instruction stringlengths 12 30k |
|---|
Second Derivative Test for Extrema:
> Let $f:\mathbb R \to \mathbb R$ be a function that is twice-differentiable on $(c-\varepsilon,c+\varepsilon)$ for some $\varepsilon >0$. Suppose $f^{\prime}\left(c\right)=0$.
>
> 1. If $f^{\prime\prime}\left(c\right)<0$, then $c$ is a strict local maximum of $f$.
>
> 2. If ... |
Second Derivative Test: Can we relax hypothesis that $f$ is twice differentiable on neighborhood? (Counterexample) |
Given:
A circle of unknown dimensions, 2 squares of a known size (x).
The squares intersect at shared vertex F, at 90º.
The squares intersect the circle at vertices A, B, and D
I'm trying to prove that vertex C is also on the circumference.
I know that A+B + circle center creates an Isosceles triangle. And th... |
how can i prove that a vertex intersect the circumference (...)? |
Given:
A circle of unknown dimensions, 2 squares of a known size (x).
The squares intersect at shared vertex F, at 90º.
The squares intersect the circle at vertices A, B, and D
I'm trying to prove that vertex C is also on the circumference.
I know that A+B + circle center creates an Isosceles triangle. And th... |
Given:
A circle of unknown dimensions, 2 squares of a known size (x).
The squares intersect at shared vertex F, at 90º.
The squares intersect the circle at vertices A, B, and D
I'm trying to prove that vertex C is also on the circumference.
I know that A+B + circle center creates an Isosceles triangle. And th... |
Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find easy counterexamples to classical statements. I hope somebody may help me to clarify. On page 69, Proposition 3.3.1.4 it states "Le... |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y$$
I cannot pursue it because of the floating point in comp... |
Given:
A circle of unknown dimensions, 2 squares of a known size (x).
The squares intersect at shared vertex F, at 90º.
The squares intersect the circle at vertices A, B, and D
I'm trying to prove that vertex C is also on the circumference.
I know that A+B + circle center creates an Isosceles triangle. And th... |
While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
Given a finite set of points in Euclidian plane such that any line passing through two of the points shall pass through at least one more of them, then all points have to be collinear.
Base case... |
I am trying to solve this problem. I got the right to left implication:
If $I$ is an intersection of prime elements, let $x\in I$. Since $x^1\in I$, we know $x\in r(I)$, and thus, $I\subseteq r(I)$. On the other hand, let $x\notin I$ such that $x^n\in I$. Then, since $I$ is an intersection of prime ideals (let them... |
Consider the function F defined on $]-\frac{\pi}{2}; \frac{\pi}{2}[$ by
$$F(x) = \int_0^x t \tan^2{t} dt$$
1. Compute $F(x)$ and $\displaystyle{\lim_{x \to \frac{\pi}{2}}} F(x)$
2. Determine the sign of $F(x)$
My attempt:
\begin{align}F(x) &= \int_0^x t \tan^2{t} dt \\
&= [t \tan{t}-t^2]_0^x - \int_0^x ... |
I am trying to solve this problem.
Let $I$ be a non trivial ideal of a ring $A$, and let $r(I)$ be its radical ring. Prove that $I=r(I)$ iff $I$ is an intersection of prime ideals.
I got the right to left implication:
If $I$ is an intersection of prime elements, let $x\in I$. Since $x^1\in I$, we know $x\in r(... |
At chemistry classes, the student receives a set of 20 names of chemical compounds. Among them there are two lipids that should be underlined. If both correct lipids are marked correctly, the student gets a point. Rightfully? Is such success trustworthy? How likely is it to get a point in this way of checking knowledge... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times (1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{1}{1 - x} - A(\frac{x}{2}) =... |
Setting: $\mathbb{R}[x]$ is the set of polynomials with real coefficients. All $f\in \mathbb{R}[x]$ are of the complex domain. $\mathbb{R}[x]|_{[a,b]}$ is the set of restrictions of functions in $\mathbb{R}[x]$. $\mathcal{C}([a,b)]$ is the set of continuous complex-valued functions on $[a,b]$.
