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If $m$ and $n$ are real numbers, $i^2 =-1$, and $(m-n)-4i=7+ni$, what is the value of $m$?
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Consider this wonderful ( think it is) identity
$$\begin{align*}
&a+b(1+a) + c(1+a)(1+b) + d(1+a)(1+b)(1+c)
+\cdots+l(1+a)(1+b)\cdots(1+k)\\
&\qquad=
(1+a)(1+b)(1+c)\cdots(1+l)-1
\end{align*}
$$
I believe there must be some beautiful applications, for example deriving some other identities, of it. Can someone... |
I have always thought that groups of individuals that spend more money(as opposed to save) have more money in the long run. My reasoning is that if a particular group spends money, then each dollar that one person spends is being used by more individuals, and thus one physical dollar, might have a value of $10/day. (sp... |
consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it
$\left[(a_1+b_1)(a_2+b_2)\cdots(a_n+b_n)\right]^{1/n}\ge \left(a_1a_2\cdots a_n\right)^{1/n}+\left(b_1b_2\cdots b_n\right)^{1/n}$
|
Let $A$ be matrix from the vector space of square $N \times N$ matrices.
With the inital information: $A^2-4A=4I$.
How does one show that $A+I$ is invertible?
(I need please a solution that involves eigenvalues)
Thank you
|
Is it possible to solve this problem:
A prince wish to marry a princess. There are 3 princesses, one is young, one is a little older and one is old. The prince is able to tell the princesses apart. One of the princesses always tells the truth, one never tells the truth and one sometimes tells the truth and sometimes no... |
Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$? Why? Thanks.
|
How can I prove that if I have $n$ eigenvectors from different eigenvalues, they are all linearly independent?
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The integral in question is $\int_{-\infty}^{+\infty} x {e}^{-\lambda x^2}dx$ where $x$ and $\lambda$ both are real numbers.
My solution:
$\int_{-\infty}^{+\infty} x {e}^{-\lambda x^2}dx = \int_{-\infty}^{0} x {e}^{-\lambda x^2}dx +\int_{0}^{+\infty} x {e}^{-\lambda x^2}dx =$
$\lim_{a\rightarrow -\infty}\int_{a}^{0} ... |
Question: Given the real numbers as a set, does it require the (non-finite) Axiom of Choice to pick out an arbitrary single element? What about if we wanted to pick out an integer? What about if we wanted to pick out "0"?
Motivation: Those who have seen some of my previous questions here know that I am terrible at ... |
So my buddy claims that if I split a chocolate bar at random into two pieces, then the expected size of the larger piece is $\frac{3}{4}$ of the bar. I can't figure out how he came up with this value...
Can someone explain this? If you can, can you provide some kind of a proof?
p.s. it would be helpful to think of this... |
How does this work exactly? If I take a simple function like f(x) := x and I graph it, I get a solid line, but if I restrict the domain to a Cantor set, then it becomes a dotted line. But does the discontinuity of the domain make the function discontinuous? I wouldn't think so because I can still find the limit of e... |
Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding $H\hookrightarrow G$ gives an embedding of Lie algebras, $\mathfrak{h}\hookrightarrow\mathfrak{g}$. Since $H$ is normal, $\m... |
Consider a ratio $\int_A f(x,a)dx/\int_B f(x,a)dx$ where $A, B \subset [0,1]$ and $a \in R$. Suppose for any $x' \in A$ and $x \in B$, $f(x',a)/f(x,a) > f(x',b)/f(x,b).$ Then can we say that $\int_A f(x,a)dx/\int_B f(x,a)dx>\int_A f(x,b)dx/\int_B f(x,b)dx$? Thank you very much.
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Does there exist a function $f$ such that it is continuous a.e. on a measurable subset
$E$ $\subseteq$ $R$ (the set of real numbers), Lebesgue integrable, but not Riemann integrable? I was thinking maybe the Dirichlet function, because it is Lebesgue integrable but not Riemann integrable, but it is nowhere continuous!... |
for example i have a code of strings which is efzau and 47 how can i define a grammar, set of terminal symbols and set of non terminal using B.N.F?
