instruction stringlengths 12 30k |
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$\int_C \frac{1}{sinz}$ where $C := \{z \in C : |z| = 1 \}$
I know $\frac{1}{sinz}$ has a pole at $z = 0$. I can apply Cauchy Residue Theorem, so I need to know the laurent series expansion of $\frac{1}{sinz}$ which I am unable to figure out.
Is there any other neat way to approach this?
Thanks in advance. |
> Given $m\in\mathbb{N},$ is there a permutation $\sigma(n)$ of
> $\{0,1,2,\ldots,m-1\},$ such that $\{ (n+\sigma(n))\pmod m: 1\leq n
\leq m \} = \{0,1,2,\ldots m-1\},\ ?$
Based on computation with Python for low numbers, the answer seems to be, "yes for prime numbers; no for non-primes". Is this really the case? ... |
Is there a permutation $\sigma(n)$ of $\{0,1,2,\ldots,m-1\},$ such that $\{(n+\sigma(n))\pmod m:1\leq n>\leq m\}=\{0,1,2,\ldots m-1\}?$ |
Yesterday I asked a [question][1] about the joint law of $(M_T, B_u)$, where $M_T = \max_{t \in [0, T]} B_t$, $B_t$ is the standard Brownian motion and $u < T$.
One of the reasons I was interested in this law is because I needed to compute $\mathbb{E} \{ \tau B_u \}$, where $\tau$ is the a.s. unique point at which t... |
> Given $m\in\mathbb{N},$ is there a permutation $\sigma(n)$ of
> $\{0,1,2,\ldots,m-1\},$ such that $\{ (n+\sigma(n))\pmod m: 0\leq n
\leq m-1 \} = \{0,1,2,\ldots m-1\},\ ?$
Based on computation with Python for low numbers, the answer seems to be, "yes for prime numbers; no for non-primes". Is this really the case... |
Is there a permutation $\sigma(n)$ of $\{0,1,2,\ldots,m-1\},$ such that $\{(n+\sigma(n))\pmod m:0\leq n\leq m-1\}=\{0,1,2,\ldots m-1\}?$ |
> Given $m\in\mathbb{N},$ is there a permutation $\sigma(n)$ of
> $\{0,1,2,\ldots,m-1\},$ such that $\{ (n+\sigma(n))\pmod m: 0\leq n
\leq m-1 \} = \{0,1,2,\ldots m-1\},\ ?$
Based on computation with Python for low numbers, the answer seems to be, "yes for prime numbers; no for non-primes". Is this really the case... |
What about this? Using matrix definition.
To find $|(XG)^2|_{max}$, the largest element of $(XG)^2$, we can express $(XG)^2$ as:
$(XG)^2$ = XG * XG
= X * G * X * G
Let's consider the $(i, j)$ th element of $(XG)^2$, denoted as $((XG)^2)_{ij}$.
$((XG)^2)_{ij}$ = $Σ_k Σ_l (X_{ik} * G_{kk} * X_{lj} * G_{ll})$
... |
//* Once again they want to lower a question with a piranha attack, putting negative scoring without giving reasons in the comments. Do it, it is clear that this is an illustrated Twitter, where only repetition of what is already known is allowed, with no room for innovation or creative questioning.
"Self referenc... |
Is this correct?
Assume that $(D_{v}f)(x)>0$ for some point $x\in \mathbb{R}^{n}$ and some vector $v\in \mathbb{R}^{n}$. Then,
\begin{align}
(D_{-v}f)(x) &= \lim_{h \to 0}\frac{f(x-hv)-f(x)}{h}\\
&= - \lim_{h \to 0}\frac{f(x-hv)-f(x)}{-h}\\
&= -(D_{v}f)(x)<0
\end{align}
I know that maybe this questi... |
Does $(D_{v}f)(x)>0$ implies $(D_{-v} f)(x)<0$? |
While trying to solve $\displaystyle\int\sqrt{14x-x^2}\;dx$, I obtained three different primitives in three different ways:
Method 1: completing the square
$c(x)=\dfrac{1}{2}\left[49\arcsin\left(\dfrac{x}{7}-1\right)+(x-7)\sqrt{14x-x^2}\right]$
Method 2: using the substitution $x=u^2$
$s(x)=49\arcsin\left(\... |
Why are these primitives containing $arcsin(x)$ equal up to a constant? |
> Given $m\in\mathbb{N},$ is there a permutation $\sigma(n)$ of
> $\{0,1,2,\ldots,m-1\},$ such that $\{ (n+\sigma(n))\pmod m: 0\leq n
\leq m-1 \} = \{0,1,2,\ldots m-1\},\ ?$
Based on computation with Python for low numbers, the answer seems to be, "yes for prime numbers; no for non-primes", although this might not... |
Why are these primitives containing $\arcsin x$ equal up to a constant? |
I would do the computations in the Laurent power series ring in the variable $x$ with coefficients in the field
$$
F=\Bbb F_2
$$
with two elements, $0,1$.
