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I came across this question on [Quora][1]:
> How many $3\times 3$ matrices over the finite field $\{0,1,2\}$ satisfy the condition $A^{-1}=(A^T)^2$?
Thanks to NumPy, with sheer [brute-force][2], I found that there are $33$ such matrices. (You may go through the code and the output.)
I believe that there is a m... |
Imagine that you have 2x conncected graphs and they have the same number of vertrices of each degree and the same number of cycles of each length how does it come up despite these facts that they are not isomorphic?
Aren't these facts obvious if they have the same number of vertrices and the same degree and the same... |
I'm currently solving an initial-boundary-value problem with Dirichlet boundary values, and are stuck on one part that I can't say what is wrong and right.
I got the following PDE
$$\begin{cases}\partial_t u(t,x)=\partial_x^2u(t,x), 0<x< \pi, 0<t\\
u(0,x)=1\\
u(t,x)=0, 0<t, x \in \partial D \end{cases}$$
Im ... |
Given the first fundamental form and the shape operator: $$g = \begin{pmatrix}
1 & 0\\
0 & \cos^2(u)
\end{pmatrix} \text{and} \,S = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},$$ I want to check, if there exists a surface $f: \Omega \rightarrow \mathbb{R^3}$ satisfying these conditions. We know that $S^T = bg^{... |
I want to calibrate the parameters $\theta$ of a known forward model $y=f(\theta, x)$, i.e., I want to identify offsets of around +- 10% from a nominal model $\theta_0$
I can not measure $y$ directly, but my measurement function $h_c$ only depends on this forward model:
$y_c = h_c(f(\theta, x))$
Let's call the ca... |
Can I use Sturm's comparison theorem to say something about the average?
Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function
$$ \ddot{x}(t) + f^2(t) x(t) = 0. $$
Let $C$ be a constant. Then, we can find $T$ such that $C^2 < f^2(t)$ for all ... |
what more can I say using Sturm's comparison theorem? |
**One-sentence summary:** I want an intuitive explanation for why closeness of probability measures (in terms of, eg., TV or Lévy-Prokhorov) implies the existence of good couplings.
**Set up.** Consider a separable metric space $(X,d)$ and the space of probability measures $\mathcal{M}(X)$. Equip $\mathcal{M}(X)$ wi... |
Actually, I don't know it's true or not. I just make observation and write simple python script that checked this for x belongs to [1, 50] and it's alright. Can somebody tell it is true or not and if it is true, how I can prove this? |
Suppose that we have a sequence of polyhedron $C^k$ that is non-increasing, i.e., we have $C^{k+1}\subseteq C^k$. Suppose moreover that
each $C^k$ is formed by the convex hull of at most $n$ points, where $n$ is a fixed constant independent of $k$, i.e., $C^k=\mbox{conv}(z_1^k,\ldots,z_n^k)$, where $\mbox{conv}$ mea... |
Suppose that $f \in K[X]$ is a separable irreducible polynomial. Let $L$ be the splitting field of $f$ over $K$. Is it true that $[L:K] = |Gal(f|K)|$?
Why I think this is because, first of all $|Aut(L|K)| = [L:K]$. Further, since if $\alpha_1, \ldots, \alpha_n$ are the roots of $f$, then $L = K(\alpha_1, \ldots, \al... |
Is the degree of the extension the same as the size of the Galois group of the polynomial? |
I am an electrical engineer, currently focused on designing a feedback network. I've attempted to model the engineering problem as a mathematical sequence and sought to derive a general expression for it. Despite numerous attempts, I have been unsuccessful thus far. The sequence is shown below
$$
A_n=A_{n-1}+k\sum_{j... |
Just gathering up the several apt comments: there is simply no assurance, no general reason, why a countable subset of $(0,1)$ (etc.) should occur as a monotone increasing sequence. The comments give counter-examples: $\{1,{1\over 2},{1\over 3},\ldots\}$ and, actually, all rationals in that interval. Trying "either mon... |
Given a Riemannian manifold $\mathcal M$ with metric $\langle \cdot, \cdot \rangle$, we define the length of a smooth curve $\gamma : I \to \mathcal M$ as
\begin{equation}
L(\gamma) = \int_I \sqrt{\langle \gamma'(t), \gamma'(t) \rangle}_{\gamma(t)} \ \text{d}t.
