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Motivation: In Spanier's Algebraic Topology we are introduced to direct and indirect limits in the first few pages. I partially understand the idea behind it but since I am an example-driven learner I'd actually like to compute some direct limits.
Question: Where can I find explicit examples of a direct (or even in... |
Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID such that $A\oplus B\cong C\oplus D$ and $A\oplus D\cong C\oplus B$. Prove that $A\cong C$ and $B\cong D$.
The only tool I have is the theorem about finitely generated modules, but I don't quite see the connection. Please Help. Thanks.
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I have the following series:
$\sum_{n=1}^{\infty} \frac{n}{\sqrt{n(n + 1)(n + 2)}}$
I must investigate if it is divergent or convergent. I have tried different approaches like the root test, but it didn't help me:(. Could someone point me in the right direction?
Best regards, Petar
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Let's say we have a value $K$. I want to find the value where
$A+B = K$ (all $A,B,K$ are integers) where $AB$ is the highest possible value.
I've found out that it's:
$a = K/2$; $b = K/2$; when $K$ is even
$a = (K-1)/2$; $b = ((K-1)/2)+1$; when $K$ is odd.
But is there a way to prove this?
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The more general version of this theorem in Munkres' 'Topology' (p. 290 - 2nd edition) states that
Given a locally compact Hausdorff space $X$ and a metric space $(Y,d)$; a family $\mathcal F$ of continuous functions has compact closure in $\mathcal C (X,Y)$ (topology of compact convergence) if and only if it is equic... |
Wikipedia tells me about Max Noether's $AF + BG$ theorem but only gives one reference and one external link. I've had a look at the MathWorld link but it seems to be an entirely geometric formulation which I'm not even sure is the same theorem. Does anyone know where I can find an algebraic statement of the theorem, to... |
I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise:
Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ matrix over $\mathbb C$ and $f(z) = \det(z-A)$ is the characteristic polynomial of $A$ then $f(A) = 0$. (This exercise wa... |
Let $\hat{f}(t) := \int_\mathbb{R} f(x) e^{-2 \pi i xt} d\lambda(x)$
I quote from my lecture notes:
Equation (3.4)
$$ sup_{t \in \mathbb{R}} | \hat{f} (t)| \leq || f ||_1$$
Remark:
In the language of normed vector spaces the inequality (3.4) means that the Fourier transform map $L^1 (\mathbb{R}) \rightarrow C_b (\mat... |
How can I find the minimum of $x^2+\frac{a}{x}$ on $\mathbb{R}_+$ without calculus?
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In differential calculus, We know that dy/dx is the ratio between the change in y and the change in x. In other words, the rate of change in y with respect to x.
Then, why is dy/dx written as d/dx ( y)?
That is, why and how d/dx is considered as an operator?
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I have some conditional questions. First, in the category of topological pairs (as used in cohomology theories), are these only pairs of the form $(X,A)$ with $A \subset X$?
If the answer to this question is yes, then my next question is: is a morphism $(X,A) \to (Y,B)$ a map $f:X \to Y$ with $f(A) \subset B$ or pai... |
Given a matrix $A$ and column vector $x$, what is the derivative of $Ax$ with respect to $x^T$ i.e. $\frac{d(Ax)}{d(x^T)}$, where $x^T$ is the transpose of $x$?
Side note - my goal is to get the known derivative formula $\frac{d(x^TAx)}{dx} = x^T(A^T + A)$ from the above rule and the chain rule.
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These four inequalities define a hollow (zero volume) cube. Is it possible to describe a hollow cube using a smaller system of (real scalar generalized-polynomial) inequalities?
\begin{eqnarray}
x^2 & \leq & 1 \\
y^2 & \leq & 1 \\
z^2 & \leq & 1 \\
(x^2 - 1)^2(y^2 - 1)^2(z^2 - 1)^2 & \leq & 0
\end{eqnarray}
EDIT: Afte... |
I have some exam questions that were left unanswered for me:
Suppose that for every $\alpha<\kappa$ there is a subset $A_\alpha$ of $\kappa$ of cardinality $\kappa$. Show that there is a subset X of $\kappa$ so that for every $\alpha<\kappa$ : |$A_\alpha\cap$ X|=|$A_\alpha$ \ X|=$\kappa$.
