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In particular, problems like $\lfloor\frac{10^{20000}}{3+10^{100}}\rfloor$. (From a putnam exam many years ago.)
Is there a general technique for attacking such problems?
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In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity
$$n! = \int_{0}^{\infty} t^n e^{-t} dt$$
, coming from the Gamma function. I have a mathematical confession to make: I have never "grokked" this identity. Why should I expect... |
Motivated by this question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern mathematics. Does it refer to the Abstract Algebra?
Edit: I got -1. If this is not a correct question, please state why. And i... |
I'm looking for clear mathematical rules on rounding a number to $n$ decimal places.
Everything seems perfectly clear for positive numbers. Here is for example what I found on math.about.com :
Rule One Determine what your rounding digit is and look to the right side of it. If that digit is $4, 3, 2, 1,$ or $0$, simply... |
Factorial is defined as
$n! = n(n-1)(n-2)\cdots 1$
But why mathematicians named this thing as FACTORIAL?
Has it got something to do with factors?
|
Which one is bigger $2^{n!}$ or $(2^{n})!$ ?
where $n\in\mathbb N$.
|
My textbook doesn't give any example of this kind of series. Could you provide some?
Trigonometric series is defined in wikipedia as :
$A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$
When
$A_{n}=\frac{1}{\pi} \int^{2 \pi}_0 f(x) \cos{nx} dx\qquad (n=0,1,2,3 \dots)$
$B_{n}=\frac{1}{\pi} \int^{2 \pi}_0 f(x)... |
I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to itself?" Call these maps endomaps. Permutations of elements of the set are a special case of endomaps: they are bijective endomaps. Pe... |
In a very interesting blog discussion at the $n$-category cafe, an anonymous poster made the following remark: "... using the dictionary between number fields and function fields, Weil suggested that G-with-adelic-entries is analogous to the group of gauge transformations of a principal G-bundle over a Riemann surface.... |
In an answer to another question I asked, Isaac suggested a book that is the standard "reform calculus" book. In a comment, I asked what the phrase "reform calculus" means, and Isaac provided a link to this summary page. It seems to be a new or revised teaching methodology. However, the page linked to doesn't provide a... |
I'm brushing up on my calculus, and geometry and trigonometry keep coming up. However, these are probably my weakest areas mathematically. Are there any good sites or free PDFs/ebooks that I can use to quickly (in a weekend, preferably) work through geometry and trigonometry and refresh myself in topics such as areas a... |
I know how to prove $\zeta (2)=\pi ^{2}/6$ by using the trigonometric Fourier series expansion of $x^{2}/4$. How can one prove the same result using the complex Fourier series of $f(x)=x$ for $0\leq x\leq 1$? Any suggestion?
|
What's the simplest formula to calculate the optimal sample size from a population size? I've been reading a little about it and they ask me stuff like confidence level or confidence interval which I have no idea about. I need to test some HTML elements in web pages to get a better idea about general properties of HTML... |
Sorry if this question is too basic. It is from Fröhlich and Taylor's "Algebraic Number Theory".
Let $E/F$ be a finite Galois extension of fields, with $G=Gal(E/F)$, and let $K$ and $L$ be two subfields of $E$, containing $F$, such that $K/F$ and $L/F$ are both Galois. Let $M=Gal(E/K)$ and $N=Gal(E/L)$ be normal subgro... |
Suppose $C \subset \mathbb{R}$ is of type $F_{\sigma}$. That is $C$ can be written as the union of $F_{n}$'s where each $F_{n}$'s are closed. Then can we prove that each point of $C$ is a point of discontinuity for some $f: \mathbb{R} \to \mathbb{R}$.
I refered this link on wiki : http://en.wikipedia.org/wiki/Thomae%27... |
Here's the question I've been thinking for $2$ days and couldn't find an answer why.
If $Y = \sqrt{X}$, where both $X$ and $Y$ are real numbers then which of the following is true?
a) $X\geq 0$; $Y\geq 0$
b) $X\leq 0$; $Y\geq 0$
c) $X\leq 0$; $Y\leq 0$
d) $X\geq 0$; $Y\leq 0$
e) Either a) or b)
Now, the answer is a) bu... |
How does one sum the series $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$
This was asked to me by a high school student, and I am embarrassed that I couldn't solve it. Can anyone give me a hint?!
|
From a 2005 Russian olympiad. Prove that there do not exist four (pairwise) different quadratic polynomials, with leading coefficient 1, such that the sum of any two of them has a single root.