\end{equation}
Why are we allowed to do this, and not ... |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
We define a complex number as a number of the form $a+ib$, and then we can equip the set of complex numbers with addition ($+$) and multiplication ($*$), these two operations have good properties, thus $\mathbb{C}$ is a field. But it seems we did not define what is the meaning of $ib$, does it mean $i*b$ even though we... |
We define a complex number as a number of the form $a+ib$, and then we can equip the set of complex numbers with addition ($+$) and multiplication ($*$), these two operations have good properties, thus $\mathbb{C}$ is a field. But it seems we did not define what is the meaning of $ib$, does it mean $i*b$ even though we... |
We define a complex number as a number of the form $a+ib$, and then we can equip the set of complex numbers with addition ($+$) and multiplication ($*$), these two operations have good properties, thus $\mathbb{C}$ is a field. But it seems we did not define what is the meaning of $ib$, does it mean $i*b$ even though we... |
[enter image description here][1]
[1]: https://i.stack.imgur.com/aUDMk.png
Hi, Can you explain to me how it is possible to calculate a derivative at the zero point **without calculating the limit at zero?**
here is translation of this question:
*Show that the zero function is differentiable on all *ℝ* w... |
This is a revision of another question found [here][1]:
Consider a criminal pondering the commission of a crime, C. It could be launching a cyber-attack or embezzling funds.
How likely is it that the attempt will succeed? Call it P(S).
How likely is it that a malefactor will attempt C in the next year? Call it P... |
How to represent difficulty and success? |
Given a Riemannian manifold $\mathcal M$ with metric $\langle \cdot, \cdot \rangle$, we define the length of a smooth curve $\gamma : I \to \mathcal M$ as
\begin{equation}
L(\gamma) = \int_I \sqrt{\langle \gamma'(t), \gamma'(t) \rangle}_{\gamma(t)} \ \text{d}t.
\end{equation}
Why are we allowed to do this, and not ... |
Let $K$ be a commutative ring and $R$ a $K$-algebra. Consider the $K$-algebra $R^e=R\otimes_K R^{op}$. The category of $R-R$-bimodules is a monoidal category with the tensor product $\otimes_R$ of $R$-bimodules and unit $R$. After identifying the category of $R-R$-bimodules with the category $R^e\text{-Mod}$ of left $R... |
Let $X \in \mathbb{R}^{d \times n}$ and $Y \in \mathbb{R}^{d \times m}$ have columns $x_{i}, y_{j} \in \mathbb{R}^{d}$ respectively. Is there any way to write the matrix
$$\sum_{i = 1}^{n}\sum_{j = 1}^{m}x_{i}y_{j}^{\top}$$
in terms of a matrix product, say via a a Kronecker product?
I have this expression in a la... |
The question is
> Examples of natural proofs by induction exercises which needs 3 or more base cases?
I think an elementary natural proof of the parity of the Somos-4
sequence is a nice exercise. Given the Somos-4 sequence of positive integers that satisfies
$ s_n s_{n-4}=s_{n-1}s_{n-3}+s_{n_2}^2 $
for all inte... |
As I understand it, Ito processes are those semi-martingales whose finite variation part is an integral (against Lebesgue measure) and whose (continuous local) martingale part is a stochastic integral against a brownian motion. But I thought all (continuous local) martingales could be written as integrals against a bro... |
What makes Ito processes special? |
$\color{black}{Background:}$
$\textbf{Exercise 14:}$
Let $p$ be a fixed prime integer and let $R$ be the set of all rational numbers that can be written in the form $\frac{a}{b}$ with $b$ not divisible by $p.$ Prove that
$(a)$ $R$ is an integral domain caontaining $\mathbb{Z}$. [Note $n=\frac{n}{1}].$
$(... |
Any single variable polynomial in $GF(2)$ is reduced to first degree $x^n=x$. So all polynomials of $n$ variables will be like $x+y+xy$ We can transform any 3-SAT as a system of polynomial equations in $GF(2)$.
