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If $p:\tilde{X}\rightarrow X$ is a regular covering space of finite degree, why is it not obvious that if two curves $\gamma$ and $\delta$ are isotopic in $\tilde{X}$ their images are isotopic in $X$? By my understanding, this is a nontrivial theorem (including the statement that the cover is possibly branched) in a p... |
Let $1 < k \in \mathbb N$ and $M = \{(z_1, z_2) \in \mathbb C^2 : k|z_1|^2 + |z_2|^2 = 1\}$. Let $S^1$ act on $M$ via $e^{i\theta}(z_1,z_2) = (e^{ik\theta} z_1, e^{i\theta} z_2)$. Then I am told that $M/S^1$ is not a manifold (observe that the action is not free as $e^{2\pi i/k}$ stabilizes the points of the form $(z... |
Let $a$, $b$ an element of $\mathbb{Z}$ with $a$ and $b$ not both zero and let $c$ be a nonzero integer. Prove that $$(ca,cb) = |c|(a,b)$$
|
I'm reading Matsumura's proof on independence of valuation (p.87, "Commutative Ring Theory"), and I understand most of the proof, except the last bit about principal ideal ring, and if anyone could help me I would greatly appreciate it:
$K$ a field, $R_1$,...,$R_n\subset K$ DVRs of $K$ not containing one another, $A=... |
Let $a_1,\ldots,a_n$ be a set of $n$ positive numbers.
Are there known lower and upper bounds on:
$\displaystyle\frac{\prod_{i} \Gamma(a_i)}{\Gamma(\sum_i a_i)}$
where $\Gamma$ is the Gamma function (a generalization of the factorial distribution)?
|
In the introduction of Hungerford's Algebra (p. 2), he gives a rather trivial example of a class that is not a set, but what is the purpose of even having this term defined? Is it useful, other than to give a name to collections of objects that are not sets?
Also, how is this term related to equivalence and congruence ... |
Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? Obviously it has one root at $z=0$. The other root is the number $W_0(1)$, where $W_0(1)$ is the principal branch of the pr... |
Problem statement:
Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one-to-one correspondence between A and C.
My thoughts:
There's a bijection between A and A (the identity function). There's a bijection between C and C (the identity function). There'... |
I ha come across a book on groups as Galois groups, and in the introduction it mentions the paper by I.R. Shafarevich which says that every solvable group can be realized as Galois groups of some Galois extensions over $\mathbb{Q}$. It, however, does not tell us where to find that paper; therefore, I would like to know... |
This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166.
And, after solving several previous exercises, I found this to be particularly difficult to solve. I am hoping only for hints, not a complete solution. And any hint is appreciated.
Also, I have made a conjecture that the numbe... |
I've been stumbling across a couple of questions in Ahlfors that fall along the lines of "If $f$ is an analytic function with property $X$, show that $f$ reduces to a polynomial."
One concrete example would be:
Show that a function which is analytic in the whole plane and has a nonessential singularity at $\infty$ red... |
Here the equivalent means:in the bundle ,B is bottom manifold(also called base space),S is section,we can also treat S as a bottom and B as a section in the same bundle.
Is it true in the vector bundle? and also in the fiber bundle?
|
Let $R$ be a commutative ring with identity and $M$ be an $R$-module. I have trouble understanding the restriction map in the definition of the sheaf of $\mathcal{O}_X$ modules. Explicitly, let $f,g\in R$ s.t. $D(g)\subseteq D(f)$, then what is the map $M_f\to M_g$. My main difficulty is understanding what kind of modu... |
Let $a_1, \ldots a_n$ be i.i.d random variables that take the value $1$ and $-1$ with equal probability. Fix a unit vector $(x_1,\ldots, x_n)$ (such that $\sum_j x_j^2 = 1 $).
Consider $v = \langle a, x\rangle$. It's easy to see that $E[v] = 0$. It's also easy to bound Pr($|v| > t$) via standard Hoeffding bounds.
Wha... |
I am not sure whether or not this is a duplicate question. I'm wondering what is an efficient way to compute
$$x \equiv a^b \bmod{n}$$
where $a,b,n \in \mathbb{Z}$ and $a,b < n$?
For example say I need to compute $13^{59} \bmod{77}$, how do I do it manually?
|
Back in college I tried to find characterizations of compact (metric) spaces that make use of Lebesgue's Covering Lemma. During the course of this, I defined the following (pretty arbitrarily named) property of subsets of topological spaces:
If $X=(X,\mathcal{O})$ is a topological space and $Y\subseteq X$, we say that... |
Hey guys,
I have this ex in Data structure course.
