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If a rectangular piece of fabric has area $2m^2$ and one side is $1m$ long how long is the other side?
|
If a rectangular piece of fabric has area $2m^2$ and one side is $1m$ long how long is the other side?
You might enjoy this
[droppings][1] xkcd. Read down to the line
Gas mileage is measured in square meters.
[1]: https://what-if.xkcd.com/11/ |
How do I control the growth rate of this dynamical system?
aₙ₊₁ = aₙ + kaₙ (2000 - aₙ)
a₀ = 4
I experimented with multiple values of *k* and i still can't find a way to control the growth.
(For additional context, is there a way such that this converges to a finite number and without this- ever exceeding 20... |
8 people total. Seated around a circular table. You can only talk to the 2 people seated directly next to you.
Each person has a connection value with each other person. eg : #1 and #3 are friendly and have a "conversation value" of 7. #1 and #2 are not friends, and have a "conversation value" of 3.
Pict... |
I'm working on a problem related to a recursive sequence, and I'm having some trouble. The sequence $S_{n}^{k}(a)$ is defined as follows:
$$S_{n}^{k}(a) =
\begin{cases}
\sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a \neq 1 \\
\sum_{i=1}^{k} \sqrt{d_i} & \text{if } n > 1, \quad k \in \mathbb{Z}^... |
I recently got an explanation of the Monty Hall problem and I thought I understood it but after giving it more thought, it still looks wrong.
Instead of using goats and doors, the example used a 52-card poker deck. The cards are facing down on a table and you need to pick one card without looking at it. Let's say th... |
[![enter image description here][1]][1]
Hi, I am trying to solve this problem but I am feeling like I am missing some prerequisites. Can help me solve this ? or at least a reference with a book where this is solved ?
$\text{Let } A \in \mathbb{M}_n(\mathbb{R}) \text{ be a symmetric and positive definite matrix, and l... |
I am reading a paper by Serre, in which at some point he say
> We extend the valuation $v$ from $K$ to $K^{sep}$; in this way we obtain a valuation on $K^{sep}$ with value group $v(K^{sep \times})=\mathbb{Q}$.
Here, $K$ is a number field complete with respect to $v$ a discrete valuation, and $K^{sep}$ is $K$'s se... |
Why is the valuation of a separable closure equal to $\mathbb{Q}$? |
Let $X$ be a manifold. Leray's Theorem for Čech Cohomology states that if a covering $\mathfrak{U}$ is acyclic $(\check{H}^q(U_{i_0}\cap\ldots U_{i_k},F)=0$ for every finite intersections of elements in $\mathfrak{U}$) for a sheaf $F$, then
$$
\check{H}^q(\mathfrak{U},F)=\check{H}^q(X,F)
$$
where the first group is... |
How do I control the growth rate of this dynamical system?
$$
\left\{
\begin{aligned}
&a_{n+1} = a_n + ka_n (2000 - a_n)\\
&a_0 = 4
\end{aligned}
\right.
$$
I experimented with multiple values of $k$ and i still can't find a way to control the growth.
(For additional context, is there a way such that th... |
Let's consider this problem:
*A bag contains 5 black marbles and 6 white ones. Marbles of the same color are indistinguishable within each others. If I draw two marbles, what is the probability they have different colors?*
This problem can be trivial if we reason like this:
$P(first = black, second=white) = P(... |
Suppose we have a fibration $F\to E \to B$ and a sequence of maps $F'\to E' \to B'$ and suppose we have a map between these two sequences, which commutes up to homotopy and which is pointwise a homotopy equivalence (i.e each map $\bullet\to\bullet', \bullet\in\{F,E,B\}$ in the homotopy commutative map is a homotopy equ... |
Are $(x_n-x_{n-1})$ and $(x_1+...+x_n)$ Schauder basis? |
Let's consider this problem:
*A bag contains 5 black marbles and 6 white ones. Marbles of the same color are indistinguishable from each other. If I draw two marbles, what is the probability they have different colors?*
This problem can be trivial if we reason like this:
$P(first = black, second=white) = P(fir... |
The "desired answer" column in the book's solution is the same as your right-hand column: it's what we're trying to construct our question so that the warrior will say. Of course it's the same as column p because that's exactly the information you're trying to get.