$x$ or $y$ is equivalent to $x+y+xy$ in $GF(2)$.
$x$ and $y$ is equivalent to $xy$ in $GF(2)$ and by ad... |
I have some confusion with the definition of unbiased estimators. Suppose you are given a random sample $X_1,...,X_n$ of real-valued random variables defined on a common probability space. A statistic is defined as $g(X_1,...,X_n)$, where $g:\mathbb{R}^n\to\mathbb{R}$ is a measurable function. If there is some paramete... |
Suppose we've removed some faces such that no two of them share an edge. Then if vertices are considered to be connected, the resulting polyhedron is still rigid, because every edge length remains fixed so we know that each hole is bounded by three edges of equal length - so it still has to be an equilateral triangle. ... |
As I understand it, Itô processes are those semi-martingales whose finite variation part is an integral (against Lebesgue measure) and whose (continuous local) martingale part is a stochastic integral against a Brownian motion. But I thought all (continuous local) martingales could be written as integrals against a Bro... |
What makes Itô processes special? |
I'm delving into variations of the classic partition problem, specifically focusing on a version that may be termed the "Minimum Product Difference Partition Problem" (MPDPP). The traditional partition problem aims to divide a set of integers into two subsets such that the difference in their sums is minimized. In cont... |
I'm delving into variations of the classic partition problem, specifically focusing on a version that may be termed the "Minimum Product Difference Partition Problem" (MPDPP). The traditional partition problem aims to divide a set of integers into two subsets such that the difference in their sums is minimized. In cont... |
It seems to me with the Sleeping Beauty problem we have the following with h = heads, t = tails, m = Monday, and tu = Tuesday.
$$
\begin{align}
P(h) &= P(t) = 1/2 \\
P(m|h) &= 1 \\
P(m|t) &= 1/2 \\
P(tu|t) &= 1/2
\end{align}
$$
Since we don't know what day it is, we want to answer the question of $P(h| m ... |
So, treat aces (A) as a 1 so that A, 2, 3, 4 is a run, while J, Q, K, A is not.
Valid examples are:
1. 9♣️, 10♣️, J♣️, 5♥️
2. 8♣️, 9♣️, 10♣️, 3♣️
3. 2♣️, 3♣️, 4♣️, K♦️
Invalid examples are
1. A♣️, 2♣️, 3♣️, 4♣️
2. 8♣️, 9♣️, 10♣️, J♣️
\
My solution so far is:
$$x = \frac{(_4C_1 × 11) × ??? }{52C4}$$
I... |
In a 52 card deck, what is the probability of getting a run of exactly 3 cards of the same suit in a 4-card hand? |
So, treat aces (A) as a 1 so that A, 2, 3, 4 is a run, while J, Q, K, A is not.
Valid examples are:
1. 9♣️, 10♣️, J♣️, 5♥️
2. 8♣️, 9♣️, 10♣️, 3♣️
3. 2♣️, 3♣️, 4♣️, K♦️
Invalid examples are
1. A♣️, 2♣️, 3♣️, 4♣️
2. 8♣️, 9♣️, 10♣️, J♣️
\
My solution so far is:
$$x = \frac{(_4C_1 × 11) × ??? }{52C4}$$
I... |
The [sleeping beauty problem][1] is a famous problem where a coin is flipped and then a subject is put to sleep. If the coin was heads they will be awoken on Monday asked what their belief is that the probability was heads. If the coin was tails they will also be awoken on Monday and asked the same question. But, they ... |
So, treat aces (A) as a 1 so that A, 2, 3, 4 is a 4-card run (not a 3-card run), while J, Q, K, A is a 3-card run.
