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We define cardinality as an equivalence relation on sets. But the class of all sets is not a set, so how do we do that? In particular, I'm interested in the proposition that equivalence classes form a partition of the initial set. It seems like it can be translated to cardinality, but I do not know how, at least in ZFC... |
My 7-year-old son was staring at the graph of tan() and its endlessly-repeating serpentine strokes on the number line between multiples of $\pi$ and he asked me the question in the title. More precisely, is the following true or false?
For any $\epsilon > 0$, there exists some $N \in \mathbb{Z}^+$ such that
$|\tan(N)... |
What are good computational geometry journals? I'm also counting geometric modeling as part of computational geometry. Are there any good journals that are considered to be non-mainstream in the US?
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I posted this on mathOverflow previously which was the wrong place to post it and I was asked to try this forum instead. Can anyone explain this in simple terms:
I met projective space via a recent class on perspective drawing, believe it or not, but I didn't know that this was the "space" we were using. I came across ... |
I'm a little confused as to the results of multiplying vectors and scalars. If I multiply a vector and a scalar, is it a vector? And if I multiply a vector and another vector, is the results also a vector? I looked at http://mathworld.wolfram.com/VectorMultiplication.html but still haven't figured it out.
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$$y=\frac{x^{5}}{6}+\frac{1}{10x^{3}}\qquad 1\leq x\leq 2$$
$$\frac{dy}{dx}=\frac{5}{6}x^{4}-\frac{3}{10x^{4}}$$
squaring this $$=\frac{25}{36}x^{8}+\frac{9}{100x^{8}}$$
Plugging into the formula $$ds=\sqrt{1+\left( \frac{dy}{dx}\right) ^{2}}$$
$$\int_{1}^{2}\sqrt{1+\frac{25}{36}x^{8}+\frac{9}{100x^{8}}}$$
Is this corr... |
Let $X$ and $Y$ be (irreducible, quasi-projective) varieties (over an algebraically closed field $k$), and $\phi: X \to Y$ be a rational map. I think I understand what it means for $\phi$ to be regular at a point $x \in X$, but I'm lost as to how to prove that $x$ is (or isn't) a regular point of $\phi$, when I am give... |
Let $(R,m)$ be commutative noetherian local ring with unity. Suppose $P$ is a finitely generated projective module over $R[X]$ of rank $n$ . Is $P$ free? If not, what is the counter example?
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OK, so I have the following polar equation:$$r = \frac{\theta}{20}.$$
And I would like to translate this a little to the right, and down from the polar origin.
Now, I figure since I know cartesian coordinate equation translations are quite simple, the best way to do this would be to convert this polar equation to a rec... |
From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8
Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ where $\Phi$ is the Euler $\Phi$-function. ( This is a well known theorem. I know of no easy solution, so don't be disappoin... |
I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is bounded.
Thank you.
|
Let $A$ be the following matrix:
$$A = \left(\begin{array}{rrr}
-1 & \hphantom{-}3 & \hphantom{-}0\\
0 & 2 & 0\\
-3 & 3 & 2
\end{array}\right).$$
I've found that the eigenvalues are -1 and 2 (multiplicity 2). However, when I try to find the eigenvectors for the eigenvalue 2, I can only find one, as the augmented m... |
Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface.
According to pg. 70 of the text Differential Geometry by Wolfgang Kuhnel, we can think of the $(h_{ij})$ as "the Hessian of matrix of a function $... |
If language L* (Kleene Star) is regular, does it imply that L is also regular?
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I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogeneous term is a constant, rather than a function of the independent variables? For example, in a hyperbolic PDE with x and t as the independent variables the eigenfunctions might be ... |
Let $A$ be a DVR in its field of fractions $F$, and $F'\subset F$ a subfield. Then is it true that $A\cap F'$ is a DVR in $F'$? I can see that it is a valuation ring of $F'$, but how do I I show, for example, that $A\cap F'$ is Noetherian?
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Is following language context-free?
Alphabet: {a,b,c,d}
L = {w | w is not in {aabbc,abc,add}}
I think it is:
{aabbc},{abc},{add} are all regular.