Then for instance for the case $m=2$ and $t=4$ the computation is, writing $A=A_{m,t}$, and $B=B_{m,t}$ for the corresponding
$A$-polynomial and $B$-polynomi... |
> Given $m\in\mathbb{N},$ is there a permutation $\sigma(n)$ of
> $\{0,1,2,\ldots,m-1\},$ such that $\{ (n+\sigma(n))\pmod m: 0\leq n
\leq m-1 \} = \{0,1,2,\ldots m-1\},\ ?$
Based on computation with Python for low numbers, the answer seems to be, "yes for prime numbers; no for non-primes", although this might not... |
I had a question I was hoping someone could answer. I'm working on this homework problem, and when graphed, the volume looks like it would be the same revolving around the x-axis or on the axis of $y=5$, but it isn't. Why is this?
Problem:
Find the volume of region R when region R is revolved about the line $y = 4... |
Suppose I have two sequences $a_n$ and $b_n$. They both start from a very high number say $25000$ and converges to zero.
If I plot them in one plot, as they start from a very large number, the plot looks like as if $a_n \approx b_n \approx 0$. But if we look closely, the difference between them is around $2$ or $3$... |
Let $(R, m_R, \kappa_R) $ a discrete valuation ring with quotient field $K$, and $K \subset L$ a finite field extension. Let $ (B, m_B, \kappa_B)$ be the integral closure of $R$ in $L$, which is as the notation suggest a dvr too.
I'm looking for such constellation $(R,B;K,L)$ satisfying following properties:
1... |
I have difficulty in understanding the solution below, and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circ... |
>Find all the prime factors of $1000027$.
I got all the factors by testing every number from $1$ to $103$, but when I try to do it using algebra, I get stuck.
My work:
$$
1000027=(100+3)(100^2-3\cdot100+3^2).
$$
How do I simplify further?
Source: Mathematics Magazine,
Vol. 23, No. 5, May - Jun., 1950,
P... |
The question asks $\iint_R (3x+4y^2)\; dA$ where $R$ is the region in the upper half plane bounded by the circles $x^2+y^2=1$ and $x^2+y^2=4$
$$\int_0^\pi \int_1^2 (3r\cos \theta + 3r^2 \sin^2 \theta) r \;dr d\theta $$
Now the question is since $r=-1$ is the same as $r=1$ by the convention if we replace the bounds o... |
Let $\{e_1, e_2, ..., e_n\}$ be the set of elements.
Let $\{s_1(e_1), s_1(e_2), ..., s_1(e_n)\}$, $\{s_2(e_1), s_2(e_2), ..., s_2(e_n)\}$ be the scores of elements after applying two ranking functions $s_1$, $s_2$.
Let $\{w\cdot s_1(e_1) + (1-w)\cdot s_2(e_1), w\cdot s_1(e_2) + (1-w)\cdot s_2(e_2), ..., w\cdot s_1(... |
How can I prove the existence or not of a surjective morphism of Riemann surfaces $\Bbb C^\times \to \Bbb C$ ? and viceversa? In case that it exists, how can I prove if there's any that can be extended to a morphism $\Bbb P^1\to \Bbb P^1$?
