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I am trying to block diagonalize a real orthogonal matrix, A. The condition is that the blocks should also be orthogonal. I found this pretty old yet [abstract paper][1] that says
"By block diagonalization methods one can obtain eigenvalues and eigenvectors while simultaneously “reducing” the size of the matrix, i.e.,... |
Let's consider that we have a variable $X$ ranging from $0.1$ to $0.5$. For simplicity, let's assume that the cumulative distribution function (CDF) is given by $F(x) \propto x^{0.5}$. However, this interval is constrained by the limited availability of experimental data. How can one extend the support of $X$ and recal... |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)= \lfloor \frac{an+b}{cn+d} \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by myself. My sol... |
In I. Daubechies' "Ten Lectures on Wavelets" it is stated in note 2 of chapter 2 (p. 51) that A. Grossmann, J. Morlet, and T. Paul's article "Transforms associated to square Integrable group representations. I. General results" can be generalized to cyclic representations. Unfortunately, Daubechies gives no proof of th... |
Suppose I consider a locally free coherent sheaf $E$ on $X$ and similarly $F$ on $Y$. Now suppose that I have a section $s$ of $E \boxtimes F$. Why does this section induce a map $H^0(X,E)^\vee \to H^0(Y,F)$? |
Is there a nice 'local' criterion for equivalence of categories? |
Here are two definitions that I have encountered:
The first, corresponding to [this Wikipedia page][1], is the following.
> _Definition._$\ $ Let $(X,d)$ be a metric space. Let $\epsilon\in\mathbb{R}^{>0}$. An **$\epsilon$-net** for $X$ is a subset $S\subseteq X$ of points such that the collection of $\epsilon$-b... |
How are $\epsilon$-nets (as in centers of $\epsilon$-ball covers) related to nets (as in topology)? |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)= \lfloor \frac{an+b}{cn+d} \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by myself. My sol... |
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = \gamma p f,\\
\\
\dfrac{dT}{dt} = \left( 1 - \dfrac{k}{T} \right) \left\{ b (1 - T) \dfrac{r^n}{f^n + r^n} - m \dfrac{f^n}{f^n + r^n} \right\},
\end{cas... |
> Create a list of all periodic points of f : $f (x) = x + sin x$ on R and
> use the graph of f to sketch its phase portrait. Identify the stable
> quantities for each of the periodic points.
What does the question mean?
"Create a list of all periodic points for f : $f (x) = x + sin x$ on $R$" ?
Then sketchi... |
How do I generate a phase-potrait (by hand)? |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by mys... |
Here are two definitions that I have encountered:
The first, corresponding to [this Wikipedia page][1], is the following.
> _Definition._$\ $ Let $(X,d)$ be a metric space. Let $\epsilon\in\mathbb{R}^{>0}$. An **$\epsilon$-net** for $X$ is a subset $S\subseteq X$ of points such that the collection of $\epsilon$-b... |
Here are two definitions that I have encountered:
The first, corresponding to [this Wikipedia page][1], is the following.
> _Definition._$\ $ Let $(X,d)$ be a metric space. Let $\epsilon\in\mathbb{R}^{>0}$. An **$\epsilon$-net** for $X$ is a subset $S\subseteq X$ of points such that the collection of $\epsilon$-b... |
I am working on part (b) of exercise 1 in Chapter 4 of Brian C. Hall's book. It states:
$\textbf{Exercise 1}$:
A $\textbf{locally integrable}$ function $\psi(x,t)$ satisfies the free Schr$\ddot{\textrm{o}}$dinger equation in the weak (or distributional) sense if for each smooth compactly supported function $\chi$, ... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I have been stuck in a seemingly trivial proof for a long time. In general, given a family of linear bounded operators $(T_i)_{i\in I}$ from Banach space $X$ to normed $Y$, if $\Vert T_n \Vert $ not uniformly bounded, I wish to find a sequence $(x_n)$ such that $T_n$ is not pointwise bounded for $x$.
