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The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta
\end{align}
So what I want to know is,... |
Following my previous question, I'm wondering how I can determine if a point is within the convex hull of two circles (a boolean value). There's no problem testing if the point is in either of the two circles, but it can also be "between" them and I don't know how to test for that.
Seeing Wolfram MathWorld's article on... |
Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise $A$ would be a discrete valuation ring.)
Is there a concrete example of such a ring $A$?
|
I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive integer, therefore there are the same number of rationals as integers. I've ignored sign-related issues, but these are eas... |
Suppose you have a normal distribution with mean=0, and stdev=1. So the expected value is 0.
Now suppose you limit the outcomes, such that no values can be below 0. So 50% of values now equal 0, and rest of distribution is still normal. Running 1000000 trials, I come out with an expected value of .4
My question ... |
I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.
Edit: The previous wording was terrible.
Given an algebraic group $G$, with Borel subgroup $B$ we can form the Flag Variety $G/B$ which is projective. I am hoping for a list of the graded ring $R$ such t... |
What is a primitive polynomial? I was looking into some random number generation algorithms and 'primitive polynomial' came up a sufficient number of times that I decided to look into it in more detail.
I'm unsure of what a primitive polynomial is, and why it is useful for these random number generators.
I'd find it pa... |
The title really say's it all, but once again is a curve's curvature invariant under rotation and uniform scaling?
|
This is a followup to this question, where I learned that curvature is invariant to rotations.
I have learned of a version of curvature that is invariant under affine transformations.
I am wondering if there a is a form of curvature between the two. Invariant under uniform scaling and rotation but not all affine trans... |
Me and my wife are often not knowing which DVD to watch. If we have two options we have a simple solution, I put one DVD in one hand behind my back and the other DVD in the other hand. She will randomly choose a hand and the DVD I have in that hand will be the one to watch.
This procedure is easy to expand to any power... |
Dot product of two vectors on plane could be defined as product of lengths
of those vectors and cosine of angle between them.
In cartesian coordinates dot product of vectors with coordinates $(x_1, y_1)$ and
$(x_2, y_2)$ is equal to $x_1x_2 + y_1y_2$.
How to prove it?
|
Given a list of coordinates of a coplanar plane $\left(pt_1, pt_2, pt_3, \cdots \right)$, how to compute the centroid of the coplanar plane?
One way to do it is to project the plane onto $XY$ and $YZ$ plane, but I don't really favor this approach as you have to check the orientation of the coplanar plane first before d... |
Let's say we have a graph, with a list of edges and vertexes $(E,V)$, all the vertexes are connected to at least one edge at one end. There are many ways a complete set of cycle basis can be found out from it.
Now the issue is, is it always possible to find a complete set of cycle basis that each edge is shared by at m... |
Proposition 0.4.5.4 in EGA appears to be a general representability theorem. It reads:
Suppose $F$ is a contravariant functor from the category of locally ringed spaces over $S$ to the category of sets. Suppose given representable sub-functors $F_i$ of $F$, such that the morphisms $F_i \to F$ are representable by open... |
You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.
But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré's Elements. This postula... |
I have a Ph.D. in computational and theoretical chemistry with advanced but field-oriented knowledge of mathematics.
I am fascinated by fractals, but I am unable to understand them from the formal point of view. To my level of understanding, they look like a graphical rendering of an ill-conditioned iterative problem, ... |
I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same thing. Similarly the method of partial fractions can be applied to rationals rather than polynomials. I find that seeing ... |
I guarantee there is an easy reference on this, but for some reason I cannot find it. If you can point me to a reference or just write a short proof for me, I would be very appreciative.
Given a graded ring $R_{\bullet}$ and a localization $R_{\bullet}^{*}$. We also have a graded $R_{\bullet}$-mod, $M_{\bullet}$.
