Counting arithmetical structures on paths and cycles

Let $G$ be a finite, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(diag\left(\mathbf{d}\right)?A)\mathbf{r}=0$, where $A$ is the adjacency matrix of $G$. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices $(diag\left(\mathbf{d}\right)?A)$). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients $\left(\genfrac{}{}{0ex}{}{2n?1}{n?1}\right)$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.     

Atomic‐Scale Observation of the Metal–Promoter Interaction in Rh‐Based Syngas‐Upgrading Catalysts

The direct conversion of syngas to ethanol, typically using promoted Rh catalysts, is a cornerstone reaction in CO₂ utilization and hydrogen storage technologies. A rational catalyst development requires a detailed structural understanding of the activated catalyst and the role of promoters in driving chemoselectivity. Herein, we report a comprehensive atomic‐scale study of metal–promoter interactions in silica‐supported Rh, Rh–Mn, and Rh–Mn–Fe catalysts by aberration‐corrected (AC) TEM. While the catalytic reaction leads to the formation of a Rh carbide phase in the Rh–Mn/SiO₂ catalyst, the addition of Fe results in the formation of bimetallic Rh–Fe alloys, which further improves the selectivity and prevents the carbide formation. In all promoted catalysts, Mn is present as an oxide decorating the metal particles. Based on the atomic insight obtained, structural and electronic modifications induced by promoters are revealed and a basis for refined theoretical models is provided.     

IDPi Catalysis

High acidity and structural confinement are pivotal elements in asymmetric acid catalysis. The recently introduced imidodiphosphorimidate (IDPi) Brønsted acids have met with remarkable success in combining those features, acting as powerful Brønsted acid catalysts and “silylium” Lewis acid precatalysts in numerous thus far inaccessible transformations. Substrates as challenging to activate as simple olefins were readily transformed, ketones were employed as acceptors in aldolizations allowing sub‐ppm level catalysis, whereas enolates of the smallest donor aldehyde, acetaldehyde, did not polymerize but selectively added a single time to a variety of acceptor aldehydes.     

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of ${?}^2\left({\mathbb{N}}^1\right)$. We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.     

Twisting the TAPPs: Bay-Substituted Non-planar Tetraazapero-pyrenes and their Reduced Anions

A new synthesis of tetraazaperopyrenes (TAPPs) starting from a halogenated perylene derivative 3,4,9,10- tetrabromo-1,6,7,12-tetrachloroperylene ( 1 ) gave access to bay-substituted TAPPs for the first time. Selective lithiation of the bromine-positions and subsequent addition of tosyl azide led to the formation of the tetraazidotetrachloroperylene ( 2 ), which was subsequently reduced by addition of sodium borohydride to the corresponding tetraaminotetrachloroperylene ( 3 ). Oxidation to its semiquinoidal form 4 and subsequent cyclization with acid chlorides gave rise to a series of bay-chlorinated TAPPs. Whereas the aromatic core of the previously studied ortho-substituted TAPPs was found to be planar, the steric pressure of the two chlorine substituents on each side leads to the twist of the peropyrene core of approximately 30 degrees, a structural feature also observed in other bay-substituted perylene derivatives. An experimental and computational analysis reveals that introducing chloride substituents at these positions leads to slightly increased electron affinities (EA) enabling the selective generation and characterization of the reduced mono-anionic radicals and closed shell di-anionic species. These anions were isolated and characterized by UV/Vis spectroscopy and EPR or NMR, respectively. Processing of the bay-chlorinated TAPPs in n-channel organic TFTs revealed electron mobilities of 0.001 to 0.003 cm² V⁻¹ s⁻¹. These reduced electron mobilities compared to the ortho-halogenated TAPPs are thought to be rooted in the less densely packed solid-state structures.     

Hamiltonian models of interacting fermion fields in quantum field theory

We consider Hamiltonian models representing an arbitrary number of spin 1 / 2 fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated \(N_\tau \) estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs.

     

On-Surface Assembly of Au-Dicyanoanthracene Coordination Structures on Au(111)

On-surface metal-organic coordination provides a promising way for synthesizing different two-dimensional lattice structures that have been predicted to possess exotic electronic properties. Using scanning tunneling microscopy (STM) and spectroscopy (STS), we studied the supramolecular self-assembly of 9,10-dicyanoanthracene (DCA) molecules on the Au(111) surface. Close-packed islands of DCA molecules and Au-DCA metal-organic coordination structures coexist on the Au(111) surface. Ordered DCA₃Au₂ metal-organic networks have a structure combining a honeycomb lattice of Au atoms with a kagome lattice of DCA molecules. Low-temperature STS experiments demonstrate the presence of a delocalized electronic state containing contributions from both the gold atom states and the lowest unoccupied molecular orbital of the DCA molecules. These findings are important for the future search of topological phases in metal-organic networks combining honeycomb and kagome lattices with strong spin-orbit coupling in heavy metal atoms.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.