{$A_\alpha | \alpha<\aleph_1... |
To calculate an average in reverse polish notation (RPN) is the following correct (assuming a-j are variables representing numbers):
a,b,+,c,+,d,+,e,+,f,+,g,+,h,+,i,+,j,+,10,/
If it is correct, are there other ways that this could be expressed in RPN?
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I am having trouble in solving the following problem.
Let $p_{n}$ denote the $n$-th prime. Then prove that $$\pi(\sqrt{p_{1}p_{2}\cdots p_{n}})>2n$$ for $n \geq 6$.
No idea how to start.
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Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals
Let
$$g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$$
where
$$\chi_{(0,1]} = \left\{\begin{array}{ll}
1&\mbox{if $x-r_n \in (0,1]$,}\\
0&\mbox{otherwise.}
\end{array}\right.$$
Show that $g$ is not continuous except possibly on a... |
I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?
I tried to play around with it:
$x = (c - by)/a$, hence $a|(c - by)$.
$a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by =... |
How does one show that the prime spectrum of a domain is irreducible in the Zariski topology?
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I am not sure where to begin on this:
Determine the images of all conic sections with a focus at the origin under the principal branch of the complex square root.
I probably have to use the formula u(x, y) + i v(x, y) = sqrt((x, y)) and view square roots via the polar form.
Any and every suggestion is appreciated. Tha... |
I was doing some thought experiments for a game project, and while considering something related to pathfinding, this problem came into my head.
Say we have an infinite plane that is covered with an even random distribution of connected points, such that the average distance of a point to any of its connected neighbour... |
Let $M$ denote the additive group of $ \mathbb{C} $. Why is $ \mathbb{C}$, as a ring, not isomorphic to $\mathrm{End}(M)$, where addition is defined pointwise, and multiplication as endomorphisms composition?
Thanks!
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This could be mathematics or computer science, but also statistical physics, so I hope it qualifies for interest.
I am aware that there were reservations about the proof $P \neq NP$, but no fatal flaws. I have followed Terence Tao's blog and Tim Gowers, both of whom have reservations, but Deolalikar is sticking with hi... |
If we have some prime $p$ and a natural number $k$, is there a formula for the largest natural number $n_k$ such that $p^{n_k} | k!$.
This came up while doing an unrelated homework problem, but it is not itself homework. I haven't had any good ideas yet worth putting here.
The motivation came from trying to figure out ... |
I need some hints for this problem from Dummit and Foote. (edit: added the full question verbatim for context)
Let $p$ be an odd prime and let $n$ be a positive integer. Use the binomial theorem to show that $(1+p)^{p^{n-1}} \equiv 1 \pmod{p^n}$ but $(1+p)^{p^{n-2}} \not\equiv 1\pmod{p^n}$. Deduce that $1+p$ is an ele... |
I can't find an example of a countable injective module over a non-Noetherian ring.
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Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that:
the product of non-adjacent vertices is constant.
the greatest common divisor of non-adjacent vertices is constant.
Below is one such hexagon. As an example, here we have that $4 \cdot 10 \cdot ... |
I got the following question by mail from someone I don't know from Adam. (Quoted in part.)
if $f(t)$ continuously diff. on $[0,1]$ and
a) $\int_0^1f(t)\ dt=0$
b) $m\le f\,'\le M$ on $[0,1]$
Prove
$\frac m{12}\le\int_0^1t\cdot f(t)\ dt\le\frac M{12}$
I suspect it might be an error
I assumed immediately that it's an e... |
How would I start in solving this recursive sequence? Thanks!
Sequence: 3, 5, -2, 7, -9, 16, -25, 41, ...
