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I'm quite new to Analytic NT, so was wondering if the following is true, and beyond being obviously true, how could we prove it rigorously line-by-line?
[See: Mertens Third theorem][1]
The function they use to "modulate the product" (which normally diverges to zero) is $\ln n$. And the constant they get is $\gt ... |
How can we formally / rigorously use Merten's third theorem with $n^2 - 1$ instead of $\ln n$? |
I’m not sure about the correctness of my assumption about the generating sets of decomposable Abelian groups, so I will be very grateful if you can tell me whether I’m right or wrong, and if I’m wrong, where are the mistakes?
There is the group G: $$G = \mathbb{Z}_{12}\oplus\mathbb{Z}_{18}$$
$$\mathbb{Z}_{12}=4\mathb... |
How can we formally / rigorously use Mertens' third theorem with $n^2 - 1$ instead of $\ln n$? |
You can also use substitution $x-3=a \ ,\ a\to0 $
$$\lim_{x\to3} \frac{(\sin(x-3))}{(x^2-9)}\\x=a+3\\\lim_{a\to 0 }\frac {\sin a}{(a+3)^2-9}=\\\lim_{a\to 0 }\frac {\sin a}{a^2+6a+9-9}=\\\lim_{a\to 0 }\frac {\sin a}{a(a+6)}=?\\$$ |
Your question has two aspects: why is the right answer right, and why is yours wrong.
The number of possible subsets in a set $S$ with $n$ elements is $2^{n}$; this collection of subsets is often referred to as the power set of $S$. The fact that $P(S)$ has $2^n$ elements is readily seen when you list the possible s... |
What is the minimum degree of polynomial with $\sqrt[n]{a}$ as a root that has integer coefficients? |
I am studying a course on Stochastic Processes and encountered the following question on Poisson Processes:
> If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k \leq n$ are non-negative integers, then prove the following:
$$ P(N(s) = k | N(t) = n) = \binom{n}{k} \Big{(} \frac{s}k \Big... |
*Context:*
[This question](https://math.stackexchange.com/questions/1827411/evaluate-int-01-log-left-fracx2-2x-4x22x-4-right-frac-mathrm)
asks to calculate a definite integral which turns out to be equal to $$\displaystyle 4 \, \text{Ti}_2\left( \tan \frac{3\pi}{20} \right) -
4 \, \text{Ti}_2\left( \tan \frac{\pi... |
I consider the first order linear ODE of the form
$$ a\left(x\right)\frac{dy}{dx}+b\left(x\right)y+c\left(x\right)=0, $$
all the functions $a,b,c$ are $C^\omega$.
**Question 1:**
Does this necessarily mean, that solutions of this ODE also belong to $C^\omega$ and so can be found with the help of method of undefi... |
Currently I am reading the first edition "Introduction to Graph Theory" by Douglas B. West.
Exercise 1.1.9 states the following:
**1.1.9.** Suppose that $G$ is a simple graph having no vertex of degree $0$ and no induced subgraph with three edges. Prove that $G$ has at most four vertices.
**Counter-examp... |
I am studying a course on Stochastic Processes and encountered the following proof exercise on Poisson Processes:
> If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k \leq n$ are non-negative integers, then prove the following:
$$ P(N(s) = k | N(t) = n) = \binom{n}{k} \Big{(} \frac{s}... |
In terms of a general understanding of Moore-Smith convergence as a whole, this motto may help:
A net is what we need
instead of a sequence,
when we are dealing with a topological space
instead of a metric space.
Gemignani, "Elementary Topology", 1967, introduces this subject with the proposition t... |
In terms of a general understanding of Moore-Smith convergence as a whole, this motto may help:
A net is what we need
instead of a sequence,
when we are dealing with a topological space
instead of a metric space.
(Not meant literally, just a gloss, so to speak.)
