Algebraic geometry over the residue field of the infinite place

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 𝔽_∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.     

On UJ-rings

UJ-rings are studied, i.e., ring in which all units can be presented in a form 1+x, for some x∈J(R). The behavior of UJ-rings under various algebraic construction is investigated. In particular, it is shown that the problem of lifting the UJ property from a ring R to the polynomial ring R[x] is equivalent to the Köthe’s problem for 𝔽₂-algebras.     

Bidentate Disilicate Framework for Bis-Grafted Surface Species

Recent advances in surface organometallic chemistry have enabled the detailed characterization of the surface species in single-site heterogeneous catalysts. However, the selective formation of bis-grafted surface species remains challenging because of the heterogeneity of the supporting surface. Herein, we introduce a metal complex bearing bidentate disilicate ligands, −OSi(O^tBu)₂OSi(O^tBu)₂O−, as a molecular precursor, which has a silicate framework adjacent to the metal (Pt) center. The grafting of the precursors on silica supports (MCM-41 and CARiACT Q10) proceeded through a substitution reaction on the silicon atoms of the disilicate ligand, which was verified by the detection of isobutene and ^tBuOH as the elimination products, to selectively yield bis-grafted surface species. The chemical structure of the surface species was characterized by solid-state NMR, and the chemical shift values of the ancillary ligands and ¹⁹⁵Pt nuclei suggested that the bidentate coordination sphere was maintained following grafting.     

Ring Expansion to 6-, 7-, and 8-Membered Benzosilacycles through Strain-Release Silicon-Based Cross-Coupling

The synthesis of silacycles is highly appealing due to their important applications in organic synthesis, medicinal chemistry, and materials chemistry. However, sila-tetralins and sila-benzosuberanes are surprisingly under-represented due to a lack of general methods to access these compounds. We successfully developed a Pd-catalyzed strain-release silicon-based cross-coupling as an unprecedented ring-expansion method, which constitutes a general route for preparing diverse sila-tetralins and sila-benzosuberanes.     

Thiete Dioxides as Templates Towards Twisted Scaffolds and Macrocyclic Structures

Thiete dioxide units have been employed as a template for further functionalization through C−H activation strategies. Using simple thiete dioxide building blocks, a new library of axially chiral molecules has been synthesized that owe their stability to electrostatic interactions in the solid state. Similar starting materials were further engaged in the formation of cyclic trimeric structures, opening the pathway to unprecedented macrocyclic ring systems.     

Homogeneous metric and matrix product codes over finite commutative principal ideal rings

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.     

Transformations and consecutivities on a king for computational applications

Aim of this work is to explore a new way to study polyhedra with minimal number of integer points, generalizing integer programming to more general algebraic structures.