instruction stringlengths 12 30k |
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Consider an homogeneous PPP in $\mathbb{R}^2$ with intensity parameter $\lambda > 0$, and denote the set of _generator seeds_ in the spatial domain as $P = \{p_1, p_2, \dots, p_N\}$. At the same time, envision a Voronoi tessellation of the planar point set such that the space is sparated into regions based on the prox... |
In fact,
$$
L^1(\Bbb{R^d}) \ast L^1(\Bbb{R^d}) = L^1(\Bbb{R^d}),
$$
which is thus in particular a vector space.
This follows from the Cohen-Hewitt factorization theorem. See [here](https://en.m.wikipedia.org/wiki/Cohen%E2%80%93Hewitt_factorization_theorem) and [here](https://www.jstor.org/stable/48662324?seq=1... |
What is the proper notation when defining certain functions?
For example, consider the function $\phi:\mathbb{Q} \rightarrow \mathbb{Z}\times\mathbb{N}$ defined by $\phi(\frac{p}{q})=(p,q)$ for any $\frac{p}{q}\in\mathbb{Q}$ such that p and q have no common divisor except 1. The way this function is defined relies o... |
Consider a square grid in $Y-Z$ plane consisting of $(L+1)^2$ points ($L$ is an even integer) centered around origin. The points in the square grid are:
$\vec{s}_{mn}= (0,md,nd)^T ,$ where $ -L/2 \leq m,n \leq L/2$
Now, consider a sphere with center $(0,0,0)$ and radius $R$ greater than $\frac{Ld}{2}$.
L... |
If $W_t$ is a standard Wiener process, Levy's Arcsine Law gives us the CDF of the random variable
$$ \int_0^1 \delta[W_t \ge 0] dt $$
(where $\delta[\text{event}]$ is $1$ if the event happens and $0$ otherwise) as $\frac{2}{\pi}\arcsin(\sqrt{x}), x\in[0,1]$. I am aware there is a generalization to $W(t)\ge\alpha$... |
Levy Arcsine Law for Brownian Bridges? |
Suppose A is a 2x2 matrix, e.g., A=(a11 a12; a21 a22), and x=(x1 x2). Suppose f() is continuous. What does "f(Ax) is supermodular" mean? For example, if f(x) is supermodular, we have the cross partials of f() with respect to x1 and x2 nonnegative. How can we similarly represent f(Ax) supermodular in terms of the second... |
What does "f(Ax) is supermodular" mean? |
Consider a square grid in $Y-Z$ plane consisting of $(L+1)^2$ points ($L$ is an even integer) centered around origin. The points in the square grid are:
$\vec{s}_{mn}= (0,md,nd)^T ,$ where $ -L/2 \leq m,n \leq L/2$ ($m$ and $n$ are integers).
Now, consider a sphere with center $(0,0,0)$ and radius $R$ greate... |
Say I have a set of sets eg. $A:=\{\{1,2\},\{2,3\}\}$. Does this formula
$$\sum_\limits{A_i \in A}\prod_{a\in A_i}x_j$$
yield $x_1x_2+x_2x_3$?
Or does it even make sense to define a sum over a set of sets? |
What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order.
Examples:
1. Quadratic Diophantine equations are always decidable, but cubic ones are not.
2. Time independent second order ODEs cannot admit chaotic solutions, but ... |
Consider the attached NFA (from Sisper's *Introduction to the Theory of Computation*, 3e) which has been deduced in order to accept precisely the language containing the string $ab$. The rest of the notation should be pretty standard as far as I know, but please let me know if not.
[![enter image description here][1... |
Why do we need empty string transitions in the NFA intended to accept this singleton language? |
Suppose we have a random sample $X=(X_1,\dots, X_n)$ which depends on $\theta$, treated as a value of the random variable $\Theta$. The posterior pdf is given by $$f_{\Theta | X}(\theta|x)=\frac{f_{X|\Theta}(x|\theta)f_\Theta(\theta)}{\int_{-\infty}^\infty f(x,\theta)d\theta}$$
where $f_\Theta$ is the prior pdf... |
Pdfs and Pmfs are mixed in one formula for MAP? |
The Question:
---
>Let $x\in\Bbb F_q$, where $\Bbb F_q$ is the field of $q=p^r$ elements, $p$ prime, $r\in\Bbb N$. Can we write
>
>$$x=a^2+b^2+c^2+d^2\tag{1}$$
>
>for $a,b,c,d\in\Bbb F_q$ such that $ad-bc=1$?