Applying the le... |
I have come across an integral that involves a hypergeometric function, which can be expressed as follows:
$$I = \int_0^1 x^{1/2}(1-x)^{\epsilon-1} {_{2}F_1}(\frac{1}{2}+\epsilon,1+\epsilon;\frac{3}{2};x) dx.$$
Here, $\epsilon$ is a small complex quantity where $|\epsilon|\ll1$.
I found an integral formula i... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I was studying **Fourier Analysis: An Introduction** by *Stein and Shakarchi*.
I came across this theorem in Section 2.2, page number 39
> **Theorem 2.1** Suppose that $f$ is an integrable function on the circle with $\hat f(n) = 0$ for all $n\in\mathbb{Z}$. Then $f(\theta_0) = 0$ whenever f is continuous at the ... |
So, I'm trying to see if my approach can show that the set $S$ is not affine and is convex with a similar argument for both cases
$S = \left\{ \alpha \in \mathbb{R}^3 : \alpha_1 + \alpha_2 e^{-t} + \alpha_3 e^{-2t}\le 1.1 \right\} \\ \text{where } t \ge 1 $
My first observation is that, given $x \in S$ I can wr... |
How to prove that $S = \left\{ \alpha \in \mathbb{R}^3 : \alpha_1 + \alpha_2 e^{-t} + \alpha_3 e^{-2t} \le 1.1 \right\} $ is convex but not affine? |
Let a sequence of functions $v_m$ that converges to $v$ in the space $L^{10/3}(\Omega)$. Additionally, $\sup\limits_{m} |v_m|<C$, where $C$ is a constant.
Can I conclude that $v_m$ converges to $v$ in $L^{\infty}(\Omega)$? |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
Prove that if $f$ is a convex and differentiable function, then for all $\mathbf{x}_1, \mathbf{x}_2$ and $\mathbf{y}$, and $\lambda_1, \lambda_2, \lambda>0$ such that $\lambda_1+\lambda_2=\lambda$, we have
$$
\left\langle\nabla f\left(\mathbf{y}\right), \lambda_1 \mathbf{x}_1+\lambda_2 \mathbf{x}_2-\lambda \mathbf{y}... |
Suppose I consider a locally free coherent sheaf $E$ on $X$ and similarly $F$ on $Y$. Now suppose that I have a global section $s$ of $E \boxtimes F$. Why does this section induce a map $H^0(X,E)^\vee \to H^0(Y,F)$? |
Are there any faster methods than long division for finding the binary expansion of a reciprocal by hand? I've noticed some patterns, such as $\frac{1}{2^n-1} = 0.\overline{[n - 1\text{ 0s}]1}$ and $\frac{1}{2^n+1} = 0.\overline{[n\text{ 0s}][n\text{ 1s}]}$, which speed up certain reciprocals, but only apply to a small... |
The **geometric mean** can be thought of as the exponential of the arithmetic mean of the logarithms of your dataset $\{a_i\}_{i=1}^n$:
$$GM = \exp\left(\dfrac{1}{n}\sum_{i=1}^{n}\ln(a_i)\right)$$
Similarly, the **standard deviation** of a dataset is the square root of the arithmetic mean of the squares of your err... |
I am currently starting to study topology through the Topology without Tears book, and I came across the Example 1.1.5 and I am not sure if I understand why is the result like it is. The example says as it follows:
> Let $\mathbb{N}$ be the set of all natural numbers and let $\tau_4$ consist of $\mathbb{N}, \emptyse... |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by mys... |
Some sources define an abelian category as a pre-abelian category where every mono is the kernel of its cokernel and epi is the conernel of its kernel. Some other sources define abelian categories as pre-abelian categories where every mono is a kernel and every epi a cokernel (not necessarily of their cokernel or kerne... |
Are these two definitions of abelian categories equivalent? |
Let $X\geq 0$ be ar real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the derivative of $$E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$ with respect to $z$. I started like this: If we have density functions $f(x)$ and $g_x(y)$ of $X$ and $Y(X)$ I calculated
\begin{align*}
E[e^{X... |
Seven people are standing in two rows, with three people in the front row and four in the back row. Among them, A and B must stand next to each other, while A and C must stand separately. How many different arrangements are there?
My thought process is like this, but the result is incorrect.