Valid examples are:
1. 9♣️, 10♣️, J♣️, 5♥️
2. 8♣️, 9♣️, 10♣️, 3♣️
3. 2♣️, 3♣️, 4♣️, K♦️
Invalid examples are
1. A♣️, 2♣️, 3♣️, 4♣️
2. 8♣️, 9♣️, 10♣️, J♣️
\
My solution so far is:
$$x = \fr... |
Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote *this* Cauchy completion of $X$. We then have a canonical isometric embedding $\iota\colon X\to\hat X$ ($x\mapsto(x, x, x, \ldots)$... |
I've been trying to understand a little of the mathematics behind FEM and I got stuck in the definition of the weak (or variational) form. All literatures I have researched so far, they just say that we should multiply the differential equation by a trial function *v*, integrate over the domain and, then, integrate by ... |
In a chemical reaction that follows the logistic equation, if the initial velocity is 0, it becomes a steady solution. However, in the equation of motion, the fact that the initial velocity is 0 does not necessarily mean that it is a steady solution.
For example,for the $$\frac{dx}{dt}(t) = 4(1-0.5x(t))x(t)$$,
unde... |
The answer [here][1] makes sense to me, in the sense that the stochastic integral does not preserve the local martingale property if the integrand is not predictable, but I am confused as to why. I have been taught the following definition of the stochastic integral:
We first define the quadratic variation for a RC... |
Why is the stochastic integral only defined for predictable integrands? |
Looking at solving:
$$\int_{0}^{2\pi}\frac{1}{\sqrt{2}-cosx}\,dx \\$$
According to integral tables (such as:https://www.math.stonybrook.edu/~bishop/classes/math126.F20/CRC_integrals.pdf) the solution is:
$$\frac{2}{\sqrt{a^2 - b^2}}atan\frac{\sqrt{a^2 - b^2}tan(x/2) }{a+b}$$
$$a=\sqrt{2}, b=-1$$
$$\lef... |
The answer [here][1] makes sense to me, in the sense that the stochastic integral does not preserve the local martingale property if the integrand is not predictable, but I am confused as to why. I have been taught the following definition of the stochastic integral:
We first define the quadratic variation for a RC... |
I have encountered many types of sigma algebra generated by a random variable. I would like to clarify my understanding of them.
The simplest one is generated by a random variable. I skip it here.
Then I have sigma algebra generated by any family of random variables. i.e. Suppose $X_i$s are rvs from $(\Omega, A)$ ... |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of
$$-\Delta u = f \,\,\,\text{in}\,U$$
$$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$
where $f \in C(\bar{U}), g\in C(\partial U).$ Prove that
$$\sup_{x \in U}|u(x)| \leq C\left(\sup_{y \in \partial U... |
I really intended for this to be a comment. You should take a look at [Why are Sobolev spaces useful?][1] There are some nice details there.
Firstly, if you had a sufficiently smooth function, then solving a PDE in the classical sense is **equivalent** to solving it in the weak sense. Next, asking why we have to mul... |
As an undergraduate engaged in applied mathematics research, specifically within mathematical programming, my work has quite some intersection with areas like algebraic geometry for polynomial optimization and the use of manifolds in manifold optimization algorithms.
Producing research is my priority, but I'm afraid... |
Let's say I have $f: \mathbf{R}^{2} \to \mathbf{R}$, and I want to know if $\lim_{(x,y) \to (0,0)} f(x,y)$ exists.
Suppose I substitute x and y for their polar forms $rcos(\theta ), rsin(\theta )$.
Then, when analysing $\lim_{r \to 0} f(r, \theta)$, I get some $\theta$-dependent trigonometric function (e.g. $1-... |
Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$.
Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an extension $\tilde f:\tilde X\to \tilde Y$ of $f$ in the sense that $\tilde f$ is Borel and $\tilde f(x) = f(x)$ for $... |
I've added a section to the bottom of my answer on a more general approach that shows how you can treat functions that can be applied to themselves within set theory, representing them as something other than sets of ordered pairs.
-----
First, in spite of the name, a $\lambda$-function isn't a function in the ma... |
How do I go from an infinite amount of instantaneous rates to an average rate? |
Is $f(x)=\lim_{m\to\infty}m(x-b)$ the same as $x=b$? |
I am trying to prove the theorem below with the Normality Lemma. In particular, I am having trouble with constructing the sets necessary to apply the Normality Lemma and was wondering if my approach is possible or not. And if not, what would be a valid approach? To clarify, I am supposed to be applying the Normality Le... |
Help with applying lemma in proving normality of a subspace? |
There are two Gaussian matrices, $A\sim N(0,\sigma_A^2\mathbf{I})$ and $B\sim N(0,\sigma_B^2\mathbf{I})$, respectively.
Their dimensions are $A\in \mathbb{R}^{L\times M}$ and $B\in \mathbb{R}^{N\times M}$, respectively.
I want to find the expectation of $|AB^H|^2$.
So, how can I express the value of $\mathb... |
Question 1: An account with an initial amount B earns compound interest at an annual effective interest rate *i*. The interest in the third year is 426 and the discount in the seventh year is 812. Find *i*.
The work for question 1 is simple: (B(1+i)^7)/(B(1+i)^3) = 812/426. This leads you to *i* = 17.5%
Question ... |
What is the difference in the wording of these questions? |
Almost nobody does this. There is a discussion in the Wikipedia article on spinor groups but it has too many mistakes and too few references. The only solid math reference I know is in Chapter 5 (freely available from author's webpage [here](https://www.math.ucla.edu/~vsv/papers/ch5.pdf)) in
<cite authors="Varadara... |
I found this question online (not homework) and wanted to check my answer.
Let
$$A_1 = \\{r: r \textit{ rational, and } \lim \text{inf } x_n <r < \lim \text{sup } x_n \\}$$
$$A_2 = \\{r: r \textit{ rational, and } \lim \text{inf } x_n >r\\}$$
$$A_3 = \\{r: r \textit{ rational, and } \lim \text{sup } x_n <r\\}$$... |
>A real-valued function $f$ defined in $(a, b)$ is said to be convex if $f( tx+ (1 - t)y) \leq tf(x) + (1 - t)f(y)$
whenever $a < x < b, a < y < b, 0 < t < 1$
I want to Prove that every convex function is continuous and I had an idea that can be used to prove this statement but my proof writing skills is not the... |
I found this question online (not homework) and wanted to check my answer.
Let
$$A_1 = \{r: r \textit{ rational, and } \lim \text{inf } x_n <r < \lim \text{sup } x_n \}$$
$$A_2 = \{r: r \textit{ rational, and } \lim \text{inf } x_n >r\}$$
$$A_3 = \{r: r \textit{ rational, and } \lim \text{sup } x_n <r\}$$
Sta... |
I found this question online (not homework) and wanted to check my answer.
Let
$$A_1 = \{r: r \textit{ rational, and } \lim \text{inf } x_n <r < \lim \text{sup } x_n \}$$
$$A_2 = \{r: r \textit{ rational, and } \lim \text{inf } x_n >r\}$$
$$A_3 = \{r: r \textit{ rational, and } \lim \text{sup } x_n <r\}$$
Sta... |
[![\[enter image description here\]][1]][1]
[1]: https://i.stack.imgur.com/aUDMk.png
Hi, Can you explain to me how it is possible to calculate a derivative at the zero point **without calculating the limit at zero?**
here is translation of this question:
*Show that the zero function is differentiable on... |
Let’s suppose I have two spheres $S_1$ and $S_2$. Let's suppose they have a non-trivial intersection, meaning its not a point. Then their intersection forms a circle.
