instruction stringlengths 22 15.8k |
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Edit:
Since I did not quite get the responses I would have liked when I asked this question four months ago, let me reformulate it slightly:
What are some examples of $\text{Isom}(M)$ and $\text{Conf}(M)$?
For example, Aaron mentions in his (very helpful) answer that $\text{Isom}(\mathbb{S}^n) \cong O(n+1)$. What ab... |
I have a functor $F\colon \mathbf{Rng}\to\mathbf{Grp}$, and a correspondence on objects which assigns to every group $F(R)$ a suitable subgroup $G_o(R)\subseteq F(R)$. Is there a way to turn $G$ into a functor, defining $G_o(R)\to G_o(S)$ via the maps I have between $F(R)$ and $F(S)$?
$$
\begin{array}{ccc}
F(R) &\to^... |
In my course there's a paragraph: Taylor polynomial with Lagrange remainder,
The paragraph starts with a theorem (I left out the constraints):
$$ ( \exists \theta \in ]0,1[)(f(a +h) = T_{f,a,n}(a + h) + \frac{h^{n+1}}{(n+1)!}D^{n+1}f(a + \theta h)) $$
With $f$ an $R-R$ function and $a$ and $a + h$ defined over an inte... |
Show that $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$.
In the LHS $\binom{n+r-1}{r}$ counts the number of ways of selecting $r$ objects from a set of size $n$ where order is not significant and repetitions are allowed. So you have $n$ people you form $r$ teams and select $r$ captains and ... |
We did in class $x^2+y^2$, which was easy, and we had for homework $2x^2+2xy+3y^2$, which I did (its values (if not square) must be divisible by form primes, or of the form $x^2+5y^2$, and clearly they can't all be the latter!).
Just to test my method I took the middle coefficient down by one - and ran into trouble...
|
So let $M^m$ be a manifold embedded into euclidean space $R^n$. Let L($\gamma$) be the
the lenght of a smooth curve $\gamma$: [0,1] --> $R^n$ which is the number
$\int_0^1 \ |d/dt \gamma(t)| \ \mathrm{d}t$ . Define the distance function
d: M x M --> [0,$\infty$) by d(p,q):=inf L ($\gamma$). The infimum is taken over ... |
For $n \geq 4$, show that $\binom{\binom{n}{2}}{2} = 3 \binom{n}{3}+ 3 \binom{n}{4}$.
LHS: So we have a set of $\binom{n}{2}$ elements, and we are choosing a $2$ element subset.
RHS: We are choosing a $3$ element subset and a $4$ element subset (each from a set of $n$ elements). But we multiply by $3$ by the multiplic... |
What is the shortest string $S$ over an alphabet of size $n$, such that every permutation of the alphabet is a substring of $S$?
Edit 2019-10-17:
This is called a superpermutation, and there is a wikipedia page that keeps track of the best results. Turns out the problem is still open.
|
My question is, how can I solve the following derivative question?
$$y = \sin(\arctan x) + \tan(\arcsin x)$$
Thanks in advance
|
Show that $\binom{2n}{2} = 2 \binom{n}{2}+n^2$.
LHS: This is the number of pairs of $2n$ distinct elements.
RHS: We can rewrite this as $2 \binom{n}{2}+ \binom{n}{1} \binom{n}{1}$. So you can divide up the $2n$ distinct elements into two subsets of $n$ elements. Then pick an element from each of these subsets to form ... |
First, define a probability distribution with density:
\begin{align*}
\newcommand\bm\boldsymbol
\newcommand{\mbl}{\mathbf{l}}
f(\mbl \mid \bm \alpha)
= \frac{1}{B(\bm \alpha + \bm 1)}\exp\left(\mbl \cdot \bm\alpha\right)
\end{align*}
where
$\mathbf l \in (-\infty,0]^k, \sum_i\exp(l_i) = 1$
I'm having trouble finishing ... |
I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted below:
The answer to our problem ($293$) is the
coefficient of $x^{100}$ in the reciprocal
of the following:
$(1-x)(1-... |
I'm not sure how well known this "riddle" is but here it goes.
3 people go to a restaurant, each buy food worth 10.00. When they're done, they give 30.00 to the waitress. She gives the money to the manager, manager says they paid too much, gives 5.00 back to the waitress. She comes back to the table, gives a dollar ... |
i am trying to figure a algorithm problem that determines the two largest number of the a series of random 10 numbers. Thanks to your help.
|
Question: What is the definition of a small class? Is there no such thing as a set which contains other sets such as {{1,2},{1},{2}, 1, 2} (ie, is this really called a "class")? What are some prototypical examples of small classes? What are some prototypical examples of large classes?