First, we consider th... |
I'm studying limits and colimits and more precisely I'm looking at forgetful functors and I'm trying to see if they preserve limits and colimits. In order to do that I first look at terminal and initial object and check if they are preserved, if not then I can conclude, but if they are preserved I don't know what else ... |
Some sources define an abelian category as
> a pre-abelian category where every mono is the kernel of its cokernel and epi is the cokernel of its kernel.
Some other sources define an abelian category as
> a pre-abelian categories where every mono is a kernel and every epi a cokernel (not necessarily of their co... |
The direct answer to your question is "no". At least if by "stable" you mean all eigenvalues of the Jacobian at $x^*$ have negative real part. (See, e.g., Section 2.9 Theorems 1 and 2 from Differential Equations and Dynamical Systems, L. Perko 3rd edition).
How then do we square this with this particular example? Th... |
> Seven people are standing in two rows, with three people in the front row and four in the back row. Among them, A and B must stand next to each other, while A and C must stand separately. How many different arrangements are there?
My thought process is like this, but the result is incorrect.
First, we consider ... |
I'm studying limits and colimits and more precisely I'm looking at forgetful functors and I'm trying to see if they preserve limits and colimits. In order to do that I first look at terminal and initial object and check if they are preserved, if not then I can conclude, but if they are preserved I don't know what else ... |
1 - ((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2 - (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3 - (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis over the sub-g... |
1- ((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2- (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3- (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis over the sub-grou... |
I want to compute a constant $\large c$ from a convergence of probability problem.
Also I have a doubt regarding the relation between the uniform and beta distribution.
The problem says -
>Suppose that $\large \{\mathrm{U_{n}}\}_{n \geq 1}$ is a sequence of iid random variables with uniform distribution ove... |
How do I compute the constant $c$ from the convergence in probability problem? |
1 ((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2 (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3 (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis over the sub-groups ... |
I had a homework question asking me to evaluate the series $1 + \frac{1}{2} - 1 + \frac{1}{3} + \frac{1}{4} - \frac{1}{2}...$
Ultimately the solution was just to combine the negative terms with the ones right before them to get a Taylor series expansion for $\ln(2)$. However, while looking for the solution I noticed... |
What does the function $f(x) = \lim \limits_{n \to \infty} H_{\lfloor xn \rfloor} - H_n$ look like? |
Suppose we have experimental data confined to a specific interval, allowing us to compute a Cumulative Distribution Function (CDF). Recognizing potential biases within the data and understanding the need to extend the interval beyond the observed data, the question arises: Is it possible to adjust the support and ref... |
I am playing a game of cards. The goal is to withdraw randomly a card until there is no more. The symbol of the last card is the winning one, ie: if the last card is a spike, spike wins. Right now the are already 10 heart drawn, 10 spikes, 11 clubs and 12 diamonds; What is the probability that clubs win ? The deck of c... |
My motivation for this question is similar to the one in [this question][1]. However, that question only asks about the possibility of an infinite $\mathbb F_2$-vector space having trivial automorphism group. What other finite automorphism groups can occur? In particular, can $\mathbb Z/(3)$ occur? (see also [this ques... |
Without choice, what can be the (finite) automorphism groups of $\mathbb F_2$-vector spaces? |
I have data with columns like this:
ENTITY Avg_Score_1 N_obs_1 Avg_Score_2 N_obs_2
With sample values:
001 | 0.997 | 900 | 1.13 | 905
002 | 0.890 | 250 | 0.96 | 251
For about 1000 unique ENTITY values, and I’m wondering how to calculate the ICC between avg_score_1 and avg_score_2 for the whole dataset.
... |
I am playing a card game. The goal is to withdraw randomly a card until there is no more. The symbol of the last card is the winning one, ie: if the last card is a spike, spike wins. Right now there are already 10 heart drawn, 10 spikes, 11 clubs and 12 diamonds; What is the probability that clubs win ? The deck of car... |
I had a homework question asking me to evaluate the series $1 + \frac{1}{2} - 1 + \frac{1}{3} + \frac{1}{4} - \frac{1}{2}...$
Ultimately the solution was just to combine the negative terms with the ones right before them to get a series expansion for $\ln(2)$. However, while looking for the solution I noticed that a... |
1- \quad ((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2-\quad (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3-\quad (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis o... |
1- ((¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R))**∨ R**
2- (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ (Q ∨ R)**∨ R**
3- (¬P ∧ ¬Q) ∨ (P ∨ R) ∨ **(Q ∨ R) ∨ (R ∨ R)**
Q:Why did we remove the double parenthesis after step number 1?