Because of closure properties(Union) R = {w | w is in {aabbc,abc,add}} is also regular.
Because of closure of complement (L is complement of R) L is also regular.
So L is r... |
Question is the following language decidable:
{(M)|given input "aaaaa" Turing machine M will perform at least 1295 steps}
I would say, yes it is. Just let the Universal Turing Machine count each step, and accept if it reaches step 1295.
But is this not a case where Rice´s theorem applies? Which means that we cannot dec... |
Given an undirected graph I'd like to color each node either black or red such that at most half of every node's neighbors have the same color as the node itself. As a first step, I'd like to show that this is always possible.
I believe this problem is the essence of the math quiz #2 in Communications of the ACM 02/20... |
The inequality
$$\zeta(s)^3 | \zeta(s + it)^4 \zeta(s + 2it)| \ge 1$$
follows from
$$3 + 4 \cos(\theta) + \cos(2 \theta) \ge 0.$$
How is that done? What is the relationship between zeta and the trigonometry?
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I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into $$\widehat{\widehat{G}}=\text{Hom}_{\mathsf{Ab}}(\text{Hom}_{\mathsf{Ab}}(G,\mathbb{C}^\times),\mathbb{C}^\times)$$
is not necessarily an isomorphism. Note that I'm looking at all homomorphisms; ... |
Suppose that $p$ is a prime. Suppose further that $h$ and $k$ are non-negative integers such that
$h + k = p − 1$.
I want to prove that $h!k! + (−1)^h \equiv 0 \pmod{p}$
My first thought is that by Wilson's theorem, $(p-1)! \equiv -1 \pmod{p}$, and $h!k!$ divides $(p-1)!$ (definition of a binomial). Where would I go f... |
I've just learned about topological quotient spaces and was wondering if anyone can help me with this example I thought of.
Let $(\mathbb{Q}, +)$ be the usual group of rational numbers for addition, likewise $(\mathbb{R}, +)$. Set $S$ to be the set of all cosets, t.i. $S=\mathbb{R}/\mathbb{Q}=\{x + \mathbb{Q} \mid x \i... |
$f$ is a linear map on a $k$-vector space $M$ with $f^n=0$ for some $n>0$. $k$ is a field.
$k^d$ is $k$-algebra generated by $x_1,\ldots,x_d$ satisfy the relation $x_i x_j + x_j x_i =0$ for all $i,j=1, \ldots, d$.
Show that $\ker(f)$ is non-zero.
Thank you!
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I am proving that
$$5n^3 + 7n^5 \equiv 0 \pmod{12}$$
It would suffice to show
$$n^3 \equiv n^5 \pmod{12}$$
How would I go about doing that?
I suppose I could just go through each $n \equiv r \pmod{12}$ with $r$ from $1$ to $11$ and show that $n^3 \equiv n^5 \pmod{12}$ for each, but that would be tedious. Surely ther... |
I'm trying to find two statistics $T_1$, $T_2$ such that $(T_1, T_2)$ is jointly sufficient for $(\lambda, \theta)$ for a random sample $X_1, \dots, X_n$ from a two parameter exponential distribution.
$f(x) = \begin{cases}
\lambda e^{-\lambda (x-\theta)}, & \theta < x < \infty, \\
0, & \text{elsewhere}.
\end{cases}$
Th... |
sorry for my bad English...
I've a system consists of many cuboids, many are adjacent to the others.
My problem is... I want to find the largest cube in this system (which may consists of many of some or all part of the cuboids)
Is there a decent algorithm for finding this ?
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I have the following Lemma (from Rotman)
Let $X$ be a space and, for $i=0,1$, let $\lambda_i^X:X \to X \times I$ be defined by $x \mapsto (x,i)$. If $H_n \left(\lambda_0^X\right)=H_n\left(\lambda_1^X\right):H_n(x)\to H_n \left(X \times I\right)$, then $H_n(f)=H_n(g)$ whenever $f$ and $g$ are homotopic.
With the proof o... |
Let us define a one-to-one function $f$ that maps binary strings of length $n$ to ternary strings of length $n$ such that if $x$ is random then $f(x)$ must be random. My question
Is there an algorithm that computes such function?
