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I'm trying to figure out the probability of a 3rd failure occurring on the 5th attempt of doing something. Let's just call the probability of success of failure P(S) or P(F), I won't put numbers as I want to actually learn.
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Question as in title. I only really need the special case where one of the open embeddings is the identity, but the more general case would be useful to know.
Edit - By product I mean: given $U\to X$ and $V\to Y$ open embeddings, is the canonical map $U\times V \to X\times Y$ an open embedding?
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I realize this is a novice question, but I'm stuck here. Suppose you have an ordered basis $B = (v_1, v_2, v_3, v_4)$ of $R^4$. My problem is how to determine a system of equations that defines, relatively to the basis $B$, the linear subspace of $R^4$ generated by the vectors $v_1-v_2+v_4$ and $-v_1+3v_3$.
Again, I r... |
I'm trying to solve an equation for B where
$2 B e^{(1-B)}=1$
I plugged it into wolfram alpha and got:
What is that $W_n$ mean? How do I intrepret this answer?
(Feel free to retag it, I wasn't entirely sure what it falls under!)
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Prove that $\gcd(a + a', b + b') = 1$ if $ab - a'b' = \pm 1$
My attempt was:
Case 1:
$ab - a'b' = 1 \implies \gcd(a, b') = 1$ and $\gcd(a', b) = 1$
Then is it sufficient to conclude that $\gcd(a + a', b + b') = 1$?
Furthermore, when we write $\pm 1$, does it mean or or and?
Thanks,
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Explain $-\int_{b}^{a}\frac{Q}{4 \pi \epsilon \bar{r}^{2}} \cdot d \bar{r}= \left[\frac{Q}{4\pi \epsilon \bar{r}}\right]_{b}^{a}$. What is the term $\frac{1}{\bar{r}^{2}} \cdot d\bar{r}$? I find it hard to think about it.
$r$ is a radius of a ball.
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i'm trying to solve this question:
Given a turing machine that is decidable by at most 50 * n^4 steps, can we build a dif algorithm that can decide it in n^4 steps?
Me and my friends thought about it, and we coulnd't get it right.
Some points we had in mind:
1. if you can reduce the overall cost by 50 somehow, couldn't... |
If $H$ is a Hilbert space of entire functions with weighted norm $||f||^{2}=\int_{R} |\frac{f(t)}{g(t)}|^{2}dt$ for some entire function $g$ (not necessary in $H$). Can we find any relation between the norm of $f$ and the norm of it's derivative? Something like:
$||f'||\leq C ||f||$ for some constant $C$. (Note: so far... |
Let $D^n$ be the closed unit disk in $\mathbb{R}^n$. I'm looking for a homeomorphism of $D^n$ to itself which is the identity on the boundary, and takes the origin to a given point $z$ in its interior. I can visualize such a function as "shifting around the mass" within $D^n$, but I can't seem to write one down.
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The upper bound for the number of moves required to solve a regular Rubik's cube has been shown to be 20.
Two questions come to mind:
Does this result have more general significance?
What are the most pressing issue with regards to Rubik's cube (or generalizations) or its group?
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I'm reading the proof of the following fact:
Let A be a ring, a an ideal of A, M an A-module. Then $(A/a) \otimes_{A} M$ is isomorphic to M/aM.
So the solution is to tensor the exact sequence $0 \rightarrow a \rightarrow A \rightarrow A/a \rightarrow 0$ with M.
This induces an exact sequence:
$a \otimes_{A} M \rightarr... |
As I understand it, to convolve $f$ and $g$ means to find $\displaystyle \int_{\mathbb R} f(a)g(t-a)da$, which is also apparently commutative, and therefore $\displaystyle \int_{\mathbb R}f(a)g(t-a)da = \displaystyle \int_{\mathbb R}f(t-a)g(a)da$
That means, if $f(a) = 1$, then $\displaystyle \int_{\mathbb R}g(t-a)da ... |
Which is a good book to refresh discrete maths fundamentals for a grad student?
It would be great if the book has short and terse explanations of concepts with lots of worked out examples(/to be worked out exercises) to set the brain rolling.
