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If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?
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For example, saying that $T$ is a k-tensor one might see $T\in 3^k(V)$, of course it's not actually a 3. It looks somewhat like Fraktur font Z: $\frak{Z}$. I couldn't detexify it, and it doesn't appear to be a standard notation for the set of tensors. Other resources seem to just use $T$.
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(Originally asked on MO by AJAY.)
What is the geometric, physical, or other meaning of the third derivative of a function at a point?
If you have interesting things to say about the meaning of the first and second derivatives, please do so.
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cross-post from stack overflow
I have a list of numbers like {2,1,1,0} and I'd like to list all permutations of that list that are not equivalent under given symmetry group. So for instance if it was a dihedral group of order 4, result would be {{2, 1, 1, 0}, {2, 1, 0, 1}}. Is there a practical way to do this when the ... |
The recent post here has led to the following question (consider $\cos(sx) = \sum\nolimits_{k = 1}^\infty {\frac{{( - 1)^k (sx)^{2k} }}{{(2k)!}}}$). I find it instructive, whether it will turn out to be easy/well-known or not.
So, suppose that $(f_k)_{k \geq 1}$ is a sequence of continuous functions on a non-compact i... |
How many ways of giving out 6 pieces of candy to 3 children if each child must receive at-least one piece ?
I think, $ 5 \choose 2$ is the answer,consistent to $ n-1 \choose r $ as this is similar to finding the number of non-zero positive integer solution of the equation $X_1 + X_2 + X_3 = 6$ ,but the correct answer... |
How to show that:
$ (-i\nabla-eA)\times(-i\nabla-eA) = (ie\nabla \times A) $
i and e are constants
A is a vector field
$\nabla$ = vector differential operator
|
I have a question about abelian groups (or rather $X$-groups):
Suppose $A$ is an abelian group (or rather an $X$-group) which is artinian and noetherian. $f$ is an endomorphism (or rather an $X$-endomorphism) of $A$, i.e. $f\in \mathrm{End}(A)$. Also know that $f(f(A))=f(A)$. Prove that $A=\ker(f)\dot{+}f(A)$, i.e. $A... |
Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
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I had this problem on my last exam in calculus, I still don't know, how to solve this, so I would appreciate your help.
Let $a_{ij} \in \mathbb{R}$ for $i,j \in \mathbb{N}$. Suppose, that for every $j$ series $S_j = \sum_{i=1}^{\infty}{a_{ij}}$ is absolutely convergent and, for every $i$ exists finite limit (limit $\no... |
I've looked but cannot find the mathematical way to represent the following:
r = Max(x1, x2, x3)
I want to mathematically show that r = max value of the set (x1, x2, x3). To make sure I'm being clear, I want to write the math formula of the code equivalent.
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I am a Software Engineer. With interest on computers I have chosen this path. But I don't know anything about great maths. I want to learn from basic to some level(which would help me in my field..like identifying solution to a problem, designing algorithms, designing shapes and some animation on them). So, please sugg... |
My algebra professor gave me this puzzle a while back. I'm pretty sure I've found the right solution; nonetheless, I wanted to share it and see if you come up with anything really nice or unexpected.
Prove that if you take a unit square and cut it into a finite number of smaller squares (in any way you can think of),... |
I seem to have a misunderstanding of how to work with a Taylor series.
Suppose I want to write $f(x)=x e^x$'s Taylor expansion of $n$ degree around $0$.
I see two ways:
1) Find the $n$th derivative of $f(x)$, this is quite easy: $f^{(n)} (x)=n e^x+x e^x$.
And from the formula I get:
$$f(x)=\sum_{k=1}^{n} \frac{x^k}... |
Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other geometric objects? For instance, why don't we just say that the Alexander invariants are a finite set of numbers? Pres... |
Here is my set up.
I am running a simulation of a gambler betting over time
Here are the givens I am using.
