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Definition: n is said to be a highly composite number if and only if $d(n)>d(m)$ for all $m<n$, where $d(n)$ denotes number of divisors of n.
Questions:
1) Are there any theorems about highest prime that divides n, assuming that n is highly composite?. It seems that the highest prime is of the order of ln(n).
2) Are th... |
You are given three non-negative integers $A$, $B$ and $C$, find a number $X$ (say) satisfy
$X^A \equiv B\pmod{2C + 1}$ and $0 \le X \le 2C$.
I am inquisitive about how to approach this one?
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Common form of system of linear equations is A*X = B, X is unknown. But how to find A, if X and B are known?
A is MxN matrix, X is column vector(N), B is column vector(M)
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Several papers give poly-time algorithms for constrained paths on labelled graphs, e.g. [1]
Quote:
Given an alphabet Σ, a (directed) graph G whose edges are weighted and Σ-labeled,
and a formal language L ⊆ Σ∗ , the formal-language-constrained shortest/simple path problem con-
sists of finding a shortest (simple) path ... |
I am helping my teenager son. So far I have been able to answer all his questions, and to solve all the problems he has had problems with. Except this one.
$\int{{3^x}{e^x}dx}$
Please, help me keep my success track and my son's confidence.
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Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an
eigenvalue of
A I
I A
where I is the 4 × 4 identity matrix?
(A) -5 (B) -7 (C) 2 (D) 1
I am not able to find out relation between eigenvalues of matrix A and of the given block matrix.
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Let $x,y,z$ be three random variables. How can you show that:
$$\operatorname{cov}(x+y,z) = \operatorname{cov}(x,z) + \operatorname{cov}(y,z)$$ by using the definition of covariance.
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My problem is:
What is the expression in $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$?
Thank you very much~
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In our analysis course, the following question came up and could, up to now, not be solved:
Let $a: \mathbb{N} \to \mathbb{C}$ be a sequence of complex numbers. What are necessary and sufficient conditions for the existence of a function $\mu: 2^\mathbb{N} \to \mathbb{C}$ satisfying the properties
$\mu(\{i\}) = a_i$
... |
How can I find $$\int \tan x \cos 2x \, dx?$$
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Looking on Internet, I mostly found this definition:
Given quantities on a cyclic domain D, first rescale the domain to [0;2$\pi$[, then, find $z_n$ the point on the unit circle corresponding to the $n$th value, and compute the average by:
$$z_m = \sum_{n=1}^N z_n$$
The average angle is then $\theta_m = \arg z_m$ and ... |
Is it true that $O(M^3 + NM^2) \, = \, O(M^3 + N)$, where $M$ and $N$ are variables of the function?
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Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$.
I can solve this question only if there is $x$ instead of $x^2$ in the exponential function $e^{-a^2x^2}$. Because in this situation i can use integral formula :-
$$\int e^{-ax} \cos sx\,dx = \frac{e^{-ax... |
I know this is a quite general question , but I've always heard about creating mathematical models in economics, social sciences, engineering,... And I would want to know what are the starting points and roadmap to understand what a mathematical model is and how to create them. Thanks
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Consider the Chernoff bound: It assumes $X_1, \ldots, X_n$ are independent Bernoulli random variables, each having probability $p > 1/2$, and seeks the probability of simultaneous occurrence of more than $n/2$ of the events.
I once heard that, in the proof of the Chernoff bound, parameters $p$ and $n$ must be constant.... |
Let $G$ be a group of permutations of $n$ objects. Let subsets $A$,$B$ of the objects be equivalent if $A=g(B)$ for some $g \in G$.
Is there a way to show that a bijection exists between the equivalence classes and the $2$-colorings of the objects that are invariant under those permutations in $G$?
So Far, I have ve... |
I am a calculus TA a bit stumped at a question a student has posed me.
What is the method for finding the sum of the series $$\sum_{n=0}^\infty\frac{1+n}{3^n}$$ If you split the numerator up, one becomes a geometric series summing to 3/2, and the other sums to 3/4 by Wolfram Alpha, but I can't explain why.
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I understand the basic idea of Conditionally Convergent (some infinitely long series can be made to converge to any value by reordering the series). I just do not understand how this could possibly be true. I think it defies common sense and seems like a clear violation of the Commutative Law.