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If I have:
\begin{align*}
N &\equiv 1 &\pmod{2}\\
N &\equiv 2 &\pmod{3}\\
N &\equiv 3 &\pmod{4}\\
&\vdots\\
N &\equiv n - 1 &\pmod{n}
\end{align*}
How could I solve for $N$? Is there any property relates to this problem?
Update
Base on Moron hint, we have:
\begin{align*}
N + 1 &\equiv 0 &\pmod{2}\\
N + 1 &\equiv 0 &... |
Possible Duplicate:
Is there an elementary proof that $∑_{k=1}^n 1/k$ is never an integer?
Hello,
Prove that $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$ is not an integer.
I tried to prove by induction on $n$, but I was stuck :(
Assume $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} = \frac{a}{b}$ for s... |
Let $V$ be a finite-dimensional irreducible representation of a finite group G over an algebraically closed field. If $V \otimes V$ has a fixed subspace acted by $G$, why should it be 1-dimensional? I really don't know how to think about this. I tried reasoning that there is a fixed space then $\phi(g)$ must have eigen... |
I've come across the term "pairwise distinct" in many research papers. But, I don't understand how it differs from just saying that the elements of a set are unique instead of saying that they are pairwise distinct.
Can someone please explain the difference, if any, to me?
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I am having a hard time understanding a condition in building a measurable function. This is an example of a measurable function from this book (p. 55). Quoting verbatim:
Consider $R_n(\omega)$, the Rademacher functions. These are measurable because they are piecewise constant; namely, for any subset $A$ of $\mathbb{R... |
Prove by element-wise:
$$A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$$
This is easy to prove by Venn Diagram, which I have already done. I do not, however, know how to prove using this technique. I know it has to do with proving each is a subset, but I do not understand the general method to prove this formally?
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I'm looking for a formula to calculate the product of all proper divisors of a number number $n$. Any idea?
Thanks,
Chan
|
For example:
$6$ has this property since proper divisors of $6$ are: $2$ and $3.
From this thread: What does the product of all proper divisors equal to?
My attempt was:
If $n = p_1^{a_1} \times p_2^{a_2} \times ... \times p_k^{a_k}$
Then $n = n^{\frac{\tau(n)}{2} - 1}$. Where $\tau(n) = (a_1 + 1) \times (a_2 + 1) \... |
I came across this post lying dormant on some online forum. I am putting it here verbatim, it seems to me worth a lot.
By Prof. S. D. Agashe, IIT Bombay
(Source: Vector Calculus, by Durgaprasanna Bhattacharyya, University
Studies Series,Griffith Prize Thesis, 1918, published by the University
of Calcutta, India, 1920,... |
This might be a dumb question but is the following true? $$\left \langle\sum_{n=1}^\infty x_n, y\right \rangle=\sum_{n=1}^\infty \langle x_n, y \rangle$$
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A 30-foot ladder rests vertically against a wall. A bug starts at the bottom of the ladder and climbs up at a rate of 3.5 feet per minute. At the same time, the foot of the ladder is being pulled along the ground at a rate of 1.5 feet per minute until the top of the ladder reaches the ground. Let $x$ be the distance o... |
Can anyone help me solve this problem. I have no idea where to even start on it. Link inside stock option problem
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Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd.
My attempt:
If $a$ is even, let $a = 2^{s}k$ for some integers $k, s$
Then, $$a^{2^{n}} + 1 = 2^{2^{n}s}\cdot k^{2^n} + 1$$ and $$a^{2^{m}} + 1 = 2^{2^{m}s}\cdot k^{2^m... |
This was a question on one of our practice midterms:
Find all invertible $n \times n$ matrices $A$ such that $$A^2 + A = 0.$$
I was told to expand $A^2$ and then solve, but that seems like a really ugly (and hard-to-generalize) solution... are there any better ones?
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I was given the problem: find the area of the region bounded by $y=1/x$, $y=x^2$, $y=0$, and $x=e$.
My approach was to break it up into two integrals, $\displaystyle \int_0^1 (x^2-0)\,dx$ and $\displaystyle \int_1^e\left(\frac{1}{x}-0\right)\,dx$.