Can someone suggest a similar refresher for graduate level algorithms?
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In my notes:
$$\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$$
where $\mu(d)$ is the Möbius function and $\sigma(d)$ is the sum of all positive divisors of $d$.
And I have no idea how they got the expression on the right hand side. Could anyone help me explain how this works?
Thanks,
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Prove that $f(n) = 2^{\omega(n)}$ is multiplicative where $\omega(n)$ is the number of distinct primes.
My attempt:
Let $a = p_1p_2\cdots p_k$ and $b = q_1q_2\cdots q_t$
where $p_i$ and $q_j$ are prime factors, and $p_i \neq q_j$ for all $1 \leq i \leq k$ and $1 \leq j \leq t$.
We will show that $2^{\omega(ab)} = 2^{... |
This idea resulted while I heard an advertisement for Sonic, where they claim to have something like 300,000 different drinks they serve.
Essentially, what they are allowing you do to is mix any soda to create your own "unique" drink.
As such, it made me curious how to calculate the number of "unique" drinks you make.
... |
Given a set of 2D points I have to find the points that when connected will form a polygon that contains all the points in the set.
A quick example: imagine you have a set S={(1,1),(2,1),(1,2),(2,2),(4,4)}.
The solution would be SolS={(1,1),(2,1),(1,2),(4,4)} as those points (when connected) form a polygon that cointai... |
I apologize in advance for what is probably an obvious question but here goes:
What structure do functions on affine schemes respect?
If we consider other types of 'spaces' we have a notion of function on them which respect structure relevant to the type of 'space'.
For a smooth manifold, we consider differential fu... |
Prove that any two consecutive terms of the Fibonacci sequence are relatively prime.
My attempt:
We have $f_1 = 1, f_2 = 1, f_3 = 2, \dots$, so obviously $\gcd(f_1, f_2) = 1$.
Suppose that $\gcd(f_n, f_{n+1}) = 1$; we will show that $\gcd(f_{n+1}, f_{n+2}) = 1$.
Consider $\gcd(f_{n+1}, f_{n+2}) = \gcd(f_{n+1}, f_{n... |
Here is the question from Rotman, verbatim:
A sequence $S'_*\stackrel{f}{\to} S_* \stackrel{g}{\to} S''_*$ is exact in Comp if and only if $S'_{n}*\stackrel{f_n}{\to} S_n \stackrel{g_n}{\to} S''_{n}$ is exact in Ab for every $n\in \mathbb{Z}$
(Note here Comp and Ab are the categories of chain complexes and abelian grou... |
I'm confused by what my solution manual is telling me.
Part a of a problem I'm doing is:
Sketch the plane curve with the given vector equation.
$r(t) = <t-2, t^2+1>, t = -1$
The solution manual basically says:
Since $(x+2)^2 = t^2 = y-1 => y = (x+2)^2 - 1$, the curve is a parabola.
How is $(x+2)^2 = t^2 = y-1?$
Anothe... |
Let $K$ be a quadratic number field, and $\mathcal{O}_K$ the ring of integers of $K$.
The map $\pi: Spec(\mathcal{O}_K) \rightarrow Spec(\mathbb{Z})$ that sends a prime ideal $\mathbb{p}$ to $\mathbb{p} \cap \mathbb{Z}$ is induced since $\mathcal{O}_K$ contains $\mathbb{Z}$. And the fiber $\pi^{-1}$ of the prime ideal ... |
I am writing up an assignment with includes many identities of Fibonacci numbers. I have made up the following notation (here $f_n$ is the number of tilings of an $n$-board by dominoes and squares - a well known interpretation of Fibonacci numbers):
Given an $n$-board, and some sequence $seq$ of dominos and squares, le... |
For programming purposes I want a function f(x,R) that given a certain seed x returns the same random value every time, in an arbitrary range R. But, I also want the output to be equally distributed. Is this mathematically possible?
For example, I can take the 1st - nth decimal places of sin(x) such that 10^n > R and d... |
Let $f$ be a continuous function , with the intial conditions: $f(0)=1$, $\frac{df}{dx} \geq f$.