Motivation: A few things I've b... |
How to input such system
into maple so that it would solve it?
BTW we some part of system will be given to us, which we do not know certainly - may be some $A_i$'s may be some $B_i$'s as constants. We do not know how much constants we will get. And we would need from given constants get a solution for other $a$'s an... |
This is a problem from Arnold's book on ODEs I cannot solve.
Prove that if $f:U\to V$ is a diffeomorphism, then the Euclidean spaces with the domains $U$ and $V$ as subsets have the same dimension.
Hint. Use the implicit function theorem.
Thanks.
|
On page 18 of Rudin's Real and Complex analysis he defines $0 \cdot \infty = 0$ and says that "with this definition the commutative, associative, and distributive laws hold in $[0,\infty]$ without any restriction".
What is not clear to me is whether the quoted statement is a justification of the definition or just a c... |
If $A$ and $B$ are two conformable matrices,and $AB = A$ and $BA = B$ then prove that A and B are idempotent matrices.
This may be trivial but I am not sure how to proceed on this.
|
What is $\displaystyle \lim_{x \to 0} \frac{\lceil x \rceil}{x}$ ? Here, $\lceil x \rceil$ is the ceiling function at $x$.
For left limit and right limit as $x\to 0$.
|
Some books say that even numbers start from $2$ but if you consider the number line concept, I think zero($0$) should be even because it is in between $-1$ and $+1$ (i.e in between two odd numbers). What is the real answer?
|
From wikipedia,
http://en.wikipedia.org/wiki/Riemann_zeta_function
"Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis."
I know the zeros are symmetrical. But what about t... |
So I had a system
#for given koefs
k:=3; n:=3;
#let us solve system:
koefSolution:= solve({
sum(a[i], i = 0 .. k) = 0,
sum(a[i], i = 0 .. k)-(sum(b[i], i = 0 .. k)) = 0,
sum(i^n*a[i], i = 0 .. k)-(sum(i^(n-1)*b[i], i = 0 .. k)) = 0
});
So I have a vector like
koefSolution := { a[0] = 7*a[2]+26*a[3]-b[1]-4*b[2]-... |
Last month, I was introduced to the card game SET. The game raises several interesting questions eg what's the probability that n randomly drawn cards contain k sets, is it possible to end with 12 (or 15) cards which contain no sets... I would like to present the solution to a question like this to undergrads, and thro... |
I'm reading a book about Calculus on my own and am stuck at a problem, the problem is
There are two circles of radius $2$ that have centers on the line $x = 1$ and pass through the origin. Find their equations.
The equation for circle is $(x-h)^2 + (y-k)^2 = r^2$
Any hints will be really appreciated.
EDIT: Here is wh... |
Show that $\displaystyle 1+ \sum\limits_{k=1}^{n} k \cdot k! = (n+1)!$
RHS: This is the number of permutations of an $n+1$ element set. We can rewrite this as $n!(n+1)$.
LHS: It seems that the $k \cdot k!$ has a similar form to $(n+1)! = (n+1)n!$ Also we can write $1 = 0!$ I think you use the mulitplication principle i... |
From the first day that I entered college, I was wondering about the relationship between some basic mathematical abstract concepts and nature. I'm going to explain them here and you may find them a little bit opinion-based, but it's not. So please guide me in case you have good scientific-based answers/examples.
Infin... |
Could someone tell me what the "square of a graph" $G^2$ is? Thanks.
|
How can I calculate average distance from center of a square to points inside the square?
|
Is the integral of $(1/x) dx$ the same as the integral of $dx/x$ ?
|
Suppose we have the following two identities:
$\displaystyle \sum_{k=0}^{n} \binom{n}{k} = 2^n$
$\displaystyle \sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$
The first says that the number of subsets of an $n$-set is $2^n$. The second says that the number of subsets of even size equals the number of subsets of odd size (... |
I want to prove that the size (cardinality) of $\mathbb{N}$ is the same as the size of the set of all ordered triples of natural numbers ; just to be more clear that $|\mathbb{N}|=|\mathbb{N}\times\mathbb{N}\times\mathbb{N}|$.
|
Say I have a time series, with actual measurements of a variable $a$ in different locations. If I want to know the average of $p$ from time $t_1$ to $t_n$, I could say that $\overline{a_{p}}=\frac{\sum\limits_{t=1}^n a_{{p}_t}}{n}$, where $p$ is the coordinate of the point of measurement. So far, so good. Let's gene... |
This article on Wikipedia states the following:
Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
I was w... |
I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where defined). My question is:
Using this definitio... |
Let M be a smooth manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that for all $\alpha, \beta$, $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1})) > 0$ (where defined). I'm struggling to understand the reason this definition is made. My question... |
Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most 4.