Doing it this way resulted in an answer of 4/3. Simply by eyeballing it... |
Given a uniformly continuous function $f(x)$ on the real numbers $\Bbb R$, then by the definition of uniform continuity this means: for any $\epsilon>0$ there exists $\delta >0$ such that $|f(x)-f(y)|<\epsilon$ whenever $|x-y|<\delta$. My question is: Given $\epsilon_{n} =\frac{1}{3^{n}}$ (for example), and $y_{n}=x_{n... |
I'm trying to formally show that if two circles $\Gamma$ and $\Gamma'$ are tangent at $A$, then $\Gamma'$ lies either entirely inside or entirely outside $\Gamma$.
The one particular case I'm struggling with is when $\Gamma$ and $\Gamma'$ lie on the same side of the tangent line $l$ at $A$. I know that $O$, $O'$ and $A... |
Let $p(z)$ be a polynomial and $C$ denote the circle $|z-a| = R$. I want to evaluate the integral over $C$ of $p(z)$ with respect to $\overline{z}$, the conjugate of $z$.
I think to start off, I should probably use a Taylor expansions centered at $a$, but I'm lost with what to do next.
Any hints on how to approach t... |
This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher)
If $0 \stackrel{id}{\to} A \stackrel{f}{\to} B \stackrel{g}{\to} C\stackrel{h}{\to} 0$ is a short exact sequence of finitely generated abelian groups, then $\ope... |
In many articles and papers I find the phrase
...discrete logarithm problem in many groups of cryptographic interest...
commonly used. I would like what groups are they exactly referring to when they mention "groups of cryptographic interest" when it comes to the discrete logarithm problem.
Thanks!
|
In order to develop a divisible test for 13, we use $1000 \equiv -1 \pmod{1001}$.
I understand the idea; however, why do we use $1001$, can we use any smaller number? For example, to test for divisible by $11$, we need to use only $10 \equiv -1 \pmod{11}$?
Thanks,
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My attempt was:
By Fermat's little theorem:
$$2^{22} \equiv 1 \pmod{23}$$
$$(2^{11})^2 \equiv 1 \pmod{23}$$
I checked with my calculator the remainder is actually $1$. However, I wonder if I can take the square root on both sides of congruence. Any idea?
Thanks,
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Let $f_n, f \colon [a,b] \to \mathbb{R}.$
Show that, if $f_n \to f$ uniformly, then the Lebesgue integrals are equal, i.e. $\int f = \lim \int f_n$.
This is clearly true for continuous functions, but how do I handle the case of non-continuous functions?
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Suppose $u$ and $v$ are measures on a measurable space $E$. Further suppose $u$ is finite and absolutely continuous with respect to v ($v(S)=0 \implies u(S)=0$).
The problem is: Show that $\forall \epsilon>0 \;\;\;\exists \delta: v(S)<\delta \implies u(S)<\epsilon.$
I've tried the following:
Fix $\epsilon>0$.
Let $S_n... |
Let $S$ be a finite group with operator + and $\pi$ be a permutation on $S$. Then what is the probability that $\pi(x) + x$ is injective over choices of $\pi$?
The concrete instantiation I'm interested in is $S=$GF$(2^n)$ for fixed $n > 0$. (Computer Scientists call this "xor on $n$-bit strings.")
For $n=1$ we have... |
Definition:
A complete system of residues modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set.
Example: The division algorithm shows that the set of $0, 1, 2, ..., m - 1$ is a complete system of residues modulo m.
What I did not understand is "congruent mo... |
Prove that if a and b are relatively prime, then
$$\sum_{n=1}^{a-1} \left\lfloor \frac{nb}{a}\right\rfloor = \frac{(a - 1)(b - 1)}{2}$$
My attempt was:
We have:
$$\sum_{i=1}^{n-1} i = \frac{n(n - 1)}{2}$$
Then,
$$\sum_{n=1}^{a-1} \left\lfloor \frac{nb}{a}\right\rfloor = \left\lfloor \frac{a(a - 1)b}{2a}\right\rflo... |
Is $2^{340} - 1 \equiv 1 \pmod{341} $?