$\Bbb C$ is identified with a chart $U_1 \subseteq \Bbb P^1$ and $\Bbb C^\tim... |
Do surjective morphisms $\Bbb C^\times \to \Bbb C$ and viceversa between Rieman surfaces exist and can they be extended to $\Bbb P^1\to\Bbb P^1$? |
Let $\mathfrak{g}$ be a finite dimensional Lie algebra and $B:\mathfrak{g} \times \mathfrak{g} \rightarrow \mathbb{K}$ be a non-degenerate invariant symmetric bilinear form. Then there exists a unique $C_{B} \in \mathfrak{U(g)}$, called the Casimir element of $\mathfrak{g}$ corresponding to $B$, such that $$\sum_{i=1}^... |
> Given $m\in\mathbb{N},$ is there a permutation $\sigma(n)$ of
> $\{0,1,2,\ldots,m-1\},$ such that $\{ (n+\sigma(n))\pmod m: 0\leq n
\leq m-1 \} = \{0,1,2,\ldots m-1\},\ ?$
Based on computation with Python for low numbers, the answer seems to be, "yes for prime numbers; no for non-primes", although this might not... |
Is there a way to solve quintics of the form $ax^5+bx^4+c=0$? |
I am studying the residue theorem applications to physics and in particular the angular sector method. I understand the concept of finding an angle such that the infinite half-segment along it acts as the real axis, but I found an integral that I do not know how to solve with this method:
> $$\int{\frac{1-x}... |
Why when we add or substract $\alpha$ to a $\sin/\cos$ of odd multiple of $\frac{\pi}{2}$ we get cofunction and when we do same thing with multiple of $\pi$ we get just same function but with sign depending on quarter? Where the transformation for $\frac{\pi}{2}$ comes from |
Why when we add or substract $\alpha$ to a $\sin/\cos$ of odd multiple of $\frac{\pi}{2}$ we get cofunction? |
Your confusion is entirely reasonable! Yes, it should come as a shock when one first learns that "convergence" of Fourier series can have different, inequivalent, senses. From a reasonable beginner's viewpoint, _of_course_ "convergence" means "pointwise". Duh. Oop, but Fourier series don't reliably do that. As some sor... |
Let $BA$ be the set of axioms of Boolean algebras.
Let $Th(BA)$ be the set of first-order sentences that are true when conjoined with the axioms in $BA$.
I believe that $Th(BA) = Th(\langle \{0,1\}, \land, \lor, \lnot \rangle)$ where $\land, \lor,\lnot$ are the usual propositional operations interpreted over $\{... |
I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf):
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/jtdN4.png
Is there a typo in the equation defining $C$? Should it be $y... |
Group law on cubic (Collinear Points) - Typo in problem? |
Do surjective morphisms $\Bbb C^\times \to \Bbb C$ and viceversa between Riemann surfaces exist and can they be extended to $\Bbb P^1\to\Bbb P^1$? |
Brownian motion on a manifold starts with constructing the heat equation problem on it. One main reference is [Stochastic Analysis On Manifolds][1].
As mentioned in Example 4.34 [Heat Equation and Brownian Motion on
Riemannian Manifolds][2], the transition density on the torus $\mathbb{T}^{n}:=\mathbb{R}^{n}/\mathb... |
> *Strict Maximum Principle - Complex Version (Gamelin; p88)* Let $h$ be a bounded complex-valued harmonic function on a domain $D$. If $|h(z)| \le M$ for all $z \in D$, and $|h(z_0)| = M$ for some $z_0 \in D$, then $h(z)$ is constant on $D$.
First I will copy the proof (then explain where I am confused)
Proof: W... |
Why when we add or substract $\alpha$ to a odd multiple of $\frac{\pi}{2}$ inside $\sin/\cos$ we get cofunction and when we do same thing with multiple of $\pi$ we get just same function but with sign depending on quarter? Where the transformation for $\frac{\pi}{2}$ comes from? |
Why when we add or substract $\alpha$ to a odd multiple of $\frac{\pi}{2}$ inside $\sin/\cos$ we get cofunction? |
I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf):
"2.11 (Group law on cuspidal cubic.) Consider the curve
$$
C:(z=x^3)\subset k^2;
$$
$C$ is the image of the bijective map $\varphi:k\rightarrow C$ b... |
I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf):
> 2.11 (Group law on cuspidal cubic.) Consider the curve
> $$C:(z=x^3)\subset k^2;$$
$C$ is the image of the bijective map $\varphi\colon k\to C$ by ... |
Given the differential equation
$$\frac{dy}{dt}=y(y-1)(y-3)$$
I found $y=1$ is a stable point while $y=0$ and $y=3$ are unstable. However, I have to evaluale
$$\lim_{t\to\infty}y(t)$$
without solving the equation and given the initial condition $y(0)=2$.