What are all types of $g$ such that:
1) $a+b$ is a root and $a+2b$ is not a root?
2) $a+2b$ is a root and $a+3b$ is n... |
Suppose $c$ is an isolated point in the domain $D$ of a function $f$.
In the delta neighbourhood of $c$, does the function $f$ have the value $f(c)$?
|
In Wikipedia, well-order is
defined as a strict total order on a set $S$
with the property that every
non-empty subset of $S$ has a least
element in this ordering.
But then later, well-order is
defined as a total order on $S$ with
the property that every non-empty
subset of $S$ has a least element in
this ordering.
... |
I have a polygon in 2D (defined by a series of Vertex $V$ with coordinates). The polygon can be convex or concave. I have $n$ number of fix points I can put inside the polygon.
The question is, how can I distribute the fix points as uniformly as possible inside the polygon?
The motivation for this question is I want t... |
How do I correctly represent the following pseudocode in math notation?
EDIT1: Formula expanded.
EDIT2: Clarification.
(a,b) represents a line segment on a 1D line. a <= b for each segment. The division show below is done as per the following T-SQL code (which I suppose could be represented as a function in the formula... |
the problem is the following:
For all $x \in \mathbb{R}$, we assign a finite set $\Phi(x) \subset \mathbb{R} - \{ x \}$. We say that a set $S \subset \mathbb{R}$ is independent if for any $x,y \in S$, $x \notin \Phi(y)$ and $y \notin \Phi(x)$, that is, if $S \cap \Phi(S) = \emptyset$. Prove that there is an independent... |
Given implicit function $F(x, y) = 0$, how can I find its asymptotes?
EDIT: Sorry, my calculations were wrong. Here is correct function:
$F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} - e$
|
Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence
which is equidistributed in $[0,1)$.
(a) Is it true that $$\frac{1}{N}\sum\limits_{n=1}^N f(x+\xi_n)$$ converges to the constant $\int_0^1 f(y) dy$
for each $x\in \mathbb{R}$ as $N\to \infty$ ?
(b) If so, is the convergence unifo... |
Can you make a simple example of an uncountable ordinal? With simple I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an uncountable ordinal, but the only proof that I know is quite complicated.
|
Given an abelian group $X$, let $F_n(X)$ denote the simplicial abelian group defined as follows:
$F_n(X)_j=0$ for all $j<n$ and $F_n(X)_j=X$ for all $j\geq n$
with the appropriate zero and identity maps between them so the normalization (normalized moore complex) gives the appropriate homology (say, $d_i=0$ for every $... |
If we only consider the natural numbers, is 3/2 undefined? If not what is the answer?
|
Problem
Given:
$$\vec r = r(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$$
$S: | \vec r | = a$, with $\hat n$ outwards
$$\vec r_0 = \frac{3a}{5}(\hat x + \hat y - \hat z) = \frac{3a}{5}(1,1,-1)$$
$$\vec F = k \frac{\vec r - \vec r_0}{ {| \vec r - \vec r_0 |}^3 }$$
Calculate:
$$\int_S \vec F \cdot \mathrm... |
Show that $f_{n-1} + L_n = 2f_{n}$.
So we need to find a $2$ to $1$ correspondence.
Set 1: Tilings an $n$-board.
Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet.
So we need to decompose a tiling of an $n$-board to a tiling of an $n-1$-board or a tiling of an $n-1$-bracelet?