But of course _what the warrior says_ isn't the sam... |
Question Q1 :
A sum of squares is $>0$ unless all the squares are zero, in which case the sum is zero.
It is known that if a quadratic expression remains always $>0$, it cannot have roots, therefore its discriminant is $<0$.
What happens in the exceptional case where all the squares are zero ? (in which case... |
Is there a known description of the set of multiplicative orders of a fixed unit $a$ modulo all divisors of $n$, i.e. of $\text{ord}_d(a)$ with $d\mid n$? It is easy to see that it is a subset of the set of all divisors of $\text{ord}_n(a)$. Indeed, if $a^m\equiv1\pmod{n}$ then $a^m\equiv1\pmod{d}$ for any $d\mid n$. S... |
I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological manifold and $E$ is another topological manifold with a surjective continuous map $\pi:E\to M$.Then we say $E$ is a fiber b... |
Why isn't the third property in the definition of vector bundles redundant? |
I have studied Hypothesis testing from [thinkstats2][1] which does simulation to estimate p value.
For e.g. In coin toss example, we run the simulation, to generate the test statistic i.e. Difference between the number of Heads and Tails. p value is the proportion of test statistics greater than observed. So intuiti... |
It is a renamed theorem the existence of a nowhere vanishing vector fields on odd dimensional spheres. Now, in his textbook, L. Tu proposes the following exercise:
>Let $x^1,y^1,\dots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$ and consider the unit sphere $S^{2n-1}$ (defined by the equation $\sum\limi... |
I have to give a formal proof of $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$.
I have so far worked out that
$$\left | \frac{\tanh{x}-1}{e^{-2x}}-(-2) \right | \iff \left | \frac{2}{1+e^{2x}} \right |$$
Which means that
$$|f(x) - L|=\frac{2}{1+e^{2x}}<\frac{2}{2x+2}=\frac{1}{x+1}<\frac{1}{x}$$
... |
> I cannot figure out how I should be allowed to use $11 \choose 2$ at
> denominator since to me there are at most $3$ different ways to get a
> group of two marbles from the $11$ namely (BB, BW, WW) and not $55$ like
> $11 \choose 2$ states.
Well, if you think about it this way, then you should feel uncomforta... |
I want to know if the map between $\mathbb{Z}$ and $\pi_3(S^3) $ given by $a \mapsto (x \mapsto x^a)$, where $S^3$ is seen as a subset of $\mathbb{H}$ with its multiplication, gives an isomorphism just like the case $\pi_1(S^1) $? |
In full generality, derived functors appear in a situation where our categories of interest admit some notion of *weak equivalences*, without those being isomorphisms per se. In homological algebra, our cochain complex categories have quasi-isomorphisms, which are such a notion of weak equivalences. These quasi-isomorp... |
It is a renamed theorem the existence of a nowhere vanishing vector fields on odd dimensional spheres. Now, in his textbook, L. Tu proposes the following exercise:
>Let $x^1,y^1,\dots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$ and consider the unit sphere $S^{2n-1}$ (defined by the equation $\sum\limi... |
> I cannot figure out how I should be allowed to use $11 \choose 2$ at
> denominator since to me there are at most $3$ different ways to get a
> group of two marbles from the $11$ namely (BB, BW, WW) and not $55$ like
> $11 \choose 2$ states.
Well, if you think about it this way, then you should feel uncomforta... |
I am trying to develop a geometry-editor and could not solve the following problem.
I have two lines with known equations (y=mx+c) intersecting at a known point P. I want to place a circle with a known radius R, so that the two lines form tangents to the circle. How do I calculate the center coordinates of the circl... |
How to find the center of a circle with two known tangent lines? |
I recently got an explanation of the Monty Hall problem and I thought I understood it but after giving it more thought, it still looks wrong.