This is one of my homework problem, prove the statement above. However, I believe it is wrong. Since $2^{10} \equiv 1 \pmod{341}$, so $2^{10 \times 34} \equiv 1 \pmod{341}$ which implies $2^{340} - 1 \equiv 0 \pmod{341}$
Any idea?
Thanks,
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Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.
$y= 27e^x$, $y= 27e^{-x}$, $x = 1$.
|
Of course there are many ways to prove this. However, I came across the following exercise (Ch. 0 #3).
Prove that:
(a) a regular surface
$S\subset \mathbb{R}^3$ is an
orientable manifold if and only if
there exists a differentiable mapping
of $N:S\rightarrow \mathbb{R}^3$ with
$N(p)\perp T_p(S)$ and $|N(p... |
N urns are assigned m balls in a stochastic process based on a Pareto distribution. The process is as follows:
X is a Pareto random variable (xminimum = 1, alpha is a parameter)
if X > N, throw the ball out
Otherwise, put the ball in urn number floor(X)
Do this until m balls have been put in urns. Then rank the urns b... |
Possible Duplicate:
“Why do I always get 1 when I keep hitting the square root button on my calculator?”
Take any positive number above or below 1. Upon performing repeated square root operations the end result is always 1. Is there a theorem in Math that can prove this rather intriguing result? Thank you.
Klim
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Let $k$ be a field and $R$ be the exterior ring over $k^d $, that is, $k$-algebra generated by elements $$x_1,\ldots,x_d,$$
where $$\ x_ix_j= - x_jx_i?$$
Is $R$ Artinian?
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I want some hints in how to solve these questions:
How can I separate two different point by polynomial in the coordinate ring $\ k[X] $?
Many thanks
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The Pareto distribution is used to say, given a particular person X, what is the pdf of his wealth.
I would like to explore the reciprocal question: Given the total amount of wealth in a population, what portion does a random person have. I conjecture that this is simply a constant times the Pareto distribution.
More ... |
What is the difference between additive and multiplicative kalman filters, as well as some other kinds?
I'm also looking for reference texts and articles that describe the algorithms, so recommendations for those as well would be preferable.
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I'm giving a presentation about orientable surfaces (as a student project) and I was wondering what I should talk about. The presentation should be 20-30 min long.
I've been thinking maybe something like this:
1) definition of a surface
2) what does orientable mean (orientable vs. oriented)
3)
And then I'm a bit lost... |
I am learning some Cryptography and I came across this exercise where I have to make the following proof (translated from German, so I hope it is accurate).
Proof the following assertion:
Let $n \in \mathbb{Z}$ be odd and $a \in \mathbb{Z}_n^*$. If $a^{\frac{n-1}{2}} \neq \pm 1 \mod n$ , then $n$ is a composite.
N... |
I have the following (directed) graph:
A --> B
the transition is labled C. Is there symbol to denote the inverse of the (A,B) transtion?
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Σ:signature
T:Σ-theory
For M:Σ-model which satisfies T, is it proper to name it “Σ,T-model” or “Σ-model satisfying T”? In comparison, in the context of Lawvere theories, any Lawvere theory L incorporates a signature as well as axioms. If we construct L from Σ and T, any L-model obviously satisfies T. Therefore “Σ,T-mod... |
Thank you~
Show that if $G$ is a finite nilpotent group, then every Sylow subgroup is normal in $G$.
I know that the normalizer of any proper subgroup of a nilpotent group contains this subgroup properly. So I think maybe I can prove the normality of the Sylow subgroup of $G$, say $P$, by showing that $N(P)=N(N(P))$,... |
What software do you use and would recommend for solving recurrence relations?
For example mathematica has RSolve:
$ a(n+1)-2 a(n) = 1$ for $a(n)$:
RSolve[a[n + 1] - 2 a[n] == 1, a[n], n]
-> a(n) = c 2^(n-1)+2^n-1
Are there any good (preferably open) alternatives?
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The answer is 2454, apparently. I am getting 1446, however. Can anyone point out my error ?
We need to partition the solution into the mutually exclusive cases of choices of 4 letters where we have no repetitions (e.g. MATH), where we have one repetition (e,g, MATT) and where we have two repetitions (e.g. MMTT). Then w... |
$f$ is continuous between $[0,1]$, and $f(0)=f(1)$.
