where is a natural number and is an arbitrary complex number. This function is used to define polynomials and power series. With this function we must have for the function to be well-behaved.

where is an integer and is a nonzero complex number.

where is a complex number is the exponential function. The at the bottom is just a symbol, though by abuse of notation some people define the number , which makes this power compatible with other power functions.

where is a positive real number and is a complex number. We can also extend this to be defined for any in a set where can be defined consistently, or even to all nonzero if we make this a multi-valued function. We can also extend the function continuously so that for all positive real numbers . However, there is no way to extend this function continuously to (0,0), so with this function is undefined.

These functions are frequently confused because in any point where two of them are defined, they give the same value. However, it is crucial to distinguish them when analyzing the expression : According to the first definition it is , but according to the rest it is undefined. Finally, applying this to your riddle, I must object to your solution as well: There is no single power function such that looks like that.

]]>]]>Problem #26

Despite its stark, last-centry plain HTML style, the TCLA list of open problems is something of a goldmine. As far as I know, it’s one of the few repositories of “Theory B” open problems, where Theory A deals with combinatorics of datatypes and complexity theory, and has avalanches of open problems, including the-problem-who-shall-not-be-named, and that several generations of computer scientists who would give 10 years of their life for a solution (I’d actually consider the offer myself).

But theory B is concerned about more philosophical issues, such as “why is writing correct software so darn hard?” or “can we write code that can be extended, but without modifying it?”. It’s nice every once in a while to have a hard, concrete, mathematical question to gnaw away at, even if the actual concrete applications of a solution might be a bit, well, nonexistent (as is tradition for juicy mathematical problems).

The TLCA list of open problems concentrates on problems in so-called type theory, which has a strong overlap with logic and “combinatory logic” (which is more about computation than logic). It’s got a few nasty problems in there, some of which have been solved, but some of which are still very open. I’ve personally got my eye on one, #9, though I don’t expect to have it nailed down anytime soon. It’s good to have stretch goals.

The one I’m going to bring up today was reminded to me by a discussion with John Baez involving descriptions of large ordinals. He was mentioning the connection with descriptions of large numbers, and I brought up the connection with

weak systems of logic, which is the essential starting point of ordinal analysis, at least historically.The connection is this: we can use our

intuitionof the well-foundedness of aconcrete“finitist” description of an ordinal to justify the consistency of some system of reasoning. In general, each ordinal corresponds to a certain logical system, and as a happy consequence, we get a well-ordering of logical systems along their consistency strength.Somewhat ironically, ordinal descriptions, invented with the goal of making consistency of logical systems intuitive, get

really hardto understand as they get large, and even reasonably weak logics have ordinals of nightmarish descriptions. Whether this means that ordinal analysis has somewhat failed at it’s task, or whether we put too much trust in our stronger logical systems is left as an exercise to the reader.Now the connection between ordinal size and theory consistency is technically fleshed out, the details are unfortunately quite strenuous, and it would be nice to have some more intuitive understanding. Now the connection between logical systems and programing languages, on the other hand, is quite simple and intuitive: proofs are programs, and statements are types. Easy peezy! It might therefore be useful to see what ordinals have to say about the programing languages themselves.

To make a long(ish) story short, the real property we want to show is that the programing language corresponding to the logic only allows

total programs, or in other words a program of type A actually defines avalueof type A after reduction/computation, rather than just an empty promise. Delivery on this promise corresponds to 1-consistency in logic: if the logic says a number exists, then it “delivers” i.e. the number “actually exists”. This implies consistency, of course.So we want to prove that certain typed programs only have finite reductions. What better tool than ordinals, which are the very definition of having only finite “computations” or decreasing sequences? It turns out to be very tough to get a “natural” mapping from programs to ordinals, which is exactly what problem 26 asks for. In theory, there always is a mapping from programs to ordinals, even smaller than omega: a program maps to the largest number of possible reductions. But this is not “natural”, because showing that this number exists involves using the usual proof that such a number exists, and that proof doesn’t involve ordinals whatsoever. Note that problem 26 restricts the question to simply typed terms, but the question remains valid for all sorts of more complicated things.

Now the mystery is a bit in defining what “natural” means. Ideally, it would be a structural-ish map on well typed terms (say, an induction on type derivations) to ordinals using not too crazy intermediate notions, or operations on ordinals. Ultimately, it comes down to a personal preference, like with the definitions of computability, where Turing’s proposal was well-received by Gödel, as opposed to the lambda calculus, which

at the timedidn’t seem to capture the intuition of general computation (so there’s a cultural component as well).There have been a few nice attempts at resolving this questions for simple types, including some old work by Turing, Howard and Tait that give

partialsolutions, but a full solution still awaits, and certainly needs to be cleaned up and simplified by whoever comes next.

https://johncarlosbaez.wordpress.com/2015/08/11/the-physics-of-butterfly-wings/

Complex! But make it simpler. Get the image up on your monitor, tweak it in whatever way you can until you like it. Now paint from it. I still use a printed image but I compare it to my monitor and paint accordingly. When I take a picture of the painting I adjust it until it looks like the painting. I still can’t guarantee that what you see on your monitor will match the painting exactly but no matter what I do I can’t guarantee that.

]]>Yes, the fact that macrolichens contain basidiomycete yeasts as well as ascomycete fungi and algae is really cool!

]]>If I haven’t overlooked something then the exact above image was not displayed in the image list. But maybe it is somewhere hidden in the hires images. That is I found

a high res image which seems to display exactly this section of Mars, but it seems to be not exactly the same image.

So the image is not from cryptic terrain, but from Russel crater -still on the southern hemissphere but a little more higher north.

The black things trees/geysers/avalanches/lichens look strange. In this high res version at Russel crater though the long ones look indeed like some matter falling down….or some long dangling lichens…However they at least follow the shape of the underlying barchan.

Looking at the black stuff and other barchans in more detail here quickly some of the most interesting images and features. The black things seem to avalanche down (if they do) on the steep sides of the slip face or rather as in this northern dunes on the slip face itself. Since there are sand avalanches on the slip face that could explain why the black lines are so straight – i.e. there is no curved terrain there. On the wind side the material seems to be spread by the wind. In this image though some of them look indeed like spewing geysers. Interestingly they prefer or may accumulate cold regions with white snow around them (see also here or here), which could make sense in the context of some sublimation process.

So some of the long black stripes seem indeed look as if they arise from black matter rolling down a hill. But as it seems not all of them. That is besides the image you posted with trees on a hill, a similar pattern is

here.

And I don’t know why this image got the name dune erosion, but here you see again that some of the long black things do not seem to follow the terrain (Look e.g. utmost top part at the red border line on the right) but rather stand strangely (inclinated) up, either as if something is spewn out (“geyser”) (but one sees no curvature due to gravitation) or as if this is some tree/fungus/lichen whatever. The spewn out material (if it is spewn out) looks sometimes strangely thicker on the ends (thats what I called falsely called “cloaking”, having a cloak and clogging and clotting in mind), but if this is from rolling down then this could be due to accumulating material at the end of the slope.

By the way there was an interesting article about only recently detected features of earthly lichens.

]]>By the way, to get LaTeX to work in comments here, follow the directions that appear in boldface above the box where you type in your comment.

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