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Jul
31
suggested suggested edit on Produce unique number given two integers
Jul
1
awarded  Commentator
Jul
1
comment Meaning of math symbol ~
Thinking on it, though the notation is seems somewhat unusual in this context (at least in my experience; ymmv), it actually makes more sense. Limits are defined on functions, so something like $\lim\frac{f(n)}{g(n)}$ only makes sense by implicitly interpreting it as the limit of another function that's defined by pointwise division. Thus, one might as well make it explicit and write $\lim\left(\frac{f}{g}\right)(n)$.
Jul
1
comment Meaning of math symbol ~
@JamesWood: the notation is standard in the sense widely understood, but unusual in the sense that it would not be typically used in this context. All $(f/g)$ means is the function defined pointwise by that ratio.
Jun
23
comment Derivative in calculus $f(t)= 7\sinh(\ln t)$
Why not just distribute the division by $t$ before differentiating?
Jun
3
comment Negative Exponents, Why?
Is the question about why $x^{-a} = 1/x^{a}$ is mathematically valid or about why so many algebra/precalculus classes insist that students always re-write all the exponents to be positive?
May
23
comment Algebra: What allows us to do the same thing to both sides of an equation?
@matth: injective means $f(a)=f(b)\Rightarrow a=b$ for all $a,b$. Since $(a)^2 = (-a)^2$, the squaring function is not injective. Graphically, injective function "pass the horizontal line test", i.e. no horizontal line intersects the graph more than once.
Mar
23
comment Something that isn't continuous can be proven to be continuous (so it is continuous - definitions - but doesn't look it!)
@Peter: In general, a function is continuous iff the preimage of every open set is open. In the sense of taking the open sets as unions of open intervals around the points (which is implied in the $\epsilon-\delta$ def.), all singletons $\{n\}$ are open--and so all subsets of $\mathbb{N}$ are open. This is an example of a discrete topology, and every function from a discrete topological space to any other topological space is continuous. If the topology on $\mathbb{N}$ is taken to be something else, not all functions from it will be continuous.
Feb
6
comment Is it faster to count to the infinite going one by one or two by two?
@EvgeniSergeev: Not in $\mathrm{ZF}$ set theory, but with some additional axiom, such as axiom of countable choice ($\mathrm{AC}_\omega$), you can. ETA: You can check the wikipedia page on the Dedekind infinite for some details.
Feb
5
answered Compute the hyperbolic angle subtended to the origin by the unit hyperbola through (ct, x) = (0, 1)
Feb
5
comment Compute the hyperbolic angle subtended to the origin by the unit hyperbola through (ct, x) = (0, 1)
The hyperbolic angle subtended by the entire hyperbola is infinite. Physically, this is related to the rapidity of the speed of light being infinite. But how to answer this depends on the starting point of "hyperbolic angle"--can one start with the parametrization $(ct,x) = (\cosh\eta,\sinh\eta)$, which makes this trivial, or must one relate to the area of a hyperbolic sector?
Jan
31
awarded  Supporter
Apr
30
comment Don't know what this means (Derivative)
Evaluate after differentiation, or any other operations.
Nov
3
comment Instance of Ehrenfest's Theorem
Note for beginning students of QM: this is in the Heisenberg picture, rather than the Schrödinger one.