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Sep
17
comment Necessary and Sufficient Conditions for $f_{xy} = f_{yx}$
You are correct. I misinterpreted what you were attempting to say. However, this is a simple restatement of the original theorem. In a sense, I'm asking that if we declare the mixed partials to be equal, then what can we deduce about $f$? Also, see my original post, edited to ask about some relevant illustrative examples.
Sep
17
comment Necessary and Sufficient Conditions for $f_{xy} = f_{yx}$
The construction of $f$ there is clever. Thank you.
Sep
17
comment Necessary and Sufficient Conditions for $f_{xy} = f_{yx}$
@Graham, nice formatting. However, the "change of order" step is only valid assuming continuity of second partials. You accidentally included a bit of your conclusion in your premise.
Sep
5
comment Calculating limits of functions explicitly
@Nir, Check my edits to see if it addresses your question. Remember x*y = y/(1/x) (except when x=0), so with limits you can often use this trick to get things in a form where L'Hopital's Rule applies.
Sep
3
comment Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$
Thank you. The recursive definition you reformulated the sum into is quite nice. I don't, however, see the 'easy induction' proof. Would you mind suppressing the laziness urge providing a little more in the way of proof? :) Also edited question to more clearly state desire of proof.
Sep
3
comment Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$
@user166967, they are just indices of the respective sums.
Nov
25
comment Characteristic properties for topological pushouts and pullbacks
@Bachmaninoff Basically, yes. These constructions work any any category as long as the relevant diagrams commute. Often these constructions correspond to more familiarly named constructions. As an exercise, try to see what pullbacks, pushouts, etc are in the category Set of sets.
Nov
19
comment Characteristic properties for topological pushouts and pullbacks
@Bachmaninoff Looks like you got it, sir. Note that it is crucial for the combined maps to all be continuous and that h be unique; I think this is what you were intending to say though.
Nov
18
comment Characteristic properties for topological pushouts and pullbacks
@Bachmaninoff Similary for quotient topologies, you are identifying elements in a parent topology P to get a quotient topology Q, so you should expect f:P->Q instead of f:Q->P, so we can write f(a)=f(b) and the like. Then since the image of f should be Q, we need f to be surjective (an epimorphism in Top). It is interesting to note that the subspace topology and quotient topology are dual to each other. Why? If you can convince yourself, then you probably can handle product and disjoint union characterizations yourself.
Nov
18
comment Characteristic properties for topological pushouts and pullbacks
@Bachmaninoff Zhen Lin is correct. For subspace topologies, you can think of a subspace A as being embedded in a bigger space B. In other words a subspace topology is characterized by a continuous injection f:A -> B between topological spaces. Note that in the category Top, injections correspond to monomorphisms.
Nov
15
comment Characteristic properties for topological pushouts and pullbacks
There is a category of topologies, Top, where the objects are topological spaces and the morphisms are continuous maps. So if you just make the necessary terminology replacements, that should get you going in the right direction.
Nov
15
comment Combinatorics : Prove - Graph Algorithm
If you're having trouble parsing it, try looking up the words and symbols you don't know. What pieces of it do and don't you understand?
Nov
15
comment Characteristic properties for topological pushouts and pullbacks
Pushouts and pullbacks can be defined in a general category, so you might be interested in looking at the category theoretic definition.
Nov
15
comment Combinatorics : Prove - Graph Algorithm
Assume G is connected and H is a spanning tree. Then could you do the problem?
Nov
15
comment Two inner products being equal up to a scalar
Cannot comment, so posting as answer. Try looking at just the algebraic properties of scalar product. Is it bilinear? Why? If so, what does this give you?
Sep
6
comment Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$
Are we sure that a_n can be unboundedly negative? Suppose my sequence is such that a_n = -(2+pi), then for the sum m=2001, there is an x such that |a_2001 * sin(2001x)| > 1+pi
Sep
2
comment Regarding the derivation of triangle inequality related inequality (undergraduate complex analysis)
Dang it, Macavity. lol
Sep
2
comment Regarding the derivation of triangle inequality related inequality (undergraduate complex analysis)
When multiplying by -1, you need to think carefully how that changes your inequality relations.
Sep
2
comment Regarding the derivation of triangle inequality related inequality (undergraduate complex analysis)
I would check your original inequality. z_1 = 1, z_2 = i seems to be a trivial counterexample.
Jun
20
comment Locating a shape defined by five points within a cloud of points
You mean identify collections of points which match your "shape" criteria? Efficiency here depends a lot on how you represent your point cloud, what extra criteria beyond "5 rigid, coplanar, non-collinear points" you're matching on etc.