Thoughts:
---
Relevant theorems include:
- **Lagrange's Four Squares Theorem:** Each natura... |
Under $ad-bc=1$, is every element of a finite field of the form $a^2+b^2+c^2+d^2$? |
How to evaluate $\int\left(\frac{\sin(x)}{2\sin(x)- x(1+\cos(x))}\right)^2dx$? |
Consider this particular instance for the $0/0$ case of the Stolz-Cesaro Theorem: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0 $$ with $a_n$ and $b_n$ two sequences of real numbers, $b_n$ strictly monotone, whereby $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = 0 \,\,\,\,\,\,\,\, (1)$$ then... |
Answers:
1. This seems OK.
2. You are relying only on compactness (Bolzano--Weierstrass), which by the way is the property you routinely use in analysis, more than the least upper bound.
3. I think your method is creative, but overtly complicated. More down-to-earth, you could use the [bisection method](https:/... |
>Is $f(x)={\log(x)}^2$ uniformly continuous on $(1,\infty)$?
I have found the above question in an old exam and was wondering how to solve it. Actually, a few friends and I have found 2 different solutions, one showing that the function is uniformly continuous, while the other showing that it isn't, and we can't fig... |
Is ${\log(x)}^2$ uniformly continuous on $(0, \infty)$? |
Of course parallel lines converge at a point at infinity in projective geometry, but visually, they appear to diverge as one gets closer and closer to the start of one's vision, i.e. they diverge behind the observer. I can't find much about this fact. How is this dealt with in the different projective spaces? Are ther... |
I'm studying the dependence of solutions of Cauchy problem to starting conditions
I have the flux function $\Phi : I \times B \rightarrow \Omega$ which maps $(x, \xi, \eta)$ to $y_{\xi \eta}(x)$ where $y_{\xi \eta}$ is the solution of the Cauchy problem
$\begin{cases} y' = f(x, y) \\
y(\xi) = \eta \end{cases} $
... |
What differentiation rule is used in this passage? |
Consider this particular instance for the $0/0$ case of the Stolz-Cesaro Theorem: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0 $$ with $a_n$ and $b_n$ two sequences of real numbers, $b_n$ strictly monotone, whereby $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = 0 \,\,\,\,\,\,\,\, (1)$$ then... |
Is ${\log(x)}^2$ uniformly continuous on $(1, \infty)$? |
>Is $f(x)={\log(x)}^2$ uniformly continuous on $(1,\infty)$?
I have found the above question in an old exam and was wondering how to solve it. Actually, a few friends and I have found 2 different solutions, one showing that the function is uniformly continuous, while the other showing that it isn't, and we can't fig... |
Obviously, being a professional group theorist, I know what the orbit-stabilizer theorem is. Or at least I thought I did.
I thought that the orbit-stabilizer theorem was that if $G$ is a finite group acting on a set $X$, with stabilizer $H$ of a point $x$, then $|G|$ is the product of $|H|$ and the length of the $G$... |
What exactly is the orbit-stabilizer theorem? |
>Is $f(x)={\log(x)}^2$ uniformly continuous on $(1,\infty)$?
I have found the above question in an old exam and was wondering how to solve it. Actually, a few friends and I have found 2 different solutions, one showing that the function is uniformly continuous, while the other showing that it isn't, and we can't fig... |
> Let $M = \text{Aut}(\mathbb{Q}(i, \sqrt{2})) \cong V_{4}$ and show that the $\prod_{\sigma \in M}(X - \sigma(\alpha))$ is equal to the minimal polynomial of $\alpha = 1 + i + \sqrt{2}$ over $\mathbb{Q}$.
**My attempt:**
In order to show that $M \cong V_{4}$ it suffices to show that $M$ is of order $4$ and not c... |
A couple of my introductory logic books appeal to modal and set-theoretic notions in building up first-order logic. (They explicitly acknowledge these connections and say, for example, that validity is a modal notion, or they speak of “sets” of propositions, using the term in an expressly technical sense.) How do these... |
Which logic is most fundamental? |
My understanding (and correct me if I'm wrong) is that it is unknown whether $\pi^e$ is algebraic or transcendental. I've also been led to believe that most open questions of this type would be solved by [Schanuel's conjecture](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture). For instance, the conjecture implies... |
Does Schanuel's conjecture imply that $\pi^e$ is transcendental? |
As $2$ and $3$ are coprime integers, riaising to the ($\frac23$)th power (equivalently, taking the ($\frac32$)th root, is esdsentially squaring and cubing.