So w... |
Struggling with yet another proof:
Prove that, for any positive integer $n: (a + b)^n \geq a^n + b^n$ for all $a, b > 0:$
I wasted $3$ pages of notebook paper on this problem, and I'm getting nowhere slowly. So I need some hints.
$1.$ What technique would you use to prove this (e.g. induction, direct, counter example... |
While reading one aptitude book, I came across this question:
$35\%$ alcohol mixed with $60\%$ alcohol to get a $50\%$ alcohol. In what ratio were they mixed?
I spent 20 minutes on this question, but I couldn't even figure out what it means! Could anyone explain this to me?
|
Some time ago Qiaochu Yuan asked about counting subsets of a set whose number of elements is divisible by 3 (or 4).
The story becomes even more interesting if one asks about number of subsets of n-element set with $r\bmod 5$ elements. Denote this number by, say, $P_n (r \bmod 5)$.
An experiment shows that for small $n$... |
Galois theory tells us that $\frac{z^{11}-1}{z-1} = z^{10} + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be $\zeta^1,\zeta^2,\ldots,\zeta^{10}$, follo... |
Prove that the equation $n^a + n^b = n^c$, with $a,b,c,n$ positive integers, has infinite solutions if $n=2$, and no solution if $n\ge3$.
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I already asked if there are more rationals than integers here...
Are there more rational numbers than integers?
However, there is one particular argument that I didn't give before which I still find compelling...
Every integer is also a rational. There exist (many) rationals that are not integers. Therefore there are ... |
What is the probability that the convex hull of $n+2$ random points on $n$-dimensional sphere contains sphere's center?
|
Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic?
Update. There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, maybe?
|
Suppose that $X$ is a topological space with a sheaf of rings $\mathcal{O}_X$. In general, the stalk at a point $p \in X$ is the direct limit of the rings $\mathcal{O}_X(U)$ for all open sets $U$ containing $p$.
Here are two questions on computing stalks - I think both should be true, since a direct limit should be som... |
This is what I know about why someone might care about fields of positive characteristic:
they are useful for number theory
in algebraic geometry, a theory of "geometry" can be developed over them, and it's fun to see how this geometry works out
Some people might read this and think, "What more could you need?" But I... |
I've been looking online (and in teaching journals) for a good introduction to Algebras of Random Variables (on an undergraduate level) and their usage, and have come up short. I know I can find the probability distribution of $h(z)$ where:
\begin{equation*}
z = x + y.
\end{equation*}
If $x$ and $y$ are from known inde... |
Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some homeomorphism? That is, when can I find $\phi:\mathbb{R}^n \to \mathbb{R}^n$ such that $f \circ \phi \in \mathbb{R}[x_1,\ld... |
This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that:
$L_4(2)$ and $L_3(4)$ both have order $20160$
$O_{2n+1}(q)$ and $S_{2n}(q)$ have the same order for $q$ odd, $n > 2$
I think this means that for each integer $g$, there are $0$, $1$ or $2$ simple ... |
This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone here can enlighten me.
Problem: Prove that the function $f : [0, 1) \to \mathbb{R}$ defined by
$\displaystyle f(x) = \log... |
How to solve this equation?
$$
x = 10^{x/10}
$$
|
The identity
$\displaystyle (n+1) \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \text{lcm}(1, 2, ... n+1)$
is probably not well-known. The only way I know how to prove it is by using Kummer's theorem that the power of $p$ dividing ${a+b \choose a}$ is the number of carries needed to add $... |
Many people are familiar with some properties of binary relations, such as reflexivity, symmetry and transitivity.
What are the commonly studied properties of ternary (3-ary) relations?
If you could provide a motivating example of why the property is interesting that would also be helpful.
|
The PCP theorem states that:
Every decision problem in NP has
probabilistically checkable proofs of
constant query complexity and
logarithmic randomness complexity.
Can anyone give an intuitive explanation of how this can be done?
Links
This is a follow up from this
question.
There is a proof here.
|
I was reading this paper on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and agent who makes the $i^\text{th}$ highest... |
Exercise $10$ in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). That is, (using $A \times B$ to mean the direct product and $A+$B to be the direct sum), whether there is an isomorphi... |
Let $O(f(n))$ be the minimum time complexity for output a real sequence $\{a_i\}$.
The input of the algorithm is a integer $n$. It output the finite sequence $a_1, a_2, \ldots, a_n$. (Clearly $f(n)\geq n$.)
What can we say about the time complexity of
\begin{equation*}
\max(a_1, a_2, \ldots, a_n)?
\end{equation*}
Is ... |
...and encoding it as a probability distribution.
Suppose we have a sequence of non-negative integers that is periodic with period $N$:
\begin{equation*}
A_{1},A_{2},...,A_{N},A_{1}...
\end{equation*}
Each $A_{k}$ takes on a value no greater than some constant $B$:
\begin{equation*}
0 \leq A_{k} \leq B
\end{equation*}... |
Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
|
This page gives a few examples of Venn diagrams for $4$ sets. Some examples:
Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could do for $<4$ sets. Yet it is doable with ellipses or rectangles, so we ... |
Question: I want to solve $0<1−an/(mb^2)e^{−r(T−t)}<1$, where $r, a, b, T, t>0$.
The solution is that either $$an\leq mb^2$$ or $$mb^2\leq an\leq mb^2e^{rT}$$ and $$t< T − (\ln(an) − \ln(mb^2 ))/r$$.
My Attempt: My thoughts are that the first part $0<1−\frac{an}{mb^2}e^{−r(T−t)}$ gives me $an \leq mb^2$ because $e^{−r... |
I am trying to understand this problem and yes this is from my assignment and I should be doing it myself, but I have been staring at it for 2 hours and not getting anywhere, so decided to post it here.
Let G be a connected cubic simple graph that contains 2 edge-disjoint spanning trees show that |G| = 4.
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Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.
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I am very weak in solving problem related to probability and permutation and combination. Can you please guide me the good content by which I can understand it better.
Thanks in advance.
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If $(\mathbb{R},+)$ is a group and $H$ is a proper subgroup of $\mathbb{R}$ then prove that $H$ is of measure zero.
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What is a set? I know that results such as Russell's paradox mean that the definition isn't as straight forward as one might expect.
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This is Problem 1.7 from Gouvea's lecture notes on deformations of Galois representations. In particular, he asks you to show that it has many subgroups of finite index which are not closed. So here's what I've got so far, which may be wrong.
I can write the compositum as F = Q[√–1, √2, √3, √5, √7 ...] (can I?) and the... |
Given the sum of first $N$ elements of sequence $A$: $S_{n} = n^{2} + 3n + 4$.
Compute $A_{1} + A_{3} + \cdots + A_{21}$.
I know this problem can be tackled by carefully calculating each value of the sequence. But I wonder what are the better ways to solve it.
Thanks in advance!
|
For the uninitiated
Morse theory, as many other early algebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of homology as 'c... |
I have to divide $2860$ by $3186$. The question gives only $2$ minutes and that division is only half part of question. Now I can't possibly make that division in or less than $2$ minutes by applying traditional methods, which I can't apply on that division anyways.
So anyone can perform below division using faster tec... |
As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms, so I want to know which Discrete Mathematics book is suitable for my needs?
|
There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"? In particular, I'm looking for reasons that I can explain to first-year calculus students.
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Possible Duplicate:
problem with inequality
How do I go about solving $0<1−\frac{an}{mb^2}e^{−r(T−t)}<1$, where a,b,T>t>0? I have been stuck here for some time now.
|
Here is a classic theorem of Herstein:
$G$ is a finite group, $M$ a maximal subgroup, which is abelian. Then $G$ is solvable.