I want to prove that there is an $a \in [0,0.5]$ such that $f(a+0.5)=f(a)$.
ok, so Rolle's theorem can be useful here, but I can't see the connection to the derivative,
(Weierstrass, Uniform continuity?) I'll be glad to instructions.
Thanks.
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In my project I'm using the value of public exponent of 4451h. I thought it's safe and ok until I started to use one commercial RSA encryption library. If I use this exponent with this library, it throws exception.
I contacted developpers of this library and got the following reply: "This feature is to prevent some att... |
the fucnction y(x) implement the Equation y''-xy=0.
In addition, know that y(0)=0 and y'(o)=1.
find the value of y(0)^(n), which means the value of 0 in the nth derivative.
Thanks.
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Update: it is not possible to reply to this question without additional information.
My comment below: "I have to agree with you that one "cannot derive (2) from (1) alone". Now it seems to me that one must consider how the sequences $v_{h,n}^{\prime }$ and $v_{h,n}$ are constructed. That is described in the first par... |
I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated when I try to follow mathematical notation. Specifically, I get frustrated trying to keep track of what each variable signifies.
As a programmer, this would be complet... |
Let there be given a set of $n$ real numbers, $\{r_i\} \subset (0,1)$; is it possible to find some conditions satisfied by a real number $m \in \mathbb{R}$ to ensure that:
$\hspace{6cm}r_1^m+r_2^m+r_3^m+...+r_n^m=1$
For example, it might be possible to define such an $m$ by an infinite series.
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I am curious if something like the Nine Lemma (http://en.wikipedia.org/wiki/Nine_lemma) is true in an arbitrary triangulated category. To be more explicit, suppose I have a map of cofiber sequences/distinguished triangles and I take the cofiber/mapping cone at each stage vertically (this gives a diagram like the diagra... |
I'm am a little bit stuck on this question, any help is appreciated.
Show that for every $n\in\mathbb{N}$, $3^{4n+2} + 1$ is divisible by $10$.
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I'm looking for a basic motivation for this topic. Meaning, why would one want to study the function field of a curve. The reason I usually give to this question is that one understands a space by understanding the functions that can be defined on it. Also, this object categorically classifies curves up to birational e... |
I am aware of the big theta notation $f = \Theta(g)$ if and only if there are positive constants $A, B$ and $x_0 > 0$ such that for all $x > x_0$ we have
$$
A|g(x)| \leq |f(x)| \leq B |g(x)|.
$$
What if the condition is the following:
$$
C_1 + A|g(x)| \leq |f(x)| \leq C_2 + B |g(x)|
$$
where $C_1, C_2$ are possibly... |
I have 3 vectors in a high-dimensional euclidean space. I would like to express the length of their sum in terms of their lengths and inner products. For two vectors, I can do it with the law of cosines:
$c^2 = a^2 + b^2 - 2ab\cos\theta$
which in my context translates to:
$||c||^2 =||a||^2+||b||^2 -2 a \cdot b$
(where... |
I have the following somewhat awkward definition of covariant differentiation along a curve:
Let $S \subseteq \mathbb{R}^N$ be a smoothly and isometrically embedded manifold, and $\alpha : I \to S$ be a smooth curve. The covariant derivative along the curve $\alpha$ of a vector field $V: I \to \mathbb{R}^N$, $V(t) \in... |
I have the following differential equation:
$$y''+y=\cos(t)\cos(2t)$$
Maybe something can be done to $\cos(t)\cos(2t)$ to make it easier to solve. Any ideas?
Thanks in advance.
|
In a random drawing with 50 entries are the odds of winning better if I:
A. Enter 1 contest 3 times
or
B. Enter 3 contests 1 time
|
I have a bunch of data, and I'd like to find the closest exponential aproximation I can to fit the points. I'm guessing there's a (relatively) straightforward way to do this.