Instead of using goats and doors, the example used a 52-card poker deck. The cards are facing down on a table and you need to pick one card without looking at it. Let's say th... |
comment about the uniform convergence of the series, whose partial sum of n terms is $S_n=\frac{nx}{1+n^2x^3}$
, for all real x.
I found that the series is pointwise convergent to 0 for all x. but for showing uniform convergence, I tried to use the definition ∀ϵ>0:∃ m∈N such that $|f_n(x)−f(x)|<ϵ \ \ \forall n>m(ϵ)... |
I have to give a formal proof of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$.
I have so far worked out that
$$\left | \frac{\tanh{x}-1}{e^{-2x}}-(-2) \right | \iff \left | \frac{2}{1+e^{2x}} \right |$$
Which means that
$$|f(x) - L|=\frac{2}{1+e^{2x}}<\frac{2}{2x+2}=\frac{1}{x+1}<\frac{1}{x... |
I have to give a formal proof of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$.
I have so far worked out that
$$ | \frac{\tanh{x}-1}{e^{-2x}}-(-2) | = | \frac{2}{1+e^{2x}} |$$
Which means that
$$|f(x) - L|=\frac{2}{1+e^{2x}}<\frac{2}{2x+2}=\frac{1}{x+1}<\frac{1}{x}$$
Which further means ... |
comment about the uniform convergence of the series, whose partial sum of n terms is $S_n=\frac{nx}{1+n^2x^3}$
, for all real x.
I found that the series is pointwise convergent to 0 for all x. but for showing uniform convergence, I tried to use the definition $∀\epsilon>0$: $∃ m∈N$ such that $|f_n(x)−f(x)|<ϵ \ \ \f... |
I'm currently delving into the foundational concepts of algebraic topology and came across the definition of CW complexes in Hatcher's "Algebraic Topology" (Chapter 0, p. 5). I must admit, I found Hatcher's exposition a bit challenging to grasp. Fortunately, I stumbled upon an alternative definition via a commutative d... |
I am trying to develop a geometry-editor and could not solve the following problem.
I have two lines with known equations (y=mx+c) intersecting at a known point P. I want to place a circle with a known radius R, so that the two lines form tangents to the circle. How do I calculate the center coordinates of the circl... |
While working on a project, I came across a paper that includes the definition $g ∈ $, where $$ is the "tetrahedral group" ("a discrete subgroup of isometries $ ⊂ O(3)$"). The paper then uses $g(z)$ as a function with a complex input, where $z$ is "the local complex coordinate on the unit 2-sphere in $^3$ obtained by s... |
Let $ G $ is a finite group. Assume that $ a_1,a_2,b_1,b_2\in G $ such that $ a_1=ga_2g^{-1} $ and $ b_1=hb_2h^{-1} $ for $ g,h\in G $. I want to ask if $ a_1b_1 $ and $ a_2b_2 $ are in the same conjugacy class, i.e. there exists $ c\in G $ such that $ a_1b_1=ca_2b_2c^{-1} $.
I find that this problem can be reduced ... |
Let $V: \mathbb R^n \rightarrow \mathbb R$ be a Lyapunov function for system $\dot{x}=f(x)$. Let $\dot V \leq 0$ hold. Do there exist conditions on th system dynamics such that we can ensure that the system converges to a steady state instead of showing an undamped oscillation?
For example, consider an LC circuit (o... |
I have to study the convergence od this integral:
$\int_{0}^{1} \frac{1}{\sqrt x - \sqrt[5]{x}} dx$
I rewrite it as:
$\int_{0}^{a} f (x) dx + \int_{a}^{1} f (x) dx$ (with $0<a<1$)
in order to deal with the problems in 0 and 1 separately.
For the first one, since $ \sqrt x = o(\sqrt [5] x )$ , for $x \... |
The [Feynmann Kac formula](https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) tell us that the solution of the PDE
$$
\frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) -V(x,t) u(x,t) + f(x,t) = 0$$
with termina... |
In the book by Görtz-Wedhorn, the following definitions are given:
> Let $Y$ be a scheme.