Probability of win = 35% which is in B1 table
Odds = 2 which is in B3 table
Kelly # (amount of current bankroll to bet each time) = 2.5% which is in D1 table
Bankroll = $10,000 which is in A6 table
I am goin... |
In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given:
$$\left( \begin{array}{ccc}
A & B \\
C & D \end{array} \right)$$
is symplectic if and only if:
$$A^TD-C^TB=I, A^TC=C^TA, D^TB=B^TD$$
but it seems wrong, since, for example:
$$\left( \begin{array}{ccc}
0 & 0 & -1 & 1 \\
0 & 0 & 0... |
The vector force field F=(yi,-xj) has a curl of -2. The acceleration of a particle in space is given by:
ax=y/m
ay=-x/m
This vector field has a divergence of 0. Will particles in this vector FORCE field move in circles or will they spiral outwards in a crazy fashion? My conundrum is this, the acceleration of the partic... |
I am trying to understand a proof, I don't understand one of the steps.
could someone shed some light why the following is true.
if $f(x)$ is a non-increasing and continuous function and $a>b$ then $\{x\colon f(x) > f(b) \} \subset \{x\colon f(x) > f(a) \} $
i don't see why this is true. any insights ?
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I was given the following question:
Calculate
$$1-\frac12+\frac14-\frac18+\frac1{16}- \frac1{32}+\frac1{64}-\cdots -\frac1{512}+\cdots;$$
and express the answer as a simple fraction.
My approach was to use the following formula: $\frac1{1-r}$ where $r$ is the common ratio. In the end I got $2/3$. Am I correct... |
I need to find out $\log_g {-1}$ in $\mathbb{Z}_n$ where $n$ is an odd prime and $g$ is a primitive root mod $n$. How do I do that?
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Question:
We say that a convex lattice polytope $P\subset \mathbb{R}^d$ is a smooth Fano polytope if:
The origin is contained in the interior of $P$
The vertices of every facet of $P$ are a $\mathbb{Z}$-basis of $\mathbb{Z}^d$
Now, suppose we have a set $V=\{v_1,\ldots,v_n\}\subset \mathbb{Z}^d$. What is an (pre... |
I was reading up on class field theory, and I have a question. On wiki (http://en.wikipedia.org/wiki/Artin_reciprocity), one formulation is that there's some modulus for which $I^c_K/i(K_{c,1})Nm_{L/K}(I^c_L)$ is isomorphic to $Gal(L/K)$.
Another formulation on the same page is:
$C_K/N_{L/K}(C_L)$ is isomorphic to $Gal... |
One of the perks of my research topic (Latin squares) is that it's somewhat possible to explain what I do to those with a fairly minimal mathematics background. I'll pose a typical question that comes up in one of my main research topics as a problem here -- I'm hoping that this will whet the appetite of someone cons... |
How many stripes can you paint on a $2$-group of fixed size?
A group of order $2^ap^b$ is solvable, by Burnside's theorem, so its chief factors are either abelian $2$-groups or abelian $p$-groups. Such a group is a zebra group if its chief factors in any chief series alternate between the $2$ and the $p$. In other ... |
It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant").
It is noteworthy that many of the named cubic curves look rather similar: the folium of Descartes, the trisectrix of Maclaurin, the (right... |
Is it true that $\frac1{n!} \int_0^\infty x^{n-k} (x-1)^k e^{-x}\,dx \approx e^{-k/n}$ when $k$ and $n$ are large integers with $k \le n$?
This quantity is the probability that a random permutation of $n$ elements does not fix any of the first $k$ elements.
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What are the formal series $f$ with non-negative integer coefficients and constant term equal to $1$ whose multiplicative inverse $1/f$ has all coefficients, apart from a finite subset, all non-positive?
In fact, assume the series converges in some disk if you want...
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And how to express $\{ x \} = x - \lfloor x \rfloor$ as function of $sin(x)$ and $sign(x)$?