While generally, using the simple $p$th power then $q$th root to calculate the ($\frac pq$)th power generally works for only positive real bases, it works in th... |
I am studying [Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan][1]. In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous and $g$ is of bounded variation. Let $a=x_0 \le x_1 \le \dots \le x_N=b$ is a partition of $[a,b]$, and $S(x_n, \xi_n) = \sum_... |
How $|S(x_n, \xi_n) - S(x'_n, \xi'_n)|$ being arbitrary small implies existence of Stieltjes-Riemann Integral? |
Suppose you have a circle two points (A and B) outside the circle in the cartesian plane. The question is to find the shortest path from A to B while touching the perimeter of the circle.
1. Is there straightforward formula to find this shortest path value (distance)?
2. How to find the point V where the the shor... |
While studying about the notion of gcd's in commutative rings, I came across a theorem which states that any two elements in a Euclidean Ring always have a gcd. While this is true, isn't this true for any arbitrary commutative ring with unit element say, R? For example, say, $a,b\in R$ then we can surely say that $1|a,... |
Why can't we say, that any two elements in an arbitrary commutative ring with unit element has a greatest common divisor(gcd)? |
While studying about the notion of gcd's in commutative rings, I came across a theorem which states that any two elements in a Euclidean Ring always have a gcd. While this is true, isn't this true for any arbitrary commutative ring with unit element say, R? For example, say, $a,b\in R$ then we can surely say that $1|a,... |
Consider this particular instance for the $0/0$ case of the Stolz-Cesaro Theorem: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0 $$ with $a_n$ and $b_n$ two sequences of real numbers, $b_n$ strictly monotone, whereby $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = 0 \,\,\,\,\,\,\,\, (1)$$ then... |
You're given the consecutive lengths of the sides of an irregular pentagon, and you want to circumscribe this pentagon (whose interior angles are unknown yet) about a circle of unknown radius.
Is there a known procedure to do this ?
The image below is one that I created in Microsoft Excel using VBA code that impl... |
I am studying *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan*. In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous and $g$ is of bounded variation. Let $a=x_0 \le x_1 \le \dots \le x_N=b$ is a partition of $[a,b]$, and $S(x_n, \xi_n) = \sum_1^N... |
Suppose you have a circle and two points (A and B) outside the circle in the cartesian plane. The question is to find the shortest path from A to B while touching the perimeter of the circle.
1. Is there straightforward formula to find this shortest path value (distance)?
2. How to find the point V where the the ... |
While studying about the notion of gcd's in commutative rings, I came across a theorem which states that any two elements in a Euclidean Ring always have a gcd. While this is true, isn't this true for any arbitrary commutative ring with unit element say, R?
My definition of greatest common divisor is valid in case o... |
Consider this particular instance for the $0/0$ case of the Stolz-Cesaro Theorem: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0 $$ with $a_n$ and $b_n$ two sequences of real numbers, $b_n$ strictly monotone, whereby $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = 0 \,\,\,\,\,\,\,\, (1)$$ then... |
Consider a linear subspace of matrices, $M \subset \textrm{Mat}_n(\mathbb{C})$, which is $\epsilon$-*approximately* closed under multiplication, i.e. for all $x,y \in M$, there exists $z \in M$ such that $\lVert{xy-z}\rVert < \epsilon \lVert x \rVert \lVert y \rVert$. Assume $M$ is closed under Hermitian adjoints. The... |
The Question:
---
>Let $x\in\Bbb F_q$, where $\Bbb F_q$ is the field of $q=p^r$ elements, $p$ prime, $r\in\Bbb N$. Can we write
>
>$$x=a^2+b^2+c^2+d^2\tag{1}$$
>
>for $a,b,c,d\in\Bbb F_q$ such that $ad-bc=1$?
Thoughts:
---
Relevant theorems include:
- **Lagrange's Four Squares Theorem:** Each natura... |
Consider a square grid in $Y-Z$ plane consisting of $(L+1)^2$ points ($L$ is an even integer) centered around origin. The points in the square grid are:
$\vec{s}_{mn}= (0,md,nd)^T ,$ where $ -L/2 \leq m,n \leq L/2$ ($m$ and $n$ are integers).