How does one show that $f(x) \geq e^x$ for every $x \geq 0$?
I tried using Taylor series
Thank you.
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I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$.
added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no Maple/Mathematica and only a rough implementation in Pari/GP for real values. That motivated to try a solution via Newton/Raph... |
Let f be a convex and bounded function, meaning there is a constant $C$, such that $f(x) < C$ for every $x$.
I need to prove that $f$ is a constant function.
Thanks!
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Let $(R,m)$ be a D.V.R with field of fraction $K$ and $L$ any finite algebraic field extension of $K$. Suppose $\bar{R}$ is the integral closure of $R$ in $L$. Then it is well known that $\bar{R}$ is a Dedekind domain and for any nonzero ideal $J$ of $\bar{R}$ the ring $\bar{R}/J$ is a finite $R$-module. My question is... |
Possible Duplicate:
(undergraduate) Algebraic Geometry Textbook Recomendations
I am interested in algebraic number theory and I am recently acquainted with the theory of valuations, which further leads to Riemann-Roch theory, and which is closely related to algebraic geometry, and the algebraic-K-theory.
Therefore, ... |
Let $f$ be a continuous function, with the initial condition: $f''>0$.
I need to prove that $f(2x)-f(x) < f(3x)-f(2x)$.
By $f''>0$ I can learn that $f'$ is monotonous increasing, and so I tried to use Lagrange or Taylor.
Thank you.
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Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
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How to prove that:
$$
\left(\sum_{i=1}^n w_i n_i \sqrt{\dfrac{y_i(1-y_i)}{n_i+1}}\right)^2 \leq \dfrac{\left(\sum_{i=1}^n w_i n_i y_i\right)\left(\sum_{i=1}^n w_i n_i (1-y_i)\right)}{(\sum_{i=1}^n w_i n_i+1)}
$$
where $w_i\geq0$, $\sum_{i=1}^n w_i=1$, $n_i>0$ and $y_i \in (0,1)$ for $i=1,\dots,n$, with $n>1$?
I have ... |
Concept of Proxy Signatures serving the main purpose of delegation is clear.
But what exactly is proxy re-signatures?
How and why is it performed? Does it use the same warrant generating and proxy signing mechanisms?
Thanks!
|
Suppose we have an algebraically closed field $F$ and $n+1$ variables $X_0, \dots, X_n$, where $n > 1$. Does there exist an irreducible homogeneous polynomial in these variables of degree $d$ for any positive integer $d > 1$? In other words, does there always exist an irreducible hypersurface of arbitrary degree?
Of... |
If $A$ is a Krull domain in its field of fractions $F$, and if $F'\subset F$ is a subfield, then $A\cap F'$ is a Krull domain. But is it necessarily a Krull domain of $F'$? That is, is $F'$ necessarily the field of fractions of $A\cap F'$?
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Suppose I have a function $f$ that is analytic on the unit disk $D = \{ z \in \mathbb{C} : |z| < 1 \}$ that is also continuous up to $\bar{D}$. If $f$ is identically zero on some segment of of the boundary (e.g. $\{ e^{it}, 0 \leq t \leq \pi/2 \}$ ), is it then true that $f$ is identically zero on the entire boundary? ... |
We have $10$ bricks, $3$ red, $2$ white, $2$ yellow, $2$ blue, $1$ black. In how many ways can these be arranged such that only $2$ red bricks are adjacent ?
We want to distribute the elements in {$RR, R$} over the spaces in arrangements of the form:
_B_B_B_B_B_B_B_
where _ represents a space (of which there are $8$),... |
Let $X_i$ be a random variable, and the probability density function of X_i is
$$P(X_i=k)=\frac{{k-1 \choose i-1}{m-k \choose n-i}}{m \choose n}$$
For more about the distribution, please refer to http://www.randomservices.org/random/urn/OrderStatistics.pdf
The mean of $X_i$ is $$E(X_i)=i\frac{m+1}{n+1}$$
I want to calc... |
I'm looking for a way to find the factor of a natural number, call it $x$, which is closest to $\sqrt{x}$.
If this question needs to be clarified, just let me know and I'll be happy to add more.