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Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=n$ the an... |
I need to solve following system of nonlinear equations.
Here are some characteristics of this system:
It consists of n equation and n variables.
Every equation is in similar form -> sum of products = constant.
The lenght of every product is the same (it will be denoted as k).
The number of elements in sum might be dif... |
This may have more to do with computing than Mathematics in its application, however this has been giving me a headache for some time and I see no other recourse than to ask...
Given a natural integer T > 9 (and probably between 1,000 and 1,000,000,000), and a choice N in {0,1,2,3,4}, provide two algorithms (encoding a... |
I am unable to understand the motivation behind the set theoretic definition of a natural number. The definition given in the book by Goldrei is as follows:
First he defines an inductive set:
A set $y$ is inductive if $\emptyset \in y$ and $x^{+} \in y$ whenever $x \in y$, where $x^{+} = x \cup \{x\}$.
So far fine.
But... |
I'm trying to make one body fly around another, using Coulomb law $F=\frac{q_1 q_2}{r^2}$ and second Newton law $ma=F$. Now I'm doing it this intuitive way:
Move body1 according to current speed and $dt$;
Calc new Coulomb force then find acceleration $a=\frac{F}{m}$;
Add acceleration to current speed;
Repeat.
This wo... |
Thanks for your attention,
I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras,
I cannot prove this sentence:
"The maps factoring through an injective object are precisely the null-homotopic maps."
Here, the object is in the... |
Let $0 \leq \alpha < \beta \leq 1$. I'm looking for an example of a Lebesgue measurable subset $E$ of $\mathbb{R}$ such that
$$\liminf_{\delta \rightarrow 0} \frac{m(E \cap (-\delta,\delta))}{2\delta} = \alpha$$
but
$$\limsup_{\delta \rightarrow 0} \frac{m(E \cap (-\delta,\delta))}{2\delta} = \beta$$
where $m$ is the... |
Consider an ellipsoid of semi-axes a, b, c (possibly prolate, b=c). I am interested in the "shadow" of this solid onto a distant plane, in a given direction d=(k,l,m) orthogonal to that plane. By shadow I mean the projected area onto the plane: each point on the surface of the ellipsoid is translated in the same direct... |
I was asked to prove:
no group can have a minimal normal subgroup isomorphic to a $\mathrm{Syl}_2(A_7)$.
I think I should find some property that $\mathrm{Syl}_2(A_7)$ has but not a minimal normal subgroup. So then I thought that it a $\mathrm{Syl}_2(A_7)$ always has characteristic subgroup, i.e. exist $G\ char\ \math... |
What is the Jacobian matrix?
What are its applications?
What is its physical and geometrical meaning?
Can someone please explain with examples?
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Is there a commutative integral domain $R$ in which:
every nonzero prime ideal $Q$ is maximal, and
for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such that $R/Q$ is a field of size $q$?
I am looking for a ring where "$q\equiv 3 \bmod 8$" describes finite fields, rather than just fi... |
I am looking for a symbol to represent the operation of taking unique values from a vector. So, say the symbol was $\theta$:
$v = [0, 0, 0, 1, 1, 1, 3, 1, 2, 0]$
$\theta(v) = [0, 1, 3, 2]$
Or is this something that just isn't defined?
Thanks
EDIT
Thanks all for your help here. Your answers and comments have led me to... |
Does there exist a bi-Invariant metric on $SL_n(\mathbb{R})$. I tried to google a bit but I didn't find anything helpful.
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Suppose $A$ is the set of all rational numbers $p$ such that $p^2 <2$ and $B$ is the set of all rational numbers $p$ such that $p^2 > 2$. We want to show that $A$ contains no largest element and $B$ contains no smallest element.
In Rudin's Principles of Mathematical Analysis, he associates $q = p- \frac{p^2-2}{p+2} = \... |
I know this seems like a juvenile question, but for some reason I can't recall how to do this.
Lets say I want to find 87 reduced by 99.75%. What is the equation to do that?
Thanks!
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I'm trying to track down the source of some wonky data. The data are response times (RTs) collected from humans using a computer keyboard. Here's a histogram of the RTs, binned to 1ms:
Obvious is a skew in the data, which is expected/known for such data. However, also noticeable is a slight "ghosting" of the image. Zo... |
Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $\inf A = -\sup(-A)$.
We know that $-A$ is bounded above. Hence $\sup(-A)$ exists. So to proceed, one would show that $\inf A \leq -\sup(-A)$ and $\inf A \geq -\sup(-A)$? Or $-\inf A >... |
Does someone know if the following is true:
Let $\mathbb{X}$ be some arbitrary Banach space.
$\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$.
Now, if the series
$$\sum_{k=1}^\infty \|x_k\|_X$$
converges, would the "more generic" series
$$\sum_{k=1}^\infty x_k$$
also converge?
If yes, ... |
I understand the principle of finite induction, but my book then mentions that there is a variant of the first where requirement b is changed to
If $k$ is a positive integer such that $1,2, \dots, k$ belong to $S$, then $k + 1$ must also be in $S$.
The sample problem is proving that the inequality about the Lucas nu... |
This problem came from a Putnam problem solving seminar.
If each person in a group of n people is a friend of at least half the people in the group, then show that it is possible to seat the n people in a circle so that everyone sits next to friends only.
My idea was to use induction on $n$; if $n$ is odd we can remo... |
Let $C: x=x(t), y=y(t), a\le t\le b$ be a $C^1$ closed curve (not necessarily simple).The isoperimetric inequality says that
$$ A\le \frac{\ell^2}{4\pi},$$ where $$A=\left|\int_C y(t)x'(t) dt\right|$$ is the area enclosed by $C$, and
$\ell=\int_a^b \sqrt{(x'(t))^2+(y'(t))^2} dt$ is the arc length of $C$.
My question... |
Fix $b>1$. Show the following:
(a) If $m,n,p,q$ are integers, $n>0, q>0$, and $r = m/n = p/q$, prove that $(b^{m})^{1/n} = (b^p)^{1/q}$. So $(b^r) = (b^m)^{1/n}$.
(b) Prove that $b^{r+s} = b^{r}b^{s}$ for $r,s \in \mathbf{Q}$.
(c) If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, there $t$ is rational ... |
I was particularly interested in the following:
When I read this proof, everything seemed fine and logical except one detail (the proof is located here).
Right after we prove, that the series $\sum_{k=1}^\infty x_{N_m} - x_{N_m+1}$ converges, there is a statement which tells us that the limit of that series (let's name... |
I completely understand how to find the likelihood functions of simple pdfs. However, how would you attempt to find the likelihood function of a pdf with a negative exponential function with two different means, where the overall mean is just the two means added together???
The method to do this would be appreciated. ... |
I'm having a little trouble grasping the definition of a topological space in terms of the open-sets. Suppose that we're defining a topology on the real line, in theory could we arbitrarily select the elements of the open-sets, so long as the set of all open sets satisfies the properties of a topological space?
|
Is it possible to find closed form of following sum: $\sum\limits_{k=1}^{n-1}\frac{x^k}{n-k}$ ?
If not then maybe there is at least some good aproximation?
|
I was sorting out the coins in my loose-change jar the other day, and the following thought crossed my head: Is it possible to deduce the number of each type of coin in this pile by simply weighing them?
The coins I were counting were Australian 5c, 10c, 20c and 50c. According to Wikipedia, they weights are:
5c 2... |
I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that
$$
\frac2{n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right)
$$
However, I was thinking that if you take $ 2\left(\frac{1}{t_n} + \frac{1}{t_{n+1}}\right) = \frac1{t_{\frac{n}2}}$ where $n$ is even.
So my logic is you can subtr... |
Another problem on combinatorics. This time I'm asking for a hint and if it is possible a general strategy when dealing with this kind of problems.
Show without using the preceding results * that the probability $p_{m}(r,n)=n^{-1}E_{m}(r,n)$ of finding exactly $m$ cells empty satisfies
$$p_{m}(r+1,n)=p_{m}(r,n)\frac{n-... |
$X$ is a random variable, which is not constant. $E[X]=0$. $E[X^4] \leq 2(E[X^2])^2$.
Let $Y$ be given by: $P(Y=E[X|X \geq 0]) = P(X \geq 0)$ and $P(Y=E[X|X \lt 0]) = P(X \lt 0)$.
Do we necessarily have $E[Y^4] \leq 2(E[Y^2])^2$?
|
How do I find generators of Octahedral group when viewing it as a subgroup of $S_8$?
Update: Alex Bartel's suggestion gives the following permutations and Cayley graph
Update 2: second suggestion by Alex gives following Cayley graph, which seems to be isomorphic to the Cayley graph of symmetric group given elsewhere... |
There are many examples in which making more "precise" predictions gives worse performance (e.g. Runge's phenomenon). My professor implied that there was a sound basis for choosing "simple" functions over complex ones in the general case, and that it had to do with information theory.
Does anyone know what he was refer... |
I have a series of tasks where when one task finishes the next task runs, until all of the tasks are done. I need to find the probability that everything will be finished at different points in time. How should I approach this? Is there a way to find this in polynomial time?
The pdfs for how long individual tasks will ... |
Fix $b>1, y>0$ and prove that there is a unique real $x$ such that $b^x = y$. So for uniqueness, $x_1 < x_2 \Rightarrow b^{x_1} < b^{x_2}$. Then consider the set $S = \{b^t: t \leq x \}$ and work from there?
Source: Chapter 1, Problem 7, Principles of Mathematical Analysis by Rudin
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I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2$, ... |
Here is the problem:
We have a square with some numbers from 1..N in some cells. It's needed to determine if it can be completed to a magic square.
Examples:
2 _ 6 2 7 6
_ 5 1 >>> 9 5 1
4 3 _ 4 3 8
7 _ _
9 _ _ >>> NO SOLUTION
8 _ _
Is this problem NP-complete? If yes, how can I prove it?
P.S.:
I kno... |
I am looking for a generalized normal distribution with mean $\mu$ and variance $\sigma^2$ but with an additional parameter for the kurtosis to add fat-tails. I can't find it - can anybody help?
|
Suppose I have an inscribed simplex which has $(n+1)$ vertices, and the diameter of the hypersphere is $d$. I have a point $x$ inside this simplex, is it true that the distance between $x$ and $x$ is nearest to vertex is not greater than $\frac{d}{2}$ ?
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Please help me in understanding the difference between this two question:
How many ways can $4$ men and $4$ women stand in a line if the women and men alternate ?
How many ways can $4$ men and $4$ women stand in a line if no two women are together ?
We employ a bit of different strategies for solving them and the answe... |
These are questions given for an assignment and I REALLY need to get a good grade so please could you guys check over them and give some pointers where I've gone wrong?
thanks.
ed: I hope it's ok to overwrite the original post:
$$\sum_0^\infty {\frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}}$$
\begin{align}
a_n &= \frac{(2x-8)^{n... |
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, by Leonard Mlodinow, Vintage Books, 2008. On p.107 Mlodinow says the chances are 1 in 3.
It seems obvious to me that the ... |
we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following thought:
Let $D = \{\text{all possible nonnull finite digit strings}\}$, $D' = \{\text{all things in D that do not start with "0"}\}$. Define a function $... |
I could use some help with proving the following irreducibility criterion. (It came up in class and got me interested.)