This ex is about big o notation, and as far I remember It means that $f_1$ and $f_2$ bound asymptotically $g_1, g_2,$ but i'm not quite sure.
The question:
$f_1(n)=O(g_1(n)), f_2(n)=O(g_2(n))$
Two functions are from and to the natural numbers.
I need to prove:
$f_1(n)... |
The following statement is an exercise in point set topology: If $E \to X$ is a covering with nonempty finite fibers and $X$ is compact, then also $E$ is compact. Now Grothendieck generalized covering theory so that in particular separable field extensions may be regarded as coverings.
Question: What is the correspondi... |
I've seen this written several places without proof, so I assume it's not difficult, but I am not getting it.
Let $\mathbb P$ be a $\kappa$-c.c. notion of forcing, and let $C\in M[G]$ be club in $\kappa$. I want to show there exists $D\in M$ such that $D\subset C$ is club in $\kappa$. Kunen suggests: let $f\in M[G],... |
How does one show that the dot product of two vectors is $ A \cdot B = |A| |B| \cos (\Theta) $?
|
I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need?
|
I have a question. Could you please help me to solve this problem?
Is it possible that $\mathbb{R}^2\setminus A$ and $\mathbb{R}^2\setminus B$ are homeomorphic, when $A$ and $B$ are non-homeomorphic closed subsets of $\mathbb{R}^2$?
|
is there any rule to differentiate the function $(a\,x)^{b\,x}$?
I've got to find the derivative of $(x^2+1)^{\arctan x}$ and Wolfram|Alpha says the answer is
$$\tan^{-1}(x) (x^2+1)^{\tan^{-1}(x)-1} \left(\frac{d}{dx}(x^2+1)\right)+\log(x^2+1) (x^2+1)^{\tan^{-1}(x)} \left(\frac{d}{dx}(\tan^{-1}(x))\right)$$
Is there ... |
I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid.
We want to model the lifetime $T$ of a website using the density function $Pr[X > t] \approx e^{-\rho t}$. Then, according to the math I am reading,
$\begin{eqnarray}
... |
LATER EDIT:
I managed to find the errors in my equations below. It was a sign mistake on one of the terms (- instead of +). Having corrected that, the method which I describe below WORKS, but DON'T USE IT ::- D. My method is convoluted and prone to mistakes (as I proved, haha). The 2 answers below offer much better cho... |
Given $a,b>0$ such that $a\ne b$ but nothing else is given (e.g. $a,b$ are not known to be coprime), I want to understand the structure of the set of solutions to the following system of two linear equations:
$a(x+y)+b(z+w)=0$
$a(z-w)+b(x-y)=0$
How can I approach such a problem? Are additional assumptions on $a,b$ help... |
Given a theory (say $T$) which satisfies the hypotheses of the Godel- Rosser theorem. How to show whether a Rosser sentence $R$ for $T$ is true for the standard interpretation ($\Bbb N$) or not?.
(I know it is true but how we can prove this claim!)
|
Where a group can consist of 1 or more items, groups don't have to be equally sized and items can be duplicates. Example - Choose 3 groups:
Items: 1 2 2 3
Groups:
(1) (2 2) (3)
(1 2) (2) (3)
(3 1) (2) (2)
(3 2) (2) (1)
Update
Following Rasmus' comment, I see that this is more technically described as finding the possi... |
let f be defined on [0,2$\pi$)
denote $f_x(y)=f(y-x)$
how do i go about showing $||f_x||_{L_p}=||f||_{L_p}$ for all p
i was trying to write out f in fourier series, and use the fact that the fourier coefficient of $f_x$ is $e^{inx}$ times the fourier coefficient of f. but i couldnt get anywhere.
|
A scheme to generate random variates distributed uniformly in $S^2=\{x\in \mathbb{R}^n \mid \|x\|_2=1\}$ is well known: generate a standard normal variate in $\mathbb{R}^n$ and normalize it to unit norm.