>
> 1. A *closed subscheme* of $Y$ is given by a closed subset $Z \subset Y$ and an ideal sheaf $\mathcal{I} \subset \mathcal{O}_Y$ such that $Z = \{x \in X \, : \, (\mathcal{O}_Y / \mathcal{I})_Z \neq 0\}$ and $(Z,(\mathca... |
I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological manifold and $E$ is another topological manifold with a surjective continuous map $\pi:E\to M$.Then we say $E$ is a fiber b... |
The [Feynmann Kac formula](https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) tell us that the solution of the PDE
$$
\frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) -V(x,t) u(x,t) + f(x,t) = 0$$
with termina... |
For $a>1$ and a fixed $N$ does $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \left(1- \frac{a}{a+N +k} \right)$ converge? |
In the book by Görtz-Wedhorn, the following definitions are given:
> Let $Y$ be a scheme.
>
> 1. A *closed subscheme* of $Y$ is given by a closed subset $Z \subset Y$ and an ideal sheaf $\mathcal{I} \subset \mathcal{O}_Y$ such that $Z = \{x \in X \, : \, (\mathcal{O}_Y / \mathcal{I})_Z \neq 0\}$ and $(Z,(\mathca... |
Take $x_n = \frac1{n^{\frac23}}$ can you see that the limit of $S_n(x_n)$ is $\infty$? |
I am currently preparing for an algebraic geometry exam and I am trying to understand how to normalize a projective curve given by a homogeneous polynomial. For example let's consider the curve given by $$xy^2-xz^2-yz^2\in\Gamma(\mathcal{O}_{\mathbb{P}_k^2}(3),\mathbb{P}_k^2)$$
From what I understand it should suffice... |
I have to give a formal proof of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$.
I have so far worked out that
$$ | \frac{\tanh{x}-1}{e^{-2x}}-(-2) | = | \frac{2}{1+e^{2x}} |$$
Which means that
$$|h(x) - L|=\frac{2}{1+e^{2x}}<\frac{2}{2x+2}=\frac{1}{x+1}<\frac{1}{x}$$
Which further means ... |
When $X$ is a topological space, any sheaf $S$ on $X$ is a connected object of the category of presheaves $\widehat{\mathcal{O}(X)}$ in the sense that
$\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A+B) \cong \mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A)+\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,B)$,
for all $A,B \i... |
Recall the projection theorem for Hilbert spaces. If $M$ is a closed subspace of a Hilbert space $H$, it stated that for each $x \in H$, there was a unique element in $M$ that minimized the norm difference $||x-y||$ over all $y \in M$. We denote this minimizing element by $P_M x$. When viewed as a function on $H$, $P_M... |
Why the projection operator $P_M$ can be written as $x \bigotimes x$? |
comment about the uniform convergence of the series, whose partial sum of n terms is $S_n=\frac{nx}{1+n^2x^3}$
, for all real x.
I found that the series is pointwise convergent to 0 for all x. but for showing uniform convergence, I tried to use the definition $∀\epsilon>0$: $∃ m∈N$ such that $|f_n(x)−f(x)|<ϵ \ \ \f... |
As we needed an axiom to take the supremum $\omega$ of the finite ordinals, how are we allowed without another axiom to take the supremum $\omega_1$ of the countable ordinals? |
Don't we need an axiom for $\omega_1$? |
In the book by Görtz-Wedhorn, the following definitions are given:
> Let $Y$ be a scheme.
>
> 1. A *closed subscheme* of $Y$ is given by a closed subset $Z \subset Y$ and an ideal sheaf $\mathcal{I} \subset \mathcal{O}_Y$ such that $Z = \{x \in X \, : \, (\mathcal{O}_Y / \mathcal{I})_Z \neq 0\}$ and $(Z,(\mathca... |
This post directly follows this [question](https://math.stackexchange.com/questions/4862992/using-euler-lagrange-equations-show-that-the-following-eom-can-be-written-as-d) and is very similar in nature:
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\... |
Consider the following Lyapunov candidate function:
$$ V(x,y,q_i) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \ \ \ \ \ (1)$$
where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$, $n\geq1$ is an arbitrary positive integer number. Suppose $x_{i}, y_{i}$ ar... |
I was wondering If $f:R--->R$ is injective , is $af$ also injective?
