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We define the following subgroups of the automorphism group of $k[x_1, \ldots, x_n]$
E(k,n) is the subgroup generated by elementary $k$-automorphisms of $k[x_1, \cdots, x_n]$ of the form $(x_1, x_2, \cdots, x_{i-1}, x_i + a, x_{i+1}, \cdots, x_n)$ where $a \in k[x_1,x_2, \cdots, x_{i-1},x_{i+1}, \cdots, x_n]$
Aff(k,n) ... |
I've stumbled upon the definition of exact sequence, particularly on Wikipedia, and noted the use of $\hookrightarrow$ to denote a monomorphism and $\twoheadrightarrow$ for epimorphisms.
I was wondering whether this notation was widely used, or if it is common to define a morphism in the general form and indicate its c... |
I'm well aware of the standard proof based on cardinality arithmetic to show that these two sets have the same cardinality and the question of defining a bijection between the two sets came up. I worked on it a little while and was wondering if there were any gaps, or a simpler method of coming up with such a bijection... |
Suppose we're given a Dihedral group $D_{30}$.
a) Find a cyclic subgroup H of order 10 in $D_{30}$. List all generators of H.
b) Let k and n be an integer such that k >= 3 and k divides n. Prove that $D_n$ contains exactly one cyclic subgroup of order k.
My attempt:
a) in $D_{30}$ we know that |r| = 30. So we can fin... |
Let $D$ be a PID, $E$ a domain containing $D$ as a subring. Is it true that if $d$ is a gcd of $a$ and $b$ in $D$, then $d$ is also a gcd of $a$ and $b$ in $E$?
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(Inspired by this question on the photography site)
Say you have N copies of the same signal, each with a layer of noise on top. You average these copies together in an attempt to reduce the effect of noise. How much noise will the average have?
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Let $\{Z_i\}$ a directed projective system of quasi-compact topological spaces with projective limit $Z$. Assume we are given open subsets $U_i \subseteq Z_i$ such that:
1) For every $i \leq j$, the preimage $(Z_j \to Z_i)^{-1}(U_i)$ is contained in $U_j$.
2) For every $i$, the preimage $(Z \to Z_i)^{-1}(U_i)$ equals $... |
I would like to apply Seifert van Kampen to a simple example taken from Wikipedia: I have $X = S^2$ and $A = S^2 - n$, where $n$ is the north pole and $B = S^2 - s$, where $s$ is the south pole.
According to my understanding, which might be wrong, Seifert van Kampen tells me that $\pi_1(X) = \pi_1(A) *_{\pi_1(A \cap B)... |
I want a convincing proof of this...
$\nabla^2 (1/|r-r'|)$ = $-4\pi \delta^3 (r-r')$
(where r and r' are vectors and $\delta^3 (r-r')$ represents the 3-D delta function...)
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Is it possible to have a general expression for the binary sequence ?
Eg.
for N= 3, and i=0, M_i = diag(0,0,0) i=1, M_i = diag(0,0,1) . . i=7, M_i = diag(1,1,1)
I need a general matrix (M_i) or expression whose arguments are N,i.
P.S: I am not looking for a software code to generate this.
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I want to take the inverse of a dot product, where both vectors have complex components. In other words, if $\textbf{A} \cdot \textbf{B} = d$, and I know $\textbf{A}$ and $d$, I want to find a $\textbf{B}$. I know that I cannot do so uniquely, which is fine; I have a procedure for creating a set of vectors that will ... |
Let's define a stair case polygon to be a polygon look like the following.
Let the lower left most point to be the origin, all edges are axis-aligned.
One can see if we remove a stair case polygon from a rectangle, we get another stair case polygon.
Let $S_k$ be some $k$ vertices stair case polygon.
Let $A(x)$ be the ... |
I am looking for an (almost everywhere) differentiable function $f(p_1,p_2,p_3,p_4)$ that given four points will give me a scale-agnostic measure for co-planarity. It is zero if the four points lie on the same plane and positive otherwise. Scale-agnostic means that, when I uniformly scale all points the planarity measu... |
Let $G$ be a finite group and $P$ is a p-Sylow subgroup of $G$.