Since $y=1$ is stable and $y=2$ is "near" of $y=1$, this l... |
I am watching a series on geometric calculus by Alan Mcdonald and in the first episodes he states that for any vector u and multivector M: $uM = u \cdot M + u \land M$
This doesn't really take into account what happens with the 0-grade i.e scalar part of M, let's call it s.
For the identity to be true either $u \... |
In geometric algebra, what is the dot product of a vector and a scalar? what is the wedge product of a vector and a scalar? |
Let $H$ be a self-adjoint operator, bounded from below, with spectrum consisting of isolated eigenvalues $E_n (E_0 < E_1 < ...)$ with finite multiplicities $d(n)$, and $Tr[exp(- \beta H)] < \infty$ for all positive $\beta$.
Furthermore, let $\lambda$ be a small positive constant, and $V$ a relatively bounded self-... |
I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf):
> 2.11 (Group law on cuspidal cubic.) Consider the curve
> $$C:(z=x^3)\subset k^2;$$
$C$ is the image of the bijective map $\varphi\colon k\to C$ by ... |
In I. Daubechies' "Ten Lectures on Wavelets" it is stated in note 2 of chapter 2 (p. 51) that A. Grossmann, J. Morlet, and T. Paul's article "Transforms associated to square Integrable group representations. I. General results" can be generalized to cyclic representations. Unfortunately, Daubechies gives no proof of th... |
so I was recently involved in a probability debate with a friend involving Scrabble: I had been selected to be Player 1. It was my turn to play (first play of the game) and I had a q and was looking for a word that would have a U in it. I was contemplating skipping my turn in the hopes that my adversary would play a U.... |
Scrabble probability of Player A playing a U word? |
How to transform a strong generating set of $G \leq S_n$ to a set that generates a proper subgroup of $G$? |
I got stuck on the following problem:
> Let $F:\mathbb{R}\to\mathbb{R}$ be a bounded, nondecreasing, and right-continuous function that satisfies $\lim_{x\to-\infty}F(x)=0$. Define a function $\mu^*:\mathcal{P}(\mathbb{R})\to[0,+\infty]$ by letting $\mu^*(A)$ be the infimum of the set of sums $\sum_{n=1}^{\infty}(F(b_... |
How to construct a concrete sequence $\{(a_i,b_i]\}$ such that the inequality $\sum_i(F(b_i)-F(a_i)) < \epsilon$ holds for all $\epsilon>0$? |
I have a $d$-dimensional finite abstract simplicial complex (ASC); that is I have a sequence of incidence matrices that define the complex. I know that any such ASC can be embedded in ${\mathbb R}^N$ for $N$ approximately the number of vertices. I'm interested in embeddings in ${\mathbb R}^d$ (or ${\mathbb R}^{d+1,d}$)... |
I recently encountered the following riddle:
Let a fair coin be flipped $N > 2$ times. Player A gets a point every time there are two heads in a row, and player B gets a point every time a tail is followed by a head, the player with more points wins. Thus, for example, for the sequence `THHH`, player A wins with 2 v... |
Let $A=\begin{pmatrix}1 & 1/2 \\ -1/2 & 0 \end{pmatrix}$. Define $(x_n)$ recursively by
$$
x_{n+1}=A x_n
$$
with some starting vector $x_0\in\mathbb R^2$. Let $\vert\cdot\vert$ be the Euclidean norm on $\mathbb R^n$. I want to show that this iteration converges linearly, which means that
$$
\lim_n \frac{\vert x_{... |
Take the partial sums
$$ \sum_{n=0}^N\frac{(-1)^{n+1}}{4n+1}=-\sum_{n=0}^N\int_0^1 (-x^4)^ndx=-\int_0^1\frac{1-(-x^4)^{N+1}}{1+x^4}dx. $$
Now,
$$ \left|\int_0^1\frac{(-x^4)^{N+1}}{1+x^4}dx\right|\leqslant\int_0^1 x^{4N+4}dx=\frac{1}{4N+5}\underset{N\rightarrow +\infty}{\longrightarrow}0 $$
so, letting $N\rightarrow... |
Studying for a final and have no clue how to go about this one.