If $f:R--->R$ is surjective , is $af$ also surjective? |
If f is injective ,is af injective?If f is surjective , is af surjective? |
I have to give a formal proof that the limit of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ is -2 as $x \rightarrow \infty$.
I have so far worked out that
$$ | \frac{\tanh{x}-1}{e^{-2x}}-(-2) | = | \frac{2}{1+e^{2x}} |$$
Which means that
$$|\frac{2}{1+e^{2x}}|≤ \frac{2}{2x+2}=\frac{1}{x+1}$$
Therefore, if given... |
I'm currently delving into the foundational concepts of algebraic topology and came across the definition of CW complexes in Hatcher's "Algebraic Topology" (Chapter 0, p. 5):
> (1) Start with a discrete set $X^0$ , whose points are regarded as $0$ cells.
> (2) Inductively, form the $n$ skeleton $X^n$ from $X^{n−1}... |
Here's one way to do it:
Suppose $F(x)=\sum_{n=0}^{\infty}a_nx^n=a_0+a_1x+a_2x^2+...$. Then $f(x)=\sum_{n=1}^{\infty}na_nx^{n-1}-x=a_1+(2a_2-1)x+3a_3x^2+...$
If these functions are to be equal, then we must have the following:
$$a_0=a_1$$
$$a_1=2a_2-1$$
$$a_2=3a_3$$
$$a_3=4a_4$$
$$a_4=5a_5$$
And so on. ... |
I have to give a formal proof that the limit of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ is -2 as $x \rightarrow \infty$.
For all ε>0 there must be a N>0 such that |h(x)-(-2)|<ε for all x>N
$$ | \frac{\tanh{x}-1}{e^{-2x}}-(-2) | = | \frac{2}{1+e^{2x}} |$$
Which means that
$$|\frac{2}{1+e^{2x}}|≤ \frac{2}{2x+2... |
I am not even sure I have a question anymore (I will just give up)... in the past month or so I have been researching cubic Bézier curves. The idea was to find a fit through data points, using piecewise Bézier curves and the Euler-Lagrange equation to minimize an "energy" proportional with the square of the length of t... |
Finding the ideal B-spline through data points using Euler-Lagrange: is it just too hard to do? |
I'm currently delving into the foundational concepts of algebraic topology and came across the definition of CW complexes in Hatcher's "Algebraic Topology" (Chapter 0, p. 5):
> (1) Start with a discrete set $X^0$ , whose points are regarded as $0$ cells.
> (2) Inductively, form the $n$ skeleton $X^n$ from $X^{n−1}... |
As we needed an axiom to take the supremum of the finite ordinals, how are we allowed without another axiom to take the supremum of the countable ordinals? |
How many positive divisors are there of 30^2024 which are not divisors of 20^2021?
I have tried many ways to try to get a pattern for this problem but I can't. I know that 30 has 8 divisors and 20 has 6 divisors, and the number of divisors in 30 not present in 20 is 4, But I just can't proceed from there. |
I quote Sakurai$^1$
> \begin{equation*} \mathscr{T}(dx^\prime)~| x^\prime \rangle = ~ |
x^\prime+dx^\prime \rangle \tag{1.194} \end{equation*}
where a possible arbitrary phase factor is set to unity by convention.
In (1.194), the operator $ \mathscr{T}(dx^\prime) $ is an infinitesimal translation operat... |
Where does the arbitrary phase factor of the infinitesimal translation operator, that appears in (1.194), on pg.40 of Sakurai, come from? |
Let $f:\mathbb R\to\mathbb R$ and $af:\mathbb R\to \mathbb R$, with $a\in\mathbb R\setminus\{0\}$, be the scalar multiple of $f$.