Show that $N(N(P))=N(P)$ where $N(P)$ is the normalizer.
Not sure where to begin... Any hints will be appreciated.
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I'm doing some exercises on group-theory but got stuck in the following question:
Let $G$ be a group such that for all $a \in G\setminus\{e\}$, $a^2 = e$. Prove that $|G|$ is even.
I tried using Lagrange's theorem but perhaps I still don't understand it fully for I was not capable of finishing it.
Could you shed some... |
I am trying to measure complexity of the following code segment
int sum = 0;
for (int i = 1; i <= n/2; i++)
{
sum++;
}
As far as I understand it can be represented by following sum $$\displaystyle\sum_{i=1}^{n/2}1$$ but I don't know how to evaluate it.
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How to solve $\int\tan^3(x)\,dx$ ? Do I use substitution?
|
What's statistically more likely to win you the UK lottery (6 numbers out of 1-49),
To play the lottery weekly or to save the money and buy multiple tickets
all in a oner?
There is a discussion in the office where a group of 6 people want to play weekly.
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My algebra books are in storage and Google is not being helpful...
In figure 6.5.5, how do I apply the angle operator to the matrix columns to determine a value for theta?
http://www.w3.org/TR/SVG/implnote.html#ArcImplementationNotes
Thanks for any pointers,
Paul
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This question came up on a pub quiz last night. The answer was parabola.
I disagreed, because I thought I remembered that angles could go off into infinity too. An angle is two rays whose endpoints are shared, so would that also count as a "bend" ? Or is there another name for this type of angle?
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Here is a question I've often wondered about, but have never figured out a satisfactory answer for. Here are the rules for the solitaire game "clock patience." Deal out 12 piles of 4 cards each with an extra 4 card draw pile. (From a standard 52 card deck.) Turn over the first card in the draw pile, and place it under ... |
This is targeted for elementary kids. One way to think is that add any number to itself zero times... Just wondering if there is a better way...
|
I tried to start by showing that $ \frac{a}{\gcd(a,b)} $ is always an integer, let's call it $d$, because $a$ is always a multiple of $(a,b)$ based on the definition of a g.c.d. I then tried to show that if $a | b$ then $a|bc$ from the theorem that states that $a|b$ implies that $a|bc$ for any integer $c$, but I cant s... |
This is a tall order, but since this site is for everything mathematical, here goes:
What are some nice games/puzzles/problems/problem-like activities that would be appropriate for a three-five year old?
I ask this because I'm not particularly interested in trying to accelerate "number literacy" or prepare for "school ... |
I want to solve system of linear equations of size 300x300 (300 variables & 300 equations) exactly (not with floating point, aka dgesl, but with fractions in answer). All coefficients in this system are integer (e.g. 32 bit), there is only 1 solution to it. There are non-zero constant terms in right column (b).
A*x = b... |
I would like to know, how do you simplify this:
$$\cos x\sin(x+y) + \sin x\cos(x+y)$$
to this:
$$\sin(2x+y).$$
Wolfram alpha says so, but how does human being do so? :)
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For some reason, I can't find a reference for $\pi_i GL(n,\mathbb C)$ nor can I figure what they are. For most Lie groups, you can get a nice fibration and use the long exact sequence in homotopy to inductively compute the homotopy groups (e.g. the fibration $SO(n-1) \to SO(n) \to S^{n-1}$). However, I can't think of... |
Let $f$ be a holomorphic function on an open, connected set $\Omega\subset \mathbb{C}$ with $z_0\in \Omega$ a fixed point, and $\{f^n\}_{n\in \mathbb{N}}$ the sequence of iterates.
I want to prove that if $|f'(z_0)|<1$ then there is a neighborhood of $z_0$ such that $\{f^n\}$ is normal in it.