(Optionally, this is what I've done so far:
If $P_i(x)=x^2+b_ix+c_i$, then $P_i+P_j$ is single-rooted iff $(b_i+b_j)^2=8(c_i+c_... |
Is there a natural way to represent all the partitions of an integer set $\{1,2,3,...,n\}$ as a matrix in the similar way permutations can be mapped to group of matrices?
|
At the beginning of Week 300 of John Baez's blog,
Baez gives a proof that the "number" of finite sets (more specifically, the cardinality of the groupoid of all finite sets, where an object in the groupoid counts as $1/n!$ if it has $n!$ symmetries) equals $e$.
He then says that this leads to a purely combinatorial pro... |
A matrix determinant (naively) can be computed in $O(n!)$ steps, or with a proper LU decomposition $O(n^3)$ steps. This assumes that all the matrix elements are constant. If, however the matrix elements are polynomials (say univariate of max order $p$) at each step of the LU decomposition an element is multiplied by an... |
Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
|
Liouville's Number is defined as $L = \sum_{n=1}^{\infty}(10^{-n!})$. Does it have other applications than just constructing a transcendental number?
(Personally, I would have defined it (as "Steven's Number" :-)) as binary: $S = \sum_{n=1}^{\infty}(2^{-n!})$, since each digit can only be "0" or "1": the correspondin... |
I have a standard linear programming problems I want to solve:
$$ \min_x f^T x \text{ such that } \left\{ \begin{aligned} A\cdot x &\le b, \\ A_{eq}\cdot x &= b_{eq}, \\ lb \le x &\le ub. \end{aligned} \right. $$
$F^T$ is a vector that doesn't involve $x$.
The problem with the above minimization is that, I don't know ... |
How to show the following identity holds?
$$
\displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1
$$
|
How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$$\newcommand{\Beta}{\mathrm{Beta}}\sum_{a=0}^n {n \choose a} \Beta(a+1, n-a+1) =... |
Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one.
You may chose the origin and orientation of the $x,y$ axes as you want.
Motivation: It can be proved [Problem of the Week, Problem No. 8 (Fall 2008 Series), Depart... |
Mathematicians, and esp. number theorists, are used to working with big numbers. I have noted on several occasions that lots of people don't have a clear understanding of big numbers as far as the real world is concerned. I recall a request for a list of all primes of less than 500 digits.
Another example is homeopat... |
How could I convert a non-uniform random variable distribution to a uniform distribution?
Is there a formula for such a transformation?
|
Could you please explain how the math behind the first answer to this stack overflow question works:
Link to the question: https://stackoverflow.com/questions/3590308/testing-if-a-line-has-a-point-within-a-triangle
Thanks.
|
I know that the product of two Gaussian functions is also Gaussian function. This is stated in Wikipedia, but I might need to cite a classical (text)book/paper stating this result. A book containing the proof is welcome. Could you name such a book?
|
Rotations in 3 dimensions are not commutative; however they are in the plane. In classical mechanics, are we allowed to say that angular momentum is a vector because particles only rotate along a single axis? Or do we make some kind of argument that you can assign angular velocity to an object because you can approxima... |
One learns evaluating indefinite integrals of the form $$ \int \frac{f(x)}{p(x)\sqrt{q(x)}} \ dx $$ where $p(x)$ and $q(x)$ are linear polynomials.
How to evaluate these integrals if $p(x)$ and $q(x)$ are quadratic or cubic in nature. Is there any method for evaluating them.
Examples of such integrals are: $\displayst... |
Wikipedia has separate pages for symmetric group and permutation group, but I don't understand what the difference between them is. A symmetric group on a set is the set of all bijections from the set to itself with composition of functions as the group action. Permutation group on a set is the set of all permutations ... |
In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to look for examples of Gröbner bases in action, but have been unable to f... |
I am going to be taking a first course in abstract algebra this fall and we are using Jacobson's Basic Algebra I for the course text. It looks like a good textbook, but it is lacking a lot of motivation (I am new to higher math), so I was wondering if there are other books to supplement my studies. Particularly histori... |
Question:
Given some natural number, we can of course split it up into various sums of other naturals (e.g. $7 = 6 + 1 = 1 + 4 + 2 = \ldots$)
More precisely, for $n \in \mathbb{N}$, we can a find distribution sequence (or partition) $s_0,\ldots,s_u \in \mathbb{N}$ with
$$\sum_{i=0}^{u}s_i = n$$
Now how can I find the p... |
I came up with this problem after discussion of taxicab geometry in math class... I thought it was a simple problem, but still pretty neat; however, I am as of yet unsure of whether my answer is correct, or logical.