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I am referring to a paper which is over here:
www.scipub.org/fulltext/jcs/jcs410879-889.pdf
The paper is not hard to read, but I think notation is rather sloppy.
I am specifically referring page 880, "Materials and Methods" in the beginning.
Elementary sets are defined as the equivalence classes of some equivalence rel... |
Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I remember that this is true but can't seem to prove it myself or find the proof anywhere.
I would be especially happy if t... |
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the entries of M are
$$m_{r,c}=\frac{eul(r,c)}{r!} $$ and $ eul(r,c)=\displaystyle\sum_{k=0}^{c}(-1)^k \binom{r+1}{k} (c+1-k)^r ... |
This is the solution to the problem. However, I don't understand how they got to the part that says "and the parallel vector is $i+4j$..". What does this mean and how did they derive that?
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Take
$$ \frac{x-a}{b-a}(b) + \frac{b-x}{b-a}(a) = x.$$
I need an interpretation of this, using concrete example.
well i don't have difficulty doing it and is not HW. I'm ust drawing a blank right now trying to interpret this. example would be on a number like 2-9 i want 6 then applying this $(4/7)(9)+(3/7)2 =6$. I'm ... |
I have an optimization problem that at first sounds quite textbook. I have a convex objective function in $D$-dimensional space that is twice differentiable everywhere and has no local optima.
Ordinarily it would be a perfect candidate for numerical Newton-Raphson methods. However, Newton-Raphson requires solving a... |
I'm reading a book called 'Rotations, Quaternions, and Double Groups' and I'm having a great deal of difficulty trying to understand the definition of bilateral rotations, bilateral binary rotation, and semiaxes.
Here is the snippet I'm confused about,
"Binary rotations and their corresponding axes, or binary axes, wi... |
Let's say I got stuck with a problem in integration, and I decided to peek at an answer, and I differentiate the answer to get the question, and then I flip from question to answer, and I have my solution ready.
But, I have many times seen that this approach does not match with the way we find answer with question, it ... |
what exactly is the meaning of "obtaining a structural result" in mathematics, for example in graph theory?
|
Give the parameterized solution for the dependent system represented by the matrix:
$$\left(\begin{array}{ccc|r}
1 & 0 & 3 & 6\\
0 & 1 & 2 & -4\\
0 & 0 & 0 & 0
\end{array}\right).$$
|
Consider a random sample from a distribution with the p.d.f. (for some $c > 0$)
\begin{equation} f(x) = cx^2 (1 − x), \text{if } 0 < x < 1; \end{equation}
\begin{equation} 0, \text{ elsewhere}. \end{equation}
What is an approximate distribution of $\bar{X}_n$ ? For a sample of size 45, find approximate mean and varianc... |
Problem statement
Five sailors were cast away in an
island, inhabited only by monkeys. To
provide food, they collected all the
coconuts that they could find. During
the night, one of the sailors woke up
and decided to take his share of the
coconuts. He divided them into 5 equal
piles and noticed that one... |
The underlying set of the product $X \times Y$ of two schemes is by no means the set-theoretic product of the underlying sets of $X$ and $Y$.
Although I am happy with the abstract definition of fiber products of schemes, I'm not confident with some very basic questions one might ask. One I am specifically thinking abou... |
If $\varphi: R^m \times R^n \to R$ is a non-degenerate bilinear map and $R$ is an integral domain then we must have $m=n$.
edit:
By "non-degenerate" bilinear map I mean that for every nonzero $m \in R^m$ there is an $n \in R^n$ such that $f(m,n)\neq 0$. The reverse also holds: for all non-zero $n \in R^n$ there is an $... |
A company produces three combinations of mixed vegetables that sell in 1Kg package. Italian style combines 0.3kg of zucchini, 0.3kg of broccoli and 0.4kg of carrots. French style combines 0.6kg of broccoli and 0.4kg of carrots. Oriental style combines 0.2kg of zucchini, 0.5kg of broccoli and 0.3kg of carrots. The compa... |
Let's say $X$ is a topological space and let's consider the categories of sheaves and presheaves of abelian groups on $X$. Suppose we have a presheaf that is a direct summand (in the category of presheaves) of a sheaf. Can we conclude that the presheaf is also a sheaf? What if we assume further that the complementary d... |
1) $(P \vee Q) \Rightarrow R$
2) $(P \Rightarrow R) \wedge (Q \Rightarrow R)$
Explain why 1 and 2 are equivalent statements.