I was wondering if $f$ is injective (respectively surjective), is $af$ also injective (respectively surjective)?
|
If $f$ is injective (resp. surjective), is $af$ injective (resp. surjective)? |
The intersection of sigma algebras is a sigma algebra
$$
\bigcap_{\alpha\in \mathcal{A}} \mathcal{X}_\alpha
$$
where $\mathcal{A}$ is an index set. However, in every definition I looked at (Billingsley, Cinlar, Evans, Bogachev, Wikipedia, Proof-Wiki) no one says if $\mathcal{A}$ is allowed to be **uncountable**, i... |
# Background
I am trying to understand the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. However, despite my efforts to decipher the paper, I can't follow what it is describing. This paper is written around complex differential geometry and topology, and appear... |
Can you explain Section 2.2 of this paper to someone who hasn't learned Differential Geometry or Topology? |
> Here I'm developing an example with 4 cards so it's not too long to write down:
>
> There are 4 cards, Ace, King, Queen, Jack (A,K,Q,J). The winning card is the Ace. Here the possible scenarios:
>
> Scenario #1: I choose A, host shows K & Q. [...]
>
> Scenario #2: I choose A, host shows K & J . [...]
>
> Sc... |
$$
Y(s)
=
K(s)-e^{-2s}K(s)
$$
where
$$K(s)=
\frac{1}{s^2(s^2+4)}
=\frac{1/4}{s^2}-\frac{1/4}{s^2+4}
$$
Using inverse Laplace table
$$
y(t)
=
k(t)
-
k(t-2)u(t-2)
$$
where
$
k(t)
=
\frac14t
-
\frac{1}{8}
\sin(2t)
$.
It follows
$y(\pi/2)=\frac14 (\pi/2)$ which is different to what is claim... |
In the preface of Lawson & Michelsohn's *Spin Geometry*, they mentioned some physics relevant to their book which I find interesting:
> The theory of Dirac had another interesting feature. In the presence of an electromagnetic field the Dirac Hamiltonian contained an additional term added on to what one might expect... |
I recently got an explanation of the Monty Hall problem and I thought I understood it but after giving it more thought, it still looks wrong.
Instead of using goats and doors, the example used a 52-card poker deck. The cards are facing down on a table and you need to pick one card without looking at it. Let's say th... |
The following properties are key to this problem,
$$K(ik)=\frac{1}{\sqrt{1+k^2}}K\left[\frac{k}{\sqrt{1+k^2}}\right]$$
$$E(ik)=\sqrt{1+k^2}E\left(\frac{k}{\sqrt{1+k^2}}\right)$$
Use the substitution $t\to t^2$,
$$I=2\int_0^1\frac{\ln(1-t^2)}{t}\left[\frac{E(t/\sqrt{1+t^2})-K(t/\sqrt{1+t^2})}{\sqrt{1+t^2}}\r... |
I'm not sure if this answers the question but I did some calculations and this is what I got:
There are three cases: square lands in one sector, two sectors or four sectors.
I calculate probabilities of each of these cases by dividing the possible area where the square's center can be to the area of entire sector.
... |
comment about the uniform convergence of the series, whose partial sum of n terms is $S_n=\frac{nx}{1+n^2x^3}$
, for all real x.
for showing uniform convergence, I tried to use the definition $∀\epsilon>0$: $∃ m∈N$ such that $|f_n(x)−f(x)|<ϵ \ \ \forall n>m(ϵ)$.
but unable to get any expression in terms of n for... |
I'm not sure if this answers the question but I did some calculations and this is what I got:
There are three cases: square lands in one sector, two sectors or four sectors.
I calculate probabilities of each of these cases by dividing the possible area where the square's center can be to the area of entire sector.
... |
Consider the following Lyapunov candidate function:
$$ V(x,y,q_i) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \tag1$$
where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$, $n\geq1$ is an arbitrary positive integer number. Suppose $x_{i}, y_{i}$ are known ... |
Question Q1 :
A sum of squares is $>0$ unless all the squares are zero, in which case the sum is zero.
It is known that if a quadratic expression remains always $>0$, it cannot have roots, therefore its discriminant is $<0$.