Source: Proofs that Really Cou... |
Does $f(n+1)/f(n)$ converge as $n\rightarrow\infty$ for $f(n)$ defined by a linear recursion, for all linear recursions?
|
The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula
$$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$
For small values of $n$, we have
$$V_1=2\qquad$$
$$V_2\approx 3.14$$
$$V_3\approx 4.18$$
$$V_4\approx 4.93$$
$$V_5\approx 5.26$$
$$V_6\approx 5.16$$
$$V_7\approx 4.72$$
It is not difficult t... |
Given $K_n$ define sequence $S_4$: writing all vertices in order of traversing it by Euler circle
How many distinct Hamilton cycles are in this sequence $S$?
|
Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function with $n\geq 2$, and define $S_{f}=\{(x_1,x_2, \ldots x_n) \in\mathbb{R}^n \mid f(x_1,x_2, \ldots x_n)=1\} = f^{-1}(1)$.
Is $S_f$ path-connected when $f$ is $C^{\infty}$? If not, what about if $f \in \mathbb{Q}[x_1,\ldots,x_n]$?
Context: I had to establish that the ... |
Let $\sigma=(1\ 3),\ \tau=(2\ 4\ 5),\ \pi=(2\ 3\ 4) \in S_{5}$. Find $\pi\circ\tau\circ\sigma$.
I know the solution is $(1\ 4\ 5\ 3)$.
What i'm doing now is writing the permutations in the "expanded" form,
for example $\tau=\begin{pmatrix}
1\ 2\ 3\ 4\ 5\\
1\ 5\ 3\ 2\ 4
\end{pmatrix}$,
and then "follow the nu... |
In the wikipedia article, two examples are given which use/ do not use the axiom of choice. They are:
Given an infinite pair of socks, one needs AC to pick one sock out of each pair.
Given an infinite collection of pairs of shoes, one shoe can be specified without AC by choosing the left one.
Aren't these equivalent ... |
Sorry for this uninteresting question but hopefully someone can provide some help.
Is there a way to simplify the following expression?
$$\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\frac{n-m-v}{n}\right)^{r}\displaystyle \frac{m}{m+v}$$
$$-\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\fr... |
How many Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ are there? The motivation is from this answer of mine on MathOverflow.
|
Let $\tau \in \mathbb{S}^1$ be such that $\tau$ is not a root of unity. Let $E_\tau$ be the quotient space $S^1/\tau^{\mathbb Z}$. Consider it as a pointed space with basepoint the equivalence class of $1$. This is a path-connected space.
What is the fundamental group of this space?
|
I have to explain the problem I am thinking first:
The solutions of $x^3+6x-20=0$ are $x = 2$, $-1+3i$ and $-1-3i$.
The cubic formula for these solutions is:
$$x = \sqrt[3]{10 + \sqrt{108}} + \sqrt[3]{10 + \sqrt{108}}$$
First $\sqrt{108} \approx -10.392304845413$, $10.392304845413$ has two values.
And $\sqrt[3]{10 + -1... |
Does anyone remember a movie about four mathematicians studying infinity, who ended up committing a suicide? It was aired online somewhere, and I can't find it (it's a documentary, if that helps).
|
I have an input array of line segments $(a,b)_1 ... (a,b)_n$. Let's call it $S$ so $(a,b)_{S_1} ... (a,b)_{S_n}$.
I have another array of the same construct, let's call it $I$, so $a,b)_{I_1} ... (a,b)_{I_n}$.
I wish to mathematically express the shortest distance of each of $I$ against $S$. In other words, the matrix ... |
One way to prove that a field $K$ has no ideals except the entire field and the trivial ideal is to note the fact that every element $x$ has an inverse. By the definition of an ideal, if $x$ is in the ideal then $x^{-1}x$ is because $x^{-1} \in K$. But now we have that 1 is in the ideal, and so again by the definition ... |
For which functions $f\colon [0,1] \to [0,1]$ is the function $g(x,y)=f(x)f(y)$ convex over $(x,y) \in [0,1]\times [0,1]$ ? Is there a nice characterization of such functions $f$?
The obvious examples are exponentials of the form $e^{ax+b}$ and their convex combinations. Anything else?
EDIT: This is a simple observatio... |
In Conceptual Mathematics 1st edition, p. 325-236, there is a sketch of a proof, but I can't carry out the complete proof.
"... This also follows from the appropriate universal mapping properties, which imply that the two composites satisfy properties that only the corresponding identity maps satisfy."
I can't figure... |
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the above statement true?
|
I asked I question here trying to obtain clarification about how to follow a hint. In spite of the fine answers I received there, the hint doesn't look very helpful. I'd like to know a hint for the following problem or a way to use the hint I already have.
