea for all those approximations for $e$ is to write some large number $N$ in two different ways, using exactly the digits from $2$ to $9$, and then take $(1+\frac{1}{N})^{N}$ as an approximation for $e... |
Let $ P = \{x \in \mathbb{R}^d:Ax \leq b\}$ be a polyhedron, where $A = \begin{bmatrix} a_1^T\\a_2^T\\...\end{bmatrix}$, then the cone hull of $(a_1,a_2,\cdots)$ is $\mathbb{R}^d$. How to prove this statement?
I think the intuition is pretty clear, but I don't know how to express them in math notations and give a st... |
This may be best answered by looking for an improvement to that formula, and explaining how to find it.
The basic idea for all those approximations for $e$ is to write some large number $N$ in two different ways, using exactly the digits from $2$ to $9$, and then take $(1+\frac{1}{N})^{N}$ as an approximation for $e... |
The following is a probability question I got during an interview:
I had two travels in the last year. At least one of them is an international travel in December, what is the probability that both travels are international? The probability of traveling in one specific month is $\frac {1}{12}$.
The correct answ... |
The following is a probability question I got during an interview:
I had two travels in the last year. At least one of them is an international travel in December, what is the probability that both travels are international? The probability of traveling in one specific month is $\frac {1}{12}$, and the probability ... |
I would like to understand this example:
the Parseval's Theorem is given by:
<pre>
\[\frac{1}{L} \int_{-L}^{L} (f(x))^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} a_n^2 + b_n^2\]
or
\[\frac{1}{L} \int_{-L}^{L} (f(x))^2 dx = 2{a_0^2} + \sum_{n=1}^{\infty} a_n^2 + b_n^2\]
What is the Parseval's Theorem ... |
I'm trying to find general properties for a half derivative operator, yet I encountered what seems to be a paradox. Let $H$ be a linear operator such that $H^2f = Df$ where $D$ is the first derivative. Then we have the following consequences
$$H0=0$$
$$H^31=DH1=HD1 \Rightarrow DH1 = H0=0 \Rightarrow H1 = c $$
$$H^... |
How is the volume not double counted in the shell method for volume of revolution? |
Does anyone have an idea if this double sum could be simplified?
$$\sum_{p=0}^\infty\sum_{q=0}^\infty \frac{(-1)^p(-1)^q}{(2p+1)(2q+1) } e^{-\pi\sqrt{(2p+1)^2+(2q+1)^2} \bar y}\sqrt{(2p+1)^2+(2q+1)^2}$$
(Context: I computed the electric potential, solution of Laplace's equation, in some geometry as a double cosi... |
I'm trying to fill in some details in Do Carmo's proof on p.149,
> If all points of a connected surface $S$ are umbilical points, then $S$ is either contained in a sphere or in a plane.
also discussed [here][1]. We have proven the local version, i.e., for every point in the surface $S$ there is a connected coord... |
I am trying to solve the following problem from a book on mathematical statistics:
*Suppose $\mathcal{P}=\{f(x|\theta_1,\theta_2)|\theta=(\theta_1,\theta_2)\in\Theta_1\times\Theta_2\}$ is a family of non-vanishing pdfs on $\mathbb{R}^d$, meaning that $f(x|\theta)\neq0$ for any $x\in\mathbb{R}^d,\theta\in\Theta_1\tim... |
Let $ P = \{x \in \mathbb{R}^d:Ax \leq b\}$ be a polyhedron, where $A = \begin{bmatrix} a_1^T\\a_2^T\\...\end{bmatrix}$, then the cone hull of $(a_1,a_2,\cdots)$ is $\mathbb{R}^d$. How to prove this statement?
I think the intuition is pretty clear, but I don't know how to express them in math notations and give a st... |
I am trying to solve the following problem from a book on mathematical statistics:
>Suppose $\mathcal{P}=\{f(x|\theta_1,\theta_2)|\theta=(\theta_1,\theta_2)\in\Theta_1\times\Theta_2\}$ is a family of non-vanishing pdfs on $\mathbb{R}^d$, meaning that $f(x|\theta)\neq0$ for any $x\in\mathbb{R}^d,\theta\in\Theta_1\tim... |
How to show that $\mu(\liminf_{n\to\infty} E_n)=0$? |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integers $c,d... |
I am an engineering and applied math undergraduate and I like reading up on the pure math behind what I study in my nonexistent free time. I’m taking probability soon and would like to supplement it by starting to read some set theory. From scouring previous posts, it seems that two names that pop up repeatedly are H... |
$\lim_{n \to \infty}\prod_{k=1}^{n}{\left ( 1+\frac{1}{2^{2^{k} } } \right )}
$
let$ a_n={\left ( 1+\frac{1}{2^{2^{k} } } \right )} $ ,
$
\left( 1-\frac{1}{2^2} \right) a_n=\left( 1-\frac{1}{2^2} \right) \left( 1+\frac{1}{2^2} \right) \cdots \left( 1+\frac{1}{2^{2n}} \right) =
1-\frac{1}{2^{2n+1}}$
$a_n=\f... |
A limit question? |
I have a real signal $f(t) $that is periodic on $[0,1]$. I sampled it uniformly $64$ times and computed FFT (in python). I was asked to find the central frequency, namely the frequency with the largest magnitude. I found the index $i$ of the largest value in the transformed signal.
How do I find the frequency (in he... |
Approximately 80,000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couples,
(a) both partners were born on April 30
I understand how to find the answer (~.45) but on an exam, what language gives away that this question would be asking for a Poisson ... |
Why do we use a Poisson distribution here rather than Binomial? |
I am trying to verify that $\text{SL}_2 (\mathbb{Z})$ is a group under matrix multiplication. The only property I am not certain on is closure under inverses.
Given $A \in \text{SL}_2 (\mathbb{Z})$, I know that $A^{-1}$ exist (it has non-zero determinant). I need to show that $A^{-1}$ has determinant $1$ and intege... |
Recently while studying Elliptic Integrals and related topics I have come across various Interesting Trigonometric Substitutions, examples given below
1.
$$\int_0^{\pi/2}\frac{1}{\sqrt{1-x^2(\sin t)^4}}\ dt=\int_0^{\pi/2}\frac{1}{\sqrt{1+x-2x(\sin v)^2}}\ dv$$
which uses the substitution
$$v=\arctan(\sqr... |
$\lim_{n \to \infty}\prod_{k=1}^{n}{\left ( 1+\frac{1}{2^{2^{k} } } \right )}
$
let$ a_n={\left ( 1+\frac{1}{2^{2^{k} } } \right )} $ ,
$
\left( 1-\frac{1}{2^2} \right) a_n=\left( 1-\frac{1}{2^2} \right) \left( 1+\frac{1}{2^2} \right) \cdots \left( 1+\frac{1}{2^{2n}} \right) =
1-\frac{1}{2^{2n+1}}$
$a_n=\f... |
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