Now, consider a sphere with center $(0,0,0)$ and radius $R$ greate... |
I have adapted this question from this question, I don't fully understand the answers given there/
https://math.stackexchange.com/questions/2753782/understanding-distribution-using-stirling-number-of-the-second-kind-inclusion-ex
**Problem**
I want to distribute $n=3$ distinguishable items to $r=3$ distinguisha... |
The problem is very vaguely phrased, but reverse engineering from the solution:
Suppose there is a block of $3$ specific boys that must be sat together, call this block $X$ and say it consists of $B_1, B_2, B_3$. Then we must sit $G_1, G_2, G_3, G_4, B_4, X$, for which there are $6!$ possible arrangements. If we w... |
Consider a square grid in $Y-Z$ plane consisting of $(L+1)^2$ points ($L$ is an even integer) centered around origin. The points in the square grid are:
$\vec{s}_{mn}= (0,md,nd)^T ,$ where $ -L/2 \leq m,n \leq L/2$ ($m$ and $n$ are integers).
Now, consider a sphere with center $(0,0,0)$ and radius $R$ greate... |
I was thinking about math and came up with the following idea. It seems very obvious but I would like to see a concrete rigorous proof (albeit possibly a one-liner) for the following.
Consider an $n-$gon. Take one random point on each the sides of the $n-$gon (these points cannot be vertices). Prove that it is impos... |
I have come across a geometry problem that states:
> Let $ABC$ be an acute triangle. Let $AM$, $BN$ and $CL$ be the medians, which intersect in the
centroid $G$. Let $M'$, $N'$ and $L'$ be the midpoints of $AG$, $BG$ and $CG$, respectively. Show that
the six points $M$, $M'$, $N$, $N'$, $L$, $L'$ all lie on a circ... |
Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable.
What does "$f(Ax)$ is supermodular" mean?
For example, if $f(x)$ is supermodular, we have the cross partials of $f()$ w... |
Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean? |
The Question:
---
>Let $x\in\Bbb F_q$, where $\Bbb F_q$ is the field of $q=p^r$ elements, $p$ prime, $r\in\Bbb N$. Can we write
>
>$$x=a^2+b^2+c^2+d^2\tag{1}$$
>
>for $a,b,c,d\in\Bbb F_q$ such that $ad-bc=1$?
Thoughts:
---
Relevant theorems include:
- **Lagrange's Four Squares Theorem:** Each natura... |
I was thinking about math and came up with the following idea. It seems very obvious but I would like to see a concrete rigorous proof (albeit possibly a one-liner) for the following.
Consider an $n-$gon. Take one random point on each the sides of the $n-$gon (these points cannot be vertices). Prove that it is impos... |
The functor $\Omega$ is the composite of a bunch of functors, which I will now describe. Write $\mathsf{A}$ for the diagram $2\to 3\leftarrow 1$. The inclusion $i_3\colon *\to\mathsf{A}, *\mapsto 3$ induces a left Kan extension $\mathrm{Lan}_{i_3}\colon\mathscr{C}\to\mathrm{Fun}(\mathsf{A},\mathscr{C})$ (informally sen... |
Given an $n\times n$ grid partially filled with the numbers $0,\ldots, n-1$, we can play a Sudoku-like game by trying to fill in the rest of the grid so that the end result is the Cayley table for a group (edit: where position $i,j$ in the table would represent the product of $i $ and $j$ in the group structure).
I... |
By minimal non-solvable group, I mean a non-solvable group whose all proper subgroups are solvable.
Let $G$ be a minimal non-solvable group. If $G$ is not a simple group, then it can be easily verified that $G$ contains only one maximal normal subgroup say $N$. Moreover, in this case we can prove that for each nontriv... |
The functor $\Omega$ is the composite of a bunch of functors, which I will now describe. Write $\mathsf{A}$ for the diagram $2\to 3\leftarrow 1$. The inclusion $i_3\colon *\to\mathsf{A}, *\mapsto 3$ induces a left Kan extension $\mathrm{Lan}_{i_3}\colon\mathscr{C}\to\mathrm{Fun}(\mathsf{A},\mathscr{C})$ (informally sen... |
A **Mal'cev** algebra is an algebra $\mathbf{A}$ with a ternary term $t$ such that $\mathbf{A} \models t(x,x,y) \equiv y, t(x,y,y) \equiv x$.
