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While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
I have an optimization problem which I hope I can formulate as a linear program. The problem involves a vector $x$ of binary decision variables (so each entry of the $x$-vector is either $0$ or $1$). The goal is to
$$\text{maximize}\ c^Tx$$
where $c$ contains positive constants that are given by the problem. Fu... |
It is emphasized in all the materials that I have seen about Hartman-Grobman theorem (including [textbooks](https://link.springer.com/book/10.1007/978-1-4612-5703-5) and [Wikipedia](https://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman_theorem)) that the topological conjugacy is only a homeomorphism but not diffeomorph... |
I asked this question on MO [here](https://mathoverflow.net/q/467524/525141)
------------
From [The influence of conjugacy class sizes on the
structure of finite groups](https://www.bing.com/ck/a?!&&p=c3f89a7137a3353aJmltdHM9MTcxMTA2NTYwMCZpZ3VpZD0xNTYyNGMzZC0xNWI3LTY5OWEtM2Y1ZS01Zjk0MTQ2NDY4NWMmaW5zaWQ9NTE5... |
I've been reading through Charles Fefferman's *The Multiplier Problem for the Ball* and going through lemma 1 sparked the following problem for me:
Let $V \in \mathbb{R}^2$ be a unit vector,
let H be the half plane defined by $H=\{x\in \mathbb{R}^2 : x\cdot V \geq 0\}$, and let $D^{r}$ be a disc centered at $rV$ wi... |
I've been reading through Charles Fefferman's *The Multiplier Problem for the Ball* and going through lemma 1 sparked the following problem for me:
Let $V \in \mathbb{R}^2$ be a unit vector,
let H be the half plane defined by $H=\{x\in \mathbb{R}^2 : x\cdot V \geq 0\}$, and let $D^{r}$ be a disc centered at $rV$ wi... |
My question originates from a short proof in Strang's LA book that shows when
$A\pmb{x}=\pmb{b}$
every vecetor $\pmb{x_r}$ in the rowspace of $A$ projects uniquely to a single vector $\pmb{b_c}$ in the column space of $A$. The proof is discussed [in this thread](https://math.stackexchange.com/questions/3585142/ho... |
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
Define by $K:=\mathbb{Q}_p(u^{1/p})$. What is the discriminant of $K$?
The discriminant of the defining polynomial $f(x):=x^p-u$ is $\Delta(f)=(-1)^{p(p-1)/2}p^pu... |
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
If have a point on an arc and other points related to it. I now want to move the point on the arc to a given target and rotate everything so that the arc-based-point has a vertical angle as the new position (like the if the arc-midpoint would now be just below (on y-axis) the target.
Example: I have an arc-midpoint ... |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have a proposition of holomorphic line bundle over $\mathbb P^n$,
> Proposition 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of ... |
According to page 6 of the paper “Ricci soliton solvmanifolds” by Lauret, the canonical inner product on $\mathbb R^n$ descends to an inner product on $V := \wedge^2 (\mathbb R^n)^* \otimes \mathbb R^n$ by the formula
$$\langle\mu,\lambda\rangle = \sum\langle\mu(e_i,e_j),\lambda(e_i,e_j)\rangle=\sum\langle\mu(e_i,e_j)... |
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
A differentiable homeomorphism that isn't a diffeomorphism? |
I have sudden confusion in matrix multiplication:
If I have
(UP)X = B, where U and P are matrix and B is a vector,
can I write UX= P^-1 B?
Thank you for great help.
|
As noted in the comments, the structure ($V_\omega$, $\in$) [satifies all the axioms][1] of ZFC except for the axiom of existence of an inductive set, and moreover every object of it is finite under all (usual?) definitions (since choice holds), so it follows that some axiom form of infinity is in fact necessary to pro... |
if X is a vector space such that dim(X)=n and $M \subset X $ is a subspace. let $B=\{b_1,b_2,...,b_m\} $ is a hamel basis for M.
then
1) $dim(M)\le dim(X) $
2) there is a set $ D=\{d_1,d_2,...,d_{m-n} \}$ such that $B \cup D$ is a hamel basis for X.
I tried to prove the first sentence like this:
Since B is... |
I have sudden confusion in matrix multiplication:
If I have
(UP)X = B, where U and P are matrix and B is a vector,
can I write UX= P^-1 B?