Let $[X]$ be the area of region $X$, and region $S_n$ be represented by the equation $|x-n|+|y-n|=k-n$ f... |
How many unique strings can be formed by the letters in "aaamottu"?
As a totally incompetent math student, I simply wrote a program to calculate it. And I ended up with $3360$ unique strings:
import itertools
unique = {}
for p in itertools.permutations(list("aaamottu"), 8):
unique[p] = True
len(unique)
I'd like... |
In making up another problem today I came across something odd. I've been thinking it over and I can't exactly place why it's true, but after running a long Python script to check, I haven't yet found a counter example.
Why is $\sum_{n=1}^{m}{n^m}\equiv 0\mod m$ true for all odd $m \ge 3$? The script showed me that eac... |
It can be seen here that the only numbers for which $n^{m+1}\equiv n \pmod{m}$ is true are 1, 2, 6, 42, and 1806. Through experimentation, it has been found that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$ is true for those numbers, and (as yet unproven) no others. Why is this true?
If there is a simple relation... |
I had a question about an instantaneous z score. The question reads as
$$P(z=2)$$
I think that this is 0, as z-score is the area under the curve and could be found by integrating the PDF over the range. As the range is 0, the area is also zero. Is that correct?
|
A train with infinitely many seats, one for each rational number, stops in countably many villages, one for each positive integer, in increasing order, and then finally arrives at the city.
At the first village, two women board the train.
At the second village, one woman leaves the train to go visit her cousin, and tw... |
The Riemann hypothesis (RH) states that all non-trivial zeros of the zeta function have real part $\frac{1}{2}$.
The zeta function is intimately connected with the Gamma function via the functional equation.
The second fact suggests that there is an equivalent form of RH which is expressed solely in terms of the Gamm... |
Calculus theory problem I'm belaboring to solve... Tried it for an hour only spinning my wheels. Any hints/nudges in the right direction would be greatly appreciated. Here's the problem:
$f$ and $g$ are functions that are differentiable for all real x. They have the following properties:
i) $f'(x) = f(x) - g(x)$
ii) $g... |
So, I know that every abelian (commutative) group $G$ is such that, for any subgroup $H$, the left cosets of $H$ in $G$ are the right cosets. I guess this is true even if $G$ is not abelian, but $H$ is (but I'm not sure). Is this enough to characterise the subgroup as abelian, or are there examples of non-abelian subgr... |
Motivation: Example. To solve a problem on evaluating the maximum of a product of $n$ real variables subject to an equality constraint on its sum $S$ ($=100$), I used the Lagrange multipliers method (which can be improved by adding the adequate conditions on the second order partial derivatives). Doing so I got a "solu... |
I'm trying to express the following function more simply:
\begin{equation*}
f(x) = (\sum_{n=3}^{x} \prod_{k=2}^{\lceil \sqrt n \rceil} \lceil \dfrac {n}{k} \rceil - \lfloor \dfrac{n}{k} \rfloor ) + 1 .
\end{equation*}
I know this is kind of a vague request, but I'm trying to get away from the ceiling and floor functi... |
In a recent (and to me, very beautiful) paper, entitled "Relational EPR", Smerlak and Rovelli present a way of thinking about EPR which relies upon Rovelli's previously published work on relational quantum mechanics (see at arxiv.org/abs/quant-ph/9609002 ).
In relational quantum mechanics, there is no non-locality,... |
Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It is known to be independent of Peano Arithemtic (PA), and in fact, was the first such purely number theoretic result. It is provable in ZFC.
One way of phrasing this is that the theory "PA + Goodstein's theorem is false" is consis... |
I was wondering if you could help...
I have Math homework, I was hoping if you could check my answer.
Find the equation of the line perpendicular to $3x+2y-4=0$ going through point $(2,-3)$.