Edit: Sorry, just included definitions. My friend think that you can say if A contains a, then because its a function there must be a b in B. And because f = g then there must be a b in D. Because g is injective there must be an a in C and therefore B ... |
I got stuck on the following problem:
> Let $F:\mathbb{R}\to\mathbb{R}$ be a bounded, nondecreasing, and right-continuous function that satisfies $\lim_{x\to-\infty}F(x)=0$. Define a function $\mu^*:\mathcal{P}(\mathbb{R})\to[0,+\infty]$ by letting $\mu^*(A)$ be the infimum of the set of sums $\sum_{n=1}^{\infty}(F(b_... |
Let $a \in (0,1)$ and a monotonically (strictly) decreasing sequence of non-negative numbers $t_n \to 0$ be given.
Define $$b_n = t_n(1+a) + a.$$
How can I show that for every $\epsilon >0$, there exists an $n_0$ such that $n \geq n_0$ implies $$b_1\times b_2 \times \cdots \times b_n \leq \epsilon?$$
Is it po... |
I'm taking a class in set theory following the text "Elements of Set Theory" by Herbert Enderton. I had a question about the construction of the naturals and the axiom of infinity.
As I understand it, the axiom of infinity says $\exists A (A \text{ Inductive})$ (assume I had already defined what it means to be an in... |
Let $(\mathcal{M}, \otimes, \mathbb{k})$ be a symmetric closed monoidal category, which in my application is the category of $dg$-modules over some commutative ring. Let $A$ be a bialgebra/bimonoid in $\mathcal{M}$. Then $A^{\vee}$ is also a bialgebra, and we have an isomorphism $A^\vee \otimes A \cong \mathcal{M}(A, A... |
$0.09 + 0.15 + 0.05 + 0.06 = 0.35$ not $0.2$.
Note that $0.35+0.65=1$ as you would hope, since the total probability should be $1$.
So you should have done the calculation $200 \times 0.35=70$ as the expected number of winners. This answers the question *"Approximately how many would you expect to win a prize... |
I have a $d$-dimensional finite abstract simplicial complex (ASC); that is I have a sequence of incidence matrices that define the complex. I know that any such ASC can be embedded in ${\mathbb R}^N$ for $N$ approximately the number of vertices. I'm interested in embeddings in ${\mathbb R}^d$ (or ${\mathbb R}^{d+1,d}$)... |
Imagine the Sturm-Liouville problem
$$
(pu')'+qu+\lambda ru=pu''+p'u'+qu+\lambda ru=0
$$
Let $h=\frac{1}{(pq)^{\frac{1}{4}}}$, $u=hv$. Deriving in the above equation and multiplying by h, we arrive at
$$
h(p(hv''+2h'v'+h''v)+p'(hv'+h'v)+qhv+\lambda rhv)=0
$$
$$
(h^2pv')'+h(ph''+p'h'+qh)v+\lambda r h^2 v=0
$$
... |
I have a $d$-dimensional finite abstract simplicial complex (ASC); that is I have a sequence of incidence matrices that define the complex. I know that any such ASC can be embedded in ${\mathbb R}^N$ for $N$ approximately the number of vertices. I'm interested in embeddings in ${\mathbb R}^d$ (or ${\mathbb R}^{d+1,d}$)... |
I have two functions:
A: y(x) = 2 exp(-0.5 (100/sigma)^2)/x
B: y(x) = (2pi sigma) exp(-0.5 (sigma x)^2)
I want to find an algebraic expression relating x to sigma, but can't figure out a way. The pertinent values are between 3<sigma<8, making 0<x<1.
