Efficient Markov-chain production of immature canines via the indecomposability theorem
In this article we discuss the theory behind the standard puppy-duplication tunnel and provide suggestions for optimization of puppy production. Directions for further research are indicated.
There exists a tunnel with the following properties. First, the tunnel has a single entrance. Second, the tunnel has two exits. Finally, whatever enters the tunnel is duplicated and emerges from both of the exits. For example, suppose P is a puppy. If P enters the tunnel, then two copies of P, which we may denote by P1 and P2 for clarity, emerge from the tunnel. Each is isomorphic to P. The exact mechanism by which the tunnel operates is not particularly important, but if a specific mechanism is required, the following will do. First construct a convergent sequence of simplicial approximations to P. Here we are assuming that puppies are compactly generated and Hausdorff. For each simplex in the approximation, apply the Hausdorff-Banach-Tarski theorem to disassemble and reassemble that simplex into two isometric simplices, each with the same volume as the original simplex. After duplicating each simplex in this manner, it is straightforward to reconstruct the simplicial approximation. Passing to the limit, we thereby construct two copies of the puppy P, as was desired.
Now it may happen that one of the duplicated puppies, say, P1, has an orbit which returns it to the entrance of the tunnel. In this case an arbitrarily large number of puppies may be produced. In general, if the puppies are limited to random walks on the floor, rather than having the ability to climb all over the furniture, then we may expect that puppies will indeed with high probability return to the tunnel. Although the probability is higher that a puppy will return to the exit of the tunnel from which it emerged rather than the entrance, by controlling the size of the tunnel we can bound the probability of recurrence to the entrance.
Note that the rate of puppy production is not optimized by this procedure. There are various obvious modifications that could be made to improve the rate of production. For example, after P enters but before either of its duplicates emerges, we may replace the tunnel with its adjunction space formed by gluing one exit to the entrance. It is, of course, important that this procedure be performed during the stated window, since we might not otherwise maintain the isomorphism class of the resulting puppies. Modifying the tunnel in this way creates a feedback loop. For each pair of puppies produced, one is immediately fed back into the system.
More impressive production rates can be obtained by chaining several tunnels in series. For example, by gluing the entrance of one tunnel to an exit of another tunnel, introducing P, then identifying one exit with the remaining entrance, we produce a puppy-production system which operates twice as fast as the previous example. Note that by performing proper surgery, namely, by preattaching a torus with sufficiently many holes, it is possible to attach multiple exits to a given entrance, boosting production rates even more. Unfortunately, such production methods are currently only theoretical, as a second tunnel is not currently known to exist, nor is it known whether one could be constructed. It would be an interesting question to determine whether other tunnels already exist, and it would be a worthwhile line of research to develop techniques for the construction of new tunnels.
Portions of this research were conducted at the Freie Universität Verrücktes. I heartily thank my hosts there for making my stay there a pleasant one. My research was conducted under the grant TLA-24B/6-2006bh12.
Prof. L. Flamshaw, Ph.D.
Director, Institute for Topology and its Applications to Canine Production
There exists a tunnel with the following properties. First, the tunnel has a single entrance. Second, the tunnel has two exits. Finally, whatever enters the tunnel is duplicated and emerges from both of the exits. For example, suppose P is a puppy. If P enters the tunnel, then two copies of P, which we may denote by P1 and P2 for clarity, emerge from the tunnel. Each is isomorphic to P. The exact mechanism by which the tunnel operates is not particularly important, but if a specific mechanism is required, the following will do. First construct a convergent sequence of simplicial approximations to P. Here we are assuming that puppies are compactly generated and Hausdorff. For each simplex in the approximation, apply the Hausdorff-Banach-Tarski theorem to disassemble and reassemble that simplex into two isometric simplices, each with the same volume as the original simplex. After duplicating each simplex in this manner, it is straightforward to reconstruct the simplicial approximation. Passing to the limit, we thereby construct two copies of the puppy P, as was desired.
Now it may happen that one of the duplicated puppies, say, P1, has an orbit which returns it to the entrance of the tunnel. In this case an arbitrarily large number of puppies may be produced. In general, if the puppies are limited to random walks on the floor, rather than having the ability to climb all over the furniture, then we may expect that puppies will indeed with high probability return to the tunnel. Although the probability is higher that a puppy will return to the exit of the tunnel from which it emerged rather than the entrance, by controlling the size of the tunnel we can bound the probability of recurrence to the entrance.
Note that the rate of puppy production is not optimized by this procedure. There are various obvious modifications that could be made to improve the rate of production. For example, after P enters but before either of its duplicates emerges, we may replace the tunnel with its adjunction space formed by gluing one exit to the entrance. It is, of course, important that this procedure be performed during the stated window, since we might not otherwise maintain the isomorphism class of the resulting puppies. Modifying the tunnel in this way creates a feedback loop. For each pair of puppies produced, one is immediately fed back into the system.
More impressive production rates can be obtained by chaining several tunnels in series. For example, by gluing the entrance of one tunnel to an exit of another tunnel, introducing P, then identifying one exit with the remaining entrance, we produce a puppy-production system which operates twice as fast as the previous example. Note that by performing proper surgery, namely, by preattaching a torus with sufficiently many holes, it is possible to attach multiple exits to a given entrance, boosting production rates even more. Unfortunately, such production methods are currently only theoretical, as a second tunnel is not currently known to exist, nor is it known whether one could be constructed. It would be an interesting question to determine whether other tunnels already exist, and it would be a worthwhile line of research to develop techniques for the construction of new tunnels.
Portions of this research were conducted at the Freie Universität Verrücktes. I heartily thank my hosts there for making my stay there a pleasant one. My research was conducted under the grant TLA-24B/6-2006bh12.
Prof. L. Flamshaw, Ph.D.
Director, Institute for Topology and its Applications to Canine Production