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In numerical mathematics, we have to approximate the value of $\pi$ using three different methods. One of the three methods is the Monte-Carlo method, i.e. I let Matlab produce $n$ random numbers within the unit square, then I counted the numbers which were within the unit circle and looked at the quotient of those two... |
I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation in trigonometry.
For a test, I wrote this,
sin(50)^2
I assumed that would be interpreted as sin(50)*sin(50)
But I was told the correct notation for this is is
sin^2 (50)
or optionally
(sin (50))^2
I'm curious if... |
Does the sum
$$\sum_{z \in \mathbb{Z}^3\setminus \{(0,0,0)\}} \left( \frac{1}{|{\bf x} - {\bf z}|^2} - \frac{1}{|{\bf z}|^2} \right)$$
converge pointwise or even uniformly for $\varepsilon < |{\bf x}| < 1-\varepsilon$?
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I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link:
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/
It seems that with a matrix multiplication $AB=C$, that the entries as scalars, are formed from the dot product computations ... |
My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be differentiable at a point x, and then differentiable in a domain. Then the focus is entirely on this latter notion. My question i... |
Is there a straightforward algorithm for fitting data to an ellipse or other conic section? The data generally only approximately fits a portion of the ellipse. I am looking for something that doesn't involve a complicated iterative search, since this has to run at interactive speeds for data sets on the order of 100s ... |
I'm having a disagreement with a client that I think is relatively simple mathematics, but I'd like a third party to confirm or dispute my assertions. Here's the situation:
We have a list of N final prices p1, ..., pN. A final price is tax-inclusive and includes two kinds of tax:
flat tax, which is the same for all it... |
Say I want to pay someone $100.00 but the service I use keeps 3% of the payment I make. Meaning they would only receive $97.00 since they would keep $3. If I increase my payment to $103 they would keep $3.09 meaning they would only receive $99.91. Is there a formula to calculate the exact extra amount I'd need to pay i... |
I was thinking about the fact that $\frac{1}{\zeta(n)}$ is the probability that $n$ randomly chosen integers are coprime, and wondered if there were “natural” ways in which $e^{-n}$ (for $n>0$) and $\frac{1}{\Gamma(n)} $(provided that $0 < \frac{1}{\Gamma(n)} < 1$) might be interpreted as probabilities with respect to ... |
I have the following exercise to do but don't know how to approach it:
Let $G$ be a graph with $n$ nodes ($n \ge 2$), and where every node has degree at least $3$. Show that $G$ has a cycle of length $\le 2\cdot \lceil\log n\rceil$. And it allows me to consider a breadth-first search tree.
First of all, what's the leng... |
I'm wondering if these matrices have a name? (I'm somehow tempted to call them subunitary but it seems to be reserved for something else.)
The matrix $M \in \mathbb{C}^{n\times n}$ is called ..., if all the singular values $\sigma_1,\dots, \sigma_n$ are strictly smaller than 1.
Note that if all the singular values a... |
I'm looking for solution to this non-homogeneus problem.
$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}=F(x,t)$ for $0<x<\pi$, $t>0$
$u(x,0)=0$
$u(0,t)=\frac{\partial u}{\partial x}(x=\pi)=0$
Does anyone know how to proceed?
|
I'm a bit rusty on my math, so please forgive me if my terminology is wrong or I'm overlooking extending a simple formula to solve the problem.
I have a collection of points in 2D space (x, y coordinates). I want to find the "average" point within that collection. (Centroid, center of mass, barycenter might be better ... |
So I picked up a couple of good undergraduate-level books over the weekend and have been working through them...
In Algebra: Chapter 0, the author of the text writes:
The prototype of the well-behaved relation is '=', which corresponds to 'the diagonal'
$$\{ (a, b) \in S \times S \, | \, a = b \} = \{(a, a) \, | \, ... |
For $\alpha, \beta \in \mathbb{Z}[\zeta_3]$ , is it true that $(\alpha \cdot \beta)'=\alpha'\cdot \beta'$?