I don't really know what t... |
The ideals $I=(X,Y)$ and $J=(X^2+Y^2)$ in $\mathbb R[X,Y]$ are such that $V(I)=V(J)$ and their radicals aren't the same contradicting the Nullstellensatz (in case it was true for arbitrary fields). However, this shouldn't be a surprise, if we look at their varieties in $\mathbb C$ we find that they aren't the same as ... |
I am already able to prove that $g \mid s$ assuming $x+y=s$ and $(x,y)=g$, but I am having some trouble showing that assuming $g \mid s$, there exists an $x$ and $y$ such that $x+y=s$ and $(x,y)=g$. So far, I've started with saying that $g \mid 0$ necessarily. Therefore we also know that $g \mid (0x+sy)$. I'd like t... |
I have been trying to read a lot of literature concerning the above subject but I've not found anything useful to help my case.
Suppose you're given a linear diophantine in $a_1,a_2,\ldots,a_k$ where $k\leq 10$, and we are asked to tell if $a_1 x_1 + a_2 x_2 + \cdots+a_kx_k = N$ has a non negative solution or not?
We a... |
Let us assume that we have an infinite line of people, and each person can either move forwards or remain at the same place. They move only one step at a time. (They are jumping from one position to the next if that position is empty). All people either move or remain still simultaneously.
Let us further assume that a ... |
I need to graph the error between a set of observed
values and an expected values.
I calculate the relative error for each point as:
(observed - expect) / expected
This error however bounces up and down because the
behaviour is sinusoidal. I want to make this error
always a positive quantity. Is there an error measur... |
I was wondering how to prove that $$\lim_{n\to \infty}\int_{1}^{n}\frac{1}{(x^{2}+1)^{n}}dx\sim \frac{1}{n\cdot 2^{n}}?.$$
This appears to be asymptotic to $\frac{1}{n2^{n}}$, but how to prove it?.
I checked with larger and larger values of n, and it does get closer and closer to
$$\frac{1}{n2^{n}}.$$
i.e
$$\int_{1... |
Please suggest me how to compute the GCD of theese really big numbers :
GCD of $2^{120547564397}-1$ and $2^{356946681940}-1$
GCD of $2^n-1$ and $n!$ where $n=3^{19}$
Thanks to Bill Dubuque's answer I understood that the first problem could be solved by the property that $gcd(f(m), f(n)) = f(gcd(m,n))$ if $ f(n) \equ... |
I was given this on a practice exam:
Decide whether each of the following expressions is valid or invalid. Justify your answers (i.e.,if invalid, give an interpretation for which the expression is false;if valid explain why the expression is true for all interpretations.
(a) $[(\forall x)P(x) \vee (\exists x)Q(x)] \imp... |
During self-study, I ran across the question of how many ways six numbers can be chosen from the numbers 1 - 49 without replacement, stipulating that no two of the numbers be consecutive.
I can obtain a simple lower bound by saying that, in the worst-case scenario, when you choose a particular number, there are now thr... |
Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over $\Re^n$?
To start with, suppose $f: \Re^2 \to \Re$, such that for all $x \in \Re$, $f(x, .)$ is concave and for all $y ... |
I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, if we let $S_n$ act on $(\mathbb{C}^N)^{\otimes n}$ by permuting tensor components, so that e.g. $P_{(1 2)} (a \otimes ... |
Forgive me if this is elementary, but my analysis is quite rusty and I'm struggling to get back up to speed. Given a convergent series $\sum_{i=1}^\infty|x_i|^p$, does it follow that $(\sum_{i=1}^\infty|x_i|^p)^\frac{1}{p}$ also converges? If so, how do they relate, i.e. is one always greater than (or equal) to the oth... |
After taking measurements on the deflection of stars during the 1919 eclipse in Borneo...
It was found that:
n = 15
Mean = 1.4824 angular units (AU)
SD = 1.349237 AU
SE = .348371 AU
A) Test the hypothesis that Einstein was correct (that the offset was 1.70 angular units as predicted by his theory). If Isaac Newton was... |
I'm talking bout this phenomenon (The third one):
I would like to know if with sound waves, happen the same as with (water) waves.
So, If two speakers were emitting the same sound, would there technically be a point where you wouldn't hear. (I guess this would only happen if you heard with only one ear).
Is this true ... |
I'm a computer science student, I'm not a mathematician, I don't know anything about number or group theory.
I'm looking at RSA, and I want to understand it.
I know what Fermats's little theorem and Euler's totient function are, and this is what I've got:
$p, q$ primes, $n=pq$.