Question:
1. How... |
Suppose $A$ is a finite cyclic group of order $n$, and $a$ is an element in $A$ of order $d$. My question is:
**Question 1:Can we find a generator $x$ of $A$ such that the element $a$ can be written in the form $x^{\frac{n}{d}}$?**
My attempt to solve this question is as follows:
Suppose $x$ is a generator of ... |
I have seen the solution to this problem where for the induction step we have:
$$2^{n+1} = 2 \cdot 2^n \gt 2 (n^2 + 4 n + 5) = (n + 1)^2 + 4(n + 1) + 5 + n^2 + 2 n \gt (n + 1)^2 + 4(n + 1) + 5$$
In this induction step we prove the inequality from left to right, my question is, is it possible to prove the inequali... |
Let $X=[0,1)$ and
\begin{align*}
\sigma\,=\,\{\varnothing\}\cup\left\{\,\bigcup_{i=1}^m\,[a_i,b_i)\,\bigg|\,m\in\mathbb N,\,a_i,b_i\in[0,1]: \ a_i<b_i\leq a_{i+1}<b_{i+1},\ \forall i\in\overline{1,m} \right\}.
\end{align*}
I want to show $\sigma$ is an algebra on $[0,1)$.
My proof :
> First, since $X=[0,1... |
Let $P$ be a statement saying there is large cardinal of some kind. For example, $P$ can be one of
0. There is a weakly inaccessible cardinal.
1. There is a Mahlo cardinal.
2. There is a weakly compact cardinal.
3. There is a measurable cardinal.
4. There is a strongly compact cardinal.
5. There is a supercompa... |
Basically I'm trying to find good starting values for algorithms that determine the roots of a polynomial (e.g. newton method). Obviously we are trying to get as close as possible to the root as we can, but how can we estimate where the roots of a polynomial lie?
Is a argument like: "If the coefficients are relative... |
Is the notion "If a polynomial has small coefficients (relative to the exponent), then it has small roots" true? |
Let $X=[0,1)$ and
\begin{align*}
\sigma\,=\,\{\varnothing\}\cup\left\{\,\bigcup_{i=1}^m\,[a_i,b_i)\,\bigg|\,m\in\mathbb N,\,a_i,b_i\in[0,1]: \ a_i<b_i\leq a_{i+1}<b_{i+1},\ \forall i\in\overline{1,m} \right\}.
\end{align*}
I want to show $\sigma$ is an algebra on $[0,1)$.
My proof :
> First, since $X=[0,1... |
Let $P$ be a statement saying there is large cardinal of some kind. For example, $P$ can be one of
0. There is a weakly inaccessible cardinal.
1. There is a Mahlo cardinal.
2. There is a weakly compact cardinal.
3. There is a measurable cardinal.
4. There is a strongly compact cardinal.
5. There is a supercompa... |
Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find easy counterexamples to classical statements. I hope somebody may help me to clarify. On page 69, Proposition 3.3.1.4 it states "Le... |
Setting: $\mathbb{R}[x]$ is the set of polynomials with real coefficients. All $f\in \mathbb{R}[x]$ are of the complex domain. $K$ is a compact subset of $\mathbb{C}$. $\mathbb{R}[x]|_{K}$ is the set of restrictions of functions in $\mathbb{R}[x]$. $\mathcal{C}(K)$ is the set of continuous complex-valued functions on ... |
Suppose I have to distribute a 2 gold bars to my 3 people. The probabilities of each person getting it is 90%, 60%, and 50%.
How can I use a random number generator to distribute the bars accordingly? |
Suppose I have to distribute 2 gold bars to 3 people. The probabilities of each person getting a bar is 90%, 60%, and 50%. Each person only gets 1 bar.