Thanks
|
Is it correct that for any uniformly, independently chosen vectors $r,s \in \mathbb{Z}_2^m$ and for any $0 \neq D \in \mathbb{Z}_2^{m \times m}$, we have that $Pr_{r,s}\left[r^T\cdot D \cdot s \neq 0 \right] \geq \frac{1}{4}$? If so, how would I go about proving this?
Thanks!
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I'm suppossed to use an example to show the following statement.
If F over K is galois but not algebraic and L is an intermediate field between K and F, then F over L is not galois.
Any help at all would be greatly appreciated.
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what is the fractional linear map maps the circles $|z-5|=3$ and $|z+5|=3$ to concentric circles? What is the general method to find such maps? What is the image of $|z+2|=2$ under the map: $z \to 1/\bar{z}$? Thank you very much.
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How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
|
Let $c_i>0$ be a scalar, $v_i\in R^d$ be a unit-length vector, $i=1,...,n$. In a previous question, it is proved that
Fact: There exist $v_i$ such that $\sum_{i=1}^n c_i v_i=0$ if and only if $c_j \le \frac{1}{2}\sum_{i=1}^n c_i, \forall j$.
We can find a solution when $c_j \le \frac{1}{2}\sum_{i=1}^n c_i, \forall j$. ... |
I read the two Wikipedia articles and it sounds like there is a relationship between the two, but I can't quite grasp it. They don't seem to be the same thing, but I can't demonstrate it in part because Lie coalgebra is defined using the exterior product, and Lie bialgebra is defined using tensor product. I'm confused ... |
Let $E$ be the splitting field of $x^6-2$ over $\mathbb{Q}$. Show that $Gal(E/\mathbb{Q})\cong D_6$, the dihedral group of the regular hexagon.
I've shown that $E=\mathbb{Q}(\zeta_6, \sqrt[6]{2})$, where $\zeta_6$ is a (fixed) primitive sixth root of unity, and thus that $[E:\mathbb{Q}]=12$.
I'm getting a little mixe... |
The following is an error correcting equation for a sidereal astrophotography tracking mount I'm building and $t$ represents the amount of time before the tracking is off by a quarter of a stepper motor step. I need to solve for $t$, but I've having trouble solving this.
I can plot it with a value of $n$ and find an... |
Question.
Find the first 5 terms of the Maclaurin Series of $\frac{\sin x}{e^x}$.
I'm trying to find it using division. Is there possibly a shorter way because every time I try to do it, I get the wrong answer:
$$
x-2x^3/3+\cdots$$
I don't really like polynomial long division.
This is how I set it up:
$$
\frac{x - x^3... |
I want/need to learn some recursion theory, roughly equivalent to parts A and B of Soare's text. This covers "basic graduate material", up to Post's problem, oracle constructions, and the finite injury priority method. Is there an alternate, more beginner friendly version of this material in another text? (At a roughly... |
When showing that $\mathbb{N}\times\mathbb{N}$ is in bijection with $\mathbb{N}$, it seems standard to give a proof by picture that shows a way to systematically weave through all the points in $\mathbb{N}\times\mathbb{N}$ and label each one as you go.
I know there is a polynomial expression for this method, given by ... |
This is an exercise from Alhfors Complex Analysis book- to show that an analytic function with a nonessential singularity at infinity must be a polynomial.
It seems like it should probably be pretty straight forward, but I must be missing something.
If it has a removable singularity at infinity then it extends to an an... |
Let $k$ be a field, let $V$ and $W$ be $k$-vector spaces of dimensions $n$ and $m$ respectively, and let $f:V\to W$ be a $k$-linear transformation. Let $\Lambda(V)$ and $\Lambda(W)$ denote the exterior algebras of $V$ and $W$ respectively. So we have
$$\Lambda(V) = \Lambda^0(V)\oplus\Lambda^1(V)\oplus\cdots\oplus\Lambd... |
Say if I define a power series over some arbitrary field $F$ as
$$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$
Then can I say:
$$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i + j} $$
|
Suppose $f$ and $g$ are two embeddings of $S^1$ in $R^3$ or in $S^3$. How do I show whether they form a link or not?