Sorry for asking such a basic question. I got this confusion suddenly before , and I presently donot have resources to confirm this.
Thank you for great help.
|
I recently encountered the following riddle:
Let a fair coin be flipped $N > 2$ times. Player A gets a point every time there are two heads in a row, and player B gets a point every time a tail is followed by a head, the player with more points wins. Thus, for example, for the sequence `THHH`, player A wins with 2 v... |
First, it should be noted that computers store [floating-point numbers](https://en.wikipedia.org/wiki/IEEE_754) in what's essentially scientific notation, with a sign bit ($+$ or $-$), fixed-width, “significand” or “mantissa”, and exponent indicating a power of two:
$$x = (-1)^sm2^p, s \in \{0, 1\}, m \in [1, 2), p ... |
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
>whether the Fourier series of $f$
converges to $f$, or, more generally, whether f is determined by its Fourier series.
This was on page 186 and then on page 187
~
If $\phi_n(x)$ is orthonormal on $[a, b]$ and if
$c_n =\int_a^b f(t) \overline{\phi_n(x)}dt $
we call $c_n$ the $n$th Fourier coefficient o... |
Is $f= \sum c_n \phi_n(x)$ in baby Rudin 8.11? |
I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let me formulate it as follows.
> Consider a linear differential equation over a differential field $k$ of characteristic zero whos... |
Let $R$ be a commutative ring with identity. Suppose the matrix algebra $\operatorname{Mat}_n (R)$ and matrices $$E_{mk}:=(a_{ij})=\begin{cases} a_{ij}=1 \ \text{if} \ i=m \ \text{and} \ j=k \\ a_{ij}=0 \ \text{otherwise} \end{cases}.$$
Now consider algebras homomorphism from the free $R$-algebra $$f\colon R\langle ... |
Is there a good presentation of the matrix algebras? |
> Should I think about elements of $H^1(X, \mathcal{O}_X^*)$ as non-vanshing global sections?
No because that would be absolutely incorrect to do. Global sections of a sheaf $\mathcal{E}$ are given by $H^0(X,\mathcal{E})$.
Elements of $H^1(X,\mathcal{E})$ are not, inherently, geometric objects. At best they can b... |
Let $p$ be a prime number. Let $V$ be a linear subspace of $\mathbb{Q}_p^n$ of dimension $d$. Is $V \cap \mathbb{Z}_p^n$ a free $\mathbb{Z}_p$-module of rank $d$? Is every $\mathbb{Z}_p$-basis for $V\cap\mathbb{Z}_p^n$ a $\mathbb{Q}_p$-basis for $V$? |
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the derivation.
$$
\zeta(3) = \frac{\pi^2}{96} \lim_{\text{N} \to \infty} \left[1444 + 15232\text{log}(2)+3456\text{log}(3... |
Let $A=\{x^2\frac{f(x)}{g(x)}:\;f(x)\in\mathbb{R}[x],\,g(x)\in\mathbb{R}[x],\,g(0)\neq0\}$.
Note that for all $r_1,r_2\in R$ you have $r_1 r_2\in A$.
Note also that for any $p(x),q(x)\in\mathbb{R}[x]$ such that neither $p(x)$ nor $q(x)$ has $x$ as a factor, we have that $A\subseteq \big<x\frac{p(x)}{q(x)}\big>$... |
In the following version of Cantor's diagonal argument, where is the assumption that the $n$-th digit of $r$ must be different from $0$ or $9$ used?
Suppose $f$ is a $1$-$1$ mapping between the positive integers and the reals.
Let $d_n$ be the function that returns the $n$-th digit of a real number.
Now, let's... |
I am reading [the paper][1] and got stuck at the formula above (5.13): why
$$\sum_{i\neq j\in[n]}\cos^2(\theta_i-\theta_j)\leq n^2/2$$
holds?
Note that this corresponds to the formula $$n^2/2\leq \|X\|_F^2$$ in the paper, where $X$ is matrix with $X_{ij}=\cos(\theta_i-\theta_j)$.