Also, my thinking process was that I could distribute the R outside the double parenthesis over the sub-grou... |
I came across a question in the book Calculus for the Practical Man. The question stated to find the maximum or minimum value of the following function:
$$f(x) = \frac{1}{4}cos^2x - sin 2x$$
I tried to find first the critical value by differentiating this function to get:
$$f'(x) = \frac{-sin2x}{4}-2cos2x$$
... |
What is the maximum and minimum value of the following function? |
> If H is a normal subgroup of a group G such that H and G/H are finitely generated, then so is G.
Proof- Let $H = \langle x_1,...,x_n \rangle$ and $G/H = \langle y_1H,...,y_mH \rangle$. Claim that $G = \langle x_1,...,x_n,y_1,...,y_m \rangle$. Note first that $\langle x_1,...,x_n,y_1,...,y_m \rangle \subset G$ by ... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I came across a question in the book Calculus for the Practical Man. The question stated to find the maximum or minimum value of the following function:
$$f(x) = \frac{1}{4}\cos^2x - \sin 2x$$
I tried to find first the critical value by differentiating this function to get:
$$f'(x) = \frac{-\sin2x}{4}-2\cos2x$... |
I was studying **Fourier Analysis: An Introduction** by *Stein and Shakarchi*.
I came across this theorem in Section 2.2, page number 39
> **Theorem 2.1** Suppose that $f$ is an integrable function on the circle with $\hat f(n) = 0$ for all $n\in\mathbb{Z}$. Then $f(\theta_0) = 0$ whenever f is continuous at the ... |
Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.
Does there exist an example of the following?
* A commutative ring with unity $R$
* A mapping $\operatorname{d}: R\operatorname{-Mod} \to \overline{\mathbb{R}}_{\geq 0}$ with the following properties:
* If $A \cong B$, $\operatorname{... |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by mys... |
I consider for a constant $m\in\mathbb{N}$ the following Diophantine equation
$$\sum_{i=1}^n x_i^4\prod_{j\neq i}x_j^2 - m = 0$$
over the hypercube $C^n=[0,m]^n$ for some $m\in\mathbb{N}$. I am trying to understand whether there are integer solutions to this equation inside $C^n$.
The basic approach would be to... |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by mys... |
Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$
Prove that $(c=0 , b < d , 0 < a \le d) \implies f$ is a surjective function
I tried to solve this Olympiad problem by mys... |
There is seemingly a difference whether one approaches the "logit" from statistics / econometrics:
$$F^{-1}(\pi) = \log\left(\frac{\pi}{1-\pi}\right) = \text{logit}(\pi)$$ (I)
Or from the machine learning perspective, where it is typically called "logistic regression":
$$F(\eta) = \frac{\exp(\eta)}{1 + \exp(\e... |
What is meant by the inverse of a CDF? Logit vs. Logistic Regression. |
> How does $\sum_{1\leq i} \frac{1}{p_i^s}$ become $\frac{1}{2^s}$?
It's simply isn't. The $\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...$ series isn't obtained by taking those sums individually but by *rearranging* their terms. First I'll explain those sum means:
$$\sum_{1\leq i} p^s_i \text{ is the sum of prime... |
It was pointed out to me recently that the polyhedron [{7,4|3}](https://www.abstract-polytopes.com/atlas/336/208/14.html) has the same automorphism group as the [Klein quartic](https://en.wikipedia.org/wiki/Klein_quartic).
Specifically {7,4|3} is:
$\langle \rho_0,\rho_1,\rho_2 \mid \rho_0^2, \rho_1^2, \rho_2^2, (... |
Does {7,4|3} have a realization on the Klein quartic? |
> How does $\sum_{1\leq i} \frac{1}{p_i^s}$ become $\frac{1}{2^s}$?