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What is
$$ \sum_{n=0}^\infty \frac{2n^7 + n^6 + n^5 + 2n^2}{n!} $$
|
Problem
Let $n = p_1.p_2.p_3 \cdots p_k.m^2$, where $p_1, p_2, p_3 \cdots p_k$ are distinct primes. Prove that n is sum of two squares if and only if $p_i$ is either 2 or $p_i \equiv 1 \pmod{4}$
For $p_i = 2$ , this is trivial case since $2m^2 = m^2 + m^2$.
For $p_i \equiv 1 \pmod{4}$, I tried to use the fact that ... |
Let $F: D \rightarrow D$ be a random function on finite domain $D$ of size $n$. It is well-known that, from any $x \in D$, iterating $F$ on $x$ traces out a sequence of values $x, F(x), F(F(x)), \ldots$ that must eventually repeat (since $D$ is finite) and resembles a Greek "$\rho$". The expected number of total elem... |
I'm looking at a specific derivation on wikipedia relevant to statistical mechanics and I don't understand a step.
$$ Z = \sum_s{e^{-\beta E_s}} $$
$Z$ (the partition function) encodes information about a physical system. $E_s$ is the energy of a particular system state. $Z$ is found by summing over all possible system... |
will the following converge if they then what are their sums
$\sum_{n=1}^\infty\left[\frac{\left(1+\frac{1}{n}\right)}{e}\right]^n$
$\sum_{n=1}^\infty\left[\frac{\left(1+{1}/{n}\right)^{n+1/2}}{e^n}\right]$
do the above series have any physical meaning (apart from fun).
|
To prove the convergence of the p-series
$$\sum_{n=1}^{\infty} \frac1{n^p}$$
for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.
I am wondering if there is a self-contained proof that this series converges which does not rely on either test.
I suspect that any proof would ... |
I'm confused about part B of the following exercise on basic integrals. I am asked to find the area of R2, the blue shaded area.
My method was to find the area of the area of the triangle (0,7), (0,0), (7,0) using $1/2bh$ or integration, then to subtract the area of the parabola between $x = 0$ and $x = 2$ (intersectio... |
Let $\Lambda$ be a Dedekind domain and $\mathcal{m}$ be a maximal ideal of $\Lambda$.
Is it possible that $\mathcal{m}=\mathcal{m}^2$?
If not, how can I prove it?
|
this is an educational question.
can someone please explain with some simple examples:
(1) what is/are the spectrum of operator
(2) where it is useful
For providing examples of spectrum of operator, please also consider the following operator, matrix (A), differential operator, integral operator, .....
thank u.
|
Can one find a number $m$ such that $\sin^2 n \geq m > 0 $ for all integers $n$?
By continuity of $\sin x$, it is enough to say that $|n - k \pi| \geq m^{\prime} > 0$ for all integers $n,k$.
Since $\pi$ can be approximated arbitrarily closely by rationals, for any $\epsilon > 0$, we can find integers $n$ and $k$ such t... |
If $X$ is a CW complex, show that there is a chain map $W_*(X) \to S_*(X)$ inducing isomorphisms in homology.
Here $W_p(X) = H_p(X^p,X^{p-1})$
Let $E$ be the CW decomposition of $X$ and let $M$ consist of one point chosen from each cell in $E$. In brief, we start with the triple $(X^k,X^k-M,X^{k-1})$, and take the long... |
How many (equivalence classes of) periodic functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$, of period $N$, satisfy
$$\sum_{i \le n \le j} f(n) \in \{-1,0,1\}$$
for all pairs of integers $(i,j)$?
I consider two functions equivalent if they are related by a simple shift, hence "(equivalence classes of)".
I wrote a comp... |
Main problem:
Let $\mathcal{L}=\left\{z\in\mathbb{C}:Re(z)<0\right\}$ be the left open half-plane of the complex plane and $\mathcal{C}=\left\{z\in\mathbb{C}:|z|<1\right\}$ be the open unit disk of $\mathbb{C}$. Let $p\in\mathbb{P}_n$ be a polynomial of degree $n\in\mathbb{N}$. Can I have a geometric representation of ... |
I am trying to show that elements of the general Möbius group generated by an affine transformation $f(z) = az+b$, the inversion map $f(z)=\frac{1}{z}$ and complex conjugation $f(z)=\overline{z}$, where $a,b \in \mathbb{C}$, are conformal maps.
