Factorizing τ-spectra of mappings

In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).     

Construction of differential equation approximations to delay differential equations

A constructive method is presented for obtaining differential equation approximations to general functional delay differential equations. It is shown that the approximating differential equation systems can be used to determine the stability of the functional differential equations     

The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szeg? polynomials, reinterpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.     

Canonical forms of 2 × 2 × 2 and 2 × 2 × 2 × 2 arrays over 𝔽2 and 𝔽3

We consider arrays of size 2?×?2?×?2 and 2?×?2?×?2?×?2 over the fields with two and three elements. We use computer algebra to determine the canonical forms of these arrays with respect to the action of the semidirect product of general linear groups with the symmetric group. For each canonical form, we determine the size of its orbit and the rank of the arrays in its orbit.     

Thermally Driven Structure and Performance Evolution of Atomically Dispersed FeN4 Sites for Oxygen Reduction

FeN₄ moieties embedded in partially graphitized carbon are the most efficient platinum group metal free active sites for the oxygen reduction reaction in acidic proton‐exchange membrane fuel cells. However, their formation mechanisms have remained elusive for decades because the Fe?N bond formation process always convolutes with uncontrolled carbonization and nitrogen doping during high‐temperature treatment. Here, we elucidate the FeN₄ site formation mechanisms through hosting Fe ions into a nitrogen‐doped carbon followed by a controlled thermal activation. Among the studied hosts, the ZIF‐8‐derived nitrogen‐doped carbon is an ideal model with well‐defined nitrogen doping and porosity. This approach is able to deconvolute Fe?N bond formation from complex carbonization and nitrogen doping, which correlates Fe?N bond properties with the activity and stability of FeN₄ sites as a function of the thermal activation temperature.     

Stack the Bowls: Tailoring the Electronic Structure of Corannulene‐Integrated Crystalline Materials

We report the first examples of purely organic donor–acceptor materials with integrated π‐bowls (πBs) that combine not only crystallinity and high surface areas but also exhibit tunable electronic properties, resulting in a four‐orders‐of‐magnitude conductivity enhancement in comparison with the parent framework. In addition to the first report of alkyne–azide cycloaddition utilized for corannulene immobilization in the solid state, we also probed the charge transfer rate within the Marcus theory as a function of mutual πB orientation for the first time, as well as shed light on the density of states near the Fermi edge. These studies could foreshadow new avenues for πB utilization for the development of optoelectronic devices or a route for highly efficient porous electrodes.     

Natural protective amyloids

Amyloidoses are a group of diseases including neurodegenerative diseases like Alzheimer's disease and also type II diabetes, spongiform encephalopathies and many others, believed to be caused by protein aggregation and subsequent amyloid fibril formation. However, occasionally, living organisms exploit amyloid fibril formation, a property inherent into amino acid sequences, and perform specific physiological functions from amyloids, in differing biological contexts. Some of these functional amyloids are natural protective amyloids. Here, we review recent evidence on silkmoth chorion protein synthetic peptide-analogues that documents the function of silkmoth chorion, the major component of the eggshell, a structure with extraordinary physiological and mechanical properties, as a natural protective amyloid. Also, we briefly discuss the reported function of other natural, protective amyloids like fish chorion, the protein Pmel17 which forms amyloid fibrils that act as templates and accelerate the covalent polymerization of reactive small molecules into melanin, the hydrophobins and the antifreeze protein from winter flounder. Molecular self-assembly is becoming an increasingly popular route to new supramolecular structures and molecular materials and the inspiration for such structures is commonly derived from self-assembling systems in biology. Therefore, a careful examination of these studies may set the basis for the exploration of new routes for the formation of novel biocompatible polymeric structures with exceptional physico-chemical properties, for potentially new biomedical and industrial applications.     

Knots,BPS States,and Algebraic Curves

We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.     

Analysis of the HO2 vibrational spectrum on an accurate ab initio potential energy surface

The complete vibrational spectrum of the HO2(X(2)A' ') radical, up to the H + O2 dissociation limit, has been determined quantum mechanically on an accurate potential energy surface (PES), based on approximately 15000 ab initio points at the icMRCI+Q/aug-cc-pVQZ level of theory. The vibrational states are found to be assignable at low energies but become more irregular as the energy approaches the dissociation limit. However, even at very high energies, regularity still exists, in sharp contrast to earlier results based on the double many-body expansion (DMBE) IV potential. Several Fermi resonances have been identified, and the spectrum is fit with a spectroscopic Hamiltonian. In addition, the vibrational dynamics is analyzed using a periodic orbit approach.