Word problems and ceers

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.     

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.     

New Perspectives for Old Clusters: Anderson–Evans Anions as Building Blocks of Large Polyoxometalate Frameworks in a Series of Heterometallic 3 d–4 f Species

A series of nine [Sb₇W₃₆O₁₃₃Ln₃M₂(OAc)(H₂O)₈]^17? heterometallic anions ( Ln₃M₂ ; Ln=La–Gd, M=Co; Ln=Ce, M=Ni and Zn) have been obtained by reacting 3 d metal disubstituted Krebs‐type tungstoantimonates(III) with early lanthanides. Their unique tetrameric structure contains a novel {MW₉O₃₃} capping unit formed by a planar {MW₆O₂₄} fragment to which three {WO₂} groups are condensed to form a tungstate skeleton identical to that of a hypothetical trilacunary derivative of the ?‐Keggin cluster. It is shown, for the first time, that classical Anderson–Evans {MW₆O₂₄} anions can act as building blocks to construct purely inorganic large frameworks. Unprecedented reactivity in the outer ring of these disk‐shaped species is also revealed. The Ln₃M₂ anions possess chirality owing to a {Sb₄O₄} cluster being encapsulated in left‐ or right‐handed orientations. Their ability to self‐associate in blackberry‐type vesicles in solution has been assessed for the Ce₃Co₂ derivative.     

A Metal–Organic Framework with Cooperative Phosphines That Permit Post‐Synthetic Installation of Open Metal Sites

PCM‐101 is a phosphine coordination material comprised of tris(p‐carboxylato)triphenylphosphine and secondary pillaring groups coordinated to [M₃(OH)]⁵⁺ nodes (M=Co, Ni). PCM‐101 has a unique topology in which R₃P: sites are arranged directly trans to one another, with a P???P separation distance dictated by the pillars. Post‐synthetic coordination of soft metals to the P: sites proceeds at room temperature to provide X‐ray quality crystals that permit full structural resolution. Addition of AuCl groups forces a large distortion of the parent framework. In contrast, CuBr undergoes insertion directly between the trans‐P sites to form dimers that mimic solution‐phase complexes, but that are geometrically strained due to steric pressure exerted by the MOF scaffold. The metalated materials are active in heterogeneous hydroaddition catalysis under mild conditions, yielding different major products compared to their molecular counterparts.     

Phase transition for absorbed Brownian motion with drift

We study one-dimensional Brownian motion with constant drift toward the origin and initial distribution concentrated in the strictly positive real line. We say that at the first time the process hits the origin, it is absorbed. We study the asymptotic behavior, ast, ofm _t , the conditional distribution at time zero of the process conditioned on survival up to timet and on the process having a fixed value at timet. We find that there is a phase transition in the decay rate of the initial condition. For fast decay rate (subcritical case)m _t is localized, in the critical casem _t is located around , and for slow rates (supercritical case)m _t is located aroundt. The critical rate is given by the decay of the minimal quasistationary distribution of this process. We also study in each case the asymptotic distribution of the process, scaled by , conditioned as before. We prove that in the subcritical case this distribution is a Brownian excursion. In the critical case it is a Brownian bridge attaining 0 for the first time at time 1, with some initial distribution. In the supercritical case, after centering around the expected value—which is of the order oft—we show that this process converges to a Brownian bridge arriving at 0 at time 1 and with a Gaussian initial distribution.     

Noyaux potentiels associés à une filtration

We study from the point of view of potential theory some operators V which are “integrals of martingales” and noteworthy the formula (I + V)⁻¹ = I − N where N is a submarkovian kernel. We give an explicit expression of N when the filtration is finite and get the general case with an usual approximation procedure. Some links are made with the matrix theory (ultrametric and Stieltjes matrices) and the graph theory (flows and capacities) when the space is finite.

Résumé

On étudie, du point de vue de la théorie du potentiel, des opérateurs V du type “intégrales de martingale”, et notamment la formule (I + V)⁻¹ = I − N où N est un noyau sous-markovien. On donne une expression explicite de N dans le cas d'une filtration finie, et on traite le cas général par un procédé d'approximation usuel. On fait le lien avec la théorie des matrices (matrices ultramétriques et de Stieltjes) et la théorie des graphes (flots et capacités) quand l'espace est fini.     

Synthesis,Crystal Structure and Magnetism Characterization of Two Coordination Polymers based on Ni(2‐mpac)2(H2O)2 as Molecular Building Block

Taking Ni(2‐mpac)₂(H₂O)₂ as molecular building block, two novel coordination polymers with the formula of [Ni(2‐mpac)₂(4,4′‐bpy)·(4,4′‐bpy)]_n ( 1 ) and [Ni(2‐mpac)₂(py)·4H₂O]_n ( 2 ) (2‐mpac = 5‐methyl‐2‐pyrazinecarboxylic acid; 4,4′‐bpy = 4,4′‐bipyridine; py = pyrazine) have been synthesized and structurally characterized by X‐ray single crystal diffraction method. In addition, elemental analysis, IR and magnetism characterization are presented.