Attempt at solution:
I converted 1 and 2 to these statements which are equivalent:
1) $(P \cup Q) \subseteq R$
2) $(P \subseteq R) \cap (Q \subseteq R)$
This now seems logical but I don't know ho... |
While solving the monkey problem, I encountered a problem that require solving this equation:
$$ax - by = c$$
And I was confused when $b < 0$. Does it hold for EEA or this require another special case?
Thanks,
Chan
|
I was reading about the Catalan numbers on wikipedia, because it seems like they show up quite a bit.
I'm reading some of the examples in the applications to combinatorics section. Some of them make sense fairly easily for me. For example, I see that the number of ways to parenthesize a product of $n+1$ terms $x_0,x_1,... |
I just started studying a small paper about the Harris Corner Detection. The problem is I don't understand how step 7 is derived from step 6. In step 7 the expression is expanded in a way that we get a structure tensor $C$ for $x$ and $y$. If one would multiply the three matrices again, I see that we would end up with ... |
I've seen Van-Kampen's theorem presented algebraically many times; and although it provides a useful method of calculation; I don't have a very clear picture for "why" it should be true. Does anyone know of a more visual argument; or even an example that makes the it easier to see?
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I know next to nothing about intuitionism, so my question is probably silly :)
As I understand from Wikipedia, intuitionism (at least finitism) doesn't 'trust' in the existence of irrational numbers, because they cannot be constructed (at least in finite number of steps). How does it deal with circles then? Or, even be... |
Suppose I have a neural network, with input variables $a,b,c,d,f,g$ and output variables $m,n,o,p,q$.
Given different input values, the neural network will output corresponding $m,n,o,p,q$.
Now I want find out the best input values which can maxmize m,n, while minimize $o,p,q$ with different weights as well. So how can... |
Let $C$ be a smooth, genus $g$ curve defined over a number field $K'$. Suppose we have a $K'$-valued point $P$ on $C$. We can view $C$ as a Riemann surface; then the space of holomorphic differential forms has dimension $g$ over $\mathbb{C}$.
It seems to me that we can choose a uniformizer $x$ at $P$ and a basis of... |
If one gets a phd in math, is it able to be admitted to another phd program in math? It is not for me, for someone else.
|
How can we prove $\bigtriangledown = \bigtriangleup E^{-1}$?
where,$\bigtriangleup \rightarrow \text{ Forward difference operator }$
$\bigtriangledown \rightarrow \text{ Backward difference operator }$
$E \rightarrow \text{ Shift operator }$ I tried to utilize the method of seperation of symbol but not quite there...
... |
let's consider regression problem. Given set of training data $\{(x_i,y_i)\}_{i=1}^N$, $x_i \in \mathbb{R}^n$ and $y_i \in \mathbb{R}$, find prediction function $y = f(x)$, e.g. in RBF regression case $y = \sum_{i=0}^m a_i g(x_i,c_i)$, where $g(x_i,c_i)$ is radial function with center $c_i$.