The proof is pretty easy, but it uses character theory (specifically, Frobenius' theorem on Frobenius groups). Is there a character-theory-free proof?
To get things going, note that we can red... |
I'm trying to understand better the following addition formula: $$\int_0^a \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^b \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_0^{a\sqrt{1-b^2}+b\sqrt{1-a^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}$$
The term $a\sqrt{1-b^2}+b\sqrt{1-a^2}$ can be derived from trigonometry (since $\sin(t) = \sqr... |
I just started learning model theory on my own, and I was wondering if there are any interesting examples of two structures of a language L which are not isomorphic, but are elementarily equivalent (this means that any L-sentence satisfied by one of them is satisfied by the second).
I am using the notation of David Mar... |
As a follow up to algebra-of-random-variables, is it possible to compute:
$z = \frac{dx}{dy}$
Where $x$ and $y$ are drawn from a known, independent probability distribution $x \in f(x)$, $y \in g(y)$? Geometrically the problem makes sense, one has a function $h(x,y) = f(x)g(y)$ that is the joint probability distributi... |
I have an application that wants controllable random functions from $\mathbb{Z}^2$ and $\mathbb{Z}^3$ to $2^{32}$ , where by controllable I basically mean seedable by some parameters (say, on the order of 3 to 5 32-bit integers) such that the same seeds will always produce the same functions. The most obvious way of d... |
Say we have a family {$A_\alpha$} of disjoint Baire spaces. Also suppose that each $A_\alpha$ is disjoint from the closure of the union of the other sets. Show that $\bigcup_{\alpha} A_\alpha$ is a Baire space.
I think we can prove this by transfinite induction. Suppose the property holds for all $\beta$ < $\alpha$, sh... |
This question was motivated by pondering this lcm identity.
Consider that $\gcd(1,6,15) = 1$, but $\operatorname{lcm}(1,6,15)=30$, but $1\cdot 6\cdot 15 = 90$. $(2,6,15)$ shows a similar phenomenon.
So what is the correct identity for $n$-ary $\gcd$/$\operatorname{lcm}$?
|
I asked this question a while back on MO :
One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$)
I am curious, as a sort of followup if anyone can suggest:
A reference where small computations are carried out? or
A specific computation to do with a small eno... |
Can someone explain to me in simple terms what a non-linear transformation is in maths?
I know some single-variable calculus, but I read it has to do with multi-variable calculus, which I'm not familiar with.
If someone could explain it in simple words, that would be helpful.
|
In a recent discussion, I came across the idea of proving a lower bound for the number of comparisons required to find the largest element in an array. The bound is $n - 1$. This is so because the set of comparisons performed by every such algorithm looks like a tournament tree, which always has $n - 1$ internal nodes.... |
How can I calculate the length of the cathetus in a triangle if I know the length of the hypotenuse and the ratio between the two catheti?
For example:
The hypotenuse is $5$cm long, and the ratio between the catheti is $4:3$ - how long are the catheti or either cathetus?
|
When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any intuitive explanations for these concepts, preferably explanations that are less dependent on knowledge of algebraic-ge... |
Stromquist's Theorem: If the simple closed curve J is "nice enough" then it has an inscribed square.
"Nice enough" includes polygons.
Read more about it here: www.webpages.uidaho.edu/~markn/squares
An "inscribed square" means that the corners of a square overlap with the curve.
I would like to suggest a counter-examp... |
A $3\times 3$ matrix $$A = \begin{pmatrix} 2 & -1 & 1\\ -2 & 3 & -2\\-1 & 1 & 0\end{pmatrix}$$
produces this characteristic equation: $\lambda^2 - 4\lambda + 3 = 0$, these eigenvalues:
$\lambda_1 = 1$, $\lambda_2 = \frac{-1 + \sqrt{13}}{2}$, $\lambda_3 = \frac{-1 - \sqrt{13}}{2}$, and these eigenvectors:
For $\lambda... |
Let $X,Y$ be $\sigma$-finite measure spaces, and let $L^p(X) \otimes L^p(Y)$ be the algebraic tensor product. The product has a natural map into $L^p(X \times Y)$ which takes $\sum a_{ij} f_i \otimes g_j$ to the function $F(x,y) = \sum a_{ij} f_i(x) g_j(y)$. A moment's thought shows that this map is well-defined. Is... |
How can I calculate $27^{41}\ \mathrm{mod}\ 77$ as simple as possible?