For example, if I have the data points (5, 9, 17, 33), the equation I'd be looking for would be f(n) = 2^x + 1, since that would fit the data po... |
Consider the string 'Let $y = f(x)$." Suppose that it occurs in some elementary context, such as when graphing the function $f$ using $x$/$y$ coordinates. How is this to be understood in predicate logic? We can't have either $x$ or $y$ be free variables, for consider the following:
Let $f:\mathbb{R}\rightarrow \math... |
Are there any restrictions on $\pi_1(S^3\backslash K)$ for a (tame) knot $K$ besides having $\pi_1^{\text{ab}}(S^3\backslash K)=\mathbb{Z}$?
So we have 1) finite presentation, 2) $H_1=\mathbb{Z}$, 3) $H_2=0$, and I saw somewhere 4) the knot group must be the normal closure of a single element.
Apparently this characte... |
Let $G$ be a group, $X$ a set. Defining an action $G\times X \to X$ is the same as defining a group morphism $\rho: G\to Sym(X)$, through the formula $g\cdot - = \rho(g)$. The morphism $\rho$ is called a permutation representation.
Now, in what little representation theory I have studied one replaces the set $X$ with a... |
Lets say you have a Riemannian Manifold $(M,g)$, and you have some given chart where $g = g_{ij} dx_i dx_j$ and you wish to compute the Christoffel symbols for the Riemannian connection in this chart. To do this involves:
1) Calculating the inverse matrix of $g$, which is not too bad in dimensions 2 and 3, but becomes ... |
I'm working on implementing Order Preserving Encryption for Numeric Data, and part of the algorithm includes approximating density of the distribution as a linear density function $f(p) = qp+r$ where $p$ exists in the distribution.
From the article:
If a distribution over $[0,ph)$ has the density function $qp + r$, wh... |
Let $a$ and $b$ be positive reals both greater than $1$. I'd like to compute the limit of the summation
\begin{eqnarray}
\lim_{x \to \infty} \frac{1}{\log_a x} \sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \left \lbrace (\log_{b} x) - k \log_b a \right \rbrace,
\end{eqnarray}
where $\{ \cdot \}$ denotes the fractional p... |
Let $F$ be an algebraic Number Field and $O_F$ its ring of integers. If $\alpha \in O_F$ then is there a criterion for when $\alpha$, $\alpha'$ are relatively prime?\
\
I am most interested in $O_F=\mathbb{Z}[\zeta_3]$ for the time being.\
\
It seems like a possible criterion could be Let $a,b\in \mathbb{Z}$ and $\alp... |
Let $\mathcal{O}_K$ be the ring of integers of some number field $K$.
It happens that $\mathcal{O}_K$ might not have unique factorization, but...
We can form the multiplicative group of ideals of $\mathcal{O}_K$
It has unique factorization
This construction doesn't seem to be a ring
Each ideal can be put into the form... |
(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic geometry study.
(2) I'm guessing a line is the variety $\mathcal{V}(f_1,\ldots,f_{n-1})$, where $f_i$ are linear homogenou... |
I am giving a talk on Euler's proof that $X^3+Y^3=Z^3$ has no solutions in positive integers. Some facts that I believe to be true are the following. For some I give proof. Please verify that my reasoning is correct and make any pertinent comments. I use the notation $\zeta=\zeta_3$.
(a) $\mathbb{Q}(\sqrt{-3})=\ma... |
How can I express that if an expression evaluates to a negative number, I assign it a value of zero? Is it possible to do this without repeating the function twice?
Here is an example
$$f(\vec{\phi})=\begin{cases}
f(\vec{\phi}) & \text{for}\; f(\vec{\phi}) \gt 0 \\\
0 & \text{for}\; f(\vec{\phi})\lt 0.
\end{cases... |
Given an algebraic monoid $M$ (by which I mean a monoid in the category of schemes), one has the group $M^\times$ of invertible elements (which by general abstract nonsense is also a scheme, see my comment below), which comes with a map of algebraic monoids $M^\times \to M$. Is this map an open immersion?
I have only ... |
Okay, this is really bugging me:
My Math book has this practice where I need to get the area and perimeter of the next Circular Crown:
$R = 3$cm , $r = 1.75$cm.
Well, I do it. But my results are simply different than the ones the book tells me (on the back of the book, there is an answers sheet). Why? The process to ... |
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