$d$ relatively prime to of $\varphi(n)=(... |
Let $k$ be a unital commutative ring, and A be a $k$-module.
Is there a homomorphism $f: A \otimes_k A \to \bigwedge^2 A$ such that $f(a \otimes a) \neq 0$ for some $a \in A$? I can take a hint :)
Motivation:
In his textbook on Lie algebra, Serre defines a Lie algebra as a k-module with a homomorphism $A \otimes_k A \t... |
I'm having a hard time understanding what exact processes I have to take to represent a function as a power-series.
I understand that $\frac{1}{1-x} = 1 + x + x^2 + x^3 + ... = \sum_{0}^\infty x^n$
So, in words, $\frac{1}{1-x}$ can be represented as the power series $\sum_{0}^\infty x^n$.
Jumping to the problem:
Repres... |
Is there a measurable set $A$ such that $m(A \cap B)= \frac12 m(B)$ for every open set $B$?
Edit: (t.b.) See also A Lebesgue measure question for further answers.
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Find an equation of the plane that passes through the line of intersection of the planes $x-y=1$ and $y+2z=3$ and is perpendicular to the plane $x+y-2z=1$.
I'm having a hard time with this. I'm very confused by this section in the book, as the definitions and equations are all over the place. Can anyone give me any hin... |
Either works when trying to find the eigenvalues, but only the former works when trying to find corresponding eigenvectors. I can understand how it makes a difference, but what I don't understand how one is supposed to "know" the former is the "correct" form, since it starts from here:
$AX = λX$
And from there, you can... |
I am creating a spreadsheet to calculate Adword formulas and I am stuck in how to calculate the Monthly Net Income for each keyword. I have created a formula which calculates it but can't figure out how to limit the Monthly Budget.
The formula I've created is this one:
Monthly Net Income = ( DailyClicks x ConversionRa... |
If $\varphi:R^{m}\to R^{n}$ is an epimorphism of free modules over a nontrivial commutative ring, does it follow that $m \geq n$?
This is obviously true for vector spaces over a field, but how would one show this over just a commutative ring?
-----Edit
Is there any way to use the following?
If $\varphi : M \to M'$ is... |
Find the value of n so that the error obtained by approximating sin(x) by the nth degree Maclaurin polynomial Tn(x) on the interval [-0.5,0.5] will be less than 5 x 10^-6.
I'm really stuck and don't even know where to start.
Any help would be appreciated.
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OK, so I am choosing 4 letters/numbers out of 10 possible letters. These letters are X,Y,LT,LB,RT,RB,U,D,L,R. The first three letters can be any one of these, but must contain at least one X or Y. The last letter can be an U,D,L, or right. Can someone please figure out the possible combinations?
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I have a question concerning the proof of theorem 3.5 in Milnor's Morse Theory.
This theorem states that if $f$ is a differentiable function on a Manifold M with no critical points, and if each $M^a = \{x\in M | f(x)\leq a \} $ is compact then $M$ has the homotopy type of a CW-complex with one cell of dimension $\lambd... |
Suppose $R$ is a commutative ring, $f\colon F_1\to F_2$ is a homomorphism of free modules, and $M$ is an $R$-module.
If $f$ is a surjective homomorphism, then $f\otimes_R \mathrm{id}_M$ is surjective.
Is it true that if $f$ is a monomorphism, then $f\otimes_R \mathrm{id}_M$ is a monomorphism? I can't think of any count... |
Define $D^+f(x) = \limsup\limits_{h\to 0^+}{\left(\dfrac{f(x+h)-f(x)}{h}\right)}$. Given the function $f(x)=\begin{cases}x\,\sin{\left(\frac{1}{x}\right)} & x\neq 0\\ 0 & x=0\end{cases}\,,$ find $D^+f(x)$.
There are also three other approximate derivatives to work out (lim inf from the right, lim sup and lim inf from t... |
I am working on a project that needs to be able to calculate the probability of rolling a given value $k$ given a set of dice, not necessarily all the same size. So for instance, what is the distribution of rolls of a D2 and a D6?