Gemignani, "Elementary Topology", 1... |
I was reading the notes "Introduction to Complex Analysis" by Michael E. Taylor, and I am stuck on the following exercise about the Fourier Transform and the space $\mathcal{A}(\mathbb{R})$ defined as follows:
$$\mathcal{A}\left(\mathbb{R}\right):= \left\{ f \in L^{1}(\mathbb{R}) \cap C(\mathbb{R}) : \ f \ \text{bo... |
In his book "Probability theory", Klenke uses the following definition of transition kernel:
[![enter image description here][1]][1]
and if in $ii)$ the measure is a probability measure for all $\omega_1$ then $K$ is called a stochastic kernel.
Later on he defines regular conditional distribution of $Y$ given ... |
I came across the following problem while studying series of functions:
> Prove that $S(x)=\sum_{n=1}^\infty \frac{\sin(2^n x)}{2^n}$ is not differentiable at integers.
I am aware that this series is commonly known as the Weierstrass function, and according to Hardy's famous work in 1917, it is a typical example ... |
I encountered a mathematical problem while studying mathematical analysis.
The problem is as follows:
> Let $n\geq 1$ and consider a polynomial $R$ of degree $2n$ in two variables that is only equal to zero at the point $(0,0)$. Additionally, let $P$ and $Q$ be polynomials of degree at most $2n-2$ in two variab... |
Consider a sine function parameterized by amplitude, frequency, phase, and constant offset. In other words,
$h(x) = a\sin(fx + p) + c\tag*{}$
where $a$, $f$, $p$, and $o$ are arbitrary real numbers. Suppose we are also given a set of $(x,y)$ points $\{\,(x_1, y_1),$ $(x_2, y_2),$ $\ldots,$ $(x_n, y_n)\,\}.$ The o... |
How many points can be fit by a single sine function? |
Given a high-dimensional dataset X containing potentially redundant features, how can we efficiently aggregate and/or select features to achieve accurate prediction of target variable Y while reducing dimensionality? (I do not want to use methods like PCA, VAE etc which reduce the dimensions but with no or very little ... |
Consider the following boundary value problem
$$
\begin{cases}
y''(x)=p(x)y'(x)+q(x)y(x)+r(x)\\
y(0)=\alpha,\quad y(1)=\beta.
\end{cases}
$$
for continuous $p(x)$, $q(x)$, $r(x)$ and positive $q(x)>0$.
According to Section 8.1 of "A friendly introduction to numerical analysis" by Brian Bradie solution exists ... |
I am interested in the following: Given N vertices, how many different(up to graph-isomorphism) hypergraphs with every hyperedge covering exactly 3 different vertices, every vertex gets covered at least once and the hypergraph is connected exist?
For N <= 2: 0 because not enough vertices
N=3: 1 just \{1,2,3\} ... |
Fix $m\in \mathbb R$.
Define $f_m :\mathbb R^2 \setminus\{(0,0)\}\rightarrow(m,m+2\pi]$
$~~$as $(x,y) \mapsto$ argument of $(x,y)$ in $(m,m+2\pi]$.
i.e $$(x,y)=\left(\cos(f_m (x,y)),\sin(f_m (x,y))\right)$$
Now if $f_m$ is continuous then its restriction to unit circle is. which is compact but its image $(m,m+... |
From my lecture notes I have that:
>The Lie algebra is introduced as a set of matrices $\theta$ that gave rise to elements $M$ of the corresponding matrix group through $M=e^{i\theta}$. However, just like a group can be thought of as an abstract set of instead of matrices, we can also give an abstract definition of th... |
Suppose you have two positive definite matrices $M$ and $N \in \mathcal{S}_{++}^n$. They induce scalar products and norms on $\mathbb{R}^n$:
\begin{align}
\lVert x \rVert_M &= \sqrt{x^\intercal M^{-1} x}, &
\lVert x \rVert_N &= \sqrt{x^\intercal N^{-1} x}.
\end{align}
Introduce the function $f:x\mapsto \lV... |
Is it possible to compute the projection of a linear system solution on a constraint set instead of computing the solution of the constrained problem? |
I have found this problem in an examination paper, and I can't find a solution.
**Problem**: Consider the following ODE
$$x''(t)=-x(t)^3+x'(t)+1$$
Prove that, for any initial values $x(0)=x_0,x'(0)=y_0$, there exists a unique *global* solution $x(t;x_0,y_0)$ defined for every $t\in \mathbb{R}$.