It's simply isn't. The $\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...$ series isn't obtained by taking those sums individually but by *rearranging* their terms. First I'll explain those sum means:
$$\sum_{1\leq i} p^s_i \text{ is the sum of prime... |
Let $X_1, \dots, X_n$ be i.i.d. random variables from a categorical distribution with parameter vector $p = (p_1, \dots, p_k)$.
Suppose we’re interested in the estimating $\vartheta := f(p)$ for some function $f : \mathbb R^k \to \mathbb R$, e.g. the true Gini-Simpson index of this distribution $\vartheta = f(p) = 1... |
I have a question about the fair use of vouchers.
The facts are as follows:
At a lottery retailer, a lottery ticket was purchased (valid for 4 consecutive weeks, for the same 6 fields [with 6 numbers chosen from 1–49 per field] ). The cost of € 29.5 for this ticket was borne by person 1, who had requested the lott... |
> How does $\sum_{1\leq i} \frac{1}{p_i^s}$ become $\frac{1}{2^s}$?
It's simply isn't. The $\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...$ series isn't obtained by taking those sums individually but by *rearranging* their terms. First I'll explain those sum means:
$$\sum_{1\leq i} p^s_i \text{ is the sum of prime... |
I have a question concerning the canonical topology of a topos.
By the definition I know, the canonical topology is given by universal effective epimorphisms as the covering sieves. Now I have read that for a topos, the canonical sieves are those that are jointly epimorphic.
But I don't know how to prove that join... |
> How does $\sum_{1\leq i} \frac{1}{p_i^s}$ become $\frac{1}{2^s}$?
It's simply isn't. The $\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...$ series isn't obtained by taking those sums individually but by *rearranging* their terms. First I'll explain those sum means:
$$\sum_{1\leq i} p^s_i \text{ is the sum of prime... |
Let a sequence of functions $v_m$ that converges to $v$ in the space $L^{10/3}(\Omega)$, Where $\Omega$ is a bounded domain. Additionally, $\sup\limits_{m} |v_m|<C$, where $C$ is a constant.
Can I conclude that $v_m$ converges to $v$ in $L^{\infty}(\Omega)$? |
> Why do we take it to be A implies B and not, say, A and B, or, other
> truth-functions?
We want to rule out the possibility that both $A$ is true and $B$ is false $\neg (A \land \neg B)$ which is often used to define logical implication:
$~~~~~~~A\implies B~~\equiv \neg(A \land \neg B)$
From this definition... |
> Why do we take it to be A implies B and not, say, A and B, or, other
> truth-functions?
We want to rule out the possibility that both $A$ is true and $B$ is false $\neg (A \land \neg B)$ which is often used to define logical implication:
$~~~~~~~A\implies B~~\equiv \neg(A \land \neg B)$
From this definition... |
How to prove $\sum_{i=1}^n\frac{(1-a_i)^n}{a_i\prod_{j\neq i}(a_j-a_i)}=\frac{1}{a_1\cdots a_n}-1$? |
According to my Textbook, a fBm Process $B_t$ has the same distribution as its increments:
$(B_{t+h}^H-B_t^H)_{t\geq 0} \stackrel{d}{=}(B_t)_{t\geq 0}$
To proof this, note that both sides are centered Gaussian Processes. Hence it is enough to show that they have the same covariance i.e.:
$Cov(B_{t+h}^H-B_t^H,... |
Let $(E,h)$ be a hermitian holomorphic vector bundle and suppose we have local trivializations $U_i$ and $U_j$ with transition maps $g_{ij}$. Let $H_i$ and $H_j$ represent the matrices of $h$ with respect to the trivializations $U_i$ and $U_j$. We have that $H_j=g^T_{ij}H_i\bar{g}_{ij}$. I'm trying to prove the followi... |
I consider for a constant $m\in\mathbb{N}$ the following Diophantine equation
$$\sum_{i=1}^n x_i^4\prod_{j\neq i}x_j^2 - m = 0$$
over the hypercube $C^n=[0,m]^n$ for some $m\in\mathbb{N}$. I am trying to understand whether there are integer solutions to this equation inside $C^n$.