Is there a similarly simple and clever procedure to simulate uniformly distributed variates on the $1$-ball $S^1=\{... |
This is a part from a homework. I solved all examples apart from this one. So the task is: We know that $3$ divides $a^2 + b^2$. Prove that $3$ divides $a$ and $3$ divides $b$. I cannot think of anything useful. I know that $a^2 + b^2 = (a + b)^2 - 2ab$, but I don't see how it can help me :(
Best regards,
Petar
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Let f(x) and g(x) be two functions
defined on some subset of the real
numbers. There are two definitions
for $f \in o(g)\mbox{ as
}x\to\infty\,$, according to
Wikipedia:
$$\lim_{x \to \infty}\frac{f(x)}{g(x)}=0.$$
for every $M > 0$, there exists a
constant $x_0$, such that $$|f(x)| \le \; M |g(x)|\mbox{ for a... |
Can every function $\psi ~\colon \mathbb{F_{p}^n} \to \mathbb{F_{p}}$ be regarded as a polynomial function for some polynomial in $\mathbb{F_{p}[x_1, \ldots,x_n]}$?
I believe this is true, but am having trouble proving it.
|
I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers.
I've recently started self-studying subgroup series, which is supposed to explain the analogy, but I'm not completely sure I understand how "a... |
I am trying to understand Apery's 1978 proof that $\zeta(3) = \displaystyle \sum_{n=1}^\infty \frac{1}{n^3}$ is irrational. The idea behind the proof is to find an 'accelerated' series for $\zeta(3)$ which converges too fast to $\zeta(3)$, thus proving that $\zeta(3)$ cannot be rational. In particular, a particular qua... |
Given the differential equation:
$$\dot{x} = y-x^2, \;\; \dot{y} = -x+y^2$$
I have to find the solutions of this differential equation which move/lie on a line. I am not quite sure how to handle this problem, I started by writing $y = mx + q$, so:
$$\dot{y} = m \dot{x} = m (y-x^2) = -x+y^2$$
Solving this equation, I ev... |
Let $X$ be a topological space.
Call $x,y\in X$ swappable if there is a homeomorphism $\phi\colon X\to X$ with $\phi(x)=y$. This defines an equivalence relation on $X$.
One might call $X$ homogeneous if all pairs of points in $X$ are swappable.
Then, for instance, topological groups are homogeneous, as well as discrete... |
This question is motivated by a step in the proof given here.
$\begin{align*}
8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\
&=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\
&\geq (n+1)^8\left(8\left(\frac{9}{10}\right)^8\right)\\
&\gt (n+1)^8 .
\end{align*}$
I had no trouble following along with the proof until I hit the s... |
Given the set of functions defined
on a subset in $\mathbb{R}$ and
taking values in $\mathbb{R}$, I was
wondering if $O$ as in $f \in O(g)$
is a total partial order? By total, I would like to know if every two functions can be compared by $O$? Is it
acting like $\geq$ on $\mathbb{R}$?
How about $\Omega$, $\Theta$, $o$... |
Possible Duplicate:
Prove $0! = 1$ from first principles
Why does $0! = 1$?
All I know of factorial is that $x!$ is equal to the product of all the numbers that come before it. The product of 0 and anything is $0$, and seems like it would be reasonable to assume that $0! = 0$. I'm perplexed as to why I have to accou... |
Show that all Eisenstein series $G_k$ can be expressed as polynomials in $G_4$ and $G_6$, e.g. express $G_8$ and $G_{10}$ in this way.
Hint: Setting $a_n := (2n+1)G_{2n+2}$, show that
$2n(2n-1)a_n = 6(2a_n + \sum\limits_{k=1}^{n-2} a_{k}a_{n-1-k})$ , $n>2$
Obviously, what I have trouble proving, is the Hint. It s... |
I've looked all over and I can't find a good proof of why the diagonals of a rhombus should intersect at right angles. I can intuitively see its true, just by drawing rhombuses, but I'm trying to prove that the slopes of the diagonals are negative reciprocals and its not working out.
I'm defining my rhombus as follows:... |
Given the sequence $\{ f(i) \}_{0 \leq i \leq n}$ and the array $\{ g_{n}(i,j) \}_{0 \leq i, j \leq n}$ such that $f(i) = \sum_{j = 0}^{n} h_{j} \, g_{n}(i,j)$ for some integers $h_{0}, \dots, h_{n}$, is there a way to write $h_{j}$ as a sum over $f(i)$ and $g_{n}(i,j)$ using the calculus of finite differences?