Also,
For $\alpha, \beta \in \mathbb{Z}[\zeta_3]$ , is it true that $(\alpha + \beta)'=\alpha'+ \beta'$?
|
In the well known matrix form of a least squares regression
where I am trying to solve for B in Y = B1X1 + B2X2 + B3
I might be given X and Y sample data as something like
$X$ =
$\begin{bmatrix}
1 & 2 \\
2 & 3 \\
\end{bmatrix}$
$Y$ =
$\begin{bmatrix}
7 \\
8 \\
\end{bmatrix}$
In such an arrangement how do... |
What is the image of $\zeta_3$ under the non-identity embedding of $\mathbb{Q}(\zeta_3)$ in $\mathbb{C}$?
|
Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious transfinite recursive definition of hyper^alpha-inaccessible ?
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Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most of them mark its formulation as an epochal moment in mathematics, it seems to me. To my uneducated self, Russell's para... |
I wonder how many different ways are there of writing the Baker-Hausdorff equation! This is a form which I recently encountered and haven't been able to figure out how it comes,
$e^ae^Xe^b = e^{[X+L_X.(b-a)+(L_X \coth L_X).(b+a)+..]}$
where $L_X$ is an operator which acts on another matrix say $f$ as,
$L_X.f = \frac{1}... |
Possible Duplicate:
How come 32.5 = 31.5?
|
This is somewhat related to the question Why doesn't Hom commute with taking stalks?.
My question is this: If $F$ and $G$ are locally free sheaves of $\mathcal{O}_X$ -modules on an arbitrary ringed space $(X,\mathcal{O}_X)$, then is the stalk of the Hom sheaf $\mathcal{H}om(F,G)$ at a point $p$ equal to $\text{Hom}_{\m... |
F is a linear functional in $V'$ a linear vector space which operates on $\phi\in V$. Show that there is a one-to one correspondence between F and $f\in V $ such that
$F(\phi)=(f,\phi)$
where $V$ is also a linear vector space. $(,)$ represents inner product. Any hint will be helpful.
Attempt: I don't even know if thi... |
My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves on. I'd like to know how to come up with the function (the teacher considers this out of scope). Even suggesting google ... |
If we have an entire function $f(z)$, can we apply the Cauchy integration formula for derivative for $f(z)$ and integrate over $\Bbb R$ instead of simple closed curve, i.e.
is this formula is true:
$$f'(a_{0}) = \frac{1}{2\pi i}\int_{\Bbb R}\frac{f(t)}{(t-a_{0})^{2}}dt$$
where $f(t)$ is just $f(z)$ with $z=t\in \Bbb R... |
Let $X$ be a finite-dimensional inner product space, and $T$ a linear operator on $X$. Let $W$ be a subset of $X$ with the following property:
If $T$ preserves norms on $W$, then $T$ is orthogonal on $X$.
One might say that the set $W$ 'detects' orthogonality.
My question is: what is the smallest number of elements th... |
I'm very interested to know how I can factorise $a^{10} +a^5 +1$ in two factors with integer coefficients. I've tried a lot but I don't have any idea how do that.
|
I read this quote attributed to VI Arnold.
"Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he studied supermanifolds, non-standard calculus or embedding theorems."
EDIT Source is "A mathematical trivium" A book of 10... |
If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then
$$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$
This identity follows from a probability calculation that you can find at the
top of page 311 in the 10th edition of Introduction to Probability Models by Sheldon Ross.
Is ... |
Is there a cis (CosX + iSinX) Button in CASIO graphic calculators ? if so where its ?
if no is there any turn around for it ?
PS. I got a CASIO fx-9860G
Thanks
|
I have a graph with edge-set E and the following SDP (page 8 here)
I'm trying to use CVXOPT to solve this problem, which asks for SDP to be expressed in vectorized form as below:
How do I go about turning my SDP into this form?
Update
From suggestion of Joachim Dahl it's easier to figure out values from the dual form... |
In the classic Coupon Collector's problem, it is well known that the time $T$ necessary to complete a set of $n$ randomly-picked coupons satisfies $E[T] \sim n \ln n $,$Var(T) \sim n^2$, and $\Pr(T > n \ln n + cn) < e^{-c}$.
