NUMB3RS Episode 416: Atomic No. 33--Wolfram Research Math Notes

Schematic visualization of the nucleus and electron cloud of the element with atomic number 33

EXT. CALSCI - MORNING

Non-Newtonian fluids

Beautiful Sunday morning on the empty campus.  CHARLIE, LARRY,
AMITA, and ALAN huddle around a pool filled with milky water.

                 ALAN
          Non-Newtonian Fluids?

                 CHARLIE
          Non-Newtonian Fluids.

                 AMITA
             (off Alan's look)
          A fluid whose viscosity is variable
          based on applied stress.

                 CHARLIE
          You slap it, it acts like a solid --

Charlie slaps the top of the water -- it looks like he's
slapping at sand on the beach.

                 CHARLIE (cont'd)
          You poke it, it's a liquid.

Charlie pokes his hand into the wet mass -- sloshing around,
proving it's liquid.

A non-Newtonian fluid is a fluid whose viscosity (i.e., resistance to flow or "stickiness") changes as a function of applied shear stress (i.e., the frictional force overcome as one "layer" of the fluid slides past another). Unlike most common fluids, non-Newtonian fluids lack well-defined values for their viscosities. Examples of fluids displaying non-Newtonian behavior include liquid polymers, corn starch in water, and even custard (though it takes a very high concentration of custard to see strong non-Newtonian behavior).

For some attractive images of non-Newtonian fluid effects, see the home page for Gareth McKinley's non-Newtonian fluid dynamics research group at MIT. The Brainiac program featured an enlightening and entertaining video of a man walking on custard. Note that in the latter, sudden application of stress (by walking briskly or trying to pull a leg out) makes the substance behave like a solid, whereas actions that produce less shear stress (such as standing still on the custard, running it through one's hands, etc.) make the custard behave like a fluid.

Riemann zeta function


                 ALAN
          This is in your lesson plan why?

                 CHARLIE
          It's an interesting study in
          transitional system analysis --

                 ALAN
          I thought you were teaching 'Advanced
          Number Theory'?

                 CHARLIE
          Well, the Riemann Zeta Function --
             (busted)
          ...yeah.

                 AMITA
          The students'll get a kick out of it.

                 ALAN
          And by 'students,' you mean teachers.

There is essentially only a single Google hit for "transitional system analysis" as a quoted phrase, so we're not sure what Charlie is referring to here.

Riemann zeta function in the complex plane

However, Charlie almost certainly would cover the Riemann zeta function in his advanced number theory class. The Riemann zeta function ζ(z) is an extremely important special function of mathematics and physics that arises in definite integration and is also intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there are important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is defined over the complex plane for one complex variable (its real part, imaginary part, and absolute square are illustrated above both as three-dimensional plots and as contour plots), and was initiated by Georg Friedrich Bernhard Riemann in his classic 1859 paper. The Riemann zeta function is implemented in Mathematica as Zeta[s].

The Riemann hypothesis, first published in Riemann's groundbreaking 1859 paper and still open today, states that the nontrivial Riemann zeta function zeros (i.e., the values of s other than s = -2, -4, -6, ... such that ζ(s) = 0) all lie on the critical line σ = Re[s] = 1/2 (where Re[z] denotes the real part of z and the critical line can easily be discerned as the vertical line on the right side of the plot above). The position of the complex zeros can be seen by plotting the contours of zero real (red) and imaginary (blue) parts, as illustrated above. The zeros (indicated as black dots) occur where the curves intersect.

Note that we have previously met both the Riemann zeta function and the Riemann hypothesis in the Season 1 Episode 106 "Prime Suspect," in which Charlie realized that Ethan's daughter had been kidnapped because Ethan was close to solving the Riemann hypothesis (which allegedly would allow the perpetrators to break essentially all internet security).

Readers may also be interested to learn that in 2000, the Clay Mathematics Institute offered a $1 million prize for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line), does not earn the $1 million award.

INT. CALSCI - CHARLIE'S OFFICE - DAY

Social and Bayesian network analysis

                 CHARLIE
          I was about to tell Don that, unless
          we have a thousand psychopaths, the
          poisoner's motives and/or actions
          should be detectable from the
          'normal' social dynamics, whatever
          those are.

FLASH CHARLIE VISION

of the CROSS SECTION of an apartment building... layers of
electrical and plumbing and cable TV line schematics laying
in...

BACK TO CHARLIE

                 CHARLIE (cont'd)
          I could use Social and Bayesian
          Network Analysis to uncover hidden
          dynamics and covert architectures of
          the cult.

                 LARRY
          Nor would I so readily discount the
          'thousand psychopaths' theory.  What
          kind of mindless cretins blind
          themselves to the gifts of science?

                 DON
          'Cretins?'

                 LARRY
             (a little hot)
          How else would a Creator speak to us?
          Can they not imagine that denying
          science is the same as defying God?

Network theory appeared previously in Episodes 216 ("Protest"; social network analysis), 310 ("Brutus"; networking theory), 316 ("Contenders"; social networks), 322 ("Under Pressure"; adaptive systems analysis), and 412 ("Power"; sexual networking models).

EXT. CALSCI - SAFE EXPERIMENTAL AREA OUTDOORS - DAY

Beer and diapers

                 CHARLIE
          Confidence is high it was not an
          accident.

                 MEGAN
          How does that change your Network
          Analysis?

                 CHARLIE
          Network Analysis was an analytic
          sledgehammer.  With the additional
          murder, we were able to graduate to a
          scalpel -- affinity analysis...

                 LARRY
          ... beer and diapers.

