A change of variables formula for mappings in

A change of variables formula for mappings in is obtained, where the usual jacobian is replaced by the determinant of the approximate differential.

     

On the number of directions determined by a pair of functions over a prime field

A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG(3,p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than pairs with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e∈Fp with the property that f(x)=cg(x)+dx+e.     

Solvent Dependency of Catalyst-Substrate Aggregation Through π-π Stacking in Photoredox Catalysis

Assemblies of photoredox catalysts and their target substrates prior to photoexcitation is a phenomenon naïvely overlooked by the majority of synthetic chemists, but can have profound influences on reactivity and selectivity in photocatalytic reactions. In this study, we determine the aggregation states of triarylamine radical cationic photocatalysts with various target arene substrates in different solvents by specifically parameterized polarizable molecular dynamics simulations. A π-stacking interaction previously implicated by more expensive, less-representative quantum calculations is confirmed. Critically, this study presents new insights on: i) the ability of solvents (MeCN vs DMF) to make or break a photocatalytic reaction by promoting (MeCN) or demoting (DMF) its catalyst-substrate assemblies, which is a determining factor for reactivity, ii) the average “lifetimes” of assemblies in solution from a dynamic simulation. We find that both in the ground state and the photoexcited state, the cationic radical assemblies remain intact for periods often higher than 60 ps, rendering them ideally suitable to undergo intra-assembly electron transfer reactions upon photoexcitation. Such aspects have not addressed by previous studies on synthetic photocatalytic reactions involving non-covalent assemblies.     

Precision Sequence-Defined Polymers: From Sequencing to Biological Functions

Precise sequence-defined polymers (SDPs) with uniform chain-to-chain structure including chain length, unit sequence, and end functionalities represent the pinnacle of sophistication in the realm of polymer science. For example, the absolute control over the unit sequence of SDPs allows for the bottom-up design of polymers with hierarchical microstructures and functions. Accompanied with the development of synthetic techniques towards precision SDPs, the decoding of SDP sequences and construction of advanced functions irreplaceable by other synthetic materials is of central importance. In this Minireview, we focus on recent advances in SDP sequencing techniques including tandem mass spectrometry (MS), chemically assisted primary MS, as well as other non-destructive sequencing methods such as nuclear magnetic resonance (NMR) spectroscopy, circular dichroism (CD), and nanopore sequencing. Additionally, we delve into the promising prospects of SDP functions in the area of cutting-edge biological research. Topics of exploration include gene delivery systems, the development of hybrid materials combining SDPs and nucleic acids, protein recognition and regulation, as well as the interplay between chirality and biological functions. A brief outlook towards the future directions of SDPs is also presented.     

Rosenhain-Thomae formulae for higher genera hyperelliptic curves

Rosenhain's famous formula expresses the periods of first kind integrals of genus two hyperelliptic curves in terms of θ-constants. In this paper we generalize the Rosenhain formula to higher genera hyperelliptic curves by means of the second Thomae formula for derivative non-singular θ-constants.     

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature     

Lower subdifferentiability in minimax fractional programming *

Minimax fractional programming problems are analyzed from the view- point of lower subdifferentiability, obtaining Kuhn-Tucker type optimality conditions. Multiobjective optimization problems with fractional objectives are also studied.