Let p be a prime. For an integer $n = p^k n_0$, where p doesn't divide $n_0$, set: $e_p(n) = k$. Let $f(x) = a_n x^n + \cdots + a_1 x + a_0$ be a polynomial with integer coefficients. If:
$e_p(a_n) =... |
How to prove that the group of order 28 which have normal subgroup of order 4 is Abelian group?
|
Given an alphabet of $k$ symbols, $\lbrace s_1,s_2, \ldots,s_k \rbrace$, how many words of length $n$, $w(n)$, can be generated knowing that,
Only two of the $k$ available symbols participate;
The symbol $s_i$ must appear exactly $n_i$ times and the symbol $s_j$ must appear exactly $n_j$ times and $n_i+n_j=n$ (the le... |
Consider the case where we have a bag of 'N' unique marbles, and we randomly select all 'S' possible sets of size 'm' from the bag (with replacement) under the following constraints:
(1) - Any set of 'm' marbles must have all unique elements, i.e. there must be no duplicate copies of a marble in any given set.
(2) - T... |
Crossposted on Mathoverflow.
Given a natural number $a$, are there infinitely many natural numbers not of the form $anm \pm m \pm n$, $n, m$ positive natural?
I give a proof that for $a=6$ the question is equivalent to the twin prime conjecture so it is known that we don't have any proof. But what about other values of... |
Suppose $z = a+bi$ and $w = u+iv$. Let $\displaystyle a = \left(\frac{|w|+u}{2} \right)^{1/2}$ and $\displaystyle b = \left(\frac{|w|-u}{2} \right)^{1/2}$. Show that $z^2 = w$ if $v \geq 0$ and $(\bar{z})^{2} = w$ if $v \leq 0$.
So $|w| = \sqrt{u^2+v^2}$. So this is just a matter of computing $(a+bi)^2$ and $(a-bi)^2$ ... |
I think it's an interesting coincidence that the locker puzzle and this puzzle about duplicate array entries (see problem 6b) have such similar solutions. Spoiler alert! Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we consider a col... |
What's the reasoning behind assigning a proposition to a set, like $A=p(x)$ and $B=q(x)$? Is it because any set of elements has some property that defines all elements in it?
$$
\begin{align*}
A \cup (B \cap C) &= \{x|p(x) \vee (q(x) \wedge r(x))\}= \\\
&= \{x| (p(x)\vee q(x)) \wedge (p(x) \vee r(x)) \} \\\
&= \{x|... |
Suppose $z$ is a complex number. Prove that there exists an $r \geq 0$ and a complex number $w$ (with $|w| = 1$) such that $z = rw$. Does $z$ uniquely determine $r$ and $w$?
Let $z = a+bi$. Then $|z| = \sqrt{a^2+b^2}$. So take $r = |z|$. It seems like one can take $w$ to be any complex number such that $|w| = 1$. So ... |
Are there any advantages or disadvantages in defining a real number in the following ways:
Definition 1
A real number is an object of the form $\lim\limits_{n \to \infty} a_n$ where $(a_n)_{n}^{\infty}$ is a cauchy sequence of rational numbers.
Definition 2 A real number is a cut in $\mathbb{Q}$.
|
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of $P$ is less than $\epsilon$.
This seems to imply that every Lebesgue measurable function on $\mathbb{R}$ is the pointwise li... |
When solving questions like these:
Let $f(x)$ be a real function. Find
$f(0.1)$ using its Taylor expansion
such that the error is less than
$10^{-3}$. Find the lowest degree of
Taylor polynomial needed.