An equivalent question, if this is any easier, is how can you take the mass function of ... |
I have a matrix-vector inner product multiplication $G = X D x$ where $D$ is a diagonal matrix. Now let's say I already know $E = Xx$. Is there a method that I can use to change $E$ into $G$ using $D$ without having to calculate $G$ in full?
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I have shown that two of the three elementary operations will not change the image of the row space of the matrix: given a row vector $\vec{v}$, $k\vec{v}$ will span the same (scalar multiplication), and switching the order of the rows won't change anything because the span of the row vectors will be the same (vectors ... |
I was just browsing through Ravi Vakil's website when i found a nice article written on what he demands from his students. Here is the webpage. (For those interested!)
http://math.stanford.edu/~vakil/potentialstudents.html
I actually found a line which caught my attention. He says:
"Mathematics isn't just about answ... |
I was wondering why an integral domain is required to be commutative and have a multiplicative neutral element.
For example the quaternions would be a non-commutative integral domain.
The interesting property is that it has no zero divisors, so why require it to be commutative and have a $1$?
In other words: what conse... |
How do I get the solution given by CRT to match another solution, e.g. the least positive?
For example say I have X = 1234. I choose mi as 5, 7, 11, 13. This satisfies the simple requirements of Mignotte's threshold secret sharing scheme. More precisely given in my example k = n = 4, and the product of any k - 1 is sma... |
How do I prove that dot product of two free vectors ( inner product in 2D and 3D vectors) does not depend on the choice of frames in which their coordinates are defined?
|
Let $R$ be a commutative ring, and let $M,N,P$ be $R$-modules. In what generality can we say that $Hom_R(M,N\otimes_R P)\cong Hom_R(M,N)\otimes_R Hom_R(M,P)$. This is true in a cartesian monoidal category, and from that, it seems like we might be able to force it if all of the modules are free (look at basis-to-basis... |
Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$
defined as composition of convolution and multiplication,
$Tf := (af) * g$ where $g$ is in $L^2$ and $a$ is a bounded function.
Can we find the spectrum of $T$?
For $a$ identically equal 1, the spectrum is the essential range of the
Fourier transf... |
I know this is a very standard question widely popular in the Internet and the Mathworld.
I myself have solved the above problem is $N^2 \log N$ avoiding floating arithmetic.However, can anyone give me a good resource/detailed explanation of how it can be solved in $O(N^2)$ using point line duality concepts.
Sorry to c... |
Let $R$ be a CRing and let $M,N$ be $R$-modules. Let $M^*:=Hom_R(M,R)$. I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces:
$M^*\otimes_R N\cong Hom_R(M,N)$.
I can give a proof of this using a basis, but in what generality can we say this kind of t... |
How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller words) but I stumble when this isn't satisfied. I'm wondering if there's a simpler way to computing all the possible permuta... |
I have the following problem:
In a triangle $ABC$ the line joining incentre and circumcentre is parallel to side $BC$. Prove that $\cos B + \cos C=1$.
Could someone help me solve it?
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I'm working with the universal and the existential quantifiers and I was wondering is there an order of operations that applies to them? What is the difference between
$(\forall x)P(x)(\exists x)Q(x)$
And
$(\exists x)Q(x)(\forall x)P(x)$
Lets say we have this relationship
(a) $[(\forall x)P(x) \vee (\exists x)Q(x)] \i... |
Firstly, I'm not a mathematician as will become evident in a quick moment. I was pondering some maths the other day and had an interesting thought: If you encased an integrable function over some range in a primitive with an easily computable area, the probability that a random point within said primitive also exists b... |
Let $f(x)$ be a decreasing, continuous function in $[0, \infty)$.
If $\int_{0}^{\infty}f(x)dx$ converges then $\int_{0}^{\infty}\sin(f(x))dx$ converges.
The only improper point is $\infty$ since $f(x)$ is continuous in $[0, \infty)$.
$f(x)$ has a bounded anti-derivate: if $\int_{0}^{\infty}f(x)dx$ converges, then ... |
I'm currently starting a course about Graph Theory, and I've been explained about the Havel & Hakimi degree sequence. However, I'm not too sure about it yet. First of all, I want to know if I understand it correctly, so I'm going to try to explain what I think the theory proves.
