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bio website bu.edu/sed/about-us/faculty/…
location Boston University, SED
age 28
visits member for 2 years, 6 months
seen 34 mins ago

Profile for Benjamin Dickman on Stack Exchange: MESE, MSE, and MO

Postdoctoral Fellow, Mathematics Education, Boston University SED.

Ph.D., Mathematics Education, Columbia University. 2014. NSFGRFP.

Dissertation: Conceptions of Creativity in Elementary School Mathematical Problem Posing. (Relevant MESE post here.)

M.Phil., Mathematics Education, Columbia University. 2014.

Fulbright Grant, Mathematics/Mathematics Education, Nanjing Normal University. 2008-2009.

Research Topic: High School Mathematics Teacher Training in China.

B.A., Mathematics, Amherst College. 2008. Honors Program.

Undergraduate Thesis: On a Theorem of Dwork. (p-adic proof of the rationality part of the Weil Conjectures.)


Have you solved any of my puzzles? If so, send me an email!
Email: bdickman[at]bu。edu


Oct
13
comment Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?
Observing that $n(n-1)$ is always even (an interesting fact on its own! cf. mathoverflow.net/a/152300/22971), you have a prime of the form $p = 4k+1$. In this light, an earlier answer and link from Lucia (mathoverflow.net/a/165441/22971) may be relevant to your question. My hunch is that the information there can be pushed through to answer your question in the negative, but I don't see how to do it. Maybe someone else (or you) will!
Oct
3
comment Why do roots of polynomials tend to have absolute value close to 1?
"The zeros of random polynomials cluster uniformly near the unit circle" www-old.newton.ac.uk/preprints/NI04017.pdf
Sep
21
comment same paper was published in the same journal twice
Sun Xiaotao's page lists only the pp. 303-343 version: math.ac.cn/index_E/Personal_Web/Sunxiaotao_E.htm
Sep
8
comment What arrangement of unit cubes minimizes surface area?
A polycube that can be written as length x width x height; among Joseph O'Rourke's pictures, only the middle one is a rectangular prism. So it would be finding the $\ell, w, h$ such that the polycube of dimensions $\ell \times w \times h$ has minimal surface area. (Meaning you can't, e.g., for $n=9$ assemble a $2 \times 2 \times 2$ polycube and stick the ninth unit cube to one of the faces as is done in J'OR's rightmost picture.)
Sep
8
awarded  Popular Question
Sep
8
comment What arrangement of unit cubes minimizes surface area?
It will take me some time to try and make sense of your last paragraph (I had not seen Young diagrams before, and "isoperimetric arguments" are not presently in my arsenal!). (1) Do you have any suggestions for what I might read to be able to grasp the last paragraph? (2) Do you see any way to modify your response to Question A so as to make sense of Question B? (Or perhaps B should be tackled using a more number theoretical approach instead of a geometric one?)
Sep
3
comment What arrangement of unit cubes minimizes surface area?
(If you are up for making another image: It would be great to see an example worked out using the approach described by Ian Agol...!)
Sep
3
comment What arrangement of unit cubes minimizes surface area?
For my own clarification (paragraph after the diagram): Given an $n$, is there an obvious way to compute $k$? I will write out my understanding so that you can tell me where I've gone wrong! In your example, $n = 8$, so the $\max$ formula for $P(2k)$ suggests making $\frac{k}{2} = 3$ will be a good idea, i.e., so that $n = 8 \leq 9$. Indeed, this value of $k=6$ satisfies the given inequality, and (since $n=8$ is even) we make a $6/2 \times (6/2 - 1) = 3 \times 2$ rectangle with a row of $8 - 6(6-2)/4 = 2$ squares added on, for the (non-unique but optimal) Young sequence $3,3,2$.
Aug
31
revised What arrangement of unit cubes minimizes surface area?
Added a note at the start to indicate that the arrangements can be assumed polycubes, and added the number-theory tag as suggested by a comment.
Aug
30
awarded  Announcer
Aug
29
comment What arrangement of unit cubes minimizes surface area?
@Michael That's a good question; certainly I mean as much in the case of Question B. For A, let us say the same (i.e., that both questions concern polycubes). I will try to clarify (along with adding a number theory tag) in my next edit.
Aug
28
awarded  Nice Question
Aug
28
comment What arrangement of unit cubes minimizes surface area?
My intuition is not much better than "try to make a cube," and so I'm not surprised to see the minimal arrangement depicted above for $2^3 + 1$. As noted in the question, it is often explored for specific cases (e.g., $n = 9$) at the pre-secondary level; but I have no good idea(s) for how to approach the general case.
Aug
26
revised What arrangement of unit cubes minimizes surface area?
Added some more background material; moved questions to start
Aug
25
comment What arrangement of unit cubes minimizes surface area?
@WlodzimierzHolsztynski For Question B I mean that the cubes, packed together, form a rectangular prism - with a length, height, and width that can be used to describe the prism in its entirety. Perhaps some sort of "imperfect" rectangular prism will (would) show up in the answer for A.
Aug
25
asked What arrangement of unit cubes minimizes surface area?
Aug
19
comment Most harmful heuristic?
@PedroTeixeira Surely this is the way to proceed! Let $y = x - 2.5$; then $x = y + 2.5$, and the equation becomes: $0 = (y + 1.5)(y + 0.5)(y - 0.5)(y - 1.5) = (y^2 - 2.25)(y^2 - 0.25)$, which holds when $y = \sqrt{2.25}$ or $y = \sqrt{0.25}$. For each square root we obtain two possible $y$-values; add back the $2.5$ to each to get the four possible $x$-values. A similar approach can be found by observing $(x-1)(x-4) = (x-2)(x-3) - 2$; now denote the LHS by $y$ so that the original equation becomes $y(y+2) = 0$; solve for $y$ using the quadratic equation, etc.
Aug
10
comment Examples of research on how people perceive mathematical objects
Does Jean Piaget's research (volume conservation tasks etc) exemplify what you are looking for here? What about Dor Abrahamson's work on proportions (youtube.com/watch?v=n9xVC76PlWc)? I'm not quite sure what you mean by mathematical objects, nor am I clear at what age/stage of mathematical development your focus lies. Probably a fair amount could be dug up from math education research on mathematical modeling, depending on how your question is interpreted...
Aug
8
comment Behaviour of power series on their circle of convergence
(Just a comment, assuming I haven't misunderstood.) To show that $\sum_{n} \frac{z^n}{n}$ converges for $C \ni z \neq 1$ (as you mention is shown in Rudin using trigonometric estimates) can be done very quickly with the Dirichlet Convergent Test, or -- even better! -- with the quick geometric proof provided here: mathoverflow.net/q/109582/22971
Jul
30
comment Examples of cubic Julia sets
@AdamEpstein Yes; I just recall this as the "most interesting" explicit cubic from the investigation. For an example of a cubic polynomial with points that don't escape, one could try $\phi(z) = \frac{1}{12}z^3 - \frac{25}{12}z + 1$, and check the forward orbit of the $5$-periodic point $z = 1$ (thereby making $0$ preperiodic). I don't think this is particularly helpful to the OP - though perhaps there is something of value in the paper - hence my leaving this as a mere comment.