How can I use a random number generator to distribute the bars accordingly? |
Suppose I have to distribute 2 gold bars to 3 people. The probabilities of each person getting a bar is 90%, 60%, and 50%. Each person only gets 1 bar. There will be 1 person who doesn't get a bar.
How can I use a random number generator to distribute the bars accordingly? |
Here is an example:
It is easy to see that
$\{[0, \alpha] \setminus F: \alpha < \omega_1, F \subset \omega_1, F \text{ finite}\}$
is a base for a topology on $\omega_1$.
Obviously, this topology is $T_1$.
For $\alpha < \omega_1$,
$\{[0, \alpha] \setminus F: F \subset [0, \alpha) , F \text{ finite}\}$ is a cou... |
Let $X=[0,1)$ and
\begin{align*}
\sigma\,=\,\{\varnothing\}\cup\left\{\,\bigcup_{i=1}^m\,[a_i,b_i)\,\bigg|\,m\in\mathbb N,\,a_i,b_i\in[0,1]: \ a_i<b_i\leq a_{i+1}<b_{i+1},\ \forall i\in\overline{1,m} \right\}.
\end{align*}
I want to show $\sigma$ is an algebra on $[0,1)$.
My proof :
> First, since $X=[0,1... |
Setting: $\mathbb{R}[x]$ is the set of polynomials with real coefficients. All $f\in \mathbb{R}[x]$ has domain $\mathbb{C}$. $K$ is a compact subset of $\mathbb{C}$. $\mathbb{R}[x]|_{K}$ is the set of restrictions of functions in $\mathbb{R}[x]$. $\mathcal{C}(K)$ is the set of continuous complex-valued functions on $K... |
Suppose I have to distribute 2 gold bars to 3 people. The probabilities of each person getting a bar is 90%, 60%, and 50%. Each person only gets 1 bar. There will be 1 person who doesn't get a bar.
How can I use a random number generator to distribute the bars accordingly?
I am currently working on creating a gam... |
Suppose I have to distribute 2 gold bars to 3 people. The independent probabilities of each person getting a bar is 90%, 60%, and 50%. Each person only gets 1 bar. There will be 1 person who doesn't get a bar.
How can I use a random number generator to distribute the bars accordingly?
I am currently working on cr... |
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be unbounded and continuous everywhere.
Define $\mathcal{B}(x,\epsilon)=(x-\epsilon,x+\epsilon)$.
Define $\Delta_{f,\epsilon}(x)=\sup\{\delta>0\:|\: f(\mathcal{B}(x,\delta))\subseteq\mathcal{B}(f(x),\epsilon))\}$
In other words, $\Delta_{f,\epsilon}(x)$ is the radius of ... |
Suppose I have to distribute 2 gold bars to 3 people. The independent probabilities of each person getting a bar is 90%, 60%, and 50%. Each person only gets 1 bar or no bar at all. 2 people will get a bar and 1 person will not get a bar.
How can I use a random number generator to distribute the bars accordingly?
... |
# Setup
Let random variables $X_1, \ldots, X_n$ be i.i.d. $N(0, \sigma^2)$.
# What I know
By Chebyshev's inequality,
$$
\frac{1}{n}\sum_{i=1}^n X^2_i \overset{P}{\to} \sigma^2.
$$
On the other hand, since
$$
\mathbb E\left[\left(\frac{1}{n}X^2_i - 0\right)^2\right] = \frac{1}{n^2}\mathbb E[X^4_i] = \frac{3\s... |
I was reading the derivation of $E = mc^{2}$ but I am stuck in a very simple minor detail. In the derivation, we have the kinetic energy (K.E.) as $\int_{0}^{s} F \, ds$, where $F$ is constant.
Then we have
\begin{align*}
\int_{0}^{s} F \, ds &= \int_{0}^{s} \frac{d(\gamma m v)}{dt} \, ds \\
... |
The question :
What is the total number of words in length $500$ above {$ a $,$ b $} such that the letter $"a"$ appears more than $"b"$?