One may... |
> Consider the function $f(x)=-x^2+6x$ with the domain $[2,5]$. Now calculate the range of the function according to its domain.
Here is my solution:
The domain is $2\leq x\leq5$ so I can wrote:
1. $2\leq x\leq5\implies4\leq x^2\leq25\implies-4\geq-x^2\geq-25\implies\boxed{-25\leq-x^2\leq-4}\ (\text{1})$
2.... |
I came across the following problem while studying series of functions:
> Prove that $S(x)=\sum_{n=1}^\infty \frac{\sin(2^n x)}{2^n}$ is not differentiable at integers.
I am aware that this series is commonly known as the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), and according to... |
Reading between the lines somewhat, it seems that the question you are *really* interested in, would be rather something like this: Suppose $(M,\varepsilon)$ is a model of set theory, $a\in M$ is a set and $C\subseteq M$ is some subset of the model with the property that each (actual) element of $C$ is an ($\varepsilon... |
I consider the first order linear ODE of the form
$$ x \frac{dy}{dx}+b\left(x\right)y+c\left(x\right)=0, $$
all the functions $b,c$ are $C^\omega$.
**Question:**
Does this necessarily mean, that solutions of this ODE also belong to $C^\omega$ and so can be found with the help of method of undefined coefficients?... |
I’m not sure about the correctness of my assumption about the generating sets of decomposable Abelian groups, so I will be very grateful if you can tell me whether I’m right or wrong, and if I’m wrong, where are the mistakes?
There is the group G: $$G = \mathbb{Z}_{12}\oplus\mathbb{Z}_{18}$$
$$\mathbb{Z}_{12}=4\mathb... |
***Question:***
Let E = $\Bbb{R}^3[X]$ an euclidian space with a dot product $<P,Q> \ = \int^1_0{P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $Vect(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector.
We have:
\begin{equation}
a + bX = p^\perp_{Vect(1,X)}(X^2) \if... |
How to calculate the distance from $X^2$ to $Vect(1,X)$? |
Let $X_t$ be a real stochastic process. I am new to stochastic processes and in some proofs I see inequalities such as:
$$|X_t - X_s| \leq C_1 \quad \forall |t-s| \leq C_2 \tag{1}$$
but $X$ is a function of $t$ and $\omega$, i.e. $X_t(\omega)$ so I am not sure what is meant by (1)? Does this mean (1) holds uniform in... |
Given sequence:
$$
\begin{cases}
x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 1\\
x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2
\end{cases}
$$
Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$
**Edit**
I've proved that $A|_{x=-1}$ is a minimum (see my answer). The only problem left is to show that $A_{... |
***Question:***
Let E = $\Bbb{R}^3[X]$ an euclidian space with a dot product $<P,Q> \ = \int^1_0{P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $span(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector.
We have:
\begin{equation}
a + bX = p^\perp_{span(1,X)}(X^2) \if... |
How to calculate the distance from $X^2$ to $span(1,X)$? |
Given sequence:
$$
\begin{cases}
x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 1\\
x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2
\end{cases}
$$
Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$
**Edit**
I've proved that $A|_{x=-1}$ is a minimum (see my answer). The only problem left is to show that $A_{... |
> For which primes $p$ do there exist $x,y\in\mathbb{Z}_p$ such that $3y^2=4x^3-10?$
This is a question in an elliptic curves course so I assume we want to transform this into an elliptic curve! My idea is that we want to birationally transform this curve into one with a Weirstrass equation, i.e. $y^2=x^3+Ax+B$.
