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The Existence of Designs via Iterative Absorption: Hypergraph 𝐹-designs for Arbitrary
We solve the existence problem for
Our main result concerns decompositions of hypergraphs whose clique distribution fulfills certain regularity
constraints. Our argument allows us to employ a ‘regularity boosting’ process which frequently enables us to satisfy these constraints
even if the clique distribution of the original hypergraph does not satisfy them. This enables us to go significantly beyond the setting
of quasirandom hypergraphs considered by Keevash. In particular, we obtain a resilience version and a decomposition result for
hypergraphs of large minimum degree.
eBook
Ellipse packing in two-dimensional celltessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law
Background: Lewis’s law and Aboav-Weaire’s law are two fundamental laws used to describe the topology of two-dimensional (2D) structures; however, their theoretical bases remain unclear. Methods: We used R package Conicfit software to fit ellipses based on the geometric parameters of polygonal cells with ten different kinds of natural and artificial 2D structures. Results: Our results indicated that the cells could be classified as ellipse’s inscribed polygon (EIP) and that they tended to form ellipse’s maximal inscribed polygon (EMIP). This phenomenon was named as ellipse packing. On the basis of the number of cell edges, cell area, and semi-axes of fitted ellipses, we derived and verified new relations of Lewis’s law and Aboav-Weaire’s law . Conclusions: Ellipse packing is a short-range order that places restrictions on the cell topology and growth pattern. Lewis’s law and Aboav-Weaire’s law mainly reflect the effect of deformation from circle to ellipse on cell area and the edge number of neighboring cells, respectively. The results of this study could be used to simulate the dynamics of cell topology during growth.
Journal Article
Experimental study of productivity enhancement in a humidification-dehumidification desalination system through various packing materials and configurations
Packaging materials significantly enhance the efficiency of the humidification—dehumidification (HDH) desalination system. The configuration of split packing materials has not been investigated, and there is a lack of studies assessing packing materials of varying heights and types. To address this deficiency, an experimental humidification-dehumidification cycle has been established utilizing packing at 30 cm split and full heights, in addition to 45 cm and 60 cm full heights. Furthermore, three types of packing materials—cellulose kraft paper, PP and PVC cellular grid, and PP trickle grid—are examined. The operational parameters include inlet water temperature and flow rates set at (50 ˚C, 60 ˚C, and 70 ˚C) and (2 kg/min, 4 kg/min, and 6 kg/min), cold water flow rates at (8 and 16 kg/min), air cycle types (closed and open), and a unified air flow rate of 1 kg/min. The efficiency of the HDH system reaches its peak when the inlet and cold water flow rates are at their maximum values of 6 and 16 kg/min, respectively, with the inlet water temperature at highest of 70 °C, utilizing cellulose kraft paper of the maximum height of 60 cm, and operating under a closed air cycle. The optimal performance of the HDH system yields a fresh water productivity of 4.2 L/h, a gained output ratio (GOR) of 0.63, humidifier and dehumidifier efficiencies of 98.7% and 84%, respectively, a recovery ratio (RR) of 1.2, a fresh water cost of $0.008/L, and a pressure drop of 0.32 Pa across the humidifier. The 30 cm split packing demonstrates improvements in productivity and GOR of 3% and 4%, respectively, when compared to the 30 cm full-height packing.
Journal Article
A characterization of 4-χρ-(vertex-)critical graphs
Given a graph G, a function c : V(G) ?1,..., k with the property that for every u?v, c(u) = c(v) = i implies that the distance between u and v is greater than i, is called a k-packing coloring of G. The smallest integer k for which there exists a k-packing coloring of G is called the packing chromatic number of G, and is denoted by ??(G). Packing chromatic vertex-critical graphs are the graphs G for which ??(G ? x) < ??(G) holds for every vertex x of G. A graph G is called a packing chromatic critical graph if for every proper subgraph H of G, ??(H) < ??(G). Both of the mentioned variations of critical graphs with respect to the packing chromatic number have already been studied [6, 23]. All packing chromatic (vertex-)critical graphs G with ??(G) = 3 were characterized, while there were known only partial results for graphs G with ??(G) = 4. In this paper, we provide characterizations of all packing chromatic vertex-critical graphs G with ??(G) = 4 and all packing chromatic critical graphs G with ??(G) = 4.
Journal Article
Characteristics of SARS-CoV-2 Transmission among Meat Processing Workers in Nebraska, USA, and Effectiveness of Risk Mitigation Measures
The coronavirus disease (COVID-19) pandemic has severely impacted the meat processing industry in the United States. We sought to detail demographics and outcomes of severe acute respiratory syndrome coronavirus 2 infections among workers in Nebraska meat processing facilities and determine the effects of initiating universal mask policies and installing physical barriers at 13 meat processing facilities. During April 1-July 31, 2020, COVID-19 was diagnosed in 5,002 Nebraska meat processing workers (attack rate 19%). After initiating both universal masking and physical barrier interventions, 8/13 facilities showed a statistically significant reduction in COVID-19 incidence in <10 days. Characteristics and incidence of confirmed cases aligned with many nationwide trends becoming apparent during this pandemic: specifically, high attack rates among meat processing industry workers, disproportionately high risk of adverse outcomes among ethnic and racial minority groups and men, and effectiveness of using multiple prevention and control interventions to reduce disease transmission.
Journal Article
Stability of the A15 phase in diblock copolymer melts
The self-assembly of block polymers into well-ordered nanostructures underpins their utility across fundamental and applied polymer science, yet only a handful of equilibrium morphologies are known with the simplest AB-type materials. Here, we report the discovery of the A15 sphere phase in single-component diblock copolymer melts comprising poly(dodecyl acrylate)−block−poly(lactide). A systematic exploration of phase space revealed that A15 forms across a substantial range of minority lactide block volume fractions (f
L = 0.25 − 0.33) situated between the σ-sphere phase and hexagonally close-packed cylinders. Self-consistent field theory rationalizes the thermodynamic stability of A15 as a consequence of extreme conformational asymmetry. The experimentally observed A15−disorder phase transition is not captured using mean-field approximations but instead arises due to composition fluctuations as evidenced by fully fluctuating field-theoretic simulations. This combination of experiments and field-theoretic simulations provides rational design rules that can be used to generate unique, polymer-based mesophases through self-assembly.
Journal Article
Growth Kinetics of Random Sequential Adsorption Packings Built of Two-Dimensional Shapes with Discrete Orientations
We studied random sequential adsorption packings constructed from rectangles, ellipses, and discorectangles, where the orientations of constituent shapes were picked from discrete sets of values with varying spacing. It allowed us to monitor the transition between the two edge cