• Lecture 18 – Chapter 2: Resource Theories

• Lecture 19 – Chapter 2: Chemistry and Scheduling

• Lecture 20 – Chapter 2: Manufacturing

• Lecture 21 – Chapter 2: Monoidal Preorders

• Lecture 22 – Chapter 2: Symmetric Monoidal Preorders

Sounds great! I’m hopelessly busy right this second, but I really look forward to seeing your applications to thermodynamics.

]]>Carlo Sparaciari, Jonathan Oppenheim and I have applied these ideas to thermodynamics. While I’m not quite convinced that our starting assumptions are “right” from the physics perspective, our results recover standard equilibrium thermodynamics and extend certain aspects of it to situations arbitrarily far away from equilibrium. More concretely, what we have considered are questions like this: what is the maximal amount of work that can be extracted from many copies of any given non-equilibrium state of a closed system? (It’s enough to consider closed systems only: if you also have a heat bath available, then you should consider that bath to be part of your given non-equilibrium state.) For example, your non-equilibrium may state consist of one thermal state at low temperature and another thermal state at high temperature, in which case our maximal work corresponds to the maximal efficiency of a heat engine operating between two large but finite thermal baths. Basically the same methods apply when you have more than two baths at different temperatures.

Currently I’m working on generalizing the resource-theoretic toolbox further and applying it to optimal transportation and randomness as a resource.

]]>http://arxiv.org/abs/1504.03661 ]]>

I have fixed the subscripts in cullina’s second comment. It would be great if this post of yours could lead to a collaboration!

I will also fix the original post so that it says has 136 vertices. Not everybody reads the comments to find errors!

]]>Very cool!! David Roberson and I are working on exploring the ordered commutative monoid of graphs in more detail. Since your observation also goes in that direction, maybe you’re interested in helping out? If so, just send me an email. (It seems that we’re both working on other projects first, so we’re likely to be quite slow.)

You’re right, I should have said 136 vertices in the original post…

In order to get the subscripts right on this blog, you can write P_{17} instead of P_17. The 7’s in cullina’s second comment all belong to the subscript.

]]>I think that I have a smaller counterexample. Let be a prime power such that and let be the Paley graph, or quadratic residue graph, on vertices. Paley graphs are vertex transitive and self-complementary, so by Theorem 12 in Lovasz’s paper “On the Shannon capacity of a graph”, . The web page at http://www.research.ibm.com/people/s/shearer/indpal.html lists independence number (which equal clique number in this case) for small Paley graphs. In particular, . Thus while and .

]]>Well done — that’s exactly right!

Having to take complementary graph is a bit of an annoyance. But there’s a reason that I work with ‘distinguishability graphs’ instead of the usual ‘confusability graphs’: converting one communication channel into another corresponds to a graph homomorphism of distinguishability graphs, but not of confusability graphs. When using the channel to communicate, we’re interested in converting the channel into a perfect channel, which distinguishes all input symbols perfectly. So we’re interested in independent sets in the confusability graph, or in cliques in the distinguishability graph. When using the same channel repeatedly, the maximal throughput of bits per channel use is described by the Shannon capacity.

]]>but some people, including Nad, might like the first part, since it gives a nice explanation of the idea of Shannon capacity.

Indeed Tom’s explanation is much better then the Wikipedia article – at least I find that. So on a first glance it seems for a triangle (a 3-cycle) in Tom’s sense the Shannon capacity would be zero, because for the triangle and for all strong products of itself the cardinality of the largest indepent set seems to be zero. However Tobias used the complementary type of graphs (for Tom’s graph an edge means “rather undistinguishable” and for Tobias an edge means “distinguishable” within a communication). I guess the complementary graph of a 3-cycle (Tobias graph) would be three distinct points (Tom graph). Those 3 points would all belong to an independent set. If I understood correctly if one takes the strong product of this graph of three points than one would have a graph with 9 seperate points taking the square root is again 3 a.s.o. so the Shannon capacity should be 3. Is that right?

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