... |
so I know parabolas are this but for line parallel to the x or y axis. is it possible to do it for any line?
taking it even further,
What about the set of all points that are equal distance to any specific parabola and any specific point?
you can take this as far as you want, I'm just curious as to what is poss... |
I've been reading about [Simpson's Paradox][1] and I understand how if you fit a regression on a per-group basis versus on the entire set, the coefficients in each group can be drastically different from coefficients on the global set because of confounding variables. However, one thought I had was whether such a princ... |
Let $X_t$ be a real stochastic process. I am new to stochastic processes and in some proofs I see inequalities such as:
$$|X_t - X_s| \leq C_1 \quad \forall |t-s| \leq C_2 \tag{1}$$
but $X$ is a function of $t$ and $\omega$, i.e. $X_t(\omega)$ so I am not sure what is meant by (1). Does this mean (1) holds uniform in... |
I am currently learning the basics of simplical homology and have been reading about gluing data and boundary maps. My question is when talking about choosing gluing data by defining map $\beta(\alpha,i)$ where $\alpha$ exists in the set of n-simplices and i describes which facet of that n simplex we are talking about,... |
The Poisson equation $\nabla^2 u = \Delta u = f$.
When $f = 0$ we obtain the Laplacian equation which to me has an intuitive interpretation with the mean-value property. However, how can we form an intuitive understanding of the Poisson equation from the function $f$? How does `f` impact `u`? |
> Let $X_t$ be a Compound Poisson Process defined by the sum of iid random variables $(Y_n)$ independent of $N$, written explicitly as:
$$ X_t := Y_1 + Y_2 + \cdots + Y_{N_t}$$
Show that there exists a function $g(a)$ such that for any $a \in \mathbb{R}$ we have that $$\mathbb E (e^{aX_t} ) = e^{g(a) t}$$
By the ... |
***Question:***
Let E = $\Bbb{R}^3[X]$ an euclidian space with a dot product $<P,Q> \ = \int^1_0{P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $span(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector.
We have:
\begin{equation}
a + bX = p^\perp_{span(1,X)}(X^2) \if... |
> Let $X_t$ be a Compound Poisson Process defined by the sum of iid random variables $(Y_n)$ independent of $N$, written explicitly as:
$$ X_t := Y_1 + Y_2 + \cdots + Y_{N_t}$$
Show that there exists a function $g(a)$ such that for any $a \in \mathbb{R}$ we have that $$\mathbb E (e^{aX_t} ) = e^{g(a) t}$$
By the ... |
> Let $X_t$ be a Compound Poisson Process defined by the sum of iid random variables $(Y_n)$ independent of $N$, written explicitly as:
$$ X_t := Y_1 + Y_2 + \cdots + Y_{N_t}$$
where $N$ is a Poisson Process.
>
>Show that there exists a function $g(a)$ such that for any $a \in \mathbb{R}$ we have that $$\mathbb E... |
***Question:***
Let $E = \Bbb{R}^3[X]$ an euclidian space with a dot product $\left\langle P,Q\right\rangle \ = \int^1_0 {P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $\operatorname{span}(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector. We have:
\begin{equatio... |
>I would like to find out what $\sum_{r=1}^{n} \pmatrix{1 & -1 \\ 0 & 2}^r$ is equal to.
I have typed this expression into WolramAlpha twice and got a different answer each time, both inconsistent.
**Attempt 1**
[![Attempt 1][1]][1]
**Attempt 2**
[![Attempt 2][2]][2]
We can see the entry in the top ... |
Why does Wolfram Alpha give two different, inconsistent answers for sum of a matrix? |
Think about $\mathbb R^3$ for a moment. Suppose you have a plane $P$ spanned by two vectors $u, v$ and you have a third vector $w$ which is *not* in the plane but is also *not* orthogonal to the plane. Then it makes sense to project $w$ onto the plane, right? But $w$ is not a linear combination of $u$ and $v$ because $... |
***Question:***
Let $E = \Bbb{R}_2[X]$ an euclidian space with a dot product $\left\langle P,Q\right\rangle \ = \int^1_0 {P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $\operatorname{span}(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector. We have:
\begin{equatio... |
Excuse my ignorance but how do you find the Galois group of an algebraic equation with integer coefficients modulo a prime number $ x^5+x+1\equiv 0\ \text{mod 5} $, say. Is there a Galois resolvent for this?
And if there is, how do you factorize an equation modulo prime, in case it hasn't got any of the p roots mo... |
> Let $X_t$ be a Compound Poisson Process defined by the sum of iid random variables $(Y_n)$ independent of $N$, written explicitly as:
$$ X_t := Y_1 + Y_2 + \cdots + Y_{N_t}$$
where $N$ is a Poisson Process.