$(*)$ We know that the total number of words it's $ 2^{500} $. By, first glance it looks a half of them. But there are subsets of words which contains the same number of letters... |
What is the total number of words in length $500$ above {$ a $,$ b $} such that the letter $"a"$ appears more than $"b"$ ( without Brute force)? |
Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $\inf \, \sigma(B) = \lambda_1^B$ are isolated eigenvalues with finite multiplicity at the bottom of the spectrum, so th... |
A number is picked uniformly and randomly from a set of five-digit natural numbers. What is the probability that at least one of the digits of the number thus picked is 0?
I solved by taking numerator as 1st digit we have 9 options, then we select 1 out of remaining 4 digits and lock it for zero so 4c1 and remaining t... |
*Let the feet of angle bisectors of triangle $ABC$ are $X,Y$ and $Z$ . The circumcircle of triangle $XYZ$ cuts off three segment from lines $AB,BC $and $CA$, let it be $XX’, YY’, ZZ’$. Prove that at least one combination exists such that sum of two segment's length is equal to third segment’s length.*
WLOG, I assume... |
I want to prove the fact that
If $W\subseteq V$ is a vector subspace of $V$ than
$\dim(W)=\dim(V)=n \implies W=V$
Without using the fact that if $\dim(V)=n$ any set of $n$ independent vectors in $V$ is necessarily a basis of $V$, neither the fact that a basis of $\mathscr{L}=(\vec{v_1},...,\vec{v_n})$ is nec... |
Here is an example:
It is easy to see that
$\{[0, \alpha] \setminus F: \alpha < \omega_1, F \subset \omega_1, F \text{ finite}\}$
is a base for a topology on $\omega_1$.
Obviously, this topology is $T_1$.
For $\alpha < \omega_1$,
$\{[0, \alpha] \setminus F: F \subset [0, \alpha) , F \text{ finite}\}$ is a cou... |
>Find $n$ when the coefficients of the $16^{th}$ and $26^{th}$ terms of $(1+x)^n$ are equal.
$16^{th}$ term coefficient: $^nC_{15}= \space ^nC_{n-15}$
$26^{th}$ term coefficient: $^nC_{25}= \space ^nC_{n-25}$
$\Rightarrow 15 = n-25 \Rightarrow n=40$ $\space$ or $\space$ $n-15=25 \Rightarrow n=40$
This is th... |
Suppose our Fourier transform is given by $\mathcal{F}(f)(\xi) := \int_{\mathbb{R}^n}e^{-i\theta\left<x,\xi\right>}f(x)dx$ for some positive $\theta > 0$. I was wondering how/if you can construct a smooth indicator function on a set such that its Fourier transform's $L^1$ norm is at most, or equal, to some given consta... |
Can you construct a smooth indicator function $f$ on a set such that $\|\hat{f}\|_{L^1}\leq C$ for some given $C > 0$? |
The question :
What is the total number of words in length $500$ above {$ a $,$ b $} such that the letter "$a$" appears more than "$b$"?
$(*)$ We know that the total number of words it's $ 2^{500} $. By, first glance it looks a half of them. But there are subsets of words which contains the same number of letters... |
Let $(x_i)_{i=1}^\infty$ be a sequence in the product space $X = \Pi_{i=1}^\infty X_i$.
I would like to know conditions such that this statement is true: $(x_i)_{i=1}^\infty$ converges to $x \in \Pi_{i=1}^\infty X_i$ $\iff$ $\pi_k(x_i)$ converges to $\pi_k(x)$ (where $\pi_k$ denote the projection to the $k$-th compo... |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
Given: $$P_1 = (x_1, y_1), P_2 = (x_2, y_2)$$
... |
Here is an example:
It is easy to see that
$\{[0, \alpha] \setminus F: \alpha < \omega_1, F \subset \omega_1, F \text{ finite}\}$
is a base for a topology on $\omega_1$.
Obviously, this topology is $T_1$.