Assume I have the centers of each sphere $P_1$ and $P_2$. Assume their radii are $r_1$ and $r_2$ respectively.
Consider this diagram:
[![ente... |
Consider having $A$, with $A^{-1} previously calculated.
Say I apply a rank 1 update on $A$ using $v = (\infty,0,0,0,0)$:
$${B} = vv^T A $$
Clearly A is no longer invertible, but the pseudo inverse still exists.
We can apply the sherman morrison formula ${\displaystyle \left(A+vv^{\textsf {T}}\right)^{-1}=... |
Take invertible matrix $A$, and set a diagonal entry of $A$ to $/infty$. Can we use $A^{-1} to find the pseudo inverse of this new matrix? |
Let $G, G'$ be smooth algebraic groups over $k$ (absolute Galois group $\Gamma$) which are etale **inner** forms of each other, that is, there exists an isomorphism $G_{k_s} \cong G'_{k_s}$ and the associated torsor for the automorphism group is in the image of the natural map $H^1(k, G/Z_G) \to H^1(k, \operatorname{Au... |
Is galois cohomology invariant under inner forms and not just pure inner forms? |
Consider having $A$, with $A^{-1}$ previously calculated.
Say I apply a rank 1 update on $A$ using $v = (\infty,0,0,0,0)$:
$${B} = vv^T A $$
Clearly A is no longer invertible, but the pseudo inverse still exists.
We can apply the sherman morrison formula ${\displaystyle \left(A+vv^{\textsf {T}}\right)^{-1}... |
Take invertible matrix $A$, and set a diagonal entry of $A$ to $\infty$. Can we use $A^{-1} to find the pseudo inverse of this new matrix? |
Consider having $A$, with $A^{-1}$ previously calculated.
Say I apply a rank 1 update on $A$ using $v = (\infty,0,0,0,0)$:
$${B} = vv^T A $$
Clearly A is no longer invertible, but the pseudo inverse still exists.
We can apply the sherman morrison formula ${\displaystyle \left(A+vv^{\textsf {T}}\right)^{-1}... |
Take invertible matrix $A$, and set a diagonal entry of $A$ to $\infty$. Can we use $A^{-1}$ to find the pseudo inverse of this new matrix? |
You can't model $\lambda$ functions as set functions because the domain of a $\lambda$ function includes that function itself. This violates the foundation axiom.
However, they clearly are some sort of arrow. Are the arrows of $\lambda$ calculus their own thing or can they be represented in some other way? |
>A real-valued function $f$ defined in $(a, b)$ is said to be convex if $f( tx+ (1 - t)y) \leq tf(x) + (1 - t)f(y)$
whenever $a < x < b, a < y < b, 0 < t < 1$
I want to Prove that every convex function is continuous and I had an idea that can be used to prove this statement but my proof writing skills is not the... |
Assuming that $X \in L^2$ then by Jensen's inequality and the fact that $\sqrt{x}$ is concave we have
$$
E(|\bar X_n - \mu|) = E\left(\sqrt{(\bar X_n - \mu)^2}\right)
\le \sqrt{\text{Var}(\bar X_n)} = \frac{\sigma}{\sqrt{n}}.
$$
The $\sqrt n$ rate is sharp, as it is obtained when the $X_i$'s are iid from a... |
I know <x²+1> has 2x²+2 but I’m confused by the fact that how (2x² + 2x + 1 + <x²+1>) = (2x + 2 + <x²+1>) ?
Thank you! |
When working with Z ₃[x] / <x²+1>, why (2x² + 2x + 1 + <x²+1>) = (2x + 2 + <x²+1>)? |
Is the De Moivre's formula only intended to be used for unknown values of the input $z$, but not to fixed values of $z$? |
Assuming $X$ is reflexive, that is the linear map $J_X:X \mapsto X^{**}$, $J_X(x)(f) = f(x)$ for all $f \in X^*$ is bijective. Is it true that $J_X^{-1}T^{**}J_X = T$?
