The exponential is a group homomorphism from the additive reals to the multiplicative positive reals. Functions do not have multiplicative “structure.”
Hi folks. Not sure if this is more of a physics question, but I'd really appreciate if someone could at least point me to a resource that can help me understand.
In a Phase Diagram, why is the gradient $dT_d/dP$ equivalent to the rate of change of the volume wrt Entropy $\Delta V/\Delta S$?
@robjohn Yes, the dual of the hypercube exists in higher dimensions. From math.ucr.edu/home/baez/platonic.html "In higher dimensions, the n-simplex is self-dual, and the dual of the n-cube is the n-dimensional cross-polytope".
I have sets with the property that if $F_1\cap F_2 = \emptyset$ then $\overline{F_1}\cap\overline{F_2} = \emptyset$. Can I get the same property for finite amount of sets from this one?
How come $D_{2n}$ acts on the set consisting of pairs of opposite vertices of a regular n-gon, if n is even?
How do I define such action?
All I know about $D_{2n}$ is that $D_{2n}:=\langle r, s: r^n=s^2=1, rs=sr^{-1}\rangle, \langle r\rangle$ is the subgroup of rotational symmetries, whereas $\langle s\rangle$ represents subgroup of reflection symmetries.
I say if $x\in D_{2n}$ then case 1: x is rotation Let (a,a') be pair of opposite vertices of n-gon and let $L_{aa'}$ be a vector connecting them. Then, $x.(a,a'):= |L_{aa'}|(\cos (r+\theta), \sin (r+\theta))$.
Something like this.
But with this, verifying that . is actually an action seems complicated.
@XanderHenderson I am looking for a finite subfamily of sets such that their intersection is empty, but closures aren't, assuming that the original family has properties 1) disjoint sets have disjoint closures, 2) they're closed under union and intersection
The quesstion:
What does the following series converge to: $$\sum_{}^{} \frac{1000 + 20n + sinh(5n) arcsin(ln(12n^{2000}-\pi{2n^{25}}+34))^{arccos\frac{1}{n}}+n^{10^{9^{8^{7^{6}}}}} - tan(n!) + 6^{n} - \sqrt{n+3}}{arccsc(ln(cos(n^{5}+\frac{12}{n}))) + sin^{2}(e^{n})(csc^{2})(6e^{2n^{2n}})(n^{24...
Hi there, question on opinion relating to notation
I'm working with a bunch of elementary symmetric polynomials, how should I denote what I'm doing similar to say, a big product, summation or repeated fraction (K notation)?
Is there something similar for a set?
So for example, e_5({}^_{i=1}^5 x_i) or something?
My proof will contain elementary symmetrics of a variable number of arguments so I'm just not sure how to write it down nicely
Or should I define a helper variable? Meh
Oops, that won't compile, remove the first hat ^ :p
When evaluating $\lim_{x \to 0-}\frac{x+1}{x^2}$, am I allowed to use substitution $t = \frac{1}{x}$ ? What about if $\lim_{x \to 0}$? Since $t \to \pm \infty$ when $x \to 0$, I'm not sure.
Also, if there is a simpler way to solve this, I would surely like to know.
Suppose that H is a subgroup of a finite group G and that H acts on G. Let $x\in G$ and suppose that O is an orbit of x under the action of H. Then the mapping from H to O given by $h\mapsto hx$ is a bijection?
The problem that I have here is $hx$. The mapping is indeed a bijection if we define $h.x:=hx$ (composition under the operation of G) then . is a bijection. However, if . is any generalized action then I don't know how to show the mapping to be 1-1.
@TedShifrin that makes sense. I came across the dual number and I've seen this $\epsilon^2 = 0, \epsilon \neq 0 $ which I do believe similar in some sense.
The proper way is to view it structurally. Don't define $i$; define $\Bbb C$
$\Bbb C$ is, up to isomorphism, the field whose underlying set is $\Bbb R^2$ with operations $+$ and $\times$ defined by $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)\times(c,d)=(ac-bd,ad+bc)$
@AkivaWeinberger I disagree about what is "proper". I agree that, in a context where you have to distinguish between the reals, the complex, the $p$-adics, the quaternions, and so on, it is best to define things structurally.
But this is, I think, bad pedagogy for folk who have never taken calculus, let alone a course which deals with structures (groups, rings, fields, etc).
Which is not to say that I disagree with you---I actually agree entirely that it makes more sense to define the complex numbers first if the concern is rigorous mathematics. But pedagogically, this can be troublesome.
I think it should still be acknowledged that, the fact that we can "add in" this number $i$ satisfying $i^2=-1$ without breaking things like associativity, is nonobvious
We create this new structure, called set of the complex numbers, out of the set of reals, and a priori we have no reason to expect that it still has all of our old properties. And then we show that it does
Well, if you believe associativity for real polynomials, then you should believe associativity for complex numbers. But you can say that explicitly
Incidentally, it's fun to see what happens if we "add in" a number $j$ satisfying $j^2=1$, but $j\ne1,-1$
What ends up happening is that $1+j$ has no multiplicative inverse
$(1+j)(1-j)=1^2-j^2=0$, so you have the product of two nonzero numbers being zero, and therefore either you have to give up that property or you have to give up the $j\ne1,-1$ condition
(In fact, a better notation for $j$ is $\pm1$. And then I've just written $(1+\pm1)(1-\pm1)=0$, or perhaps better, $(1\pm1)(1\mp1)=0$)
($e^{jx}=\cosh x+j\sinh x$ is mysterious until you realize it's $e^{\pm x}=\cosh x\pm\sinh x$)
@CroCo No, this is ostensibly a totally different notion. That is working in an algebraic construction called a quotient ring in which you introduce an object $\epsilon$ with the property that its square is $0$. Whereas I was just talking about how a general polynomial of degree $n$ (in our case $2$) has $n$ different roots (in the complex numbers, say). Now you can phrase this as quotient ring construction, too.
@Jakobian There are a few people here (@Alessandro, for example, and @Thorgott, I think) who love that stuff. I do not and did a whole career never having to think about that.
Really, I think if you transform everything into matrices. The matrices can represent the worst case of operations involved in an algorithm. Dynamic array allocation is simply changing vector spaces.
I would do it using an "adding machine" a machine that can only add or subtract constants or variables, has conditionals, loops, functions, and everything is stored in a global vector of memory. I.e. any Turing-complete language is such that its programs can be converted to this form.
There will be a slow-down for already efficient algorithms, but a speed-up for supposedly "hard" algorithms
This is because computing $A^k$ can be done with smart exponentiation (exponentiation by squaring)
Every conditional in a Turing complete language with $\gt$ can be converted into $y \gt 0$ or $y = 0$. You put your conditional expression into a variable before testing it against $0$.
I have the proof in my head, but P = NP is the conclusion.