The probability $p_{m}(r,n)$ of finding exactly $m$ cells empty... |
If $n$ is a natural number, let $\displaystyle \sigma_{11}(n) = \sum_{d \mid n} d^{11}$.
The modular form $\Delta$ is defined by $\displaystyle \Delta(q) = q \prod_{n=1}^{\infty}(1 - q^n)^{24}$.
Write $\tau(n)$ for the coefficient of $q^{n}$ in $\Delta(q)$.
I would like to know why $\tau(n) \equiv \sigma_{11}(n) \p... |
$$\int{\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}}\mathrm dx$$
This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone did) -- how would you go about it?
|
I have a background in math, but I think the following construction appears often in physics.
Is there a term for a functional which maps one smooth function $f: R^n \rightarrow R$ to another $L(f):R \rightarrow R$ which represents the integral along each level set? That is,
$L(f)(a) = \int_{f^{-1}(a)} d\mu $
wher... |
How can I find solution for $x^3 \equiv 1 \pmod p$ ($p$ a prime) efficiently?
Trivial root is $x_1 = 1$. I need to find other roots $x_2, x_3$.
|
How do we solve the recurrence $T(n) = 2T(n/3) + n\log n$?
Also, is it possible to solve this recurrence by the Master method?
|
Suppose $A$ is a linear transformation from $R^3$ to $R^3$ and $|det(A)| = 1$. I know that $A$ is volume preserving, but is it also area preserving? For example, if $a$ and $b$ are two vectors in $R^3$ that span a parallelogram, is the area of this parallelogram equal to the area of the paralellogram spanned by $A(a)$ ... |
In
Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$
Timothy Wagner gave a correct answer that was questioned for not having shown that the limit exists in the first place. My question is, are there any examples were a value of limit can be derived although the limit does not exist? Please note I am not q... |
I am referring to the Polyhedron Volume formula here:
$$V = \frac{\sum_{i=1}^k \vec{x_{i}} \cdot \hat{n_i} A_i}{3}$$
My question is, does the above formula applicable for the Polyhedron below?
The given faces for the polyhedron are:
$$f_1: n_1,n_2,n_3, n_4$$
$$f_2:n_2, n_3,n_8,n_5$$
$$f_3:n_1, n_4,n_6,n_7$$
$$f_4:n_6,... |
$$T (n) = 3T (n/5) + \log^2 n$$
i am trying to solve this recurrence i have been told that it is master methods case 1 but how do i compare $\log^2n$ with $n^ {\log_5 3}$
|
My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here is some background to make this clearer.
Suppose $f\colon \Re \mapsto \Re $ is some continuous function on $[a,b]$, the... |
After seeing this recent question asking how to calculate the following integral
$$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$
and some of the comments that suggested that it was an elliptic integral, I tried reading a little bit on the Wikipedia article about elliptic integrals.
It seems that the point is ... |
I am interested in finding out about the current status of the planned book: The Triangle Book by John H. Conway and Steve Sigur. I understand that Steve Sigur died some time back. I got no reply from Prof. Conway; is there someone who knows the fate of this project. Is it abandoned or is it still under preparation? He... |
Let's consider the sequence space $E =\mathbb R^{\mathbb N}$. If I believe in Choice, I have an isomorphism $E \simeq \mathbb R^{(\mathfrak c)}$ for some cardinal $\mathfrak c$. I further have some inequalities about $\mathfrak c$: first, $\mathfrak c = \dim \mathbb R^{\mathbb N} \leq |\mathbb R^{\mathbb N}| \leq (2^{\... |
From Rudin's Principles of Mathematical Analysis (Chapter 2, Exercise 6)
Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed.
I think I got it but my argument is a bit hand wavy:
If $x$ is a limit point of $E'$, then every neighborhood of $x$ contains some $y\in E'$, and every neighborhood o... |
How can I solve the recurrence
$$T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}\ ?$$
Please don't just write the name of the method, as I just started learning this stuff and things are a little hard for me to figure out on my own.
Thanks in advance.
|
I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (at least my head). In a sense, I still do not have the feeling that I know why they are simple and why should it be 5 and not any other number (perhaps this is only bec... |
I am aware that compactness fails on finite models, but the common counter-example uses models of arbitrary big finite size. So if we bound the size what results can we get?