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I'm working my way through Linear Algebra Done Right. To help with one proof, I want to prove the following:
Given $\mathbf{V}$, a vector space and $T$, a linear operator on it, then:
If $\mathbf{W}_1$ and $\mathbf{W}_2$ are subspaces of $\mathbf{V}$ such that:
$\mathbf{V}$ is a direct sum of $\mathbf{W}_1$ and $\math... |
The Collatz Conjecture is a famous conjecture in mathematics that has lasted for over 70 years. It goes as follows:
Define $f(n)$ to be as a function on the natural numbers by:
$f(n) = n/2$ if $n$ is even and
$f(n) = 3n+1$ if $n$ is odd
The conjecture is that for all $n \in \mathbb{N}$, $n$ eventually converges under i... |
Let's say that I have a tax of 17% over my services and I want to receive $20,000.00 regardless (after paying) the tax.
So I want something like:
$$X - 17\% = \$20,000.00$$
Where $X$ is the final value that I need to charge to receive $\$20,000$.
How do I calculate this?
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Possible Duplicate:
finding inverse of $x\\bmod y$
Hello all
Me and some friends are studying for a discrete exam and we are having some troubles finding the inverse modulus of things. The question we are suck on is "what is the inverse of 8 mod 29".
Thanks a lot!
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Draw two balls centered at the origin in R^3, where $r_1 < r_2$. What is the topological name for the boundary created? Does it even make sense to talk about a figure with an inside boundary and an outside one?
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I have a problem with solution of this limit:
$$\lim_{x\to 0}{\frac{\tan{x}-x}{x^2}}$$
Of course, it's a very easy to solve, using (twice) L'Hôpital's rule, but I need to find out, how to do this without this rule.
I stuck in this point:
$$\lim_{x\to 0}{\frac{\frac{\sin^2{x}}{x^2}\cdot\frac{1}{\cos^2{x}} - 1}{2x}}$$
... |
Ok, so I just finished a Discrete Math class where we learned about all types of proofs and counting and graphs, etc. So I know the basics, but never thought I'd need to use it in the real world (not anytime soon at least). However, a friend of mine showed me a puzzle game, and he said that it's always possible to win ... |
If we have a two lattices (partially ordered) - one for subgroups, one for factor groups, and we know order of the group we want to have these subgroup and factor group lattices, is such a group unique up to isomorphism (if exists)? Or is there a counterexample?
If that's true, are sufficient conditions on the order an... |
Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion?
Edit: I forgot to mention that the set has to be in $[0,1]$ which seemed "clear" in my head, but which obviously isn't.
Edit 2: Now it is cle... |
Anyone know how to convert from lat/lng to EPSG:27700?
|
In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to understand the theory behind it.
Examples:
Let $y$ and $x$ both be functions of $t$.
Let $z$ be a function of both $x$ and $y$... |
Is the projective closure of an infinite affine variety (over an algebraically closed field, I only care about the classical case right now) always strictly larger than the affine variety? I know it is an open dense subset of its projective closure, but I don't think it can actually be its own projective closure unless... |
in standard text books on (smooth) manifolds, for example the known series by John M. Lee or Jeffrey Lee, you either deal with continuous manifolds, or with smooth manifolds.
However, neither in these books nor in lectures I have encountered real examples when a manifold may be $C^k$, but not $C^{k+1}$.
Intuitively, I ... |
What would be the best way to calculate some sort of correlation or similarity factor between two lists of time intervals :
List1 :
1. 2010-06-06 to 2010-12-12
2. 2010-05-04 to 2010-11-02
3. 2010-02-04 to 2010-10-08
4. 2010-04-01 to 2010-08-02
5. 2010-01-03 to 2010-02-02
and List2 :
1. 2010-06-08 to 2010-12-14
2. 2010-... |
There's a question in Spivak's Calculus (I don't happen to have the question number in front of me in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, specifically the continued fraction representation of $\sqrt{2}$. Having only recently become aware of continued fractions,... |
For a two-sphere embedded in $\mathbb{R}^4$,how can you check whether or not there is an ambient isotopy to the "standard" 2-sphere (the set of points $(x,y,z,0)$ in $\mathbb{R}^4$ distance 1 from the origin)?