$y=\large\frac{-3x+4}{2}$
$y = 2$
$x=-\large\frac{2(y-2)}{3}$
$x=-3$
Therefore my equation is correct?
thanks for you help in... |
I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin's classic on "Principles of Mathematical Analysis".
Given the above context can I skip chapter 1? It seems to me that the material in chapter 1 is not as i... |
Is there any theory (analogous to Galois theory) for solving equations with irrational exponents like:
$ x^{\sqrt{2}}+x^{\sqrt{3}}=1$
?
|
I would like to know something like what's the first PDE etc.
Could you recommend book on the history of PDE/ODE?
thanks.
|
I want to fit to a dataset a power function ($y=Ax^B$). What is the best and easiest method to do this. I need the $A$ and $B$ parameters too.
I'm using in general financial data in my project, which looks like this:
8553600 458.2
17193600 373.6
25833600 694.16
34646400 738.33
44064000 817.89
54259200 10... |
I have a matrix $F=XDX^T$, where $D$ is $m\times m$ and diagonal and $X$ is $n\times m$.
Now, I compute $F^{-1}$.
Is there an efficient method to update $F^{-1}$ if $D$ is updated by $D'=D+G$, where $G$ is a sparse diagonal matrix?
|
I'm practicing some problems from past exams and found this one:
Find the n-th derivative of this function:
$$f(x)=\frac {x} {x^2-1}$$
I have no idea how to start solving this problems. Is there any theorem for finding nth derivative?
|
There are some math quizzes like:
find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$
such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$
If such $\phi$ exists (it does in this example), $\phi$ can be viewed as a "square root" of $f$ in the sense of function composition because $\phi\circ\phi = f$. Is there a genera... |
I know, fundamental group of $GL(n,\mathbb{C})$ is isomorphic to $\mathbb{Z}$. It's written in Wikipedia. Actually, I've succeed in proving this, but my proof is two pages long and very technical. I want
to find some better proofs of this fact (in order to compare to mine);
to find some book or article, using which ... |
Is there a easy way to compute the coefficients of the power series which represents
\begin{equation*}
\frac{x - x^k + x^{k+1}}{1-2x + x^k - x^{k+1}}.
\end{equation*}
I am currently solving this by assuming it has the form
\begin{equation*}
a_1x+a_2x^2+a_3x^3+\dots+a_nx^n+\dots
\end{equation*}
and solving tedious re... |
Being motivated by this post, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving transcendental equations? It seems pretty clear, intuitively, that there can be no general method; but the only reason I feel strongly about that is because I c... |
Since there is a $2$-$1$ homomorphism from $SU(2)\times SU(2)$ to $SO(4)$ there should be a way to recover the two $SU(2)$ matrices given an $SO(4)$ matrix.
I believe I could set this up as a system of equations using the map from above and solve for the coefficients of the $SU(2)$ matrices. However, I wonder if any... |
Let $S$ be some finite set. Let $Sym(S)$ be the symmetric group on $S$. Let $A$ be the subset of permutations $\sigma$ that are such that there exist $a,b,c \in S$ such that $\sigma(a) = b, \sigma (b) = c$ and $\sigma (c) =a$ and for every other element of $S$, the action of $\sigma$ is the identity. In short, let $A$ ... |
In one of my classes we discussed the ring of 2x2 matrices $M_{2}(\mathbb{Z})$. We said that its group of units was $SL_{2}(\mathbb{Z})$ which means that it is the set of 2x2 with determinant equal to $\pm$1.
Why can't we have a 2x2 matrix with entries a,b,c, and d such that $\frac{a}{ad-bc}$,$\frac{-b}{ad-bc}$,$\frac{... |
I can think of the set of bounded, continuous functions from $\mathbb R \to \mathbb R$ as a group, with composition as addition of functions. In other words, this group has the rule that the composition of two elements in the group, $f(x)$ and $g(x)$, is their point-wise sum $f(x) + g(x)$. Since this set is path conne... |
Why don't $3$-cycles generate the symmetric group? was asked earlier today. The proof is essentially that $3$-cycles are even permutations, and products of even permutations are even.
So: do the $3$-cycles generate the alternating group? Similarly, do the $k$-cycles generate the alternating group when $k$ is odd?
And d... |
What is the value of $1^i$? $\,$
|
Let there be given some conditionally convergent infinite series $S$. Then let $R$ be some real number, and $Q_k$ a rearrangement of $S$ such that the sum is equal to $R$.
