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We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $$w_{n,k,m}$$ is the number of Dyck paths of semilength $$n$$ with $$k$$ occurrences of $UD$ and $$m$$ occurrences of $UUD$, then $$w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $$w_{2k+1,k,m}$$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $$w_{n,k,m}$$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $$w_{n,k_{1},k_{2},\ldots, k_{r}}$$ counting the semilength-$$n$$ Dyck paths with $$k_{1}$$ $UD$-factors, $$k_{2}$$ $UUD$-factors, $$\ldots$$, and $$k_{r}$$ $$U^{r}D$$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $$W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$$, including real-rootedness, $$\gamma$$-positivity, and a symmetric decomposition. 

We report the synthesis of large-area, high-Ti-content, Mo 1−x Ti x S 2 alloy thin films in the 2H phase at temperature as low as 500 °C using a scalable two-step method of metal film deposition, followed by sulfurization in H 2 S. Film processing at higher temperature accelerates Ti segregation, film coarsening, and the formation of TiS 2 in the 1T phase. Crystal growth at higher temperature results in the formation of multiple binary sulfide phases, in agreement with the equilibrium phase diagram. Making highly metastable, smooth, and uniform single-phase alloy films, therefore, hinges on developing low-temperature processing. Our results are relevant to the development of technologies based on designer transition metal dichalcogenide alloys, including in photonic integrated circuits and gas sensing. 

Fluidic devices are crucial components in many industrial applications involving fluid mechanics. Computational design of a high-performance fluidic system faces multifaceted challenges regarding its geometric representation and physical accuracy. We present a novel topology optimization method to design fluidic devices in a Stokes flow context. Our approach is featured by its capability in accommodating a broad spectrum of boundary conditions at the solid-fluid interface. Our key contribution is an anisotropic and differentiable constitutive model that unifies the representation of different phases and boundary conditions in a Stokes model, enabling a topology optimization method that can synthesize novel structures with accurate boundary conditions from a background grid discretization. We demonstrate the efficacy of our approach by conducting several fluidic system design tasks with over four million design parameters. 

Efficient and correct operation of an IoT network requires the presence of a failure detector and membership protocol amongst the IoT nodes. This paper presents a new failure de- tector for IoT settings where nodes are connected via a wire- less ad-hoc network. This failure detector, which we name Medley, is fully decentralized, allows IoT nodes to maintain a local membership list of other alive nodes, detects failures quickly (and updates the membership list), and incurs low communication overhead in the underlying ad-hoc network. In order to minimize detection time and communication, we adapt a failure detector originally proposed for datacenters (SWIM), for the IoT environment. In Medley each node picks a medley of ping targets in a randomized and skewed manner, preferring nearer nodes. Via analysis and NS-3 simulation we show the right mix of pinging probabilities that simulta- neously optimize detection time and communication traffic. We have also implemented Medley for Raspberry Pis, and present deployment results. 

Abstract The making of BaZrS₃thin films by molecular beam epitaxy (MBE) is demonstrated. BaZrS₃forms in the orthorhombic distorted‐perovskite structure with corner‐sharing ZrS₆octahedra. The single‐step MBE process results in films smooth on the atomic scale, with near‐perfect BaZrS₃stoichiometry and an atomically sharp interface with the LaAlO₃substrate. The films grow epitaxially via two competing growth modes: buffered epitaxy, with a self‐assembled interface layer that relieves the epitaxial strain, and direct epitaxy, with rotated‐cube‐on‐cube growth that accommodates the large lattice constant mismatch between the oxide and the sulfide perovskites. This work sets the stage for developing chalcogenide perovskites as a family of semiconductor alloys with properties that can be tuned with strain and composition in high‐quality epitaxial thin films, as has been long‐established for other systems including Si‐Ge, III‐Vs, and II‐VIs. The methods demonstrated here also represent a revival of gas‐source chalcogenide MBE.