For $\alpha < \omega_1$,
$\{[0, \alpha] \setminus F: F \subset [0, \alpha) , F \text{ finite}\}$ is a cou... |
Given a collection of sets $\{Y_\alpha: \alpha \in A \}$ for an index set $A$. For each $\alpha \in A$, let $f_\alpha: X \to Y_\alpha$ be a function. The map $f: X \to \prod_{\alpha \in A} Y_\alpha$ defined by
$$
f(x) = \prod_{\alpha \in A} f_\alpha(x)
$$
is called the `evaluation map` where $\prod$ denotes t... |
How to proceed in order to prove the equality conditions mentioned? |
Setting: $\mathbb{R}[x]$ is the set of polynomials with real coefficients. All $f\in \mathbb{R}[x]$ has domain $\mathbb{C}$. $K$ is a compact subset of $\mathbb{C}$. $\mathbb{R}[x]|_{K}$ is the set of restrictions of functions in $\mathbb{R}[x]$. $\mathcal{C}(K)$ is the set of continuous complex-valued functions on $K... |
I am reading complex analysis by Gamelin and I am having trouble understanding the squre root function.
The principle branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this definiti... |
What is $\sqrt{-1}$? |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principle branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this definit... |
What is $\sqrt{-1}$ by the definition of The principle branch of square root function? |
Let $(x_i)_{i=1}^\infty$ be a sequence in the product space $X = \Pi_{i=1}^\infty X_i$, equipped with the product/box topology.
I would like to know conditions such that this statement is true: $(x_i)_{i=1}^\infty$ converges to $x \in \Pi_{i=1}^\infty X_i$ $\iff$ $\pi_k(x_i)$ converges to $\pi_k(x)$ (where $\pi_k$ d... |
In the plane curvature is rate of change of slope angle or rotation of tangent $\phi $ w.r.t. arc $s$, $ \tan \phi = \dfrac{dy}{dx},$ where $ ds^2= dx^2+dy^2$
This is all that is important by way of definition. Physically it is rate at which a curve turns, the rest is just manipulation:
Curvature
$$ = \fra... |
I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent.
That is, given $a,b,c \sim N(0,1)$, what is (a numerical approximation to 5 digits of) $\mathrm{Pr}(b^2 - 4 a c \geq 0)$ ?
---------... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principle branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this definit... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principle branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this definit... |
In the group $R^*$ can we find elements a and b such that $|a| = ∞$, $|b| = ∞$ and $ |ab| = 2$?
My attempt:
The group $R^*$ means the set of real numbers with zero omitted, forms a group under multiplication.
1 is the identity.
And we can't find such a & b such that, $|a| = ∞$, $|b| = ∞$ and $ |ab| = 2$.
Am ... |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y$$
I cannot pursue it because of the floating point in comp... |
Hmm... it's not as obvious as I thought
See, my first thought was this as a case of if $W$ is irrational and $a,b,c,d$ are rational and $a+bW = c+dW$ then $b=d$ and $a=c$.
Why? Because $a-c = (d-b)W$. The left hand side is rational so the right hand side must be rational. But $W$ is irrational and the *only* wa... |
I'm trying to wrap my head around the [perturbation theory paper by K.I. Martin][1]. I've made a toy Python script to try it out using float64 as 'high precision' for the reference and calculating the rest of the pixels in float32. Unfortunately, the code I've written is doing what it should be. I have a few questions ... |
solution will be
$\sin z = 10^3$
options: A z=(4n+1)pi/2+iarccosh(10^3)
B:z=(4n+1)pi/2
C:arccos(10^3)
D:none
answer first is correct
so here I first expand it like this and got two equations so how to solve these two equations and find solutions
I'm having difficulties with hyperbolic function
I have to solu... |
Recently i got interested at a [Relaxation of Collatz Conjecture](https://mathoverflow.net/questions/216358/relaxed-collatz-3x1-conjecture#:~:text=The%20Collatz%203x%2B1,2%20whenever%20x%20is%20even.). This goes just like the normal Collatz procedure, except when $x$ is even, you have the choice of applying either $x \... |
Two-player Variant of Collatz procedure? |
Let $F: \mathcal{A} \to \mathcal{B}$ be a functor and suppose that it admits a left adjoint $R: \mathcal{B} \to \mathcal{A}$. Take next an object $B \in \mathcal{B}$ and consider the object $(F\circ R \circ F \circ R)(B)$. Which is the relationship between such an object $(F\circ R \circ F \circ R)(B)$ and the object ... |
I'm reading a text (https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Projective_geometry/Chapter_2_Quadrics.pdf) to study about Quadrics in the projective space. The following is the example (1) (on Page 26).