Here's my attempt: Let $x \in X$, $y := (J_X^{-1}T^{**}J_X)(x)$. We have $(J_X^{-1}T^{**}J_X)(x) = J_X^{-1}(J_X(x)T^*) = y$. Hence $J_X(y) = J_X(x)T^... |
When working with $\mathbb Z_3[x] / \langle x^2+1\rangle$, why $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$?
I know $\langle x^2+1\rangle$ has $2x^2+2$ but I'm confused by the fact that how $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$?
Thank you! |
When working with $\mathbb Z_3[x] / \langle x^2+1\rangle$, why $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$? |
*Background:*
**Exercise 14:**
Let $p$ be a fixed prime integer and let $R$ be the set of all rational numbers that can be written in the form $\frac{a}{b}$ with $b$ not divisible by $p. Prove that
*(a)* $R$ is an integral domain caontaining $\mathbb{Z}$. [Note $n=\frac{n}{1}$].
*(b)* If $\frac{a}{b}\in R... |
Is the de Moivre's formula only intended to be used for unknown values of the input $z$, but not to fixed values of $z$? |
I am working with time series data, and applying FFT to calculate the power spectrum. The data is something like this:
$y = \sum_{k=1}^{5} \cos(10 k x)$
All the peaks in the FFT output should have identical magnitudes, similar to the pattern shown in the graph below:
[![expected FFT output][1]][1]
However,... |
Why are FFT results different from theory and how to eliminate the difference? |
I know this platform is for higher level math. This is just a high school level question but I don't know any more suitable platform to ask. Apologise in advance.
The question is differentiate the expression and multiply by x, do this twice and set x=1
(x^(n+1)-1)/(x-1)
After differentiating twice, I get:
latex... |
I know this platform is for higher level math. This is just a high school level question but I don't know any more suitable platform to ask. Apologise in advance.
The question is differentiate the expression and multiply by x, do this twice and set x=1
(x^(n+1)-1)/(x-1)
After differentiating twice, I get:
latex... |
> Prove that, regardless of the way the students choose their partners,
> there are always two students such that there are at least five other
> students who have all worked with both of them or have worked with
> neither of them.
First, it's important to understand that if a student has "worked with both of the... |
Suppose $A$ is a positive semi-definite matrix, and $x$ and $y$ are real vectors. Under what conditions does the following result hold?
$$
x'y > (<) 0 \Longrightarrow x'Ay \geq (\leq) 0
$$ |
I know this platform is for higher-level math. This is just a high school level question but I don't know any more suitable platform to ask. Apologise in advance.
The question is to differentiate the expression and multiply by $x$, do this twice and set $x=1$.
$$\dfrac{x^{n+1}-1}{x-1}$$
After differentiating twi... |
Suppose $A$ is a positive semi-definite matrix, and $x$ and $y$ are real vectors.
1. Under what conditions does the following result hold?
$$
x'y > (<) 0 \Longrightarrow x'Ay \geq (\leq) 0
$$
2. Does there exist $\varepsilon>0$ such that the following result always holds?
$$
x'y > \varepsilon (<-\varepsilon... |
*In a 52 card deck, what is the probability of getting a run of exactly 3 cards of the same suit in a 4-card hand?*
So, treat aces (A) as a 1 so that A, 2, 3, 4 is a 4-card run (not a 3-card run), while J, Q, K, A is a 3-card run.
Valid examples are:
1. 9♣️, 10♣️, J♣️, 5♥️
2. 8♣️, 9♣️, 10♣️, 3♣️
3. 2♣️, 3♣️, ... |
Let $G, G'$ be smooth algebraic groups over $k$ (absolute Galois group $\Gamma$) which are etale **inner** forms of each other, that is, there exists an isomorphism $G_{k_s} \cong G'_{k_s}$ and the associated torsor for the automorphism group is in the image of the natural map $H^1(k, G/Z_G) \to H^1(k, \operatorname{Au... |
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