Thanks.... |
Let $k\in \mathbb Z$; is there a name for the function $f(k)$ below?
$$
f(k) = \text{max}(k, 0)
$$
|
I'm working my way though a classical geometry book by Hartshorne right now, but this problem popped up in a section I'm reading. It is Problem 13.10 from Hartshorne's Geometry: Euclid and Beyond if you're curious.
Anyway, the problem states:
If $a,b\in\mathbb{Z}$, and if $a+b\sqrt{2}$ has a square root in $\mathbb{Q}... |
Shannon formally defined the amount of information in a message as a function of the probability of the occurrence of each possible message[1]. Given a universe of messages $\mathbf{M} = \{ m_1,m_2,..,m_n \}$ and a probability $\mathbf{p(m_i)}$
for the occurrence of each message, the information content of a
message i... |
Here is a rather tough integration. I think it looks like an Elliptic Integral of some sort.
$$\int_{1}^{\infty}\frac{1}{\sqrt{3x^{4}+6x^{2}-1}}dx$$
Since there are no odd terms in the quartic, I thought maybe completing the square
would be OK. But I got nowhere. I even factored it into:
$3x^{4}+6x^2-1=((2\sqrt{3}-3... |
I need to integrate this expression:
$$\int^{\pi}_0\frac{\gamma^{10} \theta^2 \sin\theta}{(\gamma^2 \theta^2 + 1)^5} d\theta.$$
I can use the fact that gamma is very large, which I think means I should rearrange the bottom line to allow an expansion, i.e.
$$\int^{\pi}_0\frac{\theta^2 \sin\theta}{(\theta^2 + \dfrac{1}{... |
I'm trying to prove that
$$ \frac{\cos(A)}{1-\tan(A)} + \frac{\sin(A)}{1-\cot(A)} = \sin(A) + \cos(A)$$
Can someone help me to get started? I've done other proofs but this one has me stumped! Just a start - I don't need the whole proof. Thanks.
|
In my Trig class we have begun working on graphing the trig functions and working with radians and I'm trying to wrap my head around them.
At the moment I'm having trouble understanding radian measures and how to find where certain points lie on the unit circle and how to know what quadrant they are in.
For example, we... |
Given two points $A =(x_1,y_1)$, $B = (x_2,y_2)$ and length $L$, how do I plot a parabolic segment of length $L$ that connects A and B? The vertex of the parabola $(p, \, q)$ should be such that $x_1 \leq p \leq x_2$ and $q \leq y_1, y_2$.
In other words, I need to draw a parabolic segment connecting $A$ and $B$ with ... |
What is the general way of finding the basis for intersection of two vector spaces in $\mathbb{R}^n$?
Suppose I'm given the bases of two vector spaces U and W:
$$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), \left(0,1,3,1\right) \right\} $$
$$ \mathrm{Base}(W) =\left\{ \left(0,-1,-2,1\right), \left(1,2,2,-2\right) ... |
The original function is $y^2 = kx^3$, and I'm being asked to find the orthogonal trajectory.
$$ 2y\frac{dy}{dx} = 3kx^2 $$
$$ \frac{dy}{dx} = \frac{3kx^2}{2y}$$
$$ \text{New }\frac{dy}{dx} = \frac{2y}{3kx^2} $$
$$ \frac{1}{2y}\,dy = \frac{1}{3kx^2}\,dx$$
$$ \frac{\ln|y|}{2} = \frac{-1}{3kx} + c$$
$$ \ln|y| = \frac{-2}... |
I am reading methods of solving recurrence relation on Wikipedia. There is one method:
Many linear homogeneous recurrence
relations may be solved by means of
the generalized hypergeometric series.
Special cases of these lead to
recurrence relations for the
orthogonal polynomials, and many
special functions... |
In Linear equations, linear ODEs and linear recurrence relations, when solving homogeneous equations, there are a subspace of solutions, and when solving inhomogeneous equations, a particular inhomogeneous solution plus the subspace of homogeneous solutions will give the set of inhomogeneous solutions. I was wondering ... |
For example:
$$7x \equiv 1 \pmod{31} $$
In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do?
Update
If I have a general modulo equation:
$$5x + 1 \equiv 2 \pmod{6}$$
What is the fastest way to solve it? My initial thought was:
$$5x ... |
I thought I had this question down, but while looking over my solution, I think I'm missing a step.
I want to show for $f_n$ the nth fibonacci number, that $f_n$ is divisible by $4$ if and only if $n$ is divisible by $6$.