When plotted on a log plot, I noticed that the intersecti... |
Let $\mathcal{A}$ be an L-structure with base set $A$ and $f$ a definable function. $$y=f(x)\Leftrightarrow \mathcal{A}\models \psi(x,y,\overline{a})$$
If we put another parameter $\overline{b}$ in place of $\overline{a}$ and we set $$B:=\{(x,y)\in A^2, \mathcal{A}\models\psi(x,y,\overline{b})\}$$
I think that the s... |
>The contest consists of a series of matches of psychological jujitsu between bots. In each match, 2 bots face off for N = 1024 turns. At the beginning of a match, each bot receives 1024 cards numbered from 0 to 1023. In each turn, both bots fight for a number of points by playing a card, so that the one who plays the ... |
Imagine the Sturm-Liouville problem
$$
(pu')'+qu+\lambda ru=pu''+p'u'+qu+\lambda ru=0
$$
Let $h=\frac{1}{(pq)^{\frac{1}{4}}}$, $u=hv$. Deriving in the above equation and multiplying by h, we arrive at
$$
h(p(hv''+2h'v'+h''v)+p'(hv'+h'v)+qhv+\lambda rhv)=0
$$
$$
(h^2pv')'+h(ph''+p'h'+qh)v+\lambda r h^2 v=0
$$
... |
Let $f\in L^{2}(0,T;L^{2}(\Omega))$ where $T$ is a finite number and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain. I'm consider the average function with respect to time that is
\begin{equation*}
f_{\kappa}(t)=\frac{1}{\kappa}\int_{t-\kappa}^{t}f(\tau) d\tau,
\end{equation*}
after extending $f$ to zero outsid... |
This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem [3086][2]. Let me start by exposing the problem in a friendly way:
You are a bus driver who will pick rockers from the Rock'n Rio festival and take them to a car parking lot, were each rocker will p... |
$\int_C \frac{1}{\sin z}$ where $C := \{z \in C : |z| = 1 \}$
I know $\frac{1}{\sin z}$ has a pole at $z = 0$. I can apply Cauchy Residue Theorem, so I need to know the laurent series expansion of $\frac{1}{\sin z}$ which I am unable to figure out.
Is there any other neat way to approach this?
Thanks in advance. |
Let $M \subset X$. Show that:
$$\overset{\circ}{M} = \bigcup_{\substack{B\subset M \\ B \text{ open}}} B$$
I tried to prove it by using the definition:
$$\overset{\circ}{M} = M \setminus \partial M \iff m \in M \land x \notin \partial M \iff x \in M \land \exists \epsilon > 0: B_{\epsilon}(x) \cap A^{C} = \emptyse... |
Let $f\in L^{2}(0,T;L^{2}(\Omega))$ where $T$ is a finite number and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain. I'm consider the average function with respect to time that is
\begin{equation*}
f_{\kappa}(t)=\frac{1}{\kappa}\int_{t-\kappa}^{t}f(\tau) d\tau,
\end{equation*}
after extending $f$ to zero outsid... |
**Context.** This question is a follow-up of [the following one.][1]
Let $n\geq 1$ be an integer, let $d\mid n$, and let $H=\langle\sigma, \tau \rangle\subset S_n,$ where $ \sigma=(1 \ 2 \cdots n), \tau=( 1 \ d+1)$.
According to an exercise of Serre's finite groups, after correction of some typo, one should have... |
How to prove that abs(sinx)=<abs(x) if X is in IR? |
Let $M \subset X$. Show that:
$$\overset{\circ}{M} = \bigcup_{\substack{B\subset M \\ B \text{ open}}} B$$
I tried to prove it by using the definition:
$$\overset{\circ}{M} = M \setminus \partial M \iff m \in M \land x \notin \partial M \iff x \in M \land \exists \epsilon > 0: B_{\epsilon}(x) \cap M^{C} = \emptyse... |
a state lottery offers a came with the following setup
Choose five (5) numbers from (0 through 9) or use the Quik Pik option and let the computer select for you.
You win by matching your numbers to the winning number(s) drawn in the EXACT ORDER AND POSITION in a single PLAY.
