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comment Intuition behind functional dependence
Yes, a continuous, differentiable function is still a function. For example, Jittorntrum's Implicit Function Theorem (J Opt Th App 1978) does not assume differentiability: if a continuous function $F(x,y)$ is locally injective in some neighborhood of a point, then $F(x,y)=0$ has a unique solution $x=G(y)$ with $G$ also continuous. I don't think there's a combinatorial proof w/o continuity assumption though.
1d
comment Intuition behind functional dependence
Check the table here: en.wikipedia.org/wiki/Relation_algebra - functional is the first item. Functions need not be differentiable or even continuous, but need to be defined everywhere ("left total", or simply "total", or "preserving elements") and the image of each element of a function is also a single element of the codomain ("single valued" or "reflecting distinctions").
Jan
25
comment Intuition behind functional dependence
Functional dependence means left-total and single-valued relation. Differential structure is not part of the definition.
Jan
22
comment Intuition behind functional dependence
What if the gradients don't exist? It's implicit in your assumption, why not mention it from the outset, eg $C^1$ or similar
Jan
21
comment Relating categorical properties of arrows
@magma: a well known categorist told me n-lab is the opposite of category theory, and I agree: whereas the implication diagrams in this question are posets/DAGs, n-lab is a sprawling graph w/o beginning or end.
Jan
21
comment Convert universal quantification to existential quantification
What if there are no students?
Jan
21
comment Relating categorical properties of arrows
@magma, edited title and added clarification in body, is that ok?
Jan
21
revised Relating categorical properties of arrows
Rephased title, clarified body
Jan
21
comment Relating categorical properties of arrows
Marco, thanks, what about split-? In categories w/ choice, "epics split" but otherwise?
Jan
19
asked Relating categorical properties of arrows
Jan
18
awarded  Popular Question
Jan
15
comment Understanding and interpreting graph spectra
Email me if you'd like to see additional examples, though at some point might even post them as Q.
Jan
15
comment Questions on fractional Laplacian graph spectra
What's your email? Unlisted in profile.
Jan
15
comment Understanding and interpreting graph spectra
Related: math.stackexchange.com/questions/179257/…
Jan
15
comment Understanding and interpreting graph spectra
The answer depends the on (1) random distribution, eg BernoulliGraphDistribution[n,p] and (2) the spectrum, eg Adjacency, signed or unsigned Laplacian - can even interpolate between these 2 extremes, and really should also consider SVD of the incidence matrix. Each of these measures a different aspect of graph topology.
Jan
2
comment Categorical description of equivalence relation generated by a relation?
Very nice, thank you.
Jan
2
comment Categorical description of equivalence relation generated by a relation?
I should have highlighted smallest and containing, which are implicit in your answer, to clarify. You also wrote "the equivalence relation generated by.." - meaning unique. Is reflexive closure of R post-composed with T*S also an equivalence? (I haven't checked). On this theme, shouldn't we also look for the largest equivalence conained in R? After all, you gave diagram constructions that could be dualized.
Jan
2
comment Categorical description of equivalence relation generated by a relation?
You're describing the reflexive, symmetric and transitive closure of R. So isn't E is the smallest equivalence relation containing R? my previous comment (re closure) is false. By the way, symmetric S and transitive T closure don't commute - the image of S*T includes that of T*S.
Jan
2
comment Categorical description of equivalence relation generated by a relation?
Understood: E can't be a closure of R. Is there an algorithm to compute E given finite R?
Jan
2
comment Categorical description of equivalence relation generated by a relation?
The transitive closure of a tolerance is an equivalence, but how can this closure of an order, which is antisymmetric, be an equivalence?