So if I have a curve lying in the $x$-$y$ plane, then its image in the $u$-... |
I would be very happy if it covers sequence and series from very basics to advanced. Thanks.:)
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Can someone enlighten me with the question in the next page:
http://www.physicsforums.com/showthread.php?p=3208664#post3208664
I am asked to find all the modular forms with weight $k$ which don't have zeros on the upper half plane.
|
Let $X$ be a proper scheme defined over an algebraically closed field of characteristic $p > 0$. Let $F : X\rightarrow X$ be the absolute Frobenius morphism. What is the dimension of $H^0(X, F_*\mathcal{O}_X)$?
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Do translations form a normal subgroup of the group of rigid motions in a general Euclidean plane with no underlying field?
This is a question that has puzzled me for the past few days. In the wikipedia article for normal subgroups it is stated that the translation group is a normal subgroup of the Euclidean group. Whe... |
Possible Duplicate:
Will moving differentiation from inside, to outside an integral, change the result?
In his book 'Surely you're joking mr Feynman' author mentions a trick he found in a book while still in high school. He calls it 'differentiation under integral sign' and claims that it helped him solve integrals... |
Solving some problem I have stumbled into the following sum :
$ \displaystyle \sum_{i=0}^{n-2} {e \choose i} (n-1-i) (1-p)^{e-i} p^i $
where $ 0 \leq e \leq {n \choose 2}$.
I am not very efficient with the evaluation of these sums so I would like to ask if there is any way to evaluate this sum or obtain a sharp lower ... |
I would like an example of a map $f:H\rightarrow R$, where $H$ is a (infinite dimensional) Hilbert space, and $R$ is the real numbers, such that $f$ is continuous, but $f$ is not bounded on the close unit ball $\{ x\in H : \|x\| \leq 1\}$.
Actually, $H$ could be replaced by any Banach space (but not just a normed space... |
The vectors $v_{1},\dots,v_{n}$ form a linearly independent set.
Addition of vector $u$ makes this set linearly dependent.
How to prove that $u \in span\{v_{1}\ldots,v_{n}\}$.
I was able to prove the reverse.
|
I would like to find out the formula for CandidateAbility used in the European PISA-test, which tests 9th grade pupil's abilities. Unfortunately the agency which publishes the results does not provide many mathematical facts. They say they use a logit-function to determine pupils' abilities in terms of percentage of co... |
Can anyone explain to me why a cycle basis hones the properties of a matroid? Especially points 2 & 3.
How can a subset of I also be a member of I? Isn't a cycle basis supposed to be consisted of cycles and cycles only? If the cycle basis consisted a triangle, wouldn't a subset of the triangle become an edge? And by po... |
Does anyone know any good source with nice examples of Hilbert class field computations? I'm trying to piece together the theory with some canonical examples.
|
I'm given the following in a homework question:
Let $G$ be a group and $k$ an algebraically closed field.
(a) Show that the action of $G \times G$ on $C_k (G)$ defined by
$$ (g_1, g_2) \varphi (x) = \varphi(g_1^{-1} x g_2) \hspace{1cm} g_1, g_2 , x \in G , \varphi \in C_k(G)$$ defines a representation $\pi$ of $G \tim... |
I have just begun to learn about topological group recently and is still not familiar with combining topology and group theory together.
I have read a useful property of discrete group on the wikipedia:
every discrete subgroup of a Hausdorff group is closed
But I have no idea how to prove it. I find that it cannot be p... |
$$\begin{align*}
|x|=x &\text{if }x\geq 0\\
|x|=-x &\text{if }x\lt 0.
\end{align*}$$
Show that $|xy|=|x||y|$.
I try to prove it as follows:
$|xy|=xy$ if $xy\geq 0$,
but $xy\geq 0$ if and only if $x\geq 0$ and $y\geq 0$.