## Example
Consider the hyperbola $xy = 1$.
Using the homogeneous coordinate ($x = x_1/x_0, y = x_... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principle branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this definit... |
$\DeclareMathOperator{\Arg}{Arg}$The times I taught complex analysis using the (good, free) [book](https://matthbeck.github.io/complex.html) of Beck, Marchesi, Pixton, and Sabalka, there was a nitpicky but useful handling polar angle:
* The _principal branch_ of argument $\Arg$ is **defined for all non-zero complex ... |
Let $a,b,c\ge 0:ab+bc+ca=3.$ Find the minimal value $$P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}.$$
I think $2+\sqrt{2}$ is desired result. $P$ attain this value when $a=b=c=1.$
I use $a+b+c\ge \sqrt{3(ab+bc+ca)}=3$ and we need to prove $$a^2b+b^2c+c^2a+abc\ge 4.$$
But it is wrong when $c=0,b=3,a=1.$
I got stuc... |
**Def:** The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^T \otimes\rho_B$
The definition can be extended to a general linear operator
**I want to prove the Partial tran... |
I want to show that for any nonnegative integers $l$ and $b$ we have
$$
\frac{l}{2^{b+1}} - 1
\leq
\left\lfloor
\frac{l-1}{2^{b+1}}
\right\rfloor.
$$
I have a proof where I wrote $l = \alpha\cdot 2^{b+1} + \beta$ with $0\leq \beta\leq 2^{b+1}$, but it's not very elegant.
Is there a more elegant way of showi... |
I was reading Sheldon Axler's Real Analysis book and he mentions the following (as does the course I am taking) :
There does not exists a function $\mu : \mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ with the following properties:
* $\mu( (a,b) ) = b-a$
* $\mu (\underset{n\in \mathbb{Z}_{+}}{\bigcup} A_n) = \u... |
**Question**: Let $q = p^{n}$ be a prime power. For which $q$ is the quadratic extension $\mathbb{F}_{q^{2}}$ of $\mathbb{F}_{q}$ of the form $\mathbb{F}_{q}(\sqrt{x})$ for $x \in \mathbb{F}_{q}$? Furthermore, for what $q$ is the cubic extension $\mathbb{F}_{q^{3}}$ of $\mathbb{F}_{q}$ of the form $\mathbb{F}_{q}(\sqrt... |
Let $\mathbb{F}$ be an arbitrary field of characteristic $\mathrm{char}(\mathbb{F})\neq 2$ and $V$ a $\mathbb{F}$-vector space. The definition of a quadratic form I am used to is a map $\varphi\colon V\to\mathbb{F}$ with the property $\varphi(\lambda v)=\lambda^{2}\varphi(v)$ such that
$$\beta_{\varphi}(v,w):=\frac{1... |
**Def:** The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^T \otimes\rho_B$
The definition can be extended to a general linear operator
**I want to prove the Partial tran... |
How can I determine the original number $x$ from its remainders $x_p$ and $x_q$, when it's divided by two relatively prime numbers $p$ and $q$, given that $\gcd(p, q) = 1$? I learned about a method using Bézout's theorem, which suggests that because $p$ and $q$ are coprime, there must be integers $X$ and $Y$ satisfying... |
How can I determine the original number $x\in[pq]$ from its remainders $x_p$ and $x_q$, when it's divided by two relatively prime numbers $p$ and $q$, given that $\gcd(p, q) = 1$? I learned about a method using Bézout's theorem, which suggests that because $p$ and $q$ are coprime, there must be integers $X$ and $Y$ sat... |
I am trying to prove the limit of 1/(\sqrt(n+2)-1)*\sin((n*\pi)/4) = 0. I have to use the limit definition to do this, but i get stuck with the sin((n*\pi)/4) part. I know how to do the 1/(sqrt(n+2)-1) part, any help? |
Let $X$ denote a vector space. Let $n:X\to\mathbb R$ be a norm over $X$ and let $s:X\to\mathbb R$ be a seminorm over $X$.