It's well known that soluti... |
It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If:
$ X_i \sim \text{Gamma}(\alpha_i, \beta) $
Then:
$ \left(\frac{X_1}{\sum_j X_j},\; \ldots\; , \frac{X_n}{\sum_j X_j}\right) \sim \text{Dirichlet}(\alpha_1,\;\ldots\;,\alpha_n) $
Problem
What hap... |
Let $A$ be a finite dimensional simple algebra over a field $k$. Denote by $K$ the center of $A$. Why the dimension of $A$ over $K$ is a square?
|
An electronics company produces three models of stereo speakers, models A, B and C, and can deliver them by station wagon, truck, or van. A station wagon holds $3$ boxes of model A, $3$ boxes of model B and $2$ boxes of model C. A truck holds $7$ boxes of model A, $6$ boxes of model B and $4$ boxes of model C. A van ho... |
How to calculate the probability that one random sample say $X_1$ will be greater than the other random sample say $X_2$ if they are from a uniform distribution with $[0,1]$ ? This is not a homework. I'm trying to solve some exercise problem and this is a part of the problem.
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Let $A$ be an invertible $n\times n$ matrix. Define the matrix of minors $\Delta(A)$ of $A$ to be the matrix whose $(i,j)$ entry is the determinant of the minor of $A$ with the $i$th row and $j$ column omitted.
Certainly, the most important property of $\Delta(A)$ is that it is instrumental in computing the inverse of... |
If $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ then $\sum_{n=1}^{\infty}(a_{n}-a_{n+2})$ converges
Using the fact that $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ I showed that $a_{n}$ is Cauchy hence $a_{n}\to L$.
If we let $b_{n}=a_{n}-a_{n+2}$ then by looking at the partial sums: $$S_{2n}=\sum_{k=1}^{2n}b_{k}=a_{2}-... |
This is Problem 3.19.3 of Dieudonné's Foundations of Modern Analysis (in my words). For $x$ a rational number, let $E_x=\{x\}\times\left[-1,0\right[$, and for $x$ an irrational number, let $E_x=\{x\}\times[0,1]$. Let $E=\bigcup_{x\in\mathbf{R}}E_x$ with the subspace topology. Show that $E$ is connected.
There is the fo... |
Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of presheaves $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is an elementary topos.
I understand the standard argument used when $\... |
Suppose we have a simply connected domain in the complex plane containing 0. I would like to show that there is no holomorphic n'th root function defined on the domain.
Using some student logic, I feel that I have to use the log function and its properties. Indeed, log is not defined at 0 so it seems I could come up w... |
In general it is not true that a vector bundle $E$ is isomorphic to its dual bundle $E^*$. But it is true when the vector bundle is the tangnet space of a manifold (at least I think it's true). How does one prove this?
|
Okay, so in my math homework I'm supposed to draw a graph of the following function:
$$f(x)=\sin x \cos x.$$
I have the solution in the textbook, but I just can't figure out how they got to that. So, if someone could please post (a slightly more detailed) explanation for this, it would be really appreciated.
I have to ... |
I have been reading about finite fields because they are used in cryptography quite a bit. However I am having some trouble conceptualizing how they work exactly. Using AES as the example. When you multiply two numbers you then repeatedly xor with 283 until some point.
Ex, my book states using Hex to binary and GF(2^8)... |
I barely remember that I could identify them by attempts. I would get the coefficient, see what numbers can divide it and put them into the Ruffini's rule.
But I don't even know what such method is called.
|
equation 1:
$-i\omega\underline \xi - 2i(\underline k \cdot \underline \Omega )\underline u = 0$
with:
$\underline \xi = i \underline k \times \underline u$
$\underline k \cdot \underline u = 0$
taking:
$i\underline k \times \text{(1)}$
derive:
$\omega = \pm 2\frac{(\underline k \cdot \underline \Omega )}{|\underline k... |
Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?
|
x is an integer, and i can write it with $\log_2 x$ bit, and, viceversa, with $n$ bit i can write a number till $2^n$..
but..
how many bits to write $\sqrt x$ ?
EDIT: the integer part!
|
I'm sure "jaggedness" isn't the right term to use here, so please correct me. I'm trying to quantify how jagged a distribution is. For example, this is moderately jagged:
distribution #1
This is really jagged:
distribution #2
Essentially, I want to quantify the number of peaks and the disparity between peaks and valley... |
So I know that you can use the Strassen Algorithm to multiply two matrices in seven operations, but what about multiplying two matrices that are exactly the same. Is there a faster way to go about doing that (ie. by reducing the number of multiplications per iteration to something less than 7) ?