>
>Show that there exists a function $g(a)$ such that for any $a \in \mathbb{R}$ we have that $$\mathbb E... |
Let A be a complex matrix of dimension n. Suppose that exists a natural number r such that A^r=c*In , where In is the identity matrix and c is a complex number. For what values of c, is A diagonalizable? I could only prove that c must be different from zero.
Sorry if I didn't write everything in the correct way, but i... |
The exercise is to show that $f(t) = t \cos \frac{\pi}{2t}$ is not rectifiable.
To show that, Tom Apostol is guiding us to consider the partition $P = \{0, \frac{1}{2n}, \frac{1}{2n-1}, ..., \frac{1}{3}, \frac{1}{2}, 1\}$, and to show the corresponding inscribed polygon $\pi(P)$ has length greater than $1 + \frac{1}... |
I need help with trying to understand the solution to $y'' + y = 5xe^{2x}$. Clearly the auxillary equation is just $m^2 + 1 = 0$ which can be solved by $m= \pm i$, thereby giving us $Acos(x) + Bsin(x)$ for our complementary function. The issue comes with the particular integral, so far, I have been taught to use the "m... |
I am reading the book Applied Fourier Analysis by Tim Olson. There I got introduced the Fourier Series and we have derived the coefficients $a_k$ and $b_k$ for the Fourier Series on $L^2[-\pi,\pi]$. The author is now setting up to introduce orthonormal expansions.
I could recreate the norm of $\sin$ and $\cos$ in $L... |
I know there were some issues with set theory that involved self-reference and famous example being that of Russell's example. Here, I have a set that I use a somewhat vague term "use" to construct but eventually answer leads to a contradiction with the property itself. What I want to ask is, can someone explain what h... |
***Question:***
Let $E = \Bbb{R}_2[X]$ an euclidian space with a dot product $\left\langle P,Q\right\rangle \ = \int^1_0 {P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $\operatorname{span}(1,X)$.
---
***Answer:***
Let $(a,b) \in \Bbb{R}^2$. $p$ is the orthogonal projector. We have:
\begin{equatio... |
Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up respectively, is?
My approach:
[Click this link][1]
But the given answer is 16/31.... |
In math, Hilbert's Hotel (e.g. https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) is a famous paradox that describes a situation where a hotel with infinitely many rooms and full capacity somehow has the ability to still accommodate more guests.
I have always found this concept difficult to unders... |
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 tried reading about the definition of a pre-additive $\infty$-category, there it is said that a pointed $\infty$-category is pre-additive if all finite products and co-products exist, and the canonical map $X\sqcup Y\longrightarrow X\times Y$ is an equivalence for any pair of objects of the $\infty$-category.
My q... |
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 $\mathcal{C}$ be an $\infty$-category, and let $X, Y$ be a pair of its objects. It is said that $\text{Mor}_{\mathcal{C}}(X,Y)$ has again the structure of an $\infty$-category.
What is this structure, how is it defined?
I guess the definition I am familiar with is saying that an $\infty$-category is a simplic... |
A projective $A$-module $P$ is called *one stably free* if $P\oplus A\simeq A^n$ for some $n\in \mathbb{Z}$. It can be seen that such projective modules are in one-to-one correspondence with the orbits of the natural $GL_n(A)$ action on the set $Um_n(A)=\{[a_1,\ldots,a_n]\in A^n\mid Aa_1+Aa_2+\dots+Aa_n=A\}$. It can be... |
I found a 'computational free' way by reading this article by [ST68](http://www.google.com.hk/url?q=https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2001/0006/0001/SDG-2001-0006-0001-a010.pdf&sa=U&ved=2ahUKEwi17rHj98GEAxWz8LsIHdVLDyIQFnoECAkQAg&usg=AOvVaw3GZ8zgfXM4z_rkSC3crSkU).
The method use Thm1.3... |
I found a 'computational free' way by reading this article by [ST68](http://www.google.com.hk/url?q=https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2001/0006/0001/SDG-2001-0006-0001-a010.pdf&sa=U&ved=2ahUKEwi17rHj98GEAxWz8LsIHdVLDyIQFnoECAkQAg&usg=AOvVaw3GZ8zgfXM4z_rkSC3crSkU).
