Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes

A concept of time-reversed entropy per unit time is introduced in analogy with the entropy per unit time by Shannon, Kolmogorov, and Sinai. This time-reversed entropy per unit time characterizes the dynamical randomness of a stochastic process backward in time, while the standard entropy per unit time characterizes the dynamical randomness forward in time. The difference between the time-reversed and standard entropies per unit time is shown to give the entropy production of Markovian processes in nonequilibrium steady states.     

Divergent Synthesis of Bioactive Dithiodiketopiperazine Natural Products Based on a Double C(sp3)−H Activation Strategy

This article provides a detailed report of our efforts to synthesize the dithiodiketopiperazine (DTP) natural products (−)-epicoccin G and (−)-rostratin A using a double C(sp³)−H activation strategy. The strategy's viability was first established on a model system lacking the C8/C8’ alcohols. Then, an efficient stereoselective route including an organocatalytic epoxidation was secured to access a key bis-triflate substrate. This bis-triflate served as the functional handles for the key transformation of the synthesis: a double C(sp³)−H activation. The successful double activation opened access to a common intermediate for both natural products in high overall yield and on a multigram scale. After several unsuccessful attempts, this intermediate was efficiently converted to (−)-epicoccin G and to the more challenging (−)-rostratin A via suitable oxidation/reduction and protecting group sequences, and via a final sulfuration that occurred in good yield and high diastereoselectivity. These efforts culminated in the synthesis of (−)-epicoccin G and (−)-rostratin A in high overall yields (19.6 % over 14 steps and 12.7 % over 17 steps, respectively), with the latter being obtained on a 500 mg scale. Toxicity assessments of these natural products and several analogues (including the newly synthesized epicoccin K) in the leukemia cell line K562 confirmed the importance of the disulfide bridge for activity and identified dianhydrorostratin A as a 20x more potent analogue.     

The commutative cohomology of nilmonoids,revisited

Semigroup Forum - A nilmonoid is a nilsemigroup N with an identity element adjoined. New properties of the canonical presentation of N yield a more precise characterization of 2-cocycles,...     

A door to model reduction in high-dimensional parameter space

Model reduction techniques such as Proper Generalized Decomposition (PGD) are decision-making tools that are about to revolutionize many domains. Unfortunately, their computation is still problematic for problems involving many parameters, for which one has to face the “curse of dimensionality”. An answer to this challenge is given in solid mechanics by the so-called “parameter-multiscale PGD”, which is based on Saint-Venant's principle. In this article, a model problem composed of up to a thousand parameters is presented, showing that the method is able to overcome the “curse of dimensionality”.     

Minoration conforme du spectre du laplacien de Hodge-de Rham

Let M ⁿ be an n-dimensional compact manifold, with n ≥ 3. For any conformal class C of riemannian metrics on M, we set , where μ _p,k (M,g) is the kth eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that . We also prove that if g is a smooth metric such that , and n = 0,2,3 mod 4, then there is a non-zero corresponding eigenform of degree with constant length. As a corollary, on a four-manifold with non vanishing Euler characteristic, there is no such smooth extremal metric.     

On a Class of Convex Sets and Functions

Given a convex subset C of ⁿ, the set-valued mapping C (where 0C is, by convention, the recession cone of C) is increasing on ₊ if and only if C contains the origin, and decreasing on ₊ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ^m with a convex subset of ⁿ to produce a convex subset of ^n+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.Mathematics Subject Classifications (2000) 26A51, 26B25, 26E25.     

On Pythagoras Theorem for Products of Spectral Triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K-homology.     

Intertwining Certain Fractional Derivatives

We obtain an intertwining relation between some Riemann–Liouville operators of order α ∈ (1, 2), connecting through a certain multiplicative identity in law the one-dimensional marginals of reflected completely asymmetric α-stable Lévy processes. An alternative approach based on recurrent extensions of positive self-similar Markov processes and exponential functionals of Lévy processes is also discussed.