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Helly's Theorem states the following: Suppose that $X_1,X_2,...,X_n$ are convex sets in $\mathbb{R}^d$, such that for any $|I|\leq d+1$, $\cap_{i\in I}X_i \neq \emptyset$. Then $\cap_{i=1}^{n}X_i \neq \emptyset$.
I'm looking for interesting problems, the solutions of which use this theorem.
Here is one example:
Let $K... |
Using the standard notation $(x,y)$ for cartesian coordinates, and $(r, \theta)$ for polar coordinates, it is true that $$ x = r \cos \theta$$ and so we can infer that
$$ \frac{\partial x}{\partial r} = \cos \theta.$$ This means that if we perturb $r$ to $r+\Delta r$, $x$ gets perturbed to approximately $x+ \cos(\thet... |
Find a 2-connected planar graph whose drawings are all topologically isomorphic, but whose planar embeddings are not all equivalent.
I think $K_{2,3}$ might be an example, but I'm not sure how to show this at all. Anything would be welcome.
|
If $0 \leq r \leq n$, is it possible to use this equation:
$$C( n, r ) = C( n - 1, r ) + C( n-1, r - 1 ).$$
The problem is I want to prove that $C( n, r )$ is integers by induction. In other words, I have to prove $C( n + 1, r )$ is integer deducted from $C( n, r )$. I thought of the equation above but I found the con... |
Struggling with some equations here. I've got a particle I need to track in an absolute reference frame, but each step I move/rotate it relative to its own reference frame. I need to track it's absolute position while moving it relatively.
I'm trying to figure out the proper set of rotation matrices. It propagates alon... |
I know it might be too easy for you guys here. I'm practicing some problems in the textbook, but this one really drove me crazy.
From $\gcd( a, b ) = 1$, I have $ax + by = 1$, where should I go from here? The extra $ac$ is so annoying. Any hint?
Thanks,
Chan
|
I'd like to say logarithmically decreasing but it does not have to decrease to zero.
An example of an increasing function which flattens out at around 4.5%.
|
I was wondering if someone could recommend good books on stochastic processes
with measure theory treatment
with not much or no measure theory
treatment
for each, it would be nice to have some books for introductory level, mid-level to comprehensive level respectively. They can be either classical or recently publis... |
I know how to solve $ax + by = c$ using Extended Algorithm. But with more than variables, I'm lost :(.
To verify if it has an integer solution is easy, since we only need to check for $\gcd(a,b,c)|d$. Other than that, how can we find an integer solution for this equation?
Thanks,
Chan
|
That
(1) Ext$^n_\mathbb{Z}(M, \mathbb{Q})=0$ for every module $M$
follows easily from the fact that
(2) $\mathbb{Q}$ is injective.
However, the only proof I have seen of the injectivity of $\mathbb{Q}$ relies on Baer's Criterion. While the proof of Baer's Criterion is not difficult, it seems stronger than the ... |
If one computes the average width of a triangle, then one gets $(s_1+s_2+s_3)/\pi$, where $s_1$, $s_2$, $s_3$ are the side lengths. I did this by brute force, using an integral which went through an interval of length $\pi$. For each angle $\theta$, the "width" $w_\theta$ of the triangle is the smallest distance one ... |
Let $X=C(S)$ where $S$ is compact. Suppose $T\subset S$ is a closed
subset such that for every $g\in C(T),$ there is an $f\in C(S)$
such that: $f\mid_{T}=g$. Show that there exists a constant $C>0$
such that every $g\in C(T)$ can be continuously extended to $f\in C(S)$
such that: $\sup_{x\in S}\left|f(x)\right|\leq C\s... |
I need to sketch a diagram which shows what happens when a 2x2 matrix is applied to a lattice. What would be the best software to perform such a thing?
|
I don't know where to begin to calculate the expectation value of the random variable $1/V$, where $V$ is a random variable with chi-square distribution $\chi^2(\nu)$.