[1]: https://arxiv.org... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(\rho)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $... |
What are the orientable prime compact three-manifolds that can be embedded in $R^4$? |
I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd appreciate some guidance on which of the two versions below is the correct way to write this, or if both versions are acc... |
I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd appreciate some guidance on which of the two versions below is the correct way to write this, or if both versions are acc... |
I am reading [the paper][1] and got stuck at the formula above (5.13): why
$$\sum_{i\neq j\in[n]}\cos^2(\theta_i-\theta_j)\geq n^2/2$$
holds?
Note that this corresponds to the formula $$n^2/2\leq \|X\|_F^2$$ in the paper, where $X$ is matrix with $X_{ij}=\cos(\theta_i-\theta_j)$.
[1]: https://arxiv.org... |
I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd appreciate some guidance on which of the two versions below is the correct way to write this, or if both versions are acc... |
Alternative approach:
Index the balls $~B_1, B_2, \cdots, B_{2n}.~$
Use the following algorithm to partner the balls. Choose the ball of lowest index, whose partner-ball has not yet been decided, and select (at random) one of the remaining (as yet unpartnered) balls to partner with.
So, for the first partneri... |
Consider an operator $\eta$ on the (tensor) product of Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_2$ such that its range is of the form $V_1\otimes V_2$. Does this imply that $\eta$ must be a product operator, i.e., of the form $\eta = A\otimes B$?
**Followup Question** (in response to @David Gao's counterexam... |
I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd appreciate some guidance on which of the two versions below is the correct way to write this, or if both versions are acc... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(\rho)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $... |
I would approach this problem by using the Lagrange multiplier method. Let the vector $k = [0,0,1]^T$, and let $ r = [x,y,z]^T $, then you cone and ellipsoid equations are
$ r^T Q_c r = 0 $ (for the cone), and
$r^T Q_e r = 1 $ (for the ellipsoid)
And you want to maximize $z = k^T r $, so define
$ G(r, \lam... |
Let $(X,\mathcal{X})$ be a metric space where assume that $X$ is countable and discrete. Show that probability measures $P_n$ converges to $P$ weakly (i.e., for all bounded continuous functions on $X$, one has $\int f dP_i\to \int f dP$) if and only if $P_n(\{x\})\to P(\{x\})$ as $n\to \infty$.
---
For one direct... |
What are the orientable prime compact three-manifolds that can be embedded in $\mathbb{R}^4$? |
I am reading [the paper][1] and got stuck at the formula above (5.13): (1) why
$$\sum_{i\neq j\in[n]}\cos^2(\theta_i-\theta_j)\geq n^2/2$$
holds?
Note that this corresponds to the formula $$n^2/2\leq \|X\|_F^2$$ in the paper, where $X$ is matrix with $X_{ij}=\cos(\theta_i-\theta_j)$.
(2) does it also hold ... |
Let G be a locally compact group and $\mathcal{H}$ a Hilbert space. Then there is a bijection between strongly continuous unitary representations of G and non-degenerate *-representations of $L^1(G)$ on $\mathcal{H}$ (see [Group algebra of locally compact group](https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_... |
Suppose for simplicity we have a plane curve $\gamma(t)=(f(t),g(t))$. I'm just curious exactly what is the property responsible for the fact that, if $R_{90}$ is the two dimensional rotation matrix by $90$ degrees counterclockwise, then $$\frac{d}{dt}\left(R_{90}\left(\gamma(t)\right)\right)=R_{90}\left(\frac{d}{dt}(\g... |
How can we know if an extension of $\sf ZF$ by some large cardinal property that results in a consistency strength beyond $0^{\#}$ is compatible with choice or not?
I mean the easiest way to know if an extension of $\sf ZF$ is compatible with choice is to construct a model of it in $L$. But if any extension goes bey... |
How to prove consistency with choice for large cardinal extensions? |
The Claim:
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
>[T]he existence of natural numbers and the fact that given a natural number $n$, there is always a successor $(n+1)$, do not imply the existence of an infinite set. You need an extra axio... |
Let $A\in\mathbb{R}^{m\times n}$ be a matrix and $\sigma_n$ be its smallest singular value. Prove that
$$\sigma_n^2=\min_{x\neq 0} \frac{x^TA^TAx}{x^Tx}.$$
From https://math.stackexchange.com/questions/2330716/proving-that-the-norm-of-a-matrix-is-bigger-or-equal-to-its-smallest-singular-v, I know that $$\sigma_n^2\le... |
Let G be a locally compact group and $\mathcal{H}$ a Hilbert space. Then there is a bijection between strongly continuous unitary representations of G and non-degenerate *-representations of $L^1(G)$ on $\mathcal{H}$ (see [Group algebra of locally compact group](https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_... |
Dagger compact closed categories are commonly said to be an abstraction of Hilbert spaces and is suppose to capture concepts such as unitary maps, scalars, basis, inner products. See for example the below Wikipedia article and the introduction in the paper by Peter Selinger.
