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comment If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$
Good observation! This follows immediately from Newton's Identities
Aug
16
answered Proving the rules of a complicated game are well defined
Aug
14
comment Volume of 3D shape with parallelogram as base
It sounds like there are (differential?) equations about elastic materials that could help pin down the shape, but even if you asked on another site where people are more likely to know the relevant equations, you'd still need to give data about the initial conditions (where are the fibers that dont move, what are the constants related to how elastic the material is,etc.). Alternatively, if you have the code that generated that image, perhaps it could be reverse engineered and/or the volume approximated with a Monte Carlo method.
Aug
14
comment Volume of 3D shape with parallelogram as base
How are the bulges defined? They don't look like pieces of a sphere. Do you have a parametric formula for the shape? What error would you allow? Etc.
Aug
14
comment Volume of 3D shape with parallelogram as base
It depends what shape it is (the picture looks close to half of the intersection of two cylinders, but you said parallelogram so we don't know how your shape differs) and what you know (are you comfortable with multivariate calculus already?).
Aug
14
comment If all derivatives are zero at a point, what does this imply?
It need not be analytic (if it were, it would be constant). Look up bump functions. If you hadn't said "differentiable nearby"n I would say it need not be continuous at any other point:combine a bump function with an example of differentiable at a point but discontinuous elsewhere.
Aug
6
answered Notation for probability distribution (capital P) and density function (lowercase p)
Aug
4
comment If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?
Mathematical "existence" is very different from both physical existence and the relevant physical theoretical possibility. Without knowing the relevant physical details myself, imagine if one restriction for the string theory were something like "the number of dimensions must be squarefree" then even if the 11d candidate were the boundary of a 12d manifold, that 12d manifold couldn't be a candidate.
Aug
4
comment If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?
"How is it possible to make compatible those restrictions with [existence of 12 dimensional object]" is similar to the question "how is it possible to reconcile the complex numbers being 2 dimensional with the existence of 3d space?" The existence of some 12d spaces has no bearing on however the string theory works out.
Aug
3
comment Are the Real numbers really Complete?
@chharvey The two properties I mentioned follow immediately from the Conway style definition (equivalence classes of pairs of sets of surrreals). And if you meant more reading about the surreals in general, there's way too much for me to put in this comment. If you're looking for a specific flavor of treatment and can't find it, maybe email me and/or post a new question.
Aug
2
comment Can a number have an uncountably infinite amount of digits?
The examples at that mathoverflow link aren't really sets where someone does something like use more than countably many digits, because nothing like digital expansions really arise. That said, sign expansions of the Surreal numbers (or a field naturally sitting inside) are pretty close to binary expansions which can involve a an expansion of uncountable length.
Jul
29
comment Notation for polynomials and equating coefficients
Can you name the paper? Without context or an example of what the paper means by this notation, I'm not sure what "given $t$" should mean.
Jul
28
comment Permutational Question
I misread the question. Your interpretation is very likely correct. Good work.
Jul
28
comment Permutational Question
AB, AC, BA, BC, CA, CB are the six permutations of two distinct letters from {A,B,C}. Could you clarify what you're unsure of?
Jul
28
comment On a metric over m-subsets of [n]
I would say this metric is just the Manhattan Distance in $\mathbb R^m$ restricted to a very special domain.
Jul
27
comment Clarification of definition of “inverse” with quaternions
Quaternion multiplication is not matrix multiplication of the corresponding vectors, as that's undefined.
Jul
27
comment The set of all real functions of a real variable
If you can argue that $\mathfrak{c}^{\mathfrak{c}}\ge2^{\mathfrak{c}}$ then that's enough. I'm not certain where the "$2^{\aleph_0 \mathfrak{c}}$" is coming from.
Jul
20
comment Can we generalize Aleph numbers to non integer values?
I don't think there is any natural way to extend $\aleph$ to non-ordinal inputs in the surreals. Since by definition $\aleph$ makes a sort of huge jump with each successor, I have no idea what, if any, would be an appropriate way to interpolate. Certainly you could interpolate linearly or any other way you want, but I doubt this can lead anywhere satisfying.
Jul
12
comment example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?
Cardinal arithmetic has addition and multiplication with an infinite cardinal reduce to max. See en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic (There the only case you have to watch out for is multiplication by $0$.)
Jul
11
comment Is Wolfram wrong about unique 3-colorability, or am I just confused?
Has anyone informed the website maintainers about the error?