March 2004 Google salutes Julia. February 3 is the birthday of Gaston Julia (18931978), one of the fathers of complex dynamics. The web search engine Google celebrated by posting a special Holiday Logo. The standard logo is embellished with fractal trimmings; the formula z_{n+1} = z_{n}^{2} + e appears in the background.
Groupoids and the leech heart. The heart of the leech Hirudo medicinalis beats to a rythm of its own. There are two disjoint tubes, running most of the length of the creature's body, each partitioned into 16 connecting chambers. In one of the tubes, all the chambers contract synchronously; in the other the contractions are staggered so as to produce traveling waves. Every 50 beats or so (about every 25 minutes), the two tubes trade roles. "This curious pattern is coordinated by a small nework of neurons at the rear of the animal," we learn from Ian Stewart in a piece ("Networking Opportunity") he contributed to the February 12 2004 Nature. The problem in network dynamics is to understand "how the network architecture produces the distinct coordinated states and switches between them." And one of the mathematical tools that has proved useful in problems like this is the theory of groupoids, "the ideal tool for describing symmetries that apply only to parts of systems." For example, network c in the figure is completely symmetrical, and is consequently acted on by the group of cyclic permutations of the nodes. Network b is not symmetrical. But if we examine the "input sets  all nodes that emit an arrow pointing to a given node" then there is a symmetry: each node has a oneelement input set, and the nodes can be partitioned into equivalence classes. For example 1 is equivalent to 4, since they both have a single input from 3. Then 2 and 5 are equivalent, since they each have an unput from the {1,4} set. Likewise 3 and 6 are equivalent; finally 7 must fall in the equivalence class of 1 and 4.
This partition into equivalence classes can be called a synchrony; in fact the equations of network b admit a solution in which equivalent nodes are synchronized. Groupoids come into the picture, Stewart tells us, when we try to work out "all robust patterns of synchrony in a network." The synchrony in network b gives rise to a traveling wave of excitations, propagating down the tail of the network. If the network were modified again so that the feedback arrow led from node 1 to itself, then all the nodes would be in a single equivalence class, and the solution to the network equation would have all nodes acting synchronously. "Qualitatively these two patterns resemble those observed in the leech heartbeat ..." Love Model Equations. The AAAS annual meeting was in Seattle last month, and the February 13 Seattle Times reported on some of proceedings. A local team of psychologists and applied mathematicians presented no less than a "mathematical formula for marital bliss." Unfortunately this formula, derived by John Gottman, James Murray, Kristin Swanson and their collaborators, is not an algorithm for achieving bliss. Rather it is a mathematical model of a relationship, based on the analysis of how a couple interacts when arguing, that can predict "with 94 percent accuracy which marriages will last and which will end in divorce." The model is a set of "coupled" firstorder ordinary differential equations. In LoveModelEquations2.pdf (available from the online Seattle Times article) Swanson spells them out:
Here I_{1} and I_{2} are piecewise linear functions (two different positive slopes, changing at 0) which encode the couple's argumentinteraction behavior. Geometrically speaking, the health of the relationship can be read off from the convexity of I_{1} and I_{2}. Both close to straight lines gives a "validating style of interaction." Both are very convex downward in conflictavoiding couples, very convex upward in volatile couples. We are not told the prognosis for a mixed marriage.
Tony Phillips

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