For the remaining of the question, refer to the definitions and results contained in Hobby and McKenzie's book [*The Structure of Finite Algebras*](https://math.hawaii.edu/~r... |
I am trying to understand the definition of spectra in stable homotopy theory.
I will focus on finite spectra.
One definition of finite spectra is as the Spanier-Whitehead category: the colimit $\lim_{\rightarrow} (Space^{w}_* \overset{\Sigma^{red}} \rightarrow Space^{w}_* \overset{\Sigma^{red}} \rightarrow \cdots... |
Let $v$ = $
\begin{align}
& \begin{bmatrix}
x_{1} \\
x_{2} \\
\vdots \\
x_{n}
\end{bmatrix}
\end{align}
∈ \Bbb R^n$ s.t $x_1, ...,x_n ≥ 0$ and $\sum_{i=1}^{n}x_i =1$. Prove that $||v||^2 ≥ \dfrac{1}{n}$. The official solution uses Cauchy-Schwart... |
I was thinking about math and came up with the following idea. It seems very obvious but I would like to see a concrete rigorous proof (albeit possibly a one-liner) for the following.
Consider an $n-$gon. Take one random point on each the sides of the $n-$gon (these points cannot be vertices). Prove that it is impos... |
I was thinking about math and came up with the following idea. It seems very obvious but I would like to see a concrete rigorous proof (albeit possibly a one-liner) for the following.
Consider an $n-$gon. Take one arbitrary point on each the sides of the $n-$gon (these points cannot be vertices). Prove that it is im... |
Say I have a set of sets eg. $A:=\{\{1,2\},\{2,3\}\}$. Does this formula
$$\sum_\limits{A_i \in A}\prod_{j\in A_i}x_j$$
yield $x_1x_2+x_2x_3$?
Or does it even make sense to define a sum over a set of sets? |
I've been reviewing some basic optimal transport concepts ([ref Peyré and Cuturi][1]). I like the notation of the Kantorovich optimal transport problem.
$$ L_C(a,b) = \min_{ P \in U(a,b) } \langle P , C \rangle = \sum_{i,j} C_{i,j}P_{i,j} $$
The optimal transport cost between discrete probability measures $a$ and... |
Since a and b are both integers, there is only a countable amount of numbers a+pi*b. Thus not every real number can be expressed as such.
But is there a way to determine if x=a+pi*b, like a fundamental property? What about if a and b are rational? Or if instead of pi you take another irrational number? |
Which numbers can be expressed as a*pi+b, with a and b being integers? |
I am trying to understand the potentially alternative definition of spectra in stable homotopy theory. I am assuming that everything is $\infty$-categorical.
I will focus on finite spectra.
One definition of finite spectra is as the Spanier-Whitehead category: the colimit $\lim_{\rightarrow} (Space^{w}_* \overset... |
Since $a$ and $b$ are both integers, there is only a countable amount of numbers $a+b\pi$. Thus not every real number can be expressed as such.
But is there a way to determine if $x = a+b\pi$, like a fundamental property? What about if a and b are rational? Or if instead of $\pi$ you take another irrational number? |
Examples of natural proofs by induction exercises which needs 3 or more base cases? |
Since $a$ and $b$ are both integers, there is only a countable amount of numbers $a+b\pi$. Thus not every real number can be expressed as such.
But is there a way to determine if $x$ can be expressed as $a+b\pi$, like a fundamental property of $x$? What about if a and b are rational? Or if instead of $\pi$ you take ... |
I'm working on a project, that takes 2 similar elements and allows you to adjust the angle of the center-line by grabbing the right side and moving it up or down.
In this scenario, lets use the key:
circles are 20mm each
centerpoint to centerpoint of the circles is 120mm total. Making the distance between the oute... |
I'm attempting to form interior structures of planets. If I know the volume percentage of each layer of the planet, how would I determine the outer radius of each layer?
Let's assume the planet is divided into three main layers: core, mantle, and crust. The core can be expressed as a sphere with some radius $R_c$. ... |
What is the relationship between radius percentage and volume percentage for a layered sphere? |
Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution
$$X_t~\xrightarrow{d}~X_\infty$$
corresponds to
$$\mu_t ~\xrightarrow{w}~ \mu_\infty,$$
(to be precise, convergence in $\sigm... |
Since $a$ and $b$ are both integers, there is only a countable amount of numbers $a+b\pi$. Thus not every real number can be expressed as such.