https://en.wikipedia.org/wiki/Dagger_comp... |
**Formula:**
$$
\begin{aligned}
\mathbb{E}(N) & =\sum_{n \geq 1} \overbrace{n}^{\sum_{m=1}^n 1} \mathbb{P}(N=n) \\
& =\sum_{1 \leq m \leq n} \mathbb{P}(N=n)=\sum_{1 \leq m} \overbrace{\sum_{m \leq n} \mathbb{P}(N=n)}^{\mathbb{P}(N \geq m)}=\sum_{1 \leq m} \mathbb{P}(N \geq m)
\end{aligned}
$$
**Question**
... |
The Claim:
---
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
>[T]he existence of natural numbers and the fact that given a natural number $n$, there is always a successor $(n+1)$, do not imply the existence of an infinite set. You need an extra... |
I think I'm being rather silly but I'm trying to see if the system of nonlinear coupled ODEs given by
$$x'=-xy$$
$$y'=-xy$$
has any non-trivial solutions. After some googling I have found that a solution to this problem is given by
$$ y(t)=c_{1}-\frac{c_{1}e^{c_{1}c_{2}}}{e^{c_{1}c_{2}}-e^{c_{1}t}}$$
$$ x(t... |
Let $x \in [0,1]$ be an *irrational algebraic number*, and $y \in [0,1]$. Is it decidable by a Turing machine (i.e. does there exist an efffective procedure) whether there exists some integer $k \geq 1$ such that
$$
kx \equiv y \mod 1?
$$
This can be thought of a question about circle rotations from dynamics: mark... |
Deciding a circle rotation reachability problem? |
> Should I think about elements of $H^1(X, \mathcal{O}_X^*)$ as non-vanshing global sections?
No because that would be absolutely incorrect to do. Global sections of a sheaf $\mathcal{E}$ are given by $H^0(X,\mathcal{E})$.
Elements of $H^1(X,\mathcal{E})$ are not, inherently, geometric objects. At best they can b... |
**First some general comments:**
I think you may be confusing the model theoretic notion of classification a la Shelah's Classification Theory with a more general (and vaguer) notion of mathematical classification.
When Shelah talks about classification, he means the classification of models of some complete firs... |
$$\sum_{n=0}^{\infty} x^{n q} \prod_{k=1}^{t} \left(\sum _{m=0}^n \frac{1}{(x^{a_t})^m}\right)$$
For $t=1$
$$ \sum_{n=0}^{\infty} x^{n q} \left(\sum _{m=0}^n \frac{1}{x^{a_1 m}}\right) = \sum_{n=0}^{\infty} x^{n q} \left( \frac{\frac{1}{x^{(n+1)a_1}}-1}{\frac{1}{x^{a_1}}-1}\right) $$ $$ = \frac{1}{ \left(\fra... |
There are a couple of things to note here:
**(1)** By quadratic reciprocity, when $a>2$, $\left( \frac{2}{p}\right)=1$, for $p$ a Pierpont prime. Thus $$2^{\frac{p-1}{2}}=1\mod p$$ for $a>2$ and thus $2$ is not a generator for $p$.
A similar result holds for $a>1$, $b>0$ and the base of $3$.
However, for th... |
There are a couple of things to note here:
**(1)** By quadratic reciprocity, when $a>2$, $\left( \frac{2}{p}\right)=1$, for $p$ a Pierpont prime. Thus $$2^{\frac{p-1}{2}}=1\mod p$$ for $a>2$ and thus $2$ is not a generator for $p$.
A similar result holds for $a>1$, $b>0$ and the base of $3$.
However, for th... |
I asked myself a question while reading the post: https://math.stackexchange.com/questions/4885552/absolute-value-of-complex-numbers-abi .
Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \mathbb R^2\to \mathbb R$ defined by $$\forall x=(\xi,\eta),y= (\xi,\eta)... |
How do you present the absolute value as a norm and derive results? |
Thank you for reading my question!
Assume we meet this problem:
$$
\min_x f(x)\\
s.t. g(x)=0, h(x)\leq 0
$$
Let's assume the solution is $(x^*,\nu^*,\lambda^*)$
Its dual problem is
$$
\max_{\nu,\lambda} l(\nu,\lambda)\\
s.t. \lambda\geq 0
$$
where $l(v,\lambda)=\min_xf(x)+\nu g(x)+\lambda h(x)$
The s... |
Is the multiplier of the primal problem equal to the solution of the dual problem if strong duality holds and the solution of primal is unique? |
In the context of a simple random walk, a claim that I find on some proofs:
When proving that a stopping time at any point $x$ is finite, it suffices to prove that the stopping time at the origin is finite.
How so ?
Is this a consequence of the independence of the sequence of stopping times and the fact that th... |
If the stopping time is finite on the origin, it is finite for any other point, why? |
I am reading [the paper][1] and got stuck at the formula above (5.13):
Let $n\geq 4$.
(1) why
$$\sum_{i\neq j\in[n]}\cos^2(\theta_i-\theta_j)\geq n^2/2$$
holds?
Note that this corresponds to the formula $$n^2/2\leq \|X\|_F^2$$ in the paper, where $X$ is matrix with $X_{ij}=\cos(\theta_i-\theta_j)$.
... |
**An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?**
**********************
*Intro__________*
On a [previous question](https://math.stackexchange.com/q/4838311/909869) I found that the following approximation could be used:
$$\ln\left(1+e^x\right)\approx \frac{x}{... |
I asked myself a question while reading the post: https://math.stackexchange.com/questions/4885552/absolute-value-of-complex-numbers-abi .
Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \mathbb R^2\to \mathbb R$ defined by $$\forall x=(\xi,\eta),y= (\xi,\eta)... |
As stated in the title, given a poset $(S,\leq)$ I think it's trivial that an unbounded subset $A \subseteq S$ is cofinal, but does the opposite implication hold?
**Definition 1.** A subset $X$ of a poset $(S,\leq)$ is said to be **cofinal** if: $\forall s \in S \; \exists x \in X \; s \leq x$.
**Definition 2.** ... |
Cofinal subset equivalent to unbounded subset? |
I asked myself a question while reading the post: https://math.stackexchange.com/questions/4885552/absolute-value-of-complex-numbers-abi .
Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \mathbb R^2\to \mathbb R$ defined by $$\forall x=(\xi,\eta),y= (\xi,\eta)... |
When I first heared about Roché’s theorem, I said to myself: "Wow, this is a great result since it gives us a method to determine approximately where the zeros of a polynomial are located". However, the other day it struck me that maybe we don't need any deep, advanced theorem like that to determine such a thing. Would... |
Do we really need Roché’s theorem? |
Let $x \in [0,1]$ be an *irrational algebraic number*, and $y \in [0,1]$ algebraic. Does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that
$$
kx \equiv y \mod 1?
$$
This can be thought of a question about circle rotations from dynamics: marking a point at $0$... |
I asked myself a question while reading the post: https://math.stackexchange.com/questions/4885552/absolute-value-of-complex-numbers-abi .
Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \mathbb R^2\to \mathbb R$ defined by $$\forall x=(\xi,\eta),y= (\xi,\eta)... |
When I first heared about Rouché’s theorem, I said to myself: "Wow, this is a great result since it gives us a method to determine approximately where the zeros of a polynomial are located". However, the other day it struck me that maybe we don't need any deep, advanced theorem like that to determine such a thing. Woul... |
Do we really need Rouché’s theorem? |
Let $n$ be a natural number. When constructing the [Goodstein sequence][1] $(n)_{k}$, we start with $(n)_{1}=n$ written in complete base $2$, and we proceed recursively. That is, if $(n)_{k}$ has been defined and it is not $0$, $(n)_{k+1}$ is obtained from $(n)_{k}$ by bumping the base by $1$ in the complete base $k+1$... |
Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema.
