Requests Thread
October 11th, 2008 by WaltIn lieu of actually finishing any of the half-finished posts, I thought I’d see if anyone has any requests for posts on particular topics. Anything on your mind?
In lieu of actually finishing any of the half-finished posts, I thought I’d see if anyone has any requests for posts on particular topics. Anything on your mind?
October 11th, 2008 at 10:08 pm
You said you knew wonderful things about Polish spaces. What are they?
October 11th, 2008 at 10:09 pm
That is - not what are Polish space, but what are those wonderful things?
October 11th, 2008 at 11:13 pm
I’m wondering about this:
Zero-sum games for continuous-time jump Markov processes in Polish spaces: discounted payoffs
Guo, Xianping
Hernández-Lerma, Onésimo
Localización: http://projecteuclid.org/euclid.aap/1189518632
Adv. in Appl. Probab. 39, no. 3 (2007), 645-668
doi:10.1239/aap/1189518632
This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. We give conditions on the game’s primitive data under which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using the extended infinitesimal operator associated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide a recursive way of computing (or at least approximating) the value of the game. Moreover, we present a `martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.
October 11th, 2008 at 11:17 pm
Also, I suppose, I wonder about:
Random Probability Measures on Polish Spaces (Stochastics Monographs)
# Hardcover: 136 pages
# Publisher: CRC; 1 edition (July 25, 2002)
# Language: English
# ISBN-10: 0415273870
# ISBN-13: 978-0415273879
In this monograph the narrow topology on random probability measures on Polish spaces is investigated in a thorough and comprehensive way. As a special feature, no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra, are made. One of the main results is a direct proof of the random analog of the Prohorov theorem, which is obtained without invoking an embedding of the Polish space into a compact space. Further, the narrow topology is examined and other natural topologies on random measures are compared. In addition, it is shown that the topology of convergence in law-which relates to the “statistical equilibrium”-and the narrow topology are incompatible. A brief section on random sets on Polish spaces provides the fundamentals of this theory. In a final section, the results are applied to random dynamical systems to obtain existence results for invariant measures on compact random sets, as well as uniformity results in the individual ergodic theorem. This clear and incisive volume is useful for graduate students and researchers in mathematical analysis and its applications.
About the Author
Hans Crauel is of the Institut fur Mathematik at Technische Universitat Ilmenau in Germany.
October 12th, 2008 at 5:14 am
Requesting the topic of one of the posts I’m avoiding finishing? You’re diabolical, John.
October 13th, 2008 at 2:51 am
I would welcome the specific topic: why/how does density functional theory (DFT) work?
Or the broader topic, when can we expect the (present-day) exponential expansion of quantum simulation capability to run out of gas?
Across broad sectors of the global economy, this ongoing expansion is becoming the “other” Moore’s Law.
Can we foresee the mathematical limits to this expansion? What the main mathematical tools (informatic, geometric, algebraic, differential, etc.) that are being deployed to study these limits?
October 13th, 2008 at 5:35 am
How about getting around to explaining the “cavalcade of normality” of statistics, as involving the Gateaux derivative?
October 22nd, 2008 at 9:29 pm
I would appreciate a post on either the irrationality of infinity or the irrationality of imaginary numbers. On the latter concept, note that the origin of the term is in a criticism of it. This is historically ironic! I don’t believe in either concept. But of course the latter, by its name, seems not to believe in itself.
October 22nd, 2008 at 9:31 pm
Oh also…. just because all those people up there fourthed and fifthed that other idea…. doesn’t mean it’s better. “Truth is not decided by majority vote.” - James Madison I think
October 22nd, 2008 at 11:40 pm
I sixth Todd’s request!
October 27th, 2008 at 3:05 pm
I second the requests for posts on Polish spaces.
Something on Sobolev spaces or Radon-Nikodym would be neat too.
October 27th, 2008 at 8:21 pm
I seventh Todd’s request. At least give a pointer to the litterature, I have been looking on and off since you first wrote that post but found nothing.