The method use Thm1.3... |
Let A be a complex matrix of dimension $n$. Suppose that exists a natural number $r$ such that $A^r=c I_n$ , where $I_n$ is the identity matrix and $c$ is a complex number. For what values of $c$, is $A$ diagonalizable? I could only prove that $c$ must be different from zero.
Sorry if I didn't write everything in th... |
Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up respectively, is?
My approach:
[![enter image description here][1]][1]
But the g... |
Salam,
let's consider the function:
$ f(x)= x^2 + y^3 -3y - 2 $
i find two critic points:
$(0,1) \& (0,-1) $
Using the definition methode with:
$h_1=x \ \& \ h_2=y-1$ to test the point$ (0,1) $ i find that:
$f(h_1,h_2+1)-f(0,0)= h_1^4 + h_2^2(h_2+3) $
we can use $h_1=0$ to find that the sign is not constant... |
Is this function has a minimum in the point (0,1)? |
Does any closed foem of this integral exists or is related to some special function integral? |
Consider a sine function parameterized by amplitude, frequency, phase, and constant offset. In other words,
$h(x) = a\sin(fx + p) + c\tag*{}$
where $a$, $f$, $p$, and $o$ are arbitrary real numbers. Suppose we are also given a set of points $\{\,(x_1, y_1),$ $(x_2, y_2),$ $\ldots,$ $(x_n, y_n)\,\}.$ The only cons... |
So I know parabolas are like this, but only for line parallel to the $x$-or $y$-axis. Is it possible to do it for any line?
Taking it even further, what about the set of all points that are equal distance to a specific parabola and a specific point?
You can take this as far as you want, I'm just curious as to wh... |
Given sequence:
$$
\begin{cases}
x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 1\\
x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2
\end{cases}
$$
Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$
**Edit**
I've proved that $\frac{dA}{dx}|_{x=-1}=0$ (see my answer). The problem now is to show that $A_{\min}... |
**Exercise.**
Suppose $T \in \mathcal{L}(V,W)$ and $c \in W$. Prove that $\{x \in V : Tx = c\}$ is either the empty set or is a translate of $\text{null} \ T$.
----------
**Source.**
Linear Algebra Done Right, Sheldon Axler, 4th edition, Exercises 3E, first part of problem 8.
----------
**Notation.**
... |
Let $(f_n)_{n \geq 1}$ be a sequence of measurable functions from $(\Omega, \mathcal F)$ to a metric space $(X, d)$. Suppose that $f$ is such that
$$ \lim_{n \rightarrow +\infty} d(f_n(\omega),f(\omega)) = 0, \ \forall \omega \in \Omega. $$
Is the function $f$ $\mathcal F$-measurable?
What I know: if the spac... |
Let's consider the function:
$$f(x,y)= x^2 + y^3 -3y - 2.$$
i find two critical points:
$(0,1) \& (0,-1) $
Using the definition method with:
$h_1=x \ \& \ h_2=y-1$ to test the point $ (0,1) $, I find that:
$f(h_1,h_2+1)-f(0,0)= h_1^4 + h_2^2(h_2+3) $.
We can use $h_1=0$ to find that the sign is not constant
... |
Does $f(x,y)= x^2 + y^3 -3y - 2$ have a minimum at the point $(0,1)$? |
Does any closed form of this integral exists or is related to some special function integral? |
Does any closed form of this integral exist or is related to some special function integral? |
Does any closed form of this integral exists or is related to some special function integral? |
Let's consider the function:
$$f(x,y)= x^2 + y^3 -3y - 2.$$
I find two critical points:
$(0,1)$ and $(0,-1) $.
Using the definition method with
$h_1=x$ and $h_2=y-1$ to test the point $ (0,1) $, I find that:
$$f(h_1,1+h_2)-f(0,1)= h_1^4 + h_2^2(h_2+3).$$
We can use $h_1=0$ to find that the sign is not constan... |
Salam, i hope you are doing well;
Let's consider the function:
$$f(x,y)= x^2 + y^3 -3y - 2.$$
I find two critical points:
$(0,1)$ and $(0,-1) $.