I approach this with strong induction. I was able to prove the backward implication by assuming th... |
A signal which is a function of a set of periodic signals with periods $p_1,\dotsc, p_n$ will have a period $P$ which is the least common multiple of all $p_i$. (If $p_i$ are relatively prime, which is the only relevant case here, then $P = \prod_i p_i$.)
Assume that each $p_i$ must be an integer and greater than some ... |
Does any one know of a particular textbook or reference that proves existence and uniquence of the ODE $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}=f(x,y)$?
Edit:
Consider the initial value problem:
$\frac{dy}{dx}=f(x,y)$, $y(x_0)=y_0$ (E)
Assume $f:D\to\mathbb{R}$ is a ... |
If $p$ is a prime integer, then $\mathbb{Z}[i]/(p)$ is isomorphic to $\mathbb{Z}_p/(x^2 + 1)$
I suspect that there is a typo, and that the $\mathbb{Z}$ on the r.h.s. should be $\mathbb{Z}[x]$ (polynomials).
Either way, I need help getting started and understanding the notation.
$\mathbb{Z}[i]$ is the Gaussian integer... |
So the question asks, given that we have a undirected graph with unique edge weights, prove that the graph has a unique minimum spanning tree.
My Proof:
If the graph has unique edge weights, we can list the edges as
$$E_1,E_2,\ldots, E_N, \text{ s.t. } E_1 < E_2 < E_3 < \cdots < E_N.$$
Then applying Kruskal's Algori... |
Using two forms of provability:
Identity Elimination/Transitivity
AnaCon: Analytical Consequence
Below, "Larger(x,y)" means "x is larger than y", "Smaller(x,y)" means "x is smaller than y", and "SameSize(x,y)" means "x is the same size as y".
What would be the best way (makes more logical sense) to prove the followin... |
In Wikipedia about difference equations, there is some description about correspondence between ODE and difference equation:
If you consider the Taylor series of
the solution to a linear differential
equation:$$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n$$ you
see that the coefficients of the
series are gi... |
I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared
$$ \pi = -i \log(-1)$$
which really bothered me. Multiplying on both sides by $2i$:
$$ 2\pi i = 2 \log(-1) = \log(-1)^2 = \log 1= 0$$
which is clearly false. I guess my error is $\log 1 = 0$ when $\log$ is complex-valued. I need to u... |
Suppose that $A=\mathbb{R}[x]$ is the polynomial ring of one variable over the real numbers and $B$ is the real vector space which is the set $B = \{nx\mid n\text{ is a real number}\}$.
Show that $A\otimes_{\mathbb{R}} B \otimes_{\mathbb{R}} A$ is a free $A \otimes_{\mathbb{R}} A$-module.
|
Calls to a customer service center
last on average $2.7$ minutes with a
standard deviation of $1.4$ minutes. An
operator in the call center is
required to answer $74$ calls each day.
Assume the call times are independent.
What is the standard deviation of the
total amount of time in minutes the
operator ... |
Say I have a complex polynomial:
$$a_0+a_1x+\cdots+a_nx^n,$$
where $a_0,\ldots,a_n$ are complex numbers.
What is the conjugate of this polynomial? How is it defined?
For example, if we have an inner product on the vector space $V$ defined by (for complex polynomials):
$$\int_0^1 p(x)\overline{q(x)}dx$$
Then how do... |
Let p be prime. For $n|p-1$, let $f_n(x)=(x-a_1)(x-a_2)\cdots(x-a_s)$ where $a_1$ through $a_s$ belong to the exponent $n \pmod{p}$ and $s=\phi(n)$. For sufficiently large x, $f_n(x)$ is positive. Prove that
$$f_n(x)=\prod_{m|n} (x^{n/m} - 1)^{\mu(m)} \pmod{p}$$
I did prove (using logs) that if we let $f(n)$ be a po... |
I was under the mistaken impression that if one could find the generating function for a sequence of numbers, you could just plug in a natural number $n$ to find the nth term of the sequence. I realize now that I was confusing this with a closed form formula.
So if that is not the case, then what is the point of gener... |
My question pertains to this link (the content of which has been included below in the most recent edit)
The ring of polynomials over a field with no linear term is not a UFD
Let $F$ be a field. Prove that the subset $R \subseteq F[x]$
consisting of all polynomials whose linear coefficient is zero is a
subring. ... |
Show that if $A$ is a retract of $X$ then for all $n \ge 0$ $$H_n(X) \simeq H_n(A) \oplus H_n(X,A)$$
So we have a retraction $r:X \to A$, which is surjective.