But is there a way to determine if $x$ can be expressed as $a+b\pi$, like a fundamental property of $x$? What about if $a$ and $b$ are rational? Or if instead of $\pi$ you t... |
Which numbers can be expressed as $a+b\pi$, with $a$ and $b$ being integers? |
I am working on an exercise in Aluffi which asks to calculate the characteristic polynomial of $$\begin{pmatrix} 6 &-10&-10\\3&-5&-6\\-1&2&3\end{pmatrix}.$$ Initially, I row reduced to $$\begin{pmatrix} 1&-2&-3\\0&1&3\\0&0&2\end{pmatrix}$$ and it is clear that the characteristic polynomial of the latter matrix is $(t-1... |
In your expression for the area of PTS, if I'm understanding right you've assumed that the length PT is 150 units, but it isn't; that's PQ and PT is shorter. So your value of $x$ is coming out slightly too small, which gives you a value of $h$ that's slightly too big.
A useful general technique for this sort of situ... |
On the third edition of Ahlfors' Complex Analysis, page 149 it states: (i) Draw a circle $C_j$ about $a_j$ of radius $<\delta_j$, and let $P_j=\int_{C_j}f(z)dz$ be the corresponding period of $f(z)$. (ii) The particular function $1/(z-a_j)$ has the period $2\pi i$. (iii) Therefore, if we set $R_j=P_j/2\pi i$, the combi... |
In this question I use the term "nested Fourier series representation" to refer to an infinite series of one or more Fourier series versus a single Fourier series. Whereas a single Fourier series is periodic, a nested Fourier series representation is not necessarily periodic. When it is periodic it can be expressed as ... |
Elemetary proof that If $n+1$ is prime in $\mathbb{Z}$, then $1+\sqrt{-n}$ is prime in $\mathbb{Z}[\sqrt{-n}]$? |
If I flip a coin $n$ times, the probability that at least one toss was heads (H) is $1-1/2^n$. Taking $n\to\infty$ gives $1$. Does this prove that if I toss a coin repeatedely I will at some point get H? |
If I flip a coin repeatedly, will I get heads at some point? |
I am studying *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan*. In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous and $g$ is of bounded variation. Let $a=x_0 \le x_1 \le \dots \le x_N=b$ is a partition of $[a,b]$, and $S(x_n, \xi_n) = \sum_1^N... |
I am interested in analytically continuing the differentiation operation on functions, and I am currently focusing on trigonometric functions. Any high schooler can conjecture that $\frac{d^a}{dx^a}sin(x) = sin(x+\pi a/2)$ applies not only for integers, but for all complex numbers; I, however, am trying to prove the st... |
Proof that $\frac{d^a}{dx^a}sin(x) = sin(x+\pi a/4)$ iff these two infinite series are equivalent? |
Consider for simplicity a real-valued random variable $X$ with finite moments $m_X(n) = E[X^n]$, $n>0$. Now suppose we wish to construct a new random variable $Y = f(X)$, where $f$ is some measureable function such that the first $N$ moments are equal, e.g., $m_Y(n) = m_X(n)$ for $n \leq N$ for some fixed number $N$. I... |
The terminology in the question is unfamiliar to me, but I _think_ what it's asking is: suppose $x=1+\cdots+n$; then what is $n$ in terms of $x$?
Well, there's a nice simple formula: $x=\frac{n(n+1)}2$. Rearrange this a little to put it in the usual form for a quadratic equation in $n$: $n^2+n-2x=0$. And apply the u... |
What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order.
Examples:
1. Quadratic Diophantine equations are always decidable, but cubic ones are not.
2. Time independent second order ODEs cannot admit chaotic solutions, but ... |
> Suppose $H = L_2(B)$ where $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B \times B$ that satisfies $|K(x,y)| \leq A|x-y|^{-d+\alpha}$ for some $\alpha > 0$ whenever $x,y \in B$. Define $$Tf(x)=\int_B K(x,y)f(y) \, dy$$
>$\bullet$ Prove that $T$ is a bounded operator on $H$.
... |
Consider a finite group $G$ with subgroups $N$ and $M$, where $N$ is normal in $G=MN$, $N \cap M = 1$, and $[M, N] \leq \Phi(N)$, where $[M, N]$ denotes the commutator subgroup of $M$ and $N$, and $\Phi(N)$ represents the Frattini subgroup of $N$. The question arises: is $M$ necessarily normal in $G$?
To explore thi... |
Since $1.681>\phi$, 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{?}{>}6.90776+06... |
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