Since $x\geq 0$ and $y\geq 0$, then
$x=|x|$ and $y=|y|$
hence from $|xy|=xy$
we have $|xy|=... |
I have been studying a particular ergodic system and it has become apparent that solutions to differential equations of the form
$M(x,y,z,t)\begin{pmatrix}
\frac{dx}{dt} \\
\frac{dy}{dt} \\
\frac{dz}{dt}
\end{pmatrix} = \begin{pmatrix}
f(x,y,z,t) \\
g(x,y,z,t) \\
h(x,y,z,t)
\end{pmatrix}$
where $M$ is a $3\time... |
Show that
\begin{equation}
\rho(\mathbf{x},\mathbf{y}) = \sqrt{\sum\limits_{i,j=1}^n a_{ij} (x_i-y_i)(x_j-y_j)},
\end{equation}
with $a_{ij} = a_{ji}$, $\mathbf{x} = (x_1,\ldots \;, x_n)$ and $\mathbf{y} = (y_1,\ldots \;,y_n)$, satisfies the triangle inequality.
$[a_{ij}]_{n \times n}$, by the way, is a symmetric, po... |
Let $f,g: \mathbb R \to \mathbb R$ be continuous. How may I show that the following are also continuous functions
$f+g$
$fg$
$f\circ g$
|
I thought I had pretty much figured out the difference between $\equiv$ and $=$. Then I came across this while reading about partial derivatives (in Colley's Vector Calculus):
$$
\frac{\partial^2f}{\partial z^2} = \frac{\partial}{\partial z} \left(\frac{\partial f}{\partial z}\right) = \frac{\partial}{\partial z} (y^2... |
I've been trying to solve a system of differential equations on Mathematica, and I wanted to know why this implementation does not work:
f[t_]:= {x[t],y[t],z[t]}
v[t_]:= f'[t]
A = {a,b,c}
DSolve[{f''[t]==k Cross[v[t],A],f[0]=={0,0,0},f,t]
If I write the system explicitly, on the other hand, it works; why is that?
By... |
According to "Concrete Mathematics" on page 434, elementary asymptotic methods show that $\displaystyle \sum_{k=1}^{n-1}\frac{k! \; k^{n-k}}{n!}$ is asymptotically $\sqrt{\frac{\pi n}{2}}$. Does anybody see how to show this?
|
Suppose that a sequence $\{x_k\}$ has a clusterpoint (or more..) $c\in\mathbb{R}$. what conclusion, if any, can be drawn about either $\liminf x_k$ or $\limsup x_k$ ?
I don't know the conclusion... I think that when the sequence has only one cluster point
then $\liminf x_k= \limsup x_k = c$. But if the sequence has m... |
Are there any functions, $f:U\subset \mathbb{R}^n \to \mathbb{R}$, with Hessian matrix which is asymmetric on a large set (say with positive measure)?
I'm familiar with examples of functions with mixed partials not equal at a point, and I also know that if $f$ is lucky enough to have a weak second derivative $D^2f$, t... |
Let $S$ be a (commutative) semigroup with distinguished element 0 such that $ab=0$ for $a,b\in S.$ Of course this is a very simple family of semigroups, defined only by their cardinality. Does it have a name? Would it be reasonable to call these nilpotent (of order 1)?
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Is it true that $\{a\}$ is closed but not open for any real number $a$?
For me, I could first let a set $A=\{a\}$,
and let $C=\{\text{collection of all subsets of }A\}$.
Since the definition of topological space requires that both the empty set and $A$ belong to $C$, thus I can conclude that $A$ must be open.
Now, $A$ ... |
In Waclaw Sierpinski's book Elementary Theory of Numbers on page 168 there is the following exercise:
"Exercises. 1. Prove that for natural numbers $n$ we have $d(n) \leq 2\sqrt{n}$," where $d(n)$ is the number of divisors of n.
As a hint right below is given: "The proof follows from the fact that of two complementary ... |
I was reviewing my old Discrete Mathematics notes, and I came across a section describing how Partial Orderings are identified. I understand this, but I can't seem to recall/find information on why partial orderings are important.