Consider the long exact sequence
$$\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \cdots$$
As $r$ is surjective we have that $H_n(X) \to H_n(X,A)$ is s... |
Problem
The Lake Shore Inn is trying to determine its break-even point. The inn has 50 rooms that are rented at $60 per night. Operating costs are as follows, in dollars
Salaries 7,200 per month
Utilities 1,500 per month
Depreciation 1,200 per month
Maintenance 300 per month
Maid Service 8 per... |
I understand how to apply the chain rule to two functions, but trip up when it comes to three or more.
Given $f~(x~)= x~\sqrt{1-x^{2}}$, I can see there are three values:
$a: 1-x^{2}$
$b: \sqrt{a}$
$c: x~\cdot b$
Now, $$f(b(a))= \frac{1}{2~\sqrt{1-x^{2}}}~\cdot~2x= x \sqrt{1-x^{2}},$$ but how do I factor in the last x ... |
On a uni-directional circular road where all vehicles travel in the same direction with varying speeds and overtaking is allowed on the entire road, what is the expected number of times
(a) a particular vehicle will overtake others?
(b) a particular vehicle will be overtaken by others?
In particular, what I want to fi... |
Consider the homogeneous space $GL(3)/GL(2) = GL(3,\mathbb{R})/GL(2,\mathbb{R})$ where $GL(2)$ fixes the first coordinate axis (so can be identified with the subgroup of $2\times 2$ blocks sitting in the 'bottom right' corner of matrices in $GL(3)$).
Is there any 'explicitly known' other homogeneous space which is isom... |
In page 76 of Atiyah's K-theory, he discussed the possibilities of extending Bott periodicity to the case when $\mathbb{S}^{2}\times X$ is replaced by a fibration over $X$ with structure group $U_{1}$. He quote an example as:
"..In particular let $$E=L\oplus 1$$ be the sum of a line-bundle $L$ and the trivial-bundle $L... |
Suppose that $p$ is a prime. Prove that the sum of the primitive roots modulo $p$ is congruent to
$\mu(p − 1) \pmod{p}$.
|
Given $f(x)= \sin(\pi x)^{2}$, find the derivative.
Using the chain rule my work is as follows: $(\sin(\pi x)^2)'$ becomes
$$2 \sin(\pi x) \cdot \frac{d}{dx}(\sin(\pi x)$$
The derivative of sin is cos, thus
$$2 \sin(\pi x) \cdot \cos(\pi x) \cdot \frac{d}{dx}(\pi x)$$
The derivative of $\pi x$ is $\pi$, and the equat... |
I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global sections ect) but I am missing the bigger picture.
Why should I care if a fun... |
Let $R$ be a ring and for $x,y\!\in\!R$, define the commutator as $[x,y]:=xy-yx$. An $r\!\in\!R$ is idempotent iff $r^2=r$.
How can one prove, that if every commutator is idempotent, then the whole ring is commutative, i.e. all commutators are zero?
|
A follow up to a previous question of mine.
I thought it was true, but according to my book $G \simeq H \oplus K$, with $G \simeq H$ does not imply $K=0$
Is there a simple counter example? In what situation is this true?
(Motivation is, from the previous question, to show that $H_n(X,A)=0$
Edit: The groups are Abelian... |
I have the following problem:
If $ \cot{C} = \frac{\sqrt{3}}{7} $, find $ \csc{C} $
From my trig identities, I know that $ \cot{\theta} = \frac{1}{\tan{\theta}} $, and $ \csc{\theta} = \frac{1}{\sin{\theta}} $, and also $ \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} $
However, I can't seem to see how to connect the... |
can anyone help me to inform the name of any book where I can get the following theorem, or give some detailed hint to solve this one:
Let $X$ and $Y$ be two random variables, if the moment generating functions of $X$ and $Y$ are equal then the probability distributions of $X$ and $Y$ will be same. In other words, if $... |
Any natural number can be expressed as the sum of three triangular numbers, or as four square numbers. The natural analog for expressing numbers as the sum of two others would apparently be the sum of two "linear" numbers, but all natural numbers are "linear", so this is rather unsatisfying. Is there a well-known spa... |
Customers arrive at a casino as a Poisson process of rate 100 customers per hour. Upon
arriving, each customer must flip a coin, and only those customers who flips heads actually enter the casino.
