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In my statistical signal modelling class we were given the following task:
For simplicity, let's assume $E(y) = E(x) = 0$. Show that if the real variables $x$ and $y$ are normally distributed, then the equation $E(y|x) = \operatorname{corr}(y,x) \cdot x/\operatorname{var}(x)$ holds.
We were also given a hint: write the... |
I've got a programming exercise I need to do, but I just can't figure out the math part.
I need to check if $3$ of $6$ lines intersect in the same point. I am given the equation $ax+by=c$, and I input random numbers in $a,b,c$. How would I go about solving this mathematically?
Thanks!
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While working on a proof showing that all functions limited to the domain of real numbers can be expressed as a sum of their odd and even components, I stumbled into a troublesome roadblock; namely, I had no clue how one divides the function into these even and odd parts.
Looking up a solution for the proof, I found th... |
How do I determine if a number is prime? I'm writing a program where a user inputs any integer and from that the program determines whether the number is prime, but how do I go about that?
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This question is taken from Dummit and Foote (14.9 #6). Any help will be appreciated:
Show that if $t$ is transcendental over $\mathbb{Q}$, then $\mathbb{Q}(t,\sqrt{t^3-t})$ is not a purely transcendental extension of $\mathbb{Q}$.
Here's what I've got so far:
Abbreviate $\sqrt{t^3-t}$ as $u$.
I've shown that the tra... |
I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in Calculus?
If you had any good methods of helping people that would be very helpful.
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I am having trouble answering the following question:
Problem
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
My Attempt
What I think they want m... |
An orthogonal matrix is a matrix $A$ over the reals such that $A^t=A^{-1}$ (its transpose is its inverse). The Frobenius norm over $n\times n$ real matrices is given by $\|A\| = \sqrt{trace(A^tA)}$.
I have come across the following claim: The distance (induced by the Frobenius norm) between any two (non equal) orthogon... |
In Direct limit, Martin rightly pointed out that my naive construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) here.
Since then, I've been lurking some of the literature about the subject and this problem of the colimit... |
a. For every real number $c$, neither $f$ nor $g$ is continuous at $c$.
b. For every real number $c$, the sum $f + g$ is continuous at $c$.
c. For every real number $c$, the product $fg$ is continuous at $c$.
Hint: consider functions defined one way on rationals and another on irrationals.
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The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their proofs are very much sophisticated.
I would like to consider two famous questions: First the "Fermat's Last Theorem" and next... |
I'm trying to find a percentage to describe how close a generated frequency distribution is to an ideal distribution, but am having a total blank. Basically I need to rate distributions
So say I have the below frequency distributions for a sample size of say 100:
A B
1 0
2 0
3 0
4 100
5 0
6 0
7 0
8 0
... |
Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group structure of G) is equal, in $\pi_1(G,e)$, to the loop $s_1*s_2$ where this product is given by first going around $s_1$ a... |
Suppose we have a $3\times 3\times 3$ cube of cubes (i. e. $27$ cubes) and consider the graph whose vertices are the cubes, and let an edge connect two iff they have touching faces. I'm quite sure there is no Hamilton path that starts at the middle cube and ends in a corner cube, though I'm not sure how to prove it wi... |
Consider the following problem:
You are given a multiset (a set with repetitions allowed) of $2n+1$ real numbers, say $S = \{r_1, \dots, r_{2n+1}\}$.
These numbers are such that for every $k$, the multiset $S - \{r_k\}$ can be split into two multisets of size $n$ each, such that the sum of numbers in one multiset is s... |
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and $d'$, in increasing order of strength:
$i$ is a homeomorphism, i.e. $d$ and $d'$ induce the same topology on $X$.
$i$ a... |
I have a series of terms $\{t_n : t_n = a_n x_n\}$, and I want to talk about the product of each term except $t_j$.
Would any of these be an appropriate way to say that? I like this:
$$\prod_{i \ne j} t_i$$
or maybe this
$$\prod_{i=1,i \ne j}^n t_i$$
Is it a normal thing to do, to put qualifiers on an index like that,... |
I have the standard logical equivalence:
$(p\rightarrow q)\wedge(q\rightarrow r)\Leftrightarrow p\rightarrow (q\wedge r)$.
Using several distributive laws I was able to get it down to:
$(\neg p\wedge\neg q) \vee (\neg p\wedge r) \vee (q\wedge r)$.
I must be missing some manipulation I can do to reduce this.
|
I have to prove that if $P$ is a $R$-module , $P$ is projective $\Leftrightarrow$ there is a family $\{x_i\}$ in $P$ and morphisms $f_i\colon P\to R$ such that for all $x\in P$
$$ x= \sum_{i\in I} f_i(x)x_i$$
where for each $x\in P$, $f_i(x)=0$ for almost all $i\in I$.
Any help?
|
I am reading the book Stochastic Simulation, Algorithms and Analysis by Asmussen. On page 39, when talking about inversion method for sampling a distribution, he gave an example:
an r.v. distributed as the overshoot X − a given X > a can be generated as $F^{-1} (U (1-F (a)) − a$,
where $U$ is a r.v. uniform in $[0,1]... |
A Blaschke product is a function of the form
$$B(z):=z^k\prod_{n=1}^{\infty}\frac{a_n-z}{1-\overline{a_n}z}\frac{|a_n|}{a_n}$$
where the $a_n$ are the non-zero zeros of $B$, and satisfie $\sum_{n=1}^{\infty}(1-|a_n|) < \infty$.
Blashke products are holomorphic and bounded by 1 on the unit disk. A well known theorem ass... |
The modular group is the group $G$ consisting of all linear fractional transformations $\phi$ of the form
$$\phi(z)=\frac{az+b}{cz+d}$$
where $a,b,c,d$ are integers and $ad-bc=1$. I have read that $G$ is generated by the transformations $\tau(z)=z+1$ and $\sigma(z)=-1/z$. Is there an easy way to prove this? In particul... |
So, I asked a question about how to find if three lines are concurrent. I built the algorithm I needed, and it was working well, until I started doubting my power of judgement. So my question:
Are two equal lines also intersecting?
For an example, $9x+9y=9$; $7x+7y+7$ ? Because now, if I would draw these lines and add... |
Let $\varphi (w)\Psi (x)$ be a solution to the heat problems. Given that $\Psi (-1)=\Psi (1)=0$
prove that it is not possible for $\Psi (x)$ and $\Psi ''(x)$ to be strictly positive for all $x$ such that
$−1 < x < 1$. You may use any facts, such as the mean value theorem.
So yes, this is homework. But we have been at... |
If the probability of an event is $\frac{1}{\infty}$ and $\infty$ trials are conducted, how many times will the event occur — $0$, $1$, or $\infty$?
|
I'm looking at the quantum group $SU_q(2)$ (over ${\mathbb C}$) and can't see why it has no zero divisors. It's clear that $M_q(2)$, the quantum $2 \times 2$ matrices have no zero divisors, but I can't seem to see why/if this extends to the quotient over $det_q - 1$. I think one can argue it from the grading on $M_q(2)... |
In Stochastic Simulation: Algorithms and Analysis by Søren Asmussen, on Page 38
A Poisson r.v. $N$ with rate $\lambda$ ($P(X = n) = e^{−\lambda} \frac{\lambda^n}{n!}$) can be constructed using the relation between the exponential distribution and the Poisson process: generate $X_1 , X_2 , ...$ as i.i.d. exponentials w... |
I'm looking for different proofs (references or sketch of main ideas) of the structure theorem for finitely generated modules over a PID.
If possible, a comparison in terms of clarity, elegance or usefulness would be appreciated.
|
I know that the product rule is generalised by Leibniz's general rule and the chain rule by Faà di Bruno's formula, but what about the quotient rule? Is there a generalisation for it analogous to these? Wikipedia mentions both Leibniz's general rule and Faà di Bruno's formula for the product and the chain rule, but ra... |
Let $A$ be a set of infinite cardinals. Assume that for every regular $\lambda$, the subset $A \cap \lambda$ of $\lambda$ is not stationary. Then I want to prove that there is an injective function $g$ on $A$ which satisfies $g(\alpha) < \alpha$ for all $\alpha \in A$.
This is an exercise in Kunen's set theory. I dont'... |
There is something in the definition of the free product of two groups that annoys me, and it's this "word" thing:
If $G$ and $H$ are groups, a word in $G$ and $H$ is a product of the form
$$
s_1 s_2 \dots s_m,
$$
where each $s_i$ is either an element of $G$ or an element of $H$.
So what is this "word" guy? Does... |
I apologise for this question, as it is no doubt very simple, but I've never been very confident with proofs. Our lecturer today (in a course related to maths but not mathematical itself) was playing with doing the modulus of powers, and used the above fact - $(x.y)\; mod\; m\; ==\; ((x\; mod\; m).y)\; mod\; m$ - to sh... |
Please suggest a suitable approach for this problem.
|
I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...):
Construct a Turing machine A which sequentially runs on all proofs in ZFC and checks if the claim "A does not halt or ZFC is inconsistent" is proved. If such a proof is found, A halts.
Note that A can kn... |
I understand that there are different types of infinity: one can (even intuitively) understand that the infinity of the reals is different from the infinity of the natural numbers. Or that the infinity of the even numbers is the same as that of the natural numbers. How many types of infinity are there? Or is this infin... |
Consider the following problem:
I got $g'(1) = f'(1) = 2$, which is correct according to the expected answer. However, I get that $a$ can be any value, as it doesn't affect anything and $g$ is differentiable no matter what $a$ is (am I missing something here?). The expected answer is that $a$ must be $2$ however.
Can... |
If we have a covering $p:H\rightarrow G$ where $G$ is a topological group, then $H$ is also a topological group. The multiplication function can be defined as follows. Consider the map $f:H\times H \rightarrow G$ which is a composition of the map $p\times p$ and the multiplication function on $G$. Choose $h\in p^{-1}(e... |
While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment.
What is a good way to think of those two terms?
|
Let $v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let $K$ be a symmetric rank-2 tensor on the 3-manifold. Then apparently the following holds:
$$\vert K_{ij} v^i v^j \vert \leq... |
How do I find a (longitude, latitude) point (any will do) on a circle where the only info I have is a (longitude, latitude) center point, and a radius measured in Feet (ft.)?
|
I'm trying to understand how the performance of financial investments are measuring when the asset has multiple points of investment.
For example, say I invest $1000 in a stock. After 6 months, it's worth $1200. At this point, I invest an additional $2000, for a total value of $3200. If after an additional 6 months, th... |
Suppose $x_{1}>0$, and consider the sequence, $\{x_{n}\}$ defined as follows: $$x_{n+1}=\log(1+x_{n}) \quad n\geq 1 $$ Find the value of $\displaystyle \lim_{n \to \infty} nx_{n}$
I am having trouble solving it. One thing is clear, that since $x_{n}>0$ and $x_{n+1} < x_{n}$, we can have a sequence which converges an $f... |
I would like to recover the probability mass function (pmf) from the characteristic function (CF) of a discrete probability distribution using Mathematica.
Ideally, I'd like to do calculations like this example (not necessarily this simple). To compute the pmf fX+X[x] of the sum of two iid discrete uniform distributio... |
The derivative of a function $f: C \rightarrow C$ with respect to $\overline{z}$ is defined as
$$ \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) $$
where $\frac{\partial}{\partial x}$ is defined as the operator which maps $u+iv$ to $u_x + iv_x$, and $\frac{\partial}{\partial y}$ ... |
In constrained Lagrangian optimization what is a general way to figure out how the optimal point varies with respect to parameters of the constraints? For example, maximize $x\cdot y$ when $x + y \lt k$, and I wanted to find $\large\frac{dx^{*}}{dk}$ and $\large\frac{dy^{*}}{dk}$ where $(x^{*},y^{*})$ is the optimal po... |
I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. My $\mathrm{sqrt}(x)$ can pick up where it left off, even if $x$ changes a bit, because it uses Newton-Raphson. But $\exp(x)$ computed using a Maclau... |
Without strenuous arithmetic.
Is there a program I can download to do so?
What are the quadratic residues modulo $5^4$ or $5^5$?
Thanks!
|
I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way:
They prove transitivity using induction.
They prove irreflexivity using induction and transitivity.
They prove asymmetry by contradic... |
This question stumped me on a recent calculus test... I would really like an explanation.
The region $\mathbf{R}$ is bounded by
\begin{equation*}
h(x) = -0.8(x-2)^2 + 8~\text{and}~p(x) = x\cos(x) - \ln(2x+2).
\end{equation*}
Find the lateral surface area of the solid formed when $\mathbf{R}$ is revolved around the li... |
Say I have a function
\begin{equation*}
f(x) = ax^3 + bx^2 + cx + d,\text{ where }a > 0.
\end{equation*}
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $k, x_0 > 0$... |
The well known formula for perfect numbers is
$$
P_n=2^{n-1}(2^{n}-1).
$$
This formula is obtained by observing some patterns on the sum of the perfect number's divisors. Take for example $496$:
$$
496=1+2+4+8+16+31+62+124+248
$$
one can see that the first pattern is a sequence of powers of $2$ that stops at $16$, the... |
I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with.
I have some basic questions that someone might be able to help with:
Is the composition of two closed geodesics itself a closed geodesic?
Is composition of ge... |
Can you advise some good books covering areas which are required for serious probability theory studies (e.g. measure theory, functional analysis)? Preferably this book should have some problem sets to work on. Thanks!
|
A problem that is an example of this is $x^2+16$. I have to know how to factor this over the set of complex numbers. How do you do that? I used to know it's just been a long time.
|
I'm stuck trying to solve an inequality of the form $\left|\frac{f(x)}{g(x)}\right| \geq 1$; specifically, $\left|\frac{2x + 5}{x + 1}\right| \geq 1$
I tried the approach that was taught for solving inequalities in general, and solved first for x when the part enclosed in absolute value signs is $\geq 1$, and then sol... |
How do we go from
\begin{equation*}
\frac{1}{1+(\frac{x}{2})+(\frac{x^2}{3})+(\frac{x^3}{4})+\dots}
\end{equation*}
to
\begin{equation*}
1-\left(\frac{x}{2}\right)-\left(\frac{x^2}{12}\right)-\dots?
\end{equation*}
If I could consolidate the series in denominator in some form, then I could use binomial expansion in ... |
I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name? Does it begin with e?
|
I was told to write the $3$ cube roots of $27$ in rectangular form how is this to be done? I know how to do cube roots on something like $5+0i$ in polar form, but how do you do it with just a number?
|
First of all, I had no idea what to title this question, so feel free to change it to something more appropriate.
I have a set defined as such:
$\{a\frown b\mid a \in A \wedge b \in \{c\frown d\mid c \in C \wedge d \in D\}\}$
Is that logically equivalent to:
$\{a\frown (c\frown d)\mid a \in A \wedge c \in C \wedge d \i... |
I have attempted to google for this, but the searching is marred by the close relationship of Jordan Algebras and Quantum Mechanics. I have been passively thinking of this question for ages, tonight I finally post it. :)
My question is fairly simple. Similar to the universal enveloping algebra for Lie algebras, we shou... |
Or, in other words, why aren't all functions surjective? Isn't any subset of the codomain which isn't part of the image rather arbitrary?
|
I am looking at Perturbation Theory by E.J. Hinch. The author introduces Van Dyke's matching rule:
(m term inner)(n term outer) = (n term outer)(m term inner)
To be used to match the terms in an inner solution to the the terms in the outer solution for some perturbed differential equation.
Hinch when comparing with in... |
Could someone please help me understand this proof given in an article by William Miller. It is supposed to follow from the prime number theorem that given $A(x)$ the sum of all primes less than or equal to $x$ and $\theta(x)$ the sum of the logarithm of all primes less than or equal to $x$,
$$A(x)\sim \frac{x^2}{2\l... |
How can one build a $3\times 3$ linear equation system ($3$ equations with $3$ variables) if the variables are known, for example, $a=1, b=2$ and $c=3$?
Thank you in advance!
|
Can we represent all algebraic structures in First-Order logic?
|
In relation to this geometric construction:
where D is the center of both circles, if the inner radius (x = length of line segments DA and DE), the angle φ = ∠CAB, and the length Δg of line segment AB are given, what is the length Δx of line segment AC?
|
If you look at this previous problem, I asked how to find the sum of all the areas between two taxicab geometry circles. However, upon learning about $\ell^p$ norms, I thought it would be pretty interesting to extend the problem to all $\ell^p$ norm circles, not just $\ell^1$ (taxicab).
If $p=1$, then the result has al... |
I was maths student but now I'm a software engineer and almost all my knowledge about maths formulas is perished.
One of my client wants to calculate optimal price for perishable products. He has come across an article named Optimal Pricing Policies for Perishable Products by Miguel F. Anjos, Russell C.H. Cheng, and Ch... |
I have a series of problems that involve events that happen one after another, and where the usual formulas like $P(\frac{A}{B})=\frac{P(AB)}{P(A)}$ don't seem to help.
Example:
In a horse race there are $3$ horses. $E12$
means horse $1$ finishes before horse $2$.
$E123$ means horse $1$ finishes before $2$,
whic... |
What are some examples of cutting-edge research involving fractals or self-similar structures?
Who's actively contributing high-quality research in this field?
|
It's said on Wikipedia page about Post's lattice that DM clone(all monotone self-dual functions) has majority function $(xy + xz + yz)$ as one of its bases. What is the proof of this fact?
|
This is mostly related to doing large modular exponentiation by hand. For example, a problem I was doing was to find the last 3 digits of $7^{9729}$; that is, find $7^{9729}\bmod{1000}$.
Using the simplest concept for Euler's theorem, I found that $7^{400}\equiv 1\pmod{1000}$, since $\varphi(1000)=400$. Using Carmichae... |
Let random variable $X\sim\text{Binomial}\left(a+b,\frac{a}{a+b}\right)$, where $a$ and $b$ are positive integers.
I'm trying to prove that $\mathbb{P}[X\geq\mathbb{E}[X]]\leq\frac{3}{4}$, which appears to be true numerically.
Does anyone have a suggestion on how to proceed?
The complication in this particular problem ... |
This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering $\mathbb{CP}^1 = \mathbb{C}\cup {\infty}$. I'll want to prove that this is an isometry, where $S^2$ has the standard angl... |
This is one of the question asked in a written test conducted by a company. The question sounded stupid to me. May be its not.
"Given the area of the coin to be 'A'. If the probability of getting a tail, head and the edge are same, what is the thickness of the coin?
|
Generally one can see that there are infinite number of solutions for this equation $$a^{2}+b^{2}=c^{2}$$ by taking multiples of the solution $3,4$ and $5$.
Can i use this as a fact to prove, that $k^{2} + (k+1)^{2}$ is a perfect square for infinitely many $k \in \mathbb{N}$? Any hints, suggestions would be helpful. I... |
Again I have trouble with some exercises in Kunen's set theory.
In the following, let $\kappa > \omega$ a cardinal. Then I want to show that
1) $|H(\kappa)| = 2^{<\kappa}$
2) $H(\kappa)=R(\kappa)$ if and only if $\kappa = \beth_{\kappa}$
In 1) the inequality $\geq$ is easy, but I cannot prove $\leq$. In 2) I have no id... |
If $\displaystyle \frac{a^n + b^n}{a^{n-1} + b^{n-1}}$ , ( $a \neq b$ and $a,b,n \in \mathbb{R}$) is the Arithmetic mean or Geometric mean between $a$ and $b$ then find the values of $n$ for both means respectively.
|
I want to show that if $F:\mathbb{R^n}\to\mathbb{R^m}$ has a derivative of rank = r at a point $p$, then there is a neighbourhood of that point where the rank of the function doesn't decrease, i.e. $\exists N=B(p, \delta)$ such that $rk(DF_x) \geq rk(DF_p) ~ \forall x\in N$
|
Find the sum to n terms of the series $\frac{1} {1\cdot2\cdot3\cdot4} + \frac{1} {2\cdot3\cdot4\cdot5} + \frac{1} {3\cdot4\cdot5\cdot6}\ldots $
Please suggest an approach for this task.
|
I'm looking for some FOSS/GPL programs (or Python libraries) for dealing with sparse matrices. I haven't found much online about these. Can someone please point me in the right direction?
|
I am reading a Set Theory book by Kunen. He presents first-order logic and claims that if a set of sentences in inconsistent, then it proves every possible sentence. Since he does not explicitly specify the inference rules, I became curious as to how fundamental this property of inconsistent systems is.
So my question ... |
Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continued fraction is given (see 'generalized contin... |
Let $d(n)$ represent the divisor function as
$d(n)=\displaystyle\sum\limits_{k|n}1$
and the divisor summatory function as
$D(x)=\displaystyle\sum\limits_{n \leq x}d(n)$
I found the following triangular representation for the values of $D(n)$
$$
\begin{array}{ccccccccc}
D(1)=&&&&&&&&& 1 &&&&&&&&&&=1\\
&\\
D(2)=&&&&&&&&... |
According A180926, the elements of the set
{$a:\exists m,n|60a=5n^2+5n=3m^2+3m$} satisfy the following recurrence relation:
$$a_{n}=\frac{62a_{n-1}+1+\sqrt{(48a_{n-1}+1)(80a_{n-1}+1)}}{2}$$
$$a_1=0$$
How this can be derived?
|
In blind experiments subjects are randomly assigned to one of groups. The most commonly used solution is to use (equivalent of) a coin toss, with the same probability to be assigned to each group.
I have read about two modifications of a procedure of random assignment in blind tests:
fair / just coin, where probabili... |
(Wade's rewording)
There is a phrase that contains $4$ words and $22$ total letters (not counting the three spaces between the words). One letter appears four times, no letter appears three times, three letters appear twice, and $12$ letters appear once each. What is this phrase?
@@$$##&&&%%%%????????????
There are $... |
Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the manifold boundary?
By "topological boundary," I'm referring to $\text{Bd } M$, which is the closure minus the interior ... |
The title and tags are a little goofy since I'm not quite sure what to call this. This may be more of a computer science question, but I thought I'd ask here too.
I have two integers $n$ and $m$ that are arbitrarily (logically) related with integers $x$ and $y$, respectively, where all values are finite and $> 0.$
I'd... |
I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?
Moreover, is there a theorem which classifies the 2-manifolds with boundary?
|
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless
Imagine there are a 100 people in line to board a plane that seats 100. The first person in line, Alice, realizes she lost her boarding pass, so when she boards s... |
I'm reading Jech's Set Theory, and in the chapter about measurable cardinals there is a theorem that if $\kappa$ is real-measurable but not measurable then it is $\le 2^{\aleph_0}$ and so and so. (Corollary 10.10)
How can a cardinal number be real-measurable without being measurable? Can't a measure be "destructed" (as... |
Can you help me realize why this is true? I can tell you that $P_i$ and $P_j$ are probabilities, i.e. $0 \leq P_i, P_j \leq 1$.
$\displaystyle \sum_{i=1}^\infty \sum_{j=1}^\infty ijP_iP_j \leq \sum_{i=1}^\infty \sum_{j=1}^\infty j^2P_jP_i$.
|
Suppose one wants to prove that a $1$-dimensional integral proper scheme over an algebraically closed field is projective. This is a step in how Hartshorne has you prove that any $1$-dimensional proper scheme (over an algebraically closed field) is projective.
The method is that first note the normalization is non-sing... |
I'm trying to figure out the limit of the sequence
\begin{equation*}
\lbrace\sum_{n=1}^{k}\frac{1}{\sqrt{k^2 + n}}\rbrace_{k=1}^\infty
\end{equation*}
I feel like I'm close to understanding it, I just need a little push.
I understand that I need to show that this sequence lies between $\lbrace \frac{k}{\sqrt{k^2 + k}}... |
Let $N$ be a normal subgroup of a finite group $G$. Let $S$ belonging to $G$ be a conjugacy class of elements in $G$, and assume that $S$ belongs to $N$. Prove that $S$ is a union of $n$ conjugacy classes in $N$, all having the same cardinality, where $n$ equals the index $[G : NCent(x)]$ of the group generated by $N$ ... |
If $S_1$, $S_2$ and $S$ are the sums of $n$ terms, $2n$ terms and to infinity of a G.P. Then, find the value of $S_1(S_1-S)$.
PS: Nothing is given about the common ratio.
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Recently, I came across this problem in my textbook. Any hints on how to solve this using dynamic programming?
"Professor Stewart is consulting for the president of a corporation that is planning a company party. The company has a hierarchical structure; that is, the supervisor relation forms a tree rooted at the presi... |
for a finite group $G$ and a trivial abelian $G$ module $A,$ there is the short exact sequence
$0 \rightarrow \mathrm{Ext}^1 (G_{ab},A) \rightarrow H^2(G,A) \rightarrow \mathrm{Hom} (H_2 (G,Z), A) \rightarrow 0$
I'm looking for a description for the connecting maps. Specially, I want to use the representation of $H_2(G... |
Greetings, dear Mathematicians!
How to implement base-9 to base-3 conversion in O(log(n)) on 4-tuple ((stateFrom, encounterWhat, Command, StateTo), where Command can be < (turn left), > (turn right), # (stop execution), or symbol from an alphabet) Turing machine, finite from the left side and infinite from the right si... |
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