I already know that $27^{60}\ \mathrm{mod}\ 77 = 1$ because of Euler’s theorem:
$$ a^{\phi(n)}\ \mathrm{mod}\ n = 1 $$
and
$$ \phi(77) = \phi(7 \cdot 11) = (7-1) \cdot (11-1) = 60 $$
I also know from using modular exponentiation that $27^{10} \mathr... |
I try to learn the theory of category from The Joy of Cats. I got stacked with the first exercise (3A a).
If we have a simple graph with one vertex and 2 nodes all we know is that in category there is 2 objects and one homomorphism (beside identities). Hence:
\[C = (\{\{x, y\}, \{1, 2\}\}, \{f, id\}, id, \cdot), f = \{... |
A discrete one-dimensional model of optical imaging looks like this:
$$I(r) = \sum_i e_i P(r - r_i)$$
Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread function that blurs each point. We can assume that $P$ is even, non-negative and has a finite extent, ie $P(x) = 0$ for... |
I am writing a Gaussian blur filter in graphics shader code. I want to make the blur parameterized by radius from the users perspective. The best method I can figure to do this is to pick a suitable stopping point for y, say .001, and solve for the variance to plug into the normal distribution function that will achi... |
I've sampled a real world process, network ping times. The "round-trip-time" is measured in milliseconds. Results are plotted in a histogram:
Ping times have a minimum value, but a long upper tail.
I want to know what statistical distribution this is, and how to estimate its parameters.
Even though the distribution i... |
Are there any examples of solving for the global maximum of a non-differentiable function where you:
Construct a series of differentiable functions that approach the non-differentiable function in the limit
Show the maximum of each differentiable function converges to some value, which is thus your answer.
For all I ... |
I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron. By distinct I mean they are not congruent by some symmetry
of the icosahedron or dodecahedron (respectively). So they do not make the same sequence of angular turns. For example (as Gerhard corrected ... |
Say $P \sim Q$ ($P$ and $Q$ are «projectively equivalent») iff there is a projective transformation $f$ such that $f(P) = Q$. Then $\sim$ is an equivalence relation. I read that the space of inscribed $n$-gons modulo projective equivalence has dimension $n-3$. Why is this? Also, are there any related results?
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Whenever I think about how a function behaves, I always try to identify a general pattern of behavior with some common numbers (somewhere between 5 and 100 maybe) and then I try to see if anything interesting happens around 1, 0 and into negative numbers if applicable.
If that all works out, I essentially assume that I... |
Suppose I have a point $x \in \mathbb{R}^n$ on an n-sphere. Suppose I divide the n-sphere into 4 sections (I think this makes sense in $n$ dimensions), how do I know which section $x$ lies on?
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Denote by $D(a_1,\dots,a_n)$ the product $\prod_{j>i}(a_j-a_i)$. Assuming that $a_i$ are integers s.t. $a_1\le a_2\le\dots\le a_n$, prove that $D(a_1,...,a_n)/D(1,...,n)$ is the number of Gelfand-Zetlin triangles (that is, triangles consisting of $\frac{n(n+1)}2$ integers, s.t. each number is greater it's lower-left ne... |
Let $p$ be a prime. If $$\frac {p-1}{4} $$ and $$\frac {p+1}{2} $$ are also primes then prove that $p=13$.
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If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. $B(X,Y)$ is not complete?