This upper bound is better than the one given by the Chebyshev inequality, which would be roug... |
I'm learning some group cohomology from the third section in Serre's Local Fields, and I'm up to the section on change of group. If $f:G'\rightarrow G$ is a homomorphism of groups and $A$ is a $G$-module, there is an induced $G'$-module structure given by $s'\cdot a=f(s')\cdot a$, for $s'\in G'$ and $a\in A$ (this is S... |
I'm having an issue with a large amount of time stamped data for work. Each data point has a value and a time stamp.
We are going to be displaying this data on a graph, but we need a way to reduce the amount of points in the series due to the restriction on visible points on the graph.
Eg. A week time range will return... |
$N$ points are to be put on sides of a square. What's the maximum number of triangles which are formed by joining those points?
Hints would be appreciated.
|
The fundamental group $\Pi_1(G)$ of a connected graph $G$ is defined to consist of all loops
(i.e., closed paths) based at a given fixed basepoint/vertex $g \in G$ as elements,
and concatenation as the group operation. (for $G$ connected, we can show that the
group is independent of the choice of basepoint; giv... |
$\int_{0}^{2}\frac{1}{(x+2)(x-1)}$
Will I be able to use partial fractions on the indefinite integral and then evaluate my answer as f(2)-f(0)?
Any help is appreciated as always. :)
EDIT: I looked this up and apparently the integral doesn't converge so now I need help finding where I went wrong.
I had solved for 1 ... |
I just learned about modular numbers on wikipedia, such as $17 \equiv 3\pmod{7}$.
So what is infinity $\pmod{n}$? It can't very well be all the numbers at once, so what happens?
|
I solved a problem in Calculus 2 exercise lesson and I do not fully understand it.
The task is to show that cross derivates differ in sign ( $f_{\text{xy}}(0;0)=-1$ and $f_{\text{yx}}(0;0)=1\, $ )
The function is:
$ f(x,y)=\left\{
\begin{array}{cc}
\frac{x y \left(x^2-y^2\right)}{x^2+y^2} & x^2+y^2\neq 0 \\
0 & x^2+... |
This appeared on my professor's test review, and its taken me hours to, surprise surprise, get the wrong answer. Could someone help me with the method I should be using to solve this?
$$y^{\prime\prime}+y=\tan x$$
|
Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to $\frac{1}{2}(f(x^+)+f(x^-))$. I have seen a proof in Zygmund's book but I don't understand it as it apparently requires... |
Let $\{ f_n \}$ be a sequence in a Hilbert space $L^2(\mathbb{R}^d)$. We say that this sequence converges weakly to an element $f \in L^2$ if $\langle f_n, g \rangle \to \langle f,g \rangle$ for every $g \in L^2$ (where $\langle \cdot,\cdot \rangle$ denotes the inner product on $L^2$). By definition, we are given that ... |
I am trying to review my differential Geometry. In J.Lee's Smooth Manifolds, there is an exercise in which one has to show that the product of smooth covering maps is a smooth covering map.
A smooth covering map is a smooth cover $\pi: M' \rightarrow M$, where
$M, \, M'$ are smooth manifolds, and in which every $p$ in... |
Really, the curve in question is the polar plot $ r = cos( K * \theta) $, where $K$ is any irrational number (I use $\pi$), but the transformation to a parametric one on $x$ and $y$ with domain $t$ is an unsurprising one.
It would appear that the curve is confined to the unit circle, and also that it never repeats -- t... |
The following problem has come up in my research and I don't have the tools (i.e., I don't know Algebraic Geometry, especially over $\mathbb{R}$) to solve it.
Consider two subsets $X$ and $Y$ of $M_n(\mathbb{R})^k$ with $k\geq 3$. The subset $X$ consists of all $k$ tuples of n x n matrices $(A_1,..., A_k)$ such that f... |
There's a problem I can't figure out in my homework. I can't really understand what it's asking. Maybe someone can help.
A meteor enters the Earth's atmosphere
and burns up at a rate that, at each
instant, is proportional to its
surface area. Assuming that the meteor
is always spherical, show that the
radius... |
I am interested in somehow characterizing the zeros of the following function:
$P(z) = \sum_{i=1}^n \frac{\alpha_i}{z+\lambda_i}$,
with $\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_n=0$ and $\sum_{i=1}^n \alpha_i = 0$.