Could somebody help me?
|
I hope this doesn't turn out to be a silly question.
There are lots of nice examples of continuous bijections $X\to Y$ between topological spaces that are not homeomorphisms. But in the examples I know, either $X$ and $Y$ are not homeomorphic to one another, or they are (homeomorphic) disconnected spaces.
My Questio... |
OK, so I know that the Lebesgue measure of the ternary Cantor set is $0$.
However, in class the prof briefly mentioned that if we build a Lebesgue-Stieltjes measure $\mu$ out of the Cantor function, then $\mu(C) = 1$.
I don't understand why this is true, can someone please explain?
|
I am having some trouble understand essential prime implicants. So if a minterm is not covered by another overlapping rectangle, then that is an EPI. However, if we make a K-map for $f(x,y,z)=xy+xz'+y'z$, we have minterms m4, m6, and m7 not covered by overlapping rectangles. If what I did is correct, then there should ... |
I was reading about the Fano plane, the smallest possible projective plane. After playing around with it, it seems that any projective plane of 7 points will be isomorphic to the Fano plane.
However, I've always been troubled with showing isomorphisms between sets of points and lines, because it seems like any functio... |
If a Cat has 4 kittens, and in 6 months those kittens have 4 kittens, and so on for 7 years, such that 14 batches of kittens are produced, from the one kitten. I produce the summation:
$$\sum_{x=1}^{14} 4^x$$
Where added up looks like $4 + 16 + 64 + \cdots +4^{14}$.
Is there some kind of trick in maths for working this... |
I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of functions $f_n$ is convergent to $f$ if and only if $f_n$ is pointwise-convergent to $f$.
My intuition tells me that this is t... |
Let $C_*$ be a chain complex of abelian groups.
Is it true that $H_i(C_*\otimes \mathbb{Z}/p)=0$ for all $i$ if and only if $H_i(C_*\otimes \mathbb{Z}_p)=0$ for all $i$, where $\mathbb{Z}_p$ is localization of $\mathbb{Z}$ away from $p$?
And I want to know that the precise definition of localization of $\mathbb{Z}$ aw... |
If I had a recursive function (f(n) = f(n-1) + 2*f(n-2) for example), how would I derive a formula to solve this? For example, with the Fibonacci sequence, Binet's Formula can be used to find the nth term.
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Let $F$ be an extension field of $K$. let $L$ and $M$ be intermediate fields, with both finite algebraic extensions of $K$.
Suppose {$a_1,...,a_n$} is a basis for $L$ over $K$ and {$b_1, ...,b_m$} is a basis for $M$ over $K$. Show that {$a_ib_j$} is a spanning set for the field $LM$ ($LM$ is the smallest field betwe... |
Bonjour,
J'ai rencontré le problème suivant dans le livre "Real and Functional Analysis" de Lang, au chapitre $3$. J'explique d'abord le contexte, puis j'en viendrai à la question précise.
Il faut démontrer que les fonctions de la forme $e^{-x}p(x)$, où $p$ est un polynôme, sont denses dans l'ensemble des fonctions co... |
Consider this Fibonacci equation:
$$f_{n+1}^2 - f_nf_{n+2}$$
The problem asked to write a program with given n, output the the result of this equation. I could use the formula
$$f_n = \frac{(1+\sqrt{5})^n - ( 1 - \sqrt{5} )^n}{2^n\sqrt{5}}$$
However, from mathworld, I found this formula Cassini's identity
$$f_{n-1}f_... |
Recently, I was going over a curious recreational math paper titled "On the Diagonal Queens Domination Problem". The main result of the paper is establishing the minimum value $diag(n)$ of number of queens needed to be kept on the diagonal of a $n \times n$ chessboard so that they attack all the squares on the board.
T... |
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