First, I've shamelessly stolen the follo... |
I'm returning from an exam on group-theory and there were 2 questions I couldn't solve (and still can't), so I'm asking here for any hint you could possibly give.
Let G be a group and H and K subgroups such that $|H| = n$, $|K| = m$ and $gcd(n, m) = 1$. Show that $H \cap K = \{e\}$.
I wish I could show you some of my... |
What is the relation between homogeneous spaces and principal bundles. I've been reading the two definitions and am left confused as to whether one is a subset of the other or whether no such relation exists.
|
In his book Lectures on the theory of algebraic numbers, Hecke says that the content of the quadratic reciprocity theorem, formulated and proved entirely in terms of rationals (integers) points beyond the domain of rational numbers.
He is talking about the algebraic numbers but how is it seen that it points elsewhere?
|
the question is exactly "why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?" Is this simply because in the normalization process we can have many irreducible component?
|
How could one prove the following?
Find a finite set of 3-connected graphs from which all can be constructed iteratively by the following operation, or show that no such set can exist. The operation consists of adding a new vertex to the graph H constructed so far and joinig it by t least three edges to some subdivisio... |
If it true that for every $b$-normal number (where the base $b,b^2,b^3,\ldots$ digits are asymptotically equiprobable) that the first differences of the base-$b$ digits, interpreted as a (signed) base-$b$ expansion, form a $b$-normal number?
It seems 'obviously' true but I wanted to check.
This is actually a practical... |
Let $\mathbb{H}$ be a Hilbert space on which $\Phi$ is a bounded linear functional. Define $f : \mathbb{H} \rightarrow \mathbb{R} : x \mapsto ||x||^2 + \Phi (x)$. Prove that each (nonempty) closed convex $K\subset \mathbb{H}$ contains a unique point at which the restriction $f|K$ is minimized.
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I've been thinking about this particular problem and I'm stumped.
For all $n g\geq 5$, show that there exists a graph, $G = \langle V,E \rangle$ such that all vertices of $V$ have degree of $4$.
I've tried using Havel-Hakami with a degree sequence of n-fours (ie: $[4,4,4,4,4,4,\ldots]$), but there were too many cases. ... |
I gave my trigonometry students the following example: Solve $\triangle ABC\ $ , where AC=0.923, AB=.387, and $\measuredangle A\ = 43.33^\circ\ $. First I found BC using the law of cosines, then I found $\measuredangle B\ $ using the law of sines, and $\measuredangle C\ $ using the sum of the interior angles of the t... |
Consider the quotient space obtained by identifying the north and south pole of $S^2$. I think the fundamental group should be infinite cyclic, but I do not know how to prove this.
If it is infinite cyclic, would this and $S^1$ be an example of two spaces which have isomorphic fundamental groups but are not of the sam... |
1.Let $A \in GL_n(\mathbb{C})$. Show that $\det(I+A)=1+\operatorname{tr}(A)+ \epsilon(A)$ where Modulas of epsilon(A) by norm of A=0 as A tends to 0,for any matrix norm.
If I define J(A)= det(A) for A is in GLn(C),then J is differentiable at all such A and that,if H is in Mn(C),then J'(A)(H)=det(A)tr(A*H). where A* is ... |
Again, this is a homework question, so (for the sake of learning) I'd be happy I get subtle hints only.
Anyhow, the setup is as follows: "$K=\mathbb{Q}(\alpha)$ is of degree n over $\mathbb{Q}$, where $\alpha$ is in the ring of integers $O_K$ of $K$, let the index of $\alpha$ be the index of $[O_K:\mathbb{Z}[\alpha]]$.... |
I am studying valuation theory on the way to local class field theory, and the texts I have looked at immediately focus on discrete valuations in developing the theory of nonarchimedean valuations. Why? Are there nondiscrete nonarchimedean valuations? If so, why do we ignore them? (it is true that if a field is locall... |
I asked myself a question during my first year and never find any interesting clue about it.