So how can I find
the PMF of N? Let's say for the
number of customers who arrive
between 5PM to 7PM.
the pdf of T? The ... |
In elementary number theory, I find the following notations being used interchangeably, which leads me to have many ambiguous assumptions:
$\mathbb{Z}_p^\times$;
$\mathbb{Z}_p$,
where p is any integer. What's the difference between them?
One more question to add to the kitty: what do they mean when they say "a su... |
I have a computational problem that I am solving that ultimately relies on either the areas (for $\mathbb{R}^2$) or the volumes (for $\mathbb{R}^3$ and above) for a pair of cropped objects. We can think of these cropped objects as closed, concave (but sometimes convex), orientable polytopes with finite surface area an... |
How can I compute of the $2\pi$-periodic function whose restriction to the range $[-\pi,\pi]$ is:
$$ \begin{cases} 1 && 0 < x < \pi \\ 0 && -\pi < x < 0 \end{cases} $$
|
Greetings,
I've been reading Maclane's "Homology" and ran into the following question:
Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon \operatorname{R-Mod} \to \operatorname{S-Mod}$, that is additive exact and faithful. We also have an left adjoint functor of $\Psi$... |
I read a little proposition that any quantity obtainable from rational numbers by a finite number of operations $+$, $-$, $\cdot$, $\div$, $\sqrt{{}}$ can be put in a standard form $\frac{p}{q}\cdot A$, where $\frac{p}{q}\in\mathbb{Q}$ and $A$ is an expression of only integers and $+$, $-$, $\cdot$, and $\sqrt{{}}$.
I... |
I am trying to export a matrix in a file using Mathematica.
I get the matrix and then use
Export["myfile.dat", MyMatrix, "???Format???"]
There are many formats. I want the one that prints the matrix in this way:
{{0,1,0,....},{0,0,1,...},...}
That is the classic Mathematica format....
How can I do this?
Which one is ... |
This is Exercise III.2.8 of Bourbaki's Theory of Sets.
An ordered set $E$ is said to be ramified if, for each pair of elements $x,y$ of $E$ such that $x<y$, there exists $z>x$ such that $y$ and $z$ are not comparable. $E$ is said to be completely ramified if it is ramified and has no maximal elements.
(a) Let $E$ be an... |
I am trying to find a way to calculate the probability that a user will pass a test at a certain certification authority, based on the user's history with answering questions using our service. Looking at a single question at a time (I know how to combine the probabilities), I have the following data:
pass - The perce... |
Question: Let’s call a real number $x$ approachable if there exists a Turing machine $M$ such that $M(1), M(2),\dots$ is a sequence than converge to $x$. If we can prove (edit: In ZFC) that the sequence converge, we say that $x$ is provably approachable. Is every approachable number also provably approachable?
Clearifi... |
What is the value of $$\int_{[0,1]^2} \!\!\!dx\,dy\,\mathcal{P} \frac{\log(x)}{x-y},$$
where $\mathcal{P}$ denotes Cauchy's principal value. I solved this once for a homework, but I can neither remember the answer nor reproduce it right now... (bad memory)
Any help or comment is highly welcome.
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Which textbook is good for introductory group theory?
|
Let $W$ be a real Hilbert space of dimension $n$ and $V$ a Hilbert subspace of dimension $m$. Assume that $f_1,\cdots,f_k$ are points in $W$ such that the following holds:
there exists $\sigma>0$ such that $dist(f_i,V)\geq\sigma$ for $i=1,2,\cdots,k$,
there exists $0<\delta\leq\sigma$ such that the balls $B(f_i,\delta... |
Is there an easy way to construct, on the same filtered probability space, a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ?
|
Can anyone say how to relate Stone-Weierstrass approximation theorem in this problem? Given that $\mu$ and $\nu$ are two probability measure on the real line, if for all bounded continuous functions $f$, $\int f d \mu = \int f d \nu$, then $\mu=\nu$?
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Hi I really need help in how to determine sub fields $L$ of $\mathbb{C}$ over $\mathbb{Q}$ and finding $[L:\mathbb{Q}]$. The question is:
Find subfields $L$ of $\mathbb{C}$ that are splitting fields over $\mathbb{Q}$, find $[L:\mathbb{Q}]$
$$\begin{array}{ll}
\mathrm{(i)} & t^8 + 1 \\
\mathrm{(ii)} & t^4 -4t^2 +1\... |
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