Is $Q_k$ unique? In other words, is there some other rearrangement $W_k$ for which the sum of $S$ is also $R$?
I believe the answer is that $Q_k$ ... |
To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would back it up.
Wikipedia's entries on the zero vector seems to agree with that intuition, but as always, one shouldn't bli... |
Consider the following data points: $1, 1, 2, 3, 4$
I understand that...
the average is the total of the numbers divided by the count of numbers in the set, the median is the central value based on location in the set, the mode is the value occurring most often in the set, and the midrange is highest and lowest values ... |
I'm relatively new to ring theory so this is probably a simple question.
Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.
I am wondering if we can rearrange these properties and always get implication or produce finite rings wit... |
Is anything known about these integrals? Textbook suggestions are welcome
\begin{equation*}
f(n,p)=\int_{x=-0.5}^p \frac{n!}{x!(n-x)!} dx,
\end{equation*}
$n>0, p\le n+0.5$.
For instance, as $n$ grows $2^n-f(n,n+1/2)$ seems to be positive and monotonically decreasing...how would I find the limit?
ListPlot[Table[2^n - ... |
i. If it rains next Sunday, then Andy will only go to the football game if Bob does.
ii. If Andy will only go to the foot ball game on Sunday if it rains, then Bob will definitely go to the Sunday football game regardless of the weather.
My gut feeling is they are not equivalent. I chose the following three predicates:... |
I tried to find some reference on the net but couldn't find a good one. What is the advantage of pivoting in LU decomposition over regular LU decomposition? Is it something to do with matrix singular or not?
|
I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is a prime ideal in $\mathbb Z$. Isn't $(p)$ being a prime ideal the very definition of $p$ being a prime element?
(I know t... |
Let's take a look at the following integrals :
1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -\frac 1 2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= -\frac 1 2 \zeta(2)$
2) For $c<1$ $\displaystyle \int\limits_{0}^{\frac{\pi}{2}} \arcsin(c \cos{x}) \ dx = \frac{c}{1^2} + \frac{c}{3... |
Say we have a set of symmetric $n \times n$ matrices $M_i$ for $1 \leq i \leq k$, elements in $\mathbb{R}$. Suppose that for every $\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_k) \in \mathbb{R}^k$ we have that the kernel of
\begin{equation*}
M_{\boldsymbol{\lambda}} = \sum_i \lambda_i M_i
\end{equation*}
is n... |
I have to get all combinations of a six digit number where each digit is unique. Its clear that this combination will not have a 0 at the start as it will not be a valid six digit number. Help me on how to attack this problem.
|
Consider the piecewise constant function $\psi:I=[-1,1] \rightarrow \mathbb{R}$ given by
$$\psi(x) = \begin{cases} \psi_1 & x \leq 0, \ \psi_2 & x > 0 \end{cases}$$
for some constants $\psi_1, \psi_2 \in \mathbb{R}$. I would like to evaluate the integral
$$\int_I \delta(x) \psi(x) dx$$
where $\delta$ is the Dirac del... |
Suppose you have $100$ coins. $96$ of them are heavy and $4$ of them are light. Nothing is known regarding the proportion of their weights. You want to find at least one genuine (heavy) coin. You are allowed to use a weight balance twice. How do you find it?
Assumptions:
Heavy coins all have the same weight; same for... |
This is problem taken from Problem 4.
I couldn't find the solution anywhere and I am curious to see a solution for this problem, as I can at least comprehend the question and it seems that the mechanism for the solution involved will be somewhat understandable.
|
Suppose $\Omega$ is a domain of the complex plane (i.e. an open and connected subset of the plane). Suppose $f$ is holomorphic on $\Omega$, and $f$ is not identically zero.
Suppose $f$ has a holomorphic logarithm on $\Omega$, which means that there is a function $g$ holomorphic on $\Omega$ such that $e^g=f$. Then it is... |
Do you have any favorite open problem?
Let me mention one of my favorites. Let $A(\mathbb{T})$ be the Wiener algebra, that is, the linear space of absolutely convergent Fourier series on the unit circle $\mathbb{T}$ $(=\mathbb{R}/2\pi\mathbb{Z}=(-\pi,\pi])$ carrying the norm
$$ f\mapsto \|f\|=\sum_{n\in\mathbb{Z}}|\ha... |
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra.