1. $\forall x \ (0 \neq S ( x ))$
2. $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
3. $\forall x \ (x + 0 = x )$
4. $\forall x, y \ (x + S( y ) = S( x + y ))$
5. $\forall x \ (x \cdot... |
Is there a problem if I don't use $0$ in Peano arithmetic? |
Let $x \in [0,1]$ be an *irrational algebraic number*, and $y \in [0,1]$ algebraic. Does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that
$$
kx \equiv y \mod 1?
$$
This can be thought of a question about circle rotations from dynamics: marking a point at $0$... |
Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema.
1. $\forall x \ (0 \neq S ( x ))$
2. $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
3. $\forall x \ (x + 0 = x )$
4. $\forall x, y \ (x + S( y ) = S( x + y ))$
5. $\forall x \ (x \cdot... |
There is a combinatorial way of doing this, for those who know basic graph theory, that uses far less number theory as well. Indeed, we need just the following:
**Lemma 1:** *Let $C$ be a cycle and let $A$ be a subset of the vertices of $C$ s.t. $A$ is more than half the vertices of $C$. Then at least two vertices i... |
Let $x \in [0,1]$ be an *irrational algebraic number*, and $y \in [0,1]$ algebraic. Does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that
$$
kx \equiv y \mod 1?
$$
This can be thought of a question about circle rotations from dynamics: marking a point at $0$... |
How do I find all the solutions to
> $2006=(x+y)^{2}+3x+y$
Where $x,y \in \mathbb{Z}$
One such solution I found was $(x,y)=(1000,-998)$. I did it by the following,
$2006+3y=(x+y)^{2}+3(x+y)+y$
$2(1003+y)=(x+y)(x+y+3)$.
Which upon solving the system, we will get $x=1000$ and $y=-998$.
How can I co... |
I'm doing practice problems in a book for studying purposes and I'm just stumped on how to solve this one. I've tried everything I can think of.
Water from source to destination can go through 8 different pumping stations (see Figure). Each of the 16 pipes lets the water through with probability p. What's the probab... |
What's the probability that water will reach the destination? |
Is it legal to chain derivative operators? |
I'm doing practice problems in a book for studying purposes and I'm just stumped on how to solve this one. I've tried everything I can think of.
Water from source to destination can go through 8 different pumping stations (see Figure). Each of the 16 pipes lets the water through with probability p. What's the probab... |
Let $X$ be a compact metric space with a homeomorphism $f$· The dynamical system transitive, but not minimal. Also Let $Intran(f)$ be the set of all points whose orbit is not transitive. How to show that $Intran(f)$ is dense in $X$?
I have tried by contradiction. If that happens maybe both the the set of transitive ... |
I'm studying Newton's method for solving a system of non-linear equations.
A system of non-linear equations, whose (one of the) solutions is ($x^*$, $y^*$):
\begin{equation*}
\left\{
\begin{alignedat}{3}
% R & L & R & L & R & L
f(x,y) = 0 \\
g(x,y) = 0
\end{alignedat}
\right.
\end{equation*}
Ca... |
How to choose initial (x0,y0) point to approximate a solution of a system of non-linear equations using newton method? |
I'm studying Newton's method for solving a system of non-linear equations.
A system of non-linear equations, whose (one of the) solutions is ($x^*$, $y^*$):
\begin{equation*}
\left\{
\begin{alignedat}{3}
% R & L & R & L & R & L
f(x,y) = 0 \\
g(x,y) = 0
\end{alignedat}
\right.
\end{equation*}
Ca... |
I am currently engaging with the following hypergeometric function as a result of attempting to find [a solution for this probability problem for $n$ number of dice][1]:
$$_2F_1\left (\frac{n+k}{2}, \frac{n+k+1}{2};k+1;\frac{4b(d-e+1)}{d^2}\right ); \{n,k,b,d,e\}\in \mathbb{N}^+, z\in \mathbb{R},b < e \leq d$$
I'... |
I am currently trying to learn geometric algebra with Dorans book. I tried to do excercise 2.2., but I get stuck at the formulation of the exercise:
By expanding the bivector $a \wedge b$ in terms of geometric products, prove
that it anticommutes with both a and b, but commutes with any vector
perpendicular to the... |
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