One of the first thing you learned in integral calculus is that $\int fg \neq \int f \int g$ in general. One can ask the following question : When do we have $\int_I fg = \int_I f \int_I g$ ? Can I find a maximal subset of $CM(... |
Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous map. Show that if $Y$ is compact then the function $g: X \rightarrow \mathbb{R}$ defined by $g(x) = \inf \{f(x,y): y \in Y\}$ is also continuous.
No clue here. Can you please help?
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Is there a way of go from $a^3+b^3$ to $(a+b)(a^2-ab+b^2)$ other than know the property by heart?
|
My situation is as follows: I have a symmetric positive semi-definite matrix L (the Laplacian matrix of a graph) and a diagonal matrix S with positive entries $s_i$.
There's plenty of literature on the spectrum of $L$, and I'm most interested in bounds on the second-lowest eigenvalue, $\lambda_2$.
Now the thing is tha... |
the only proof I know of that $\mathbb{R}^n$ cannot be homeomorphic to $\mathbb{R}^{n+1}$ requires quite a lot of mathematical apparatus, namely singular homology theory with its exact sequences and pretty abstract setting, as well as some previous (though admittedly not too difficult) calculations of the homology grou... |
I'm trying to simplify:
$$\frac{x^2}{x^2-4}-\frac{x+1}{x+2}$$
but I can't get to the answer:
$$\frac{1}{x-2}$$
How to do it?
|
In our matrices class we were given a problem that I'm having trouble with.
Let $E$ and $H$ be subspaces of $\mathbb{C}^n$ such that $\mathbb{C}^n = E \oplus H$. Construct a projection $\mathbf P$ such that $R(\mathbf P) = E$ and $R(\mathbf I − \mathbf P) = H$. Hint: set $F = H ^\perp$ , take a basis $\mathbf v_1, \dot... |
The fraction:
$$\frac{y^2-x^2}{x-y}$$
should simplify to the answer:
$$-(x+y)$$
but the best I could do was expand it to:
$$\frac{(y+x)(y-x)}{x-y}$$
What's next step?
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How does one find all integer solutions to this equation: $$ \biggl\lfloor{\frac{x}{1!}\biggr\rfloor} + \biggl\lfloor{\frac{x}{2!}\biggr\rfloor} + \cdots +\biggl\lfloor{\frac{x}{10!}\biggr\rfloor}=1001$$
Can't think about methods for solving this. Though i did something i am not sure of it.
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I'm working a little program that converges on vector-based approximations of raster images, inspired by Roger Alsing's genetic Mona Lisa. (I started on this after his first blog post two years ago, played a bit, and it's been on the back burner. But I'd like to un-back-burner it because I have some interesting further... |
I have the expression (for some $k$ and $r$ natural numbers):
$\sum_{l=0}^r {l \choose k}$.
Is there a way to bound this expression using a polynomial of degree which is linear in $k$ (or polynomial in $k$)?
I am pretty sure that each $l \choose k$ is a polynomial of the form $O(l^k)$.
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This is from a post in sci.math that did not get a full answer; I may repost it for the OP there:
I am interested on the issue I read in another site of when an embedding from a closed set extends into a homeomorphism, i.e., if $C$ is closed in $X$, and $f:C \rightarrow Y$ is an embedding, when can we extend $f$ to $F... |
please bear with me as I'm not a mathematician and this is difficult to word properly. :]
I need the equation for a logarithmic spiral (let's call it $S(\theta)$) that meets certain constraints for a music visualizer I'm working on. Let's call the arc-length of the spiral $A(\theta)$. I'm looking for a spiral that mee... |
The notation of my teacher confuses me in the title. If x is the same on the both sides, it seems trivial. So I suspect the x is different so it becomes:
$x_{1}\leq x_{2}$ for every $x_{1} \in S$
so what it would mean is that every $x_{1} \in S$ has a maximum element $x_{2}$ that does not need be in $S$. Attacking the ... |
Has anyone experienced using TI-89 BASIC programming to solve math problems? I want to create an array using TI-89 but I don't know how?
Update
I tried both
{0, 0, 0, 0}->A
and
[0, 0, 0, 0]->A
Basically, I want to create a boolean array to check for duplicate digits of a number.
Update 1
dup(n)
Prgm
{0,0,0,0,0,... |
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