Assume we have an infinite set of sentences $\Sigma$, of a language of first order logic $\mathcal{L}$. If there is a natural number $n$ such tha... |
This is probably just elementary, but I don't know how to do it. I would like to find the minimum value of $a$ such that $x^a \geq \ln(x)$ for all $x > 0$. Numerically, I have found that this minimum value lies between 0.365 and 0.37 (i.e., $x^{0.37} > \ln(x)$ for all $x > 0$, but $x^{0.365}$ is not). Is there any anal... |
I just asked this question on ask.metafilter, and it was suggested that I ask here. Though I'm talking about coding something up, this question is about the math behind it, not the implementation.
We have done analysis in the past where we've computed an approximation for Gaussian curvature of a surface in Cartesian c... |
I was wondering when in history did people start use the $\sqrt{}$ sign for square root, what did they use before, and why this curious nomenclature is adopted.
|
Let $\tau: E \to E$ be a measure-preserving transformation of the measure space $(E, \mathcal{E}, \mu)$, i.e. $\mu(\tau^{-1}(A)) = \mu(A)$ for all $A \in \mathcal{E}$. Let $\mathcal{E}_\tau = \{ A \in \mathcal{E} : \tau^{-1}(A) = A \}$. In my lecture notes, it is claimed that a measurable function $f$ is invariant (i.e... |
Does anyone know how to factorize the following expression:
$$4x^4+12x^{10/3} y^{2/3}+33x^{8/3} y^{4/3}+46x^2 y^2+33x^{4/3} y^{8/3}+12x^{2/3} y^{10/3}+4 y^4$$
?
|
The (spherical) loxodrome, or the rhumb line, is the curve of constant bearing on the sphere; that is, it is the spherical curve that cuts the meridians of the sphere at a constant angle. A more picturesque way of putting it is that if one wants to travel from one point of a (spherical) globe to the antipodal point (sa... |
How to find the intersection between a set of two ranges of number.
let me explain the question with an example,
{2,3} {3,8} would result to 0
{2,5} {3,8} would result to 2
{3,6} {3,8} would result to 3
{4,5} {3,8} would result to 1
{4,5} {3,8} would result to 1
{8,9} {3,8} would result to 0
Note:This could be easily ... |
The following problem is basically from Miranda's "Algebraic Curves and Riemann Surfaces", which I am reading on my own; if there are any rules against posting textbook problems, my apologies!
Let $X$ be a smooth projective curve defined by the homogeneous polynomial $F(x,y,z)=0$, with $\deg F = d \geq 3$. Let $f(x,y) ... |
I've just started learning about rings. Rings are one additive abelian group strung together, through the associative law, with another structured operation.
Couldn't we continue stringing together operations in this manner (a multi-operation associative law)? Would what I'm thinking be encompassed by the ring definiti... |
I am working on a model which predicts the total final score (sum of the scores of both teams) of certain sports matches. I have checked the predictions of the model against the actual scores of matches for the last several years. The error in the predicted total score to the actual toal score is approximately normally... |
I've asked this question at SO, but only answer I got is a non-answer as far as I can tell, so I would like to try my luck here.
Basically, I'm looking for a better-than-naive algorithm for the constructions of polygons out of union of many polygons, each with a list of Vertex $V$. The naive algorithm to find the unio... |
From Wikipedia's Weierstrass Factorization Theorem, I learned that every entire function can be represented as a product involving its zeroes. Examples are the sine and cosine function. The Riemann zeta function, however, is not entire.
Let us assume the Riemann Hypothesis. Can $\zeta(s)$ be represented by an infinite... |
Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.
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Broadly I would like to know what is the connection between homogeneous binary quartic forms and elliptic curves.
I see that the invariants on the first space under the action of $SL(2,\mathbb{C})$ are precisely the functions $g_2$ and $g_3$. Also given such a quartic form on $\mathbb{C}^2$, if one thinks its 0-set as... |
Is there a good online community for discussing optimization models? The ones I have found don't seem to have a critical mass for active discussions.
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Let $p$ be a prime and $H$ a subgroup of a finite group $G$. Let $P$ be a p-sylow subgroup of G. Prove that there exists $g\in G$ such that $H\cap gPg^{-1}$ is sylow subgroup of $H$.
I have no idea how to do this, any hints?
Note: Originally it was unclear if the problem was for possibly infinite groups or just fini... |
Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 \rbrace$ and factorize them, we will get
$$\begin{align} 1&=2^0\times 3^0\times 5^0\times 7^0\times 11^0\times 13^0\ti... |
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