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I received my Ph.D in mathematics from Ohio State in 1994 under the direction of Gerald Edgar and have been a professor of mathematics at The University of North Carolina - Asheville since 1997. In recent years, I've also worked as a part-time consultant to Wolfram Research focusing on development of mathematical content for WolframAlpha.


Aug
10
comment My first partial differential equation attempt
@Dmoreno I'm glad you liked it. I generate the image with Mathematica. I generated a list of images and then exported that to an animated GIF. The command to generate the list of images was something like so: Table[Plot[Exp[-Pi^2 t] Sin[Pi*x], {x, 0, 1}, PlotRange -> {0, 1}], {t, 0, 0.7, 0.01}].
Aug
6
comment Get position of a point with known distance between other points
Well, there's a fairly obvious system of equations to write down. Do you have a specific example?
Aug
6
comment Non-Trivial Self-Inverse Analytic Function In The Complex Plane
This is a well studied problem on the Riemann sphere - namely the complex plane plus infinity. In that context, the only holomorphic, self-inverse functions are the Mobius functions, $z\rightarrow (az+b)/(cz+d)$. In the everywhere analytic case, $c\neq 0$ and you're left with your two examples.
Aug
6
comment Analytic Function In The Complex Plane Which Always Gives Real Values
You can use the Cauchy-Riemann equations to prove fairly easily that, if $f$ is defined and analytic for all $z$ in the complex plane, then $f'=0$ on the complex plane. It follows that $f$ must be constant.
Aug
6
comment Optimization Software
You need to ask a much more precise question. If you're using Mathematica, you might consider asking on mathematica.se but, again, you must reference a specific example. Is there any chance you're using the Minimize command command when NMinimize would be more appropriate?
Aug
2
comment interpreting triple integrals
You could make the question more clear with a specific example.
Aug
1
comment Limit of a Discrete Dynamical System
@user54738 Many software packages provide the ability to do numerical computations to much higher precision that just the 16 decimal digits provided by the CPU. I used Mathematica which provides significance tracking so that you can actually measure how precise your computations are. As a result, you can have confidence of your results. To see the need for this sort of thing, try iterating $f(x)=4x(1-x)$ from $x_0=(2+\sqrt{3})/4$ at machine precision. It will appear to converge to $0.75$ (correctly) but, after around 40 iterates, it moves away from $0.75$, even though $3/4$ is fixed.
Jul
17
comment Prove $f(x) < 0 \forall x$
@Mr.T Please take a look at our FAQ on homework and pay particular attention to the part labeled "Why don't you provide a complete answer to my question?". Now I don't know for sure that this is a homework assignment but, even if it is not, it's clearly at that level and I wouldn't want to rob you of the joy of discovery. :) In that context, the response is appropriate. Incidentally, you should upvote all answers that provide at least some assistance, though you might choose to accept a later answer, if it is forthcoming.
Jul
17
comment System of non-linear ODE's
By "analytically", you mean to find closed form expressions for $x(t)$ and $y(t)$? Mathematica is unable to find such expressions and I see no reason to think that such expressions exist. On the other hand, you can "analytically" find that the origin and $(p/3,p/12)$ are equilibria, use linear algebra to classify the behavior at those points, and finally see how these fit into the vector field determined by the system.
Jul
10
comment Bessel's Equation
Yes, I'm sure that your textbook refers to your differential equation as Bessel's Equation. I'm saying that there is a special function, called Bessel's Function, which solves Bessel's Equation. The link I provided explains this in detail.
Jul
10
comment Bessel's Equation
Using a Bessel function, perhaps?
Jul
2
comment Lecture notes ready for $\LaTeX$
@DaveL.Renfro If your looking for TeX documents, you can use the filetype modifier in your Google search, for example: fractal filetype:tex.
Jul
2
comment a system of finite difference equations
What do you mean by "solve" exactly? Clearly, $x_t=y_t=0$ for all $t$ is a stable equilibrium for the system and I'm sure there's at least one more equilibrium, which might or might not be stable. Do you mean more than this?
Jul
1
comment Calculate the length of curve
@Sacheo You're welcome, though I wouldn't be so fast to accept this answer - I think it's missing something and someone else may spot it.. You can always upvote without accepting.
Jun
25
comment Integrate $\dfrac{\tan{3x}}{\cos{3x}}\mathrm{d}x$
No, the integrand simplifies to $\sin(3x)/\cos^2(3x)$. Now use $u$-subs.
Jun
17
comment If a function has a inverse that is well defined is it a bijection
Your function is certainly a bijection between $X$ and $\text{range}(f)$. It will be a bijection between $X$ and $Y$ only if $f$ is surjective, i.e. $\text{range}(f)=Y$.
Jun
16
comment Spectral radius and Dominant Eigenvalue
@Ranade Correct - such a matrix does not have a dominant eigenvalue.
Jun
16
comment Spectral radius and Dominant Eigenvalue
@user1551 Maybe - Every time I've seen the term 'dominant eigenvalue', it's uniqueness is essential. The power method is a classic example. More to the point, the uniqueness is an essential part of the answer as stated.
Jun
16
comment Spectral radius and Dominant Eigenvalue
@lavkush No problem! I was typing up an answer when yours came through. You hit the crux so I just added some context.
Jun
16
comment Spectral radius and Dominant Eigenvalue
@user1551 False