I have proved quite easily that the sum of two norms is a norm itself. I would like to understand what happens if I sum a norm with a seminorm.
Does one obtain a norm or a seminorm?
Accord... |
Is the sum between a norm and a seminorm still a norm? |
Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \leq (\sum_{i=1}^n x_iy_i)^2$$
The proposed inequality has a certain "Cauchy-Schwarz" flavour, but from the lower bound side of the s... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this defini... |
What is $\sqrt{-1}$ by the definition of The principal branch of square root function? |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
What is $\sqrt{-1}$? and what is $i$? (definition of The principal branch of square root function) |
We have to prove that $2^{7+n}\gt(9+n)^2+1$ which is true for $n=0$ (since $128\gt82)$. Assuming that
$2^{7+n}\gt(9+n)^2+1$ for $n$ and prove us that $2^{8+n}\gt(10+n)^2+1$.
One has $2^{7+n}\gt(9+n)^2+1\Rightarrow 2^{8+n}\gt 2((9+n)^2+2$
Is it $2((9+n)^2+2\gt(10+n)^2+1$ for $n$ integer positive?
Yes it is. I... |
I've been looking at the inequality:
$$\left(1-\frac{3}{5}\right)\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)+\left(1+3\cdot\frac{3}{5}\right)\frac{3\sqrt[3]{xyz}}{x+y+z}\geq4$$
Which proved to be surprisingly resistant to more 'elegant methods' or I'm just oblivious to something obvious.
Buffalo way works aft... |
I should probably study the classical viewpoint first. I haven't yet and I will eventually but let us stick to $\infty$-operads for this question. I'm following the [lecture notes from Hebestreit-Wagner][1]. Let me recall the terminology first.
One can define an $\infty$-operad $\mathbb{A}\mathrm{ssoc} \to \mathbb{\... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
I am trying to prove by induction that $$\sum_{k=1}^n k^2(k+1) = \frac{1}{12} n(n+1)(n+2)(3n+1)$$
I proved the base case for n=1 easily, however when proving the inductive case for (n+1), I am met with too complex of an expression. I go from $$\sum_{k=1}^{n+1} k^2(k+1) = (\sum_{k=1}^n k^2(k+1)) + (n+1)^2(n+2)$$
A... |
I want to show that for any nonnegative integers $l$ and $b$ we have
$$
\frac{l}{2^{b+1}} - 1
\leq
\left\lfloor
\frac{l-1}{2^{b+1}}
\right\rfloor.
$$
I have a proof where I wrote $l = \alpha\cdot 2^{b+1} + \beta$ with $0\leq \beta\leq 2^{b+1}$, but it's not very elegant, I say.
Let $l = \alpha\cdot 2^{b+1... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Let $V\subset \subset U\subset \mathbb{R}^n$. Can we construct a Lipschitz cut-off function $f$ such that
$$
f=0 \quad \text{on} \quad V ,\quad f=-1 \quad \text{on} \quad \mathbb{R}^n\setminus U,\quad f^2-|f'|\geq -C \quad \text{on} \quad U\setminus V
$$
hold？
Could you please give me some help with details? T... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.