(if someone can rephrase the question so its more clear that would be great. I didn't quite nail the transl... |
I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values:
Sample
Population
I didn't change it, but I kept getting the wrong results for the standard deviation. When I changed it to "Population" type,... |
IF $A$,$B$,$C$ are finite sets then,
Number of elements in exactly one of sets $A$,$B$,$C$:$$n(A)+n(B)+n(C)-2 \times n(A \cap B)-2 \times n(A \cap C)-2 \times n(C \cap B) + 3 \times n(A \cap B \cap C)$$
Number of elements in exactly two of the sets $A$,$B$,$C$: $$n(A \cap B)+(A \cap C)+n(C \cap B) - 3 \times n(A \cap B... |
I have two images. One is the reference image and the other is the rotated version of the reference image. I would like to find what is the angle by which the second image is rotated with respect to the first one.
i want this to work automatic.
|
What is the integral of a $k$-dimensional (multivariate) Gaussian over a convex $k$-polytope?
Here I am specifically interested in $k\in\{2,3\}$, but insight on the general problem would also be appreciated. I am guessing that there is no closed form in the general case, but what if we restrict the number of vertices... |
How many symmetric relations are there on an $n$-element set?
Thank you.
|
If $K_1$ and $K_2$ are subfields of a pre-chosen $\overline{\mathbb{Q}_p}$, and if they're both unramified at $p$, and $[K_1:\mathbb{Q}_p]=[K_2:\mathbb{Q}_p]$, does that imply that $K_1=K_2$?
My intuition says that this is true because all that's happening is that we're extending the residue field, and there's only one... |
During a test I was given the following question:
What is $$10+15+20+25+...+1490+1495+1500=?$$
After process of elimination and guessing, $ came up with an answer. Can anyone tell me a simple way to calculate this problem without having to try to actually add the numbers?
|
Taken from Rudin's Principles of Mathematical Analysis, 3rd edition (Chapter 1, Exercise 16):
Suppose $k\geq 3, x,y\in\mathbb{R}^k, |x-y|=d>0$, and $r>0$. Prove:
(a) If $2r > d$, there are infinitely many $z\in\mathbb{R}^k$ such that $|z-x|=|z-y|=r$
(b) If $2r=d$ there is exactly one such $z$.
(c) If $2r < d$, there is... |
Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous positio... |
how can be prove that $\max(f(n),g(n)) = \Theta(f(n)+g(n))$
though the big O case is simple since $\max(f(n),g(n)) \leq f(n)+g(n)$
edit : where $f(n)$ and $g(n)$ are asymptotically nonnegative functions.
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Let $G$ be a set of elements and $*$ a binary operation defined on the elements of $G$. Then $G$ is a group with respect to $*$ if the following holds:
$a*(b*c) = (a*b)*c$ for all $a,b,c \in G$.
There exists a left identity in $G$.
For each $a \in G$, there exists a left inverse.
We could replace "left" with "right."... |
Are there any good algorithms which can be used to construct a prime greater than $n$, for arbitrary $n$?
There are some brute force approaches: for example, factoring $n!+1$. However, I'm looking for something significantly more efficient; preferably, polynomial in $n$.
Also, if anyone knows of a proof showing a bound... |
I keep trying to solve it but I seem to be making an error somewhere:
$$4-\left|\frac{5y}{3}+4\right| > \frac25$$
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What values of $Z_1$ and $Z_2$ make the five regions of the unit circle, shown below, equal in area? $\overline{Z_1}$ and $\overline{Z_2}$ are conjugates of $Z_1$ and $Z_2$; in other words they lie directly across the real axis from their counterparts.
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I'm asking in the spirit of these two questions: can bitwise operations (AND, OR, XOR) be expressed in terms of other (more familiar?) functions?
I had been playing around with the bitwise operations in Mathematica a while back, and was at first struck by this identity for bitwise NOT: ~ n == -1 - n (I'll use C-ish not... |
Differential equations of the form $M\,dx+N\,dy=0$ such that $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ are said to be exact, because the left hand side of our equation is the exact differential of some function f, and the DE has a solution of the form $f(x,y)=c$, where c is a constant. Standard text... |
What is a "simpler" formula for
$$\sum_{3}^{n} \frac{(k-1)(k-2)(k-3)}{6}$$
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