What is the significance of knowing whether a set is partially ordered?
|
How to compute a linear fractional transformation that maps a circle to a given circle? For example, let $C_1$ be the circle $|z+(2+i)|=1$ and $C_2$ be the circle $|z-5|=7$. How to find a linear fractional transformation that maps $C_1$ to $C_2$? Thank you very much.
|
I'm not 100% sure if I set up the following problem right. Once I have the problem set up I know how to solve it. (this is a homework problem)
The problem goes like this:"A company can use plastic, aluminum, or a combination of the two to make one of its products. The company cannot use more than 20 units of plastic an... |
I was looking for approaches on how to adequately interpolate the values for a continuous 3D function for which I have the exact values in a 3D grid of equidistant points. I found that linear interpolation in 3D works well but I lose some precision. So I found Gauss interpolation, mentioned here
http://mathworld.wolfra... |
Is there any geometric interpretation for the following second partial derivative?
$$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$
In particular, I'm trying to understand the determinant from second partial derivative test for determining whether a critical point is a minima/maxima/saddle points:
$$D(a, b) = ... |
Help please
Let $(a_{n})_{n>o}$ and $(b_n)_{n>o}$ be sequences of complex numbers and
suppose that:
(i) The sequence of partial sums $S_{m}=\sum_{n=0}^{m}a_{n}$ is bounded
(ii) $\mathrm{lim}b_{n}=0$
(iii) The sum $\sum_{n=1}^{\infty} |b_{n} - b_{n-1}|$ is finite.
Prove that the series $\sum a_{n}b_{n}$ is convergent. ... |
On my homework today, we had to show that Tr(n,R), the translational group on R^n, is a normal subgroup of M(n,R), the group all Isometries on R^n. In fact, the quotient group M(n,R)/Tr(n,R) is isomorphic to O(n,R), the real orthogonal group.
What homomorphism can have Tr(n,R) as its kernel?
This is easily undrestandbl... |
I'm trying to understand how to handle power series who use floor or ceiling functions in their general term.
For example, the power series $\displaystyle{\sum_{k \geq 1} \left\lfloor \frac{2^k}{(k+1)^2}\right\rfloor}x^k$ is supposed to have a radius of convergence of $\frac{1}{2}$ but I fail to see how to handle this.... |
Possible Duplicate:
Does .99999… = 1?
Example 1.6 on page no 9 in "Elementary Real Analysis" by Bruckner say that " the interval (0,1) has no maximum and minimum as 0 and 1 do not belong to the set."
My doubt is if it is an interval, all the numbers between 0 and 1 are included in the set so, .9999..... is maximum, ... |
I am posed with the following question.
Let $\left(V:F\right)$ be a two dimensional vector space.
$\beta = \{x_1, x_2\}$ is a basis of $V$ and $\beta^* = \{\phi_1, \phi_2\}$ is the dual basis of $V$. If $ \beta^{'} = \{x_1+2x_2\, 3x_1+4x_2\} $ is also a basis of $V$, find the dual basis of this in terms of $\phi_1$ ... |
As you already know, the interior of a circle is represented by an inequality. For example,
$$x^2+y^2\leq1$$
for the unit circle. Today I was thinking by myself and I wondered if there is a curve that could represent every point inside of a circle. Maybe with a spiral like this,
If you can't represent it perfectly wi... |
Given an invertible $3\times 3$ matrix:
$A = \begin{pmatrix}
1 & 2 & 2 \\
1 & 2 & -1 \\
-1 & 1 & 4
\end{pmatrix}$
I am trying to find $f(x)$ from $F[x]$ such that $A^{-1}=f(A)$. To do so, I want to use the result of a previous question, which says that $f(A)$ is invertible if and only if $f$ and the minimal polynom... |
Give an example on $\Omega = \{a, b, c\}$ in which
$E(E(X|F_{1})|F_{2}) \neq E(E(X|F_{2})|F_{1})$
--
Obviously X is a random variable and $F_{1}$ and $F_{2}$ are sigma-algebras... but I'm not even sure how to get started on the actual example. Any help is appreciated.
Thanks.
|
Let $i_{q0} : M\rightarrow M\times N$, $i_{q0}(p) = (p, q0)$ be a mapping between smooth manifolds. I need some hints to show that it is $C^{\infty}$.
I have so far... Let $(U,\phi)$ and $(V,\psi)$ be charts about $p$ and $i_{q0}$, and let $r^{i}$ be the $ith$ coordinate function on Euclidean space. Then we need to ... |
Suppose we have a bunch of objects with value, all of which are a power of $2$.
So $1, 2, 4, 8, 16, ...$
We are given some value $x \ge 1$ and we want to find the minimum amount of objects we can use that add up to this $x$.