Using the definition method with
$h_1=x$ and $h_2=y-1$ to test the point $ (0,1) $, I find that:
$$f(h_1,1+h_2)-f(0,1)= h_1^4 + h_2^2(h_2+3).$$
We can use $h_1=0$... |
Let's consider the function:
$$f(x,y)= x^2 + y^3 -3y - 2.$$
I find two critical points:
$(0,1)$ and $(0,-1) $.
Using the definition method with
$h_1=x$ and $h_2=y-1$ to test the point $ (0,1) $, I find that:
$$f(h_1,1+h_2)-f(0,1)= h_1^4 + h_2^2(h_2+3).$$
We can use $h_1=0$ to find that the sign is not constan... |
A key to understanding Hilbert's Hotel is the idea of "countable infinity". In intuitive terms, a set is countably infinite if it is the "same size" as the set of natural numbers $\mathbb{N} = \{1, 2, 3, \ldots\}$.
----------
How do we determine if two sets are the same size, especially infinite ones? Let's ... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
I want to find the domain and range of the function $y = \frac{\sqrt{x^2 + 4x + 3}}{x-5}$. I found the domain to be $(-\infty,-3] \cup [-1, 5) \cup (5, \infty)$. I also found the inverse of the function:
$x = \frac{10y^2+4 \pm \sqrt{192y^2+4}}{2(y^2-1)}$ which is defined when $y \not= 1$ and $y \not= -1$. For this i... |
Does any closed form of this integral exists or is related to some special function integral? |
**General problem:**
Let $X$ and $Y$ be processes taking values in $\mathbb{R}^n$ which solve the Stratonovich SDEs
$$\partial X_t = \sigma(X_t) \partial W_t$$
$$\partial Y_t = \xi(Y_t) \partial B_t,$$
and further assume, for all $t\geq 0$, $X_t \stackrel{(d)}{=} Y_t$ and that $\sigma \sigma^T = \xi \xi^T$ identica... |
The phrase "prebase" it a bit unusual, most frequently one uses the word *[subbase](https://en.wikipedia.org/wiki/Subbase)*.
Given a set $X$ and a set $\Sigma$ of subsets of $X$, there is a coarsest topology $\mathcal T(\Sigma)$ containing $\Sigma$. It is called the *topology generated by $\Sigma$*. Explicitly it ca... |
I don't quite understand $p$-adic series normalization.
I know that a $p$-adic series $\eta\in\mathbb{Q}_p$ is normalized if it satisfies:
$$\eta=\sum_{i\in\mathbb{N}_{>\nu}}a_ip^i$$
Where $a_i\in\mathbb{Z}_p$ belongs to the integers modulo $p$, and $\nu\in\mathbb{N}$ is any starting index.
Now, a general series $... |
I don't quite understand $p$-adic series normalization.
I know that a $p$-adic series $\eta\in\mathbb{Q}_p$ is normalized if it satisfies:
$$\eta=\sum_{i\in\mathbb{N}_{>\nu}}a_ip^i$$
Where $a_i\in\mathbb{Z}_p$ belongs to the integers modulo $p$, and $\nu\in\mathbb{N}$ is any starting index.
Now, a general series $... |
A bit of geometry:
$y=-x^2 +6x$ with $x\in [2,5];$
$y=-(x^2-6x)=-(x-3)^2+9$;
This is a parabola opening downward with maximum at $x=3$: $y_{max} =9;$
$y_{min} =-(5-3)^2+9=5$, and we are done.
(Why is $x=5$ chosen for the minimum?)
|
Let's consider the function:
$$f(x,y)= x^2 + y^3 -3y - 2.$$
I find two critical points:
$(0,1)$ and $(0,-1) $.
Using the definition method with
$h_1=x$ and $h_2=y-1$ to test the point $ (0,1) $, I find that:
$$f(h_1,1+h_2)-f(0,1)= h_1^2 + h_2^2(h_2+3).$$
We can use $h_1=0$ to find that the sign is not constan... |
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