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In DotA, there is a character called "Axe". Every time he is attacked, he has a chance to spin ($17\%$) his blade and deal damage based on what level the skill is, $100 / 125 / 150 / 175$ damage for levels $1, 2, 3$, and $4$. When the spin activates, it triggers the cooldown of $0.7 / 0.65 / 0.6 / 0.55$ seconds, so tha... |
Somewhat related to this question, but almost infinitely more basic.
A Confession
I am, should classification prove essential, a differential geometer and a topologist by inclination and by training: as an undergraduate I shunned any ring that wasn't $\mathbb{Z}_n$ or a ring of differential operators and held close the... |
Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations?
I am aware of Hilbert's founda... |
I want to try to emulate what this application can do:
alt text http://img530.imageshack.us/img530/5756/bezier.png
Given the red round dots (from the mouse) it is able to solve for the bezier handles given that the tension of the curve is set to $0.6$. How could I solve for the bezier handles? Is there some way I could... |
Let $C$ be a plane curve given by the set $\{(x,y) \in \mathbb{R}^2: \, P(x,y) = 0 \}$ and define $\omega=\frac{\mathrm{d}x}{y}$. Then, it is
$$\int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega$$
(with $\oplus$ being addition on a group on the curve) a theorem?
I'm quite sure this is w... |
Wikipedia sez:
The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the "childish $\ln$ notation," whi... |
I came across the statement that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations.
How exactly is Yoneda-lemma a generalization of Cayley`s theorem? Can Cayley's theorem be deduced from Yoneda lemma, is it a generalization of a particular case ... |
In group cohomology theory, we know that the cohomological dimension $cd(G)$ for a profinite group is a fundamental numerical invariant. We say $G$ is of virtual cohomological dimension $n$ (denote it by $vcd(G)$) if there exists an open subgroup $H$ such that $cd(H)=n$.
It seems that $vcd(G)$ is not always a fixed in... |
I've been trying to find answer to this question for some time but in every document I've found so far it's taken for granted that reader know what $\mathbf ℝ^+$ is.
|
I've been experimenting with the summation of polynomials. My line of attack is to treat the subject the way I would for calculus, but not using limits.
By way of a very simple example, suppose I wish to add the all numbers between $10$ and $20$ inclusive, and find a polynomial which I can plug the numbers into to get ... |
Related to this question.
Note that I'm using the geometer definition of an $n$-sphere of radius $r$, i.e.$ \\{ x \in \mathbb{R}^n : \|x\|_2 = r \\} $
Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $2^k$ orthants by $k$ axis-aligned hyperplanes (note, $k ... |
I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to start with the definition of one, yet many "official" definitions I have found too convoluted (or poorly written) for genera... |
Let $f$ be a function defined on an open interval $(a,b)$ which has continuous derivatives of order $1,2, \dots , n-1$, and there is a point $c \in (a,b)$ such that $f(c), f^{{1}}(c), \dots , f^{(n-1)}(c)$ are all $0$. However $f^{n}(c)$ is not $0$ and we can assume that $f^{n}(x) > 0$ for $x \in (a,b)$, however no as... |
Classes can be considerd as "larger" than sets in the sense that any set is a class.
Is there mathematical object which is "larger" than classes ?
|
I needed (for my research) to solve a Diophantine equation, in particular,
$$ 2 a + 3 b + 4 c + 5 d = 12 .$$
And I could easily solve it
(for example, on solution is $a=2, b=1, c=0, d=1$).
But this made me wonder if such equations, with their coefficients increasing sequences of
natural numbers, are a special case of D... |
Not a math student, so forgive me if the question seems trivial or if I pose it "wrong". Here goes...
Say I'm flipping a coin a n times. I am not sure if it's a "fair" coin, meaning I am not sure if it will come up heads and tails each with a propability of exactly 0.5. Now, if after n throws it has come up heads exact... |
What are the best algebraic geometry textbooks for undergraduate students?
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