Knot theory was discussed in an intro topology course I took and I'm wondering about further generalizations o... |
The adelics seem counter-intuitive. I wonder how they came up originally, and what was the immediate reward for introducing them. What was the politics of introducing the adelics into mathematical mainstream?
|
Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.
One Dimension - Clearly this structure is infinitely long.
Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$
Three Dimensions - Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}... |
I'm having some trouble proving the following statement: let $g:S^{2n-1} \to S^{2n-1}, f: S^{2n-1} \to S^n$. Then $H(f\circ g) = \deg g H(f)$ where $H(f)$ is the Hopf invariant. The definition I am using for Hopf invariant is as follows: let $C_f = D^{2n} \sqcup_f S^{n}$ where $D^{2n}$ is the $2n$ disk which we atta... |
I have a random variable X that is a mixture of a binomial and two normals (see what the probability density function would look like (first chart))
and I have another random variable Y of similar shape but with different values for each normally distributed side.
X and Y are also correlated, here's an example of data ... |
Let $S$ be a ruled surface over a curve of genus $g$. Is it possible to compute the second Chern class of $S$ in terms of $g$?
|
Just wondering:
How to solve ultrahyperbolic PDEs? Is there any analytical solution for linear ultrahyperbolic PDEs?
If there are only numerical solutions, are the solutions' behavior similar to those of nonlinear eqns? I mean, like an anharmonic oscillator, chaotic but deterministic?
Information of reference books i... |
I've always had the following silly(?) doubt about convergent sequences.
Given a finite sequence $A_{1:n}=(a_1,a_2\ldots ,a_n)$, define its reverse as $rev(A) = (a_n,a_{n-1},..a_1)$. Further, if $A$ is infinite, define its reverse as the limit of the reverses of its initial subsequences, i.e. as the limit of the se... |
Is there an algorithm to find out all possible resistance values of series, parallel, and series-parallel arrangements given $n$ identical resistors, $R$? All of them must be used.
This might even extend to differently-valued resistors, but I'll just focus on identical resistors.
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I am studying QR decomposition.
Could you explain the geometric intuition for what the Householder transformation does in that context, and why it's sometimes referred to as the Householder reflection.
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I've been looking for less trivial examples of computing Ext than finitely generated abelian groups, which tends to be the standard example (and often the only example). Here's an interesting exercise I found in some notes:
Let $M = \mathbb{C}[x,y] / (x,y), N = \mathbb{C}[x,y] / (x-1)$. My question is how to compute $\... |
Concerning this question of mine which involves telescoping in the solution, I was wondering if it is possible to express
$$\ln(x)=\lim_{n\to\infty}n\left(x^{\frac1{n}}-1\right)$$
and
$$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$
as infinite telescoping products.
Is it possible?
Thanks.
EDIT:
The infinite prod... |
Can a polynomial size CFG over large alphabet describe any of these languages:
Each terminal appears $0$ or $2$ times
Word repetition $\{www^* \mid w \in \Sigma^*\}$ (word repetition of an arbitrary word $w$)
"Polynomial size" Context Free Grammar is defined as Chomsky Normal Form polynomial in $|Σ|$.
References to p... |
The infinite-length sequence $x_1[n]$ defined by
\begin{multline}
x_1[n]=
\begin{cases}
\dfrac{1}{n}& \text{if $n \geq $1},\
0& \text{if $n \leq $0}.
\end{cases}
\end{multline}
has an energy equal to
$\mathcal{E _x {_1}} = \sum^\infty_{n=1}(\dfrac{1}{n})^{2}$
which converges to $\pi^2/6$ indicating that $x_1[n]$ has... |
One partition of 8 is 5 + 3, but if we then partition each of the 5 and 3 we could get (3+2) + (2+1), and then partition again to get ((2+1)+(1+1)) + ((1+1)+1) and finally (((1+1)+1)+(1+1)) + ((1+1)+1). 5+3 could also be expanded as (4+1)+(2+1), then ((2+2)+1)+((1+1)+1), then (((1+1)+(1+1))+1)+((1+1)+1).
This question ... |
I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is not because 5 is in position 5. I know that the number of total permutations is $n!$. Is there a formula for how many ... |
Suggest an algorithm for factorizing an integer such that each of its factor is square-free.