The basic approach would be to... |
If 4D space is represented by quaternions with real component being time and imaginary components describing 3D space, will the area in this 3D space be oriented? Will volume be negative? And will this 4D space metric be [+---] ds^2=dt^2-dx^2-dy^2-dz^2? |
In a round robin tournament with $n\geq 10$ teams, each match ends in either a win or loss. What is the minimum number of losses to ensure that a team is out of contention of top 2 spots?
**My attempt:** assign them ranks 1 to $n$. There are a total of $\frac{n(n-1)}{2}$ matches and hence as many wins. So, the rank ... |
Let $\langle A,R \rangle$ and $\langle B,S \rangle$ be two well-orders, and let $\text{pred}(A,x,R) := \{y \in A \;|\; yRx\}$ and similarly for $\text{pred}(B,z,S)$.
It is claimed that one of the following must hold:
$1$. $\langle A,R \rangle \cong \langle B,S \rangle$.
$2$. $\exists y \in B\left(\langle A,R... |
This question concerns the theory developed in Winning Ways for Your Mathematical Plays. The relevant Volume 1 can be found online [here.](https://annarchive.com/files/Winning%20Ways%20for%20Your%20Mathematical%20Plays%20V1.pdf)
I'm unclear about the intuition behind the authors' "Bypassing Reversible Moves" result,... |
I am tasked with finding whether the function $()=\tan(/(-1))$ can be developed into a Laurent series around $=1$, and if so what is the radius of convergence and what is the residue?
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Today, my professor at university gave us a problem to solve. We need to prove that if a connected graph G has at least 4 vertices and contains a vertex of degree at most 2, G cannot be 3-connected.
I understand that a graph is 3-connected if it has at least 4 vertices and remains connected even if we remove any tw... |
According to Wolfram Alpha, this number is not prime, but it does not display prime factors of this number as it normally does for other numbers so I am not so sure about it. Several other factorization calculators I tried also failed to find prime factors of this number. The main question I want to ask is this: Is i... |
I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \boxtimes v$ for any $v \in V$, and extended linearly over $T(V)$. I've encountered the following formula that expresses ... |
I have to find the number of zeros at the end of the number
$$5^5 \times 10^{10} \times 15^{15} \times \dots \times 120^{120} \times 125^{125} $$
So, First I know that the zeros are formed by the multiplication of 2 with 5 and 2 appears less number of times than 5 than we should count the number of 2.
Even though ... |
Number of zeros at the end of $5^5 \times 10^{10} \times 15^{15} \times \dots \times 120^{120} \times 125^{125} $? |
This answer ended up being kind of long. So if you want to see the process, jump to the summary at the end.
#### Denominators of the form $2^n - 1$
When we write a real number $x \in [0, 1]$ as a sequence of bits $(b_1, b_2, \dots)$, where each $b_i \in \{0, 1\}$, what we are really doing is expressing it as the ... |
I consider for a constant $k\in\mathbb{N}$ the following Diophantine equation
$$\sum_{i=1}^n x_i^4\prod_{j\neq i}x_j^2 - k = 0$$
over the hypercube $C^n=[0,m]^n$ for some $m\in\mathbb{N}$. I am trying to understand whether there are integer solutions to this equation inside $C^n$.
The basic approach would be to... |
Let $F$ be a field containing all $n^{th}$ roots of unity. A *Kummer Extension* over $F$ is a Galois extension of the form $E=F[r_1,...,r_m]$ where $r_i^n\in F$ for all $i=1,...,m$.
From Kummer theory, the groups $$\operatorname{kum}\left(E/F\right)=\left(F^{1/n}\right)^\times/F^\times\simeq \operatorname{Gal}(E/F)... |
Is there polynomial P(x) withbinteger coefficiants that satisfies P(2)=4 and P(6)=6? |
I spent some time learning about the standard proof of Goodstein theorem using ordinals, and its strength. In order to remove the question from the unanswered list I will give an answer. I would appreciate feedback if my understanding of the concepts is erroneous.
The answer to the first questions is yes, it can be ... |
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