Furthermore, the values $\alpha_i$ are in general non-integer and may even be complex. The pol... |
Let $X_1,\dots,X_n$ be Gaussian random variables with mean $\mu$ and variance $\sigma^2$. What is known about the distribution of
$$\frac{\sqrt{\frac1{n-1}\sum_{i=1}^n(X_i-\bar X)^2}}{\bar X} $$
with $\bar X = \frac 1n \sum_{i=1}^n X_i$?
Does this distribution have a name? I am especially interested in its cumulative d... |
I am trying to find all $1$- or $2$-dimensional Lie algebras $\mathfrak{a}$ up to isomorphism. This is what I have so far:
If $\mathfrak{a}$ is $1$-dimensional, then every vector (and therefore every tangent vector field) is of the form $cX$. Then, by anti-symmetry, and bilinearity:
$$
[X,cX] = c[X,X] = -c[X,X] = 0 \... |
I've been looking in vain (most books I came across give identities involving sums or recurrence relations, but do not give much attention to fixed values) for a reference to the following identity:
$$S(n,n-3)={n \choose 2}{n \choose 4},$$
where $S(n,k)$ is the unsigned Stirling number of the first kind. Is this well-k... |
The curved line is a part of a circle with a center at the lower right corner.
What would be the name of the shaded region?
Thanks
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I've got 64 images of 96x192 pixels. I have to arrange them on a rectangle. I need the height and width of that rectangle to be powers of two. Given the dimensions of my images, they don't perfectly 'fit' on it (I can not rotate or cut the images into smaller ones); there is always some 'wasted space'.
¿What are the d... |
I'm confused about the different types of homology groups. For instance, on Wikipedia (Orientability) they write $H_n (M, \partial M; \mathbb{Z} )$.
On the other hand, where they explain the construction of homology groups, they write $H_n (X) := ker(\partial_n) / im(\partial_{n + 1})$. (here)
Are these two the same?
T... |
We have to choose 4 types of soup from a supermarket's 10 varieties, such that at least 2 are identical. In how many ways can we do this ?
We have 10 varieties of soup like so {v1, .. v10}. We can partition our choices into those with 4 identical (e.g. {v1, v1, v1, v1}), those with 3 identical (e.g. {v1, v1, v1, v2}), ... |
Say I have three constants a, b and c (all > 0) and three variables x, y and z (all >= 0). I want to find values for x, y, z which maximise the lowest of the following:
ax - y - z
-x + by - z
-x -y + cz
How can I find these values?
EDIT: just corrected expressions -- variables and constants were wrong way around
EDIT... |
I am taking a course in linear algebra at the moment, and the book I have uses $1\times n$ matrices, $n\times 1$ matrices and $n$-tuples to represent vectors. In condition I have been taught that $1\times n$ and $n\times 1$ matrices are vectors.
What, then, is the difference between an $n$-tuple and a $1\times n$ matr... |
Does there exist a reduced quadratic equation with solutions equal to both the $\sqrt{2}$ and $\sqrt{3}$?
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Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$
$${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$
This is again a $p$-adic integer, and we can define
$$\alpha_1:=\sum_{n=0}^\infty \overline{{\alpha \choose n}} p^n$$
and
$$\alpha_2:=\sum_{n=0... |
Simplify $\sum\limits_{k = 0}^n {{2^k}\tan ({2^k}x)}$ which $k \in \{ 0,1,...,n + 1\} ,{2^k}x \notin \{ 0,\frac{\pi }{2}\}$
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The following is alpha of t times x:
$$\alpha(t)x$$
The following is alpha times t times x:
$$\alpha(t)x$$
My instructor had one interpretation, I used the other. ;)
Is there an easy or standard way to signify that we're talking about a function rather than a variable multiplication here?
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There is a square table and 2 persons are sitting on each side of it so there are 8 persons in total ... how many total number of permutation is possible?
Does circular permutation rules applies here?
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