I want to show that to obtain the minimum amount of objects, I have to pick the biggest possibl... |
Background
The use of tools from algebraic topology to study simplicial complexes coming from point cloud data has been thoroughly discussed in the papers of Carlsson, Zomorodian, Ghrist, Edelsbrunner, Harer and more. Computing homology can be achieved in cubic time (in simplicies) and using persistent homology one can... |
X and Y are independent random variables. f is a function with E|f(X,Y)| <
∞. show E(f(X,Y)|X) = E(f(X,Y))
|
Suppose I have a function $f_n$ = $\frac1n$ $\chi_{[-n,n]}$. I read somewhere that this function is certainly uniformly convergent but not convergent in measure. How is that? Can I get a better explanation? I would really appreciate some input on this.
|
Which functions satisfy $\forall n,x,y (x \equiv y \pmod n \implies f(x) \equiv f(y) \pmod n )$ ?
I know polynomials do; are there others?
|
I'm in differential equations without having taken higher level calculus first
I understand differentiation and integration pretty well but I get surprised calculus-wise periodically.
For example, how is it that $\int{e^{xt}dt} = \frac{e^{xt}}{x} + c$? I must be missing something!
|
Today in Coding/Cryptography class, we were talking about basic definitions, and the professor mentioned that for a set $A=\left \{ \left. a, b, \dots, z \right \} \right.$ (the alphabet) we can define a set $A^{*}=\left \{ \left. a, ksdjf, blocks, coffee, maskdj, \dots, asdlkajsdksjfs \right \} \right.$ (words) as a s... |
let $1 = \frac{1}{p} + \frac{1}{q}$ as usual. Let $f \in L^p, g \in L^q$ be vector fields from $\mathbb R^n$ to itself.
Assume $div f = 0$ and there exists a function $G$ s.t. $\nabla G = g$. Then $f \cdot g \in \mathcal H^1$ is a Hardy space function.
Do you know where I can find a proof of this conclusion? I am aware... |
I know the order of the group is the number of elements in the set. For example the group of $U_{10}$ (units of congruence class of 20) has order 4.
Major Edit, kinda changed the question.
Lets say my element $a$ has a finite order $n$. Then what is the order of $a, a^2, a^3...a^{11}$?
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There is a proposition that states the following: Assume $E$ has finite measure. Let {$f_n$} be a sequence of measurable functions that converges pointwise a.e. on $E$ to $f$ and $f$ is finite a.e. on $E$. Then {$f_n$} converges in measure to $f$ on $E$. How do you show that this fails if $E$ has infinite measure?
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This is a more practical question. In Pari/GP I have difficulties to use mateigen for some real matrices because of extreme long computation time and frequently "missing eigenspace" due to too low float precision. Say, the (real) matrix M is $48x48$ (a modified Vandermonde matrix), float precision of $1200$ decimal dig... |
This is an exercise from Remmert's Theory of Complex functions.
Let $D\subset \mathbb{C}$ be a domain and $f:D\rightarrow \mathbb{C}$ a real-differentiable function. Assume that the following limit exists:
$ \mathrm{lim}_{h\rightarrow 0} \left| \frac{f(c+h) - f(c)}{h} \right|.$
Show that either $f$ or $\overline{f}$... |
It is well-known that if you have an inverse system of abelian groups $(A_n)$ (this works in several other nice categories) in which all the maps are surjective (or at least satisfy the Mittag-Leffler Condition), and if you have a short exact sequence of inverse systems $0\to (A_n)\to (B_n)\to (C_n)\to 0$, then taking ... |
How can I compute Cohomology of complex projective plane $CP^2$?
Any magic like the one here?
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From the book,
Suppose $p \equiv 1 \pmod{4}$, then by law of quadratic reciprocity, we have:
$$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) $$
Next, if $p \equiv 2 \pmod{3}$, then $p \equiv 5 \pmod{12}$
Hence, $$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) = -1$$
How do they get those Legendre fraction ... |
The following problem was asked in the CMO 2011 and I'd be interested in finding various solutions for it. Here's the problem:
Fix a positive integer $d$, then for any integer $k$ there exists a positive integer $n$ and integers $\epsilon_i$ where $\epsilon_i = \pm 1$ for $i=1 \ldots n$ such that:
$$k = \sum_{i=1}^n \... |
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