                 MEGAN
          Okay -- I can't wait to hear this one --

                 LARRY
          It's a classic -- and likely
          apocryphal -- story --

ENTER AUDIENCE VISION --
Of a SUPERMARKET AISLE -- STACKS OF DIAPERS --

                 LARRY (cont'd)
          -- about a supermarket that wanted to
          know what product sold most often
          with diapers...

-- then SIXPACKS OF BEER appear beside the diapers --

                 LARRY (cont'd)
          ... only to discover that the answer
          was beer.  They subsequently
          discovered that the purchases were
          most often made closer to the weekend
          -- by men --

-- several 25-35 year old men appear in the aisle, filling
their carts --

                 LARRY (cont'd)
          -- from which they inferred that
          wives left the responsibility for
          picking up diapers to their husbands,
          who consolidated this duty into their
          weekend beer run.

BACK TO SCENE --

                 LARRY (cont'd)
          Using this information -- so the
          story goes -- the store began placing
          the diapers by the beer coolers,
          which led to increased sales.

                 CHARLIE
          Using affinity analysis and
          association rule mining, we
          determined it's highly probable that
          the crimes were committed by a
          partnership.  So we reexamined the
          relationships and behaviors of your
          various suspects -- and kept coming
          back to Audrey.

                 MEGAN
          You think she's responsible for all
          of this?

                 CHARLIE
          More likely, someone's been sending
          her out for the diapers.

Affinity analysis is a technique used in data analysis and data mining to identify correlations between actions and purchasing habits among individuals and groups. A common example of affinity analysis is the practice of shopping-cart and past purchase-history analysis used by many online businesses in order to make recommendations for other purchases or promotions. Such sophisticated mining techniques can also be superior to traditional approaches in retail knowledge discovery, such as frequent-buyer programs. In fact, affinity analysis can be used to enhance store arrangement, provide targeted coupons for competing products, and so on.

The prototypical (but possibly apocryphal) example of affinity analysis for product arrangement in a retail outlet is the observation that beer and diaper purchases tend to be correlated. While this is an initially surprising fact, it can be explained, as indicated by Charlie, by presuming that male shoppers asked to perform parental shopping duties frequently combine this chore with the purchase of mild intoxicants for use when done with child-duty.

T.J. Blischok considered another unexpected affinity in shopping patterns, namely an association between bottled juice and cold remedies. Another (but perhaps also apocryphal) example is tuna and toothpaste; as discussed by Noe Gutierrez, it seems that people who eat tuna are more prone to brush their teeth right after finishing their meal.

Affinity analysis also finds applications in other retail situations. A salient example is mentioned in a paper by H. L. Lee and S. Whang, who discuss Seven-Eleven of Japan's policy of adjusting its store layout and product placement multiple times every day in order to reflect the changing purchase patterns at different hours of the day. This in turn enables customers to easily find their favorite items. Furthermore, according to a research article by Yen-Liang Chen, Jen-Ming Chen, and Ching-Wen Tung, recent marketing research has suggested that in-store environmental stimuli (such as shelf-space allocation and product display) not only have significant influence on consumer buying behavior, but may in fact also induce additional demand.

INT. FBI - TECH ROOM - LATER

K-optimal pattern discovery

Charlie and Larry work at the clear board.

                 CHARLIE
          It's not working -- Craig/Audrey and
          Craig/Susan have low support values,
          and Craig alone has low predictive
          value.
              (beat)
          Maybe we need to circle back around --
          try K-optimal pattern discovery.

K-optimal pattern discovery is a data mining technique used to attempt to identify patterns that optimize a set of user-specified criteria.

References

Anonymous. "Data Mining Is More than Beer and Diapers." Chain Store Age 74, June 1998.

Blischok, T. J. "Every Transaction Tells a Story." Chain Store Age Executive with Shopping Center Age 71, 50-56, 1995.

Borin, N.; Farris, P. W.; and Freeland, J. R. "A Model for Determining Retail Product Category Assortment and Shelf Space Allocation." Decision Sciences 25, 359-384, 1994.

Brainiac. "John Tickle Walks on Custard."

Chen, Y.-L.; Chen, J.-M.; and Tung, C.-W. "A Data Mining Approach for Retail Knowledge Discovery with Consideration of the Effect of Shelf-Space Adjacency on Sales." Decision Support Systems 42, 1503-1520, 2006.

Clay Mathematics Institute. "Riemann Hypothesis." http://www.claymath.org/millennium/Riemann_Hypothesis/

Gutierrez, N. "Demystifying Market Basket Analysis." DMReview.com. October 2006. http://www.dmreview.com/specialreports/20061031/1067598-1.html

Han, J.; Wang, J.; Lu, Y.; and Tzvetkov, P. "Mining Top-K Frequent Closed Patterns without Minimum Support." In Proceedings of the International Conference on Data Mining, pp. 211-218, 2002.

Lee, H. L. and Whang, S. "Demand Chain Excellence." Supply Chain Management Review, 40-46, March/April 2001.

Liu, B.; Hsu, W.; and Ma, Y. M. "Integrating Classification and Association Rule Mining." KDD-98, pp. 80-86, 1998.

McKinley, G. "Gareth McKinley's Non-Newtonian Fluid Dynamics Research Group." http://web.mit.edu/nnf.

Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859.

Webb, G. I. "OPUS: An Efficient Admissible Algorithm for Unordered Search." Journal of Artificial Intelligence Research 3, 431-465, 1995.

Webb, G. I. and Zhang, S. "K-Optimal Rule Discovery." Data Mining and Knowledge Discovery 10, 39-79, 2005.

Wikipedia. Affinity analysis, Non-Newtonian fluid.