What happens in the exceptional case where all the squares are zero ? (in which case... |
Is there a closed-form solution for the recurrence relation $x_{t+1} = x_{t}^\alpha + 1$, where $\alpha \in (0,1)$? If it were $x_{t+1} = x_{t}^\alpha$, then taking the Log of both sides would make it linear and easy to solve. However, the same thing does not work in the example. Taking the exponent of both sides does ... |
Here's the standard proof that the reals are uncountable:
* Suppose that the reals are countable.
* Let $a_m$ be the $m$th real number in the interval $[0,1)$, and let $a_{m,n}$ be the $n$th digit after the decimal place.
* Consider the number where the $n$th digit after the decimal place is $9-a_{n,n}$, call this... |
I quote Sakurai$^1$
> \begin{equation*} \mathscr{T}(dx^\prime)~| x^\prime \rangle = ~ |
x^\prime+dx^\prime \rangle \tag{1.194} \end{equation*}
where a possible arbitrary phase factor is set to unity by convention.
In (1.194), the operator $ \mathscr{T}(dx^\prime) $ is an infinitesimal translation operat... |
Given a set of $n$ states $V = \{ s_1, s_2, \ldots, s_n \}$, and a complete digraph $G = (V, A)$ where $A = \{ (a,b) \mid (a,b) \in V^2\; \text{and}\; a \neq b \}$, I'm interested in finding cyclic trails through $G$ which traverse every element of $A$ exactly once.
I've found a number of different ways to construct... |
How many ways of traversing every arc of a complete digraph exactly once from a given starting vertex are there? |
A Riemannian manifold $(M,g)$ is a manifold equipped with a metric tensor ($g$) that is both symmetric and positive definite. Now if the metric tensor is symmetric but not positive definite, then it's a *pseudo-Riemannian* manifold. But what the case of a manifold equipped with a metric tensor $(M,g)$ such that $g$ is ... |
> This diagram is easy to verify on $S^1$ and $S^2$.
No. You did not define what $\alpha$ is and what attaching maps (vertical arrows) are. It is impossible to deduce anything from that diagram, it is incomplete in the particular case.
There are infinitely many CW structures on a sphere, two of them are "standard... |
I made the following claim and gave it a proof during my winter break.
------------
Given four hypothesis
1. $X=\mathbb{N},Y$ is a metric space.
2. $\{f_n\}$ uniformly converge to $f$.
3. $\lim_{k\to \infty}f(k)$ exists.
4. $\lim_{n\to \infty}\lim_{k\to \infty}f_n(k)$ exists.
We have
\begi... |
I have to give a formal proof that the limit of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ is -2 as $x \rightarrow \infty$.
For all ε>0 there must be a N>0 such that |h(x)-(-2)|<ε for all x>N
$$ | \frac{\tanh{x}-1}{e^{-2x}}-(-2) | = | \frac{2}{1+e^{2x}} |$$
Which means that
$$|\frac{2}{1+e^{2x}}|≤ \frac{2}{2x+2... |
Question Q1 :
A sum of squares is $>0$ unless all the squares are zero, in which case the sum is zero.
It is known that if a quadratic expression remains always $>0$, it cannot have roots, therefore its discriminant is $<0$.
What happens in the exceptional case where all the squares are zero ? (in which case... |
Given an ordered sequence $S_n = s_1, s_2, ..., s_n$ we define its cost as the sum of absolute deviations from any median:
$$ cost(S_n) = \sum_{i=1}^{n} \left| s_i - median(S_n) \right|\text{, where } median(S_n) = s_{\left\lceil \frac{n}{2}\right \rceil} \text{ or } s_{\left\lceil \frac{n+1}{2}\right \rceil}. $$
B... |
I want to prove the following inequality using Maclaurin series (for all $\mathbb{R}$).
$$1 - \frac{1}{2} x^2 \leq \cos x$$
I have tried:
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{\sin c}{5!} x^5 \geq 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^5}{5!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} \left... |
> This diagram is easy to verify on $S^1$ and $S^2$.
No. You did not define what $\alpha$ is and what attaching maps (vertical arrows) are. It is impossible to deduce anything from that diagram, it is incomplete in the particular case.
There are infinitely many CW structures on a sphere, two of them are "standard... |
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