Thanks,
|
Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total Chern class provides a natural map $\mathrm{Vect}(B) \to H^*(B)$ for every such space $B$. By the multiplicativity propert... |
Given two positive rational number $\frac{a_1}{b_1}$ and $\frac{a_3}{b_3}$ (written in lowest terms) such that $$\frac{a_1}{b_1} < \frac{a_3}{b_3},$$
I want to find a rational number
$\frac{a_2}{b_2}$ such that $$\frac{a_1}{b_1} < \frac{a_2}{b_2} < \frac{a_3}{b_3}$$
and $a_2$ and $b_2$ are the smallest possible numb... |
I'm trying to compute the variational distance between two negative exponential distributions. If $a$ denotes the value of $x$ where both curves are equal, then:
$$\int_0^{+\infty}\left|\lambda_1e^{-\lambda_1x}-\lambda_2e^{-\lambda_2x}\right|dx$$
$$=-\int_{0}^a(\lambda_1e^{-\lambda_1x}-\lambda_2e^{-\lambda_2x})dx+\int... |
Let $f(x)$ be two times differentiable in $[a, b]$. The Mean Value Theorem tells us that there's such a point $a\lt c\lt b$, such that ${f}'(c)=\frac{f(b)-f(a)}{b-a}$.
If $\forall x \in [a, b], {f}''(x)\gt 0$ and $f''(x)$ is strictly increasing then $c\gt \frac{a+b}{2}$.
I'm having a hard time figuring this one out. Mo... |
This is a quick (not to scale) drawing of what I mean: the floor gradually slopes from the wall to the drain at about 7/16( 0.4375) inch per foot from the short sides of the wall (the 7 ft from wall to the drain) and exactly 3/16(.1875") of a inch per foot on the long sides ( the 16ft from the wall to the drain).
So th... |
Why is $\int_{-1}^1 x =\frac{x^2}{2}$ rather than $x^2$?
|
Suppose $f$ is analytic on the domain $D = \{z \big| |z|>1\}$ in $\mathbb{C}$. I know that if the line integral of $f$ along any closed path is 0 then $f$ has an anti-derivative. Is this also a necessary condition.
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Two players in a dice game to see who can roll a total of 60 first, taking turns each rolling 2 dice.
For one player, one die is 4 sided and one is 6 sided. Therefore the average roll for this player will be a 6.
The other player has two 6 sided dice. So this players average roll will be a 7.
If they take turns, eac... |
I have this equation $3-x-x^2$ which I want to convert into the form $a^2 - y^2$, so that then I can suppose $y$ = $a\sin\theta$, differentiate and then integrate it. But I can't convert that equation into this $a^2 - y^2$ form.
P.S. this is a homework question. My question here is not to solve this for me, but tell me... |
How to prove $\{t\} \notin t$ using the axiom of foundation (aka axiom of regularity):
$A = \emptyset \vee \exists x \in A \forall y \in x : y \not\in A$
|
I'm trying to understand a bit about chern classes/chern characters (in the algebraic setting, for varieties over C say) and was hoping to find some notes describing some well-known properties of ch (e.g. the relation between the support of a coherent sheaf and the component of ch of the same dimension).
I'm really hop... |
What is the difference between Gödel's completeness and incompleteness theorems?
|
I've recently studied characteristic functions in my probability course and I can't get why we define it to be the Fourier transform of the distribution (if the random variabile is continuous).
I mean that if $X$ is a random variable, $\varphi_X (t) = \mathbb{E}(e^{i t X}) = \int_{-\infty}^{+\infty} e^{i t x}f_X(x) dx... |
First,
to understand my question, checkout this one:
Calculating percentages for taxes
Second,
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final result. I was wondering:
Why multiply $20,000 by 100 and then divide by 83? I know this is the rule of three, but, I can't un... |
I am looking for the the method of constructing the midpoint of two points in spherical geometry. The only tools allowed for the construction are a pair of spherical compasses and a spherical ruler.
In Eclidean geometry constructing the midpoint is relatively easy. We are looking for the midpoint of points A and B. We ... |
I haven't found any useful method to solve the following problem: Prove that if $x,y,z\in\mathbb{Z}$ and $x^3+y^3=3z^3$ then $xyz=0$.