I was wondering if anyone knew a nice intuitive motivation for the study of dual spaces and whether or n... |
This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I am looking for is other WAYS to obtain the solution. I've been told there exists a solution using only arithmetic, but... |
When implicitly finding the derivative of:
$xy^3 - xy^3\sin(x) = 1$
How do you find the implicit derivative of:
$xy^3\sin(x)$
Is it using a triple product rule of sorts?
|
How does one sum the given series: $$ \frac{1}{2n+1} + \frac{1}{2} \cdot \frac{1}{2n+3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{1}{2n+5} + \frac{ 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{1}{2n+7} + \cdots \ \text{ad inf}$$
Given, such a series, how does one go about solving it. Getting an Integral Representation seems t... |
Can someone please point me in the direction of any theory on graphs where the edge weights are not scalars but represent some relation between the nodes that is a simple function of a single variable (simple, say piecewise linear).
In particular, I'm interested in various basic graph properties and also thinking of t... |
From some reading, I've noticed that $\sup(\emptyset)=\min(S)$, but $\inf(\emptyset)=\max(S)$, given that $\min(S)$ and $\max(S)$ exist, where $S$ is the universe in which one is working. Is there some inherent reasoning/proof as to why this is? It seems strange to me that an upper bound of a set would be smaller than ... |
I was musing about a particular limit,
$L = \prod\limits_{n > 0} \bigl(1 - 2^{-n}\bigr)$:
we may bound 0.288 < L < 0.308, which we may show by taking the logarithm:
$\ln(L) = \ln \bigl( \frac{315}{1024}\bigr) + \sum\limits_{n > 4} \ln\bigl(1 - 2^{-n}\bigr) > \ln\bigl(\frac{315}{1024}\bigr) - \sum\limits_{n > 4} 2^{-n... |
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions to get students to think more deeply about the material. The first one I asked was this:
1) Recall that a function $f: ... |
I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a few people out there will tell me how they go about learning math so I can try out their systems.
I need to know basic ... |
Can we find a polynomial $p(x) \in \mathbb{R}$ such that $\text{deg}\ p(x)>1$ and which satisfies $$p^{2}(x)-1=p(x^{2}+1)$$ for all $x \in \mathbb{R}$.
This question can be very well identified with my previous question.
|
For the equation
$\frac{\partial^2}{{\partial}x^2}(IE\frac{{\partial^2}u}{{\partial}x^2}) = \mu(\frac{{\partial^2}u}{{\partial}t^2})$
with $E$ a function of x, derive two ODE's by separation of variables.
I greatly appreciate your help/advice. I'm pretty unsure with how to handle derivatives like this. This is my best... |
I have the following well-known exercise in cardinal arithmetics:
If $\kappa, \lambda$ are cardinals such that $\lambda$ is infinite, then $\kappa^{<\lambda}$ equals the supremum of the $\kappa^{\theta}$, where $\theta < \lambda$ is a cardinal.
1) Isn't this false for $\kappa=1$? The left side is $\lambda$ and the righ... |
Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q \cap N = 0$.
I've got just a short question what maximality means here: Does it mean that every other $Q'$ with $Q' \ca... |
The presentation of the dihedral group $D_{2n}$ is
$$D_{2n}= \langle r,s \mid r^{n}=s^{2}=1, rs=sr^{-1} \rangle.$$
Why is it incorrect to conclude that $r=s=1$ ? In other words why doesn't this group presentation describe the trivial group.
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Let $R$ be the coordinate ring of an affine complex variety (i.e. finitely generated, commutative, reduced $\mathbb{C}$ algebra) and $M$ be an $R$ module.
Let $s\in M$ be an element, such that $s\in \mathfrak{m}M$ for every maximal ideal $\mathfrak{m}$. Does this imply $s=0$?
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Possible Duplicate:
How can I write an equation that matches any sequence?
I'm learning sequences right about now, and I'm having a really hard time finding the formula for a given sequence of numbers. I'm worried that, at exam time I'll be spending as much time finding the formula as I am now.
For example, one pro... |
first question on math.stackexchange :)
I'm studying for a Cryptography - Communication Security exam, and it involves a certain quantity of number theory - finite field theory, so be warned: this is my first encounter with these topics, and you'll have to be extra-clear with me :)
I thought I was doing pretty well wit... |
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