Match all FIVE of your numbers in E... |
Let $f\in L^{2}(0,T;L^{2}(\Omega))$ where $T$ is a finite number and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain. I'm consider the average function with respect to time that is
\begin{equation*}
f_{\kappa}(t)=\frac{1}{\kappa}\int_{t-\kappa}^{t}f(\tau) d\tau,
\end{equation*}
after extending $f$ to zero outsid... |
Imagine the Sturm-Liouville problem
$$
(pu')'+qu+\lambda ru=pu''+p'u'+qu+\lambda ru=0
$$
Let $h=\frac{1}{(pq)^{\frac{1}{4}}}$, $u=hv$. Deriving in the above equation and multiplying by h, we arrive at
$$
h(p(hv''+2h'v'+h''v)+p'(hv'+h'v)+qhv+\lambda rhv)=0
$$
$$
(h^2pv')'+h(ph''+p'h'+qh)v+\lambda r h^2 v=0
$$
... |
$$\overline{\langle x,y\rangle } = {\langle \overline{x},\overline{y}\rangle } ?$$
Hello, first time asker here!
I have looked online for proofs of the above statement. Whilst some stackexchange questions imply that the above statement is true, I have not been able to find a proof.
[Why conjugate when switc... |
It seems that you already know that the space is completely regular. So all you need to prove is that $X$ is closed in it. Because then it is a simple matter of applying the definition of being completely regular to $(0,0)$ and $X$.
So why is $X$ closed? The simplest way is to show that its complement $X^c$ is open.... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
So essentially for a school project I've been given a scatterplot with an array of points shaped in a rough sinusodial wave, and I need to plot a line of best fit manually.
So far I have all the values, except for c (horizontal/phase shift). I want to find the best way to get the most accurate value. so far my equ... |
So essentially for a school project I've been given a scatterplot with an array of points shaped in a rough sinusodial wave, and I need to plot a line of best fit manually.
So far I have all the values, except for c (horizontal/phase shift). I want to find the best way to get the most accurate value. so far my equ... |
So essentially for a school project I've been given a scatterplot with an array of points shaped in a rough sinusodial wave, and I need to plot a line of best fit manually.
So far I have all the values, except for c (horizontal/phase shift). I want to find the best way to get the most accurate value. so far my equ... |
If we are conditioning X given the trivial sigma algebra then we get the expectation of X, its proof is trivial but intuitively what does this case represent ? |
Intuitively, why does the conditional expectation of X given the trivial sigma algebra equals the expected value of X itself? |
So essentially for a school project I've been given a scatterplot with an array of points shaped in a rough sinusodial wave, and I need to plot a line of best fit manually.
So far I have all the values, except for c (horizontal/phase shift). I want to find the best way to get the most accurate value. so far my equ... |
It seems that you already know that the space is completely regular. So all you need to prove is that $X$ is closed in it. Because then it is a simple matter of applying the definition of being completely regular to $(0,0)$ and $X$.
So why is $X$ closed? The simplest way is to show that its complement $X^c$ is open.... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
I'm cross-posting this from Mathoverflow. Since I'm asking for recent developments, it seems best to have answers in both sites.
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The website [Formalizing 100 Theorems](http://www.cs.ru.nl/~freek/100/) by [Freek Wiedijk](http://www.cs.ru.nl/~freek/) contains a list of some theorems that were chosen at some poi... |
Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is strictly increasing with respect to $t$ on $(0,1)$, where
$$h(t)=\frac{1-t+F(t)+t(1-f(t))}{(\phi-t)(1-f(t))}-\frac{F(1)}{... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
What is the relationship between the Laplace equation and the Wave equation? |
I have two functions:
A: y(x) = 2 exp(-0.5 (100/sigma)^2)/x
B: y(x) = (2pi sigma) exp(-0.5 (sigma x)^2)
I want to find an algebraic expression relating x to sigma, but can't figure out a way. The pertinent values are between 10<sigma<50, and 0<x<1.
When plotted on a log plot, I noticed that the intersectio... |
So essentially for a school project I've been given a scatterplot with an array of points shaped in a rough sinusodial wave, and I need to plot a line of best fit manually.
So far I have all the values, except for c (horizontal/phase shift). I want to find the best way to get the most accurate value. so far my equ... |
For back-ground, I'm not well-schooled in higher-dimensional geometry, but I'm currently learning statistical data-science in which many methods rely on the properties of higher-dimensional space.
In Support-Vector Machine (SVM) clustering, the goal is to find a hyper-plane which will cleanly divide the data-points ... |
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