Source: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=382377
|
There's a puzzle where you have 3 houses and 3 utilities. You must draw lines so that each house is connected to all three utilities, but the lines cannot overlap. However, I'm fairly sure that the puzzle is impossible. How is this proved?
|
Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be uniformly continuous with $g(0)=0,c\geq 0, c \in \mathbb{R}$. Show: $$\exists a\geq 0 \in \mathbb{R}: \forall x \in \mathbb{R}: |g(x)| \leq a \cdot |x|+c$$
I could also say $g(x) \in \mathcal{O}(x)$.
Notes: I could not make up any counterexample so I guess it could be tru... |
The solution for the number of distributions leaving none of the $n$ cells empty (with unlike cells and $r$ unlike objects) is given by
$$A(r,n)=\sum_{\nu=0}^{n-1}(-1)^{\nu}\binom{n}{\nu}(n-\nu)^{r}$$ (although I have seen the same expression summing from $\nu=0$ to $n$).
By the way, if anyone knows which one is corre... |
Linear Algebra - True or False if $Ay = B$ and $Az=B$ then y = z? Why or why not? you may give a counter example to prove it false.
|
I am looking at the following problem:
Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
$-1<x_1=x_2<0$
$x_1<-1<x_2<0$
$-1<x_1<0<x_2$
Let $a=3m-2$, $\Delta... |
Is the intersection of a decreasing sequence of countably many stationary subsets of $\omega_1$ always stationary?
It seems elementary, but I fail to find the answer in textbooks.
|
I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the application of lattice point methods. (The audience consists of junior and senior faculty across the mathematical disciplines.) A ... |
Is it true or false that a group of order $12$ always has a normal $2$-Sylow subgroup?
I have a hunch it is false.
|
Consider an sequence of unique values. Each value is composed of a surname and a first name in that order of significance. Based on that, I can define a complete ordering of values in the sequence.
However, I can also usefully define less fine-grained equivalence classes of items that have a "consistent" ordering, but ... |
This may seem like an overly trivial question, but I've just recently become confused about Langrange's 'prime' notation for derivatives (for example $f'(x)$).
I know for sure that $f'(x) = \frac{\delta f(x)}{\delta x}$.
But suppose we replace x with an expression, like 2x+1. Do we write $f'(x^2+1) = \frac{\delta f(x^2... |
I am mostly asking about terminology here.
Background A generalized linear eigenvalue problem is of the form $Ax=\lambda B x$, where $A$ and $B$ are square matrices, $\lambda$ is the eigenvalue and $x$ is the eigenvector. We can write this in the form that makes it evident that it is linear in $\lambda$: $(A-\lambda B)... |
Let $F(n)$ be $n$-th Fibonacci number.$$F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3 \text{ and so on. }$$Given a positive integer $n \gt 2$,Find the smallest prime number $P$ such that $P$ divides $F(n)$ but it does not divide any $F(k)$ smaller than $F(n)$ ?
Now,my question: Is it possible to find the answer wit... |
Suppose we have a Hadamard matrix $H$ with rows $H_1,H_2,\ldots$ and look at the following system
$$H_j \cdot \mathbf{x} \le 1 \ \forall j$$
$$\sum_i x_i=1$$
Is there a name for polytopes defined in this way?
|
The invariant subspace problem states something along the following lines: Let $H$ be a Banach space of dimension greater than one. Does every bounded linear operator $T:H \to H$ have a non-trivial closed $T$-invariant subspace?
Now my question is: Why is this a big problem? What does the solution give us? It has been ... |
Fiction "Division by Zero" By Ted Chiang
I read the fiction story "Division by Zero" By Ted Chiang
My interpretation is the character finds a proof that arithmetic is inconsistent.
Is there a formal proof the fiction can't come true? (I don't suggest the fiction can come true).
EDIT: I see someone tried
|
Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what probability distributions can exist over the positive real numbers (it seems that this would be quite limited, given that such a... |
I recently took my Honors Pre-Calc. exam. Wanting to get as many points as I possibly could, I circled and skipped several problems. However, I forgot to do this one, and wanted to know if I'm on the right track, because the answer I get for x doesn't work correctly. I've come up with a separate solution that comes clo... |
Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution?
The Brauer–Fowler results show that if a finite group has no non-identity normal subgroups of odd order (so is core-free), then its order is bounded by a function of the size of a few cent... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equ... |
I'm investigating an idea in cryptography that requires irreducible polynomials with coefficients of either 0 or 1 (e.g. over GF[2]). Essentially I am mapping bytes to polynomials. For this reason, the degree of the polynomial will always be an integer multiple of 8.
I would like to build a table of such irreducible po... |
In numerical analysis, I was asked whether Bairstow's algorithm convergence rate is quadratic.
My initial feeling was that it does, since it is essentially Newton's method for a system of non-linear equations, and Newton's method converges quadratically in one dimension (When f is from R to R).
A quick search on the in... |
This is part of Exercise III.1.16(c) of Bourbaki's set theory.
Let $E$ be a lattice. The condition
$(x\vee y)\wedge(z\vee(x\wedge y))=(x\wedge y)\vee(y\wedge z)\vee(z\wedge x)$
for all $x,y,z\in E$ implies that $E$ is distributive.
I already know that
$(x\vee y)\wedge(y\vee z)\wedge(z\vee x)=(x\wedge y)\vee(y\wedge... |
Let me try to split the question in a few parts,
I would like to understand the claim that all non-degenerate bilinear symmetric forms are equivalent over the complex while for the reals they can be distinguished by the signature. Hence there is just one Clifford algebra over $\mathbb{C}^n$ but over $\mathbb{R}^n$ the... |
This question came up when trying to trying to find a $3\times3$ real matrix $A$ such that
$Ax$ is nonzero for nonzero $x$
$Ax$ is orthogonal to $x$ for any $x$ in $\mathbb{R}^3$
We know such a matrix cannot exist because $A$ must have an eigenvalue (thus there is some $x$ such that either $Ax = 0$ or $Ax$ is para... |
I'm not so well up on my math, but I was wondering if the set of "[EDIT: finitely] decimal expressible" real numbers $R$ between $[0,1]$ is countable?
Wikipedia gives the definition:
A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$
It would seem to m... |
Is there a commutative integral domain $R$ in which:
every nonzero prime ideal $Q$ is maximal, and
there are maximal ideals $Q$ with $R/Q$ of sizes $3$, $11$, and $27$?
This doesn't happen with number rings of Galois extensions of $\mathbb{Q}$, as far as I know, since what happens to $3$ seriously happens to $3$. ... |
Suppose that $g$ is a bijection on the real line, and $g^{-1} \circ f \circ g$ is a $C^\infty$ function whenever $f$ is $C^\infty$. It seems howlingly obvious that this can only happen if $g$ is itself $C^\infty$. But I can't figure out how to prove it.
Can you help me, Internets?
(Context: I want to show that in a ... |
The infinite series...
$\pi/4 = 1 - 1/3 + 1/5 - 1/7 ...$
...is very intriguing to me and seems like a crazy coincidence (its relationship to $\pi$). Is it actually crazy or does it have an easy to explain, logical reasoning behind it that would make it seem not so magical?
|
I was given the question:what is 9+99+999+9999+...+999..99(30 digits) After noticing a trend, I came with the conclusion that the answer would be 28 1's 080. Can anyone confirm my answer and give a reason as to why?
|
Imagine the varying frequency generator, which feeds this frequency to the oscilloscope.
Oscilloscope is turned to show signal spectra with FFT.
The image on the screen of the oscilloscope is as expected: single peak, traveling from 1 kHz to 50 kHz. Of course, my generator is not ideal, it gives a number of harmonics w... |
I have an old Spanish CG book that calls Perlin Noise a "fractal structure". After reading this I couldn't deny it or confirm it. Is it a fractal structure? What would it Hausdorff dimension be?
|
$G = \{ x,y \mid x^{2^{n+1}} = 1, y^4 = 1, xy = yx^{−1} \}$
$H = \{ x,y \mid x^{2^{n+1}} = 1, y^4 = 1, xy = yx^{2^n−1}\}$
Can these groups of order $2^{n+3}$ and nilpotency class $n+1$ be distinguished by any reasonable (computable) set of invariants?
While for any set of reasonable invariants, there is surely a pa... |
Let $X$ be a CW complex. The (isomorphism classes of) complex line bundles on $X$ are classified by the homotopy classes of maps $X \to \mathbb{CP}^\infty$, that is by the elements of $H^2(X, \mathbb{Z})$.
It is also true that the tensor product of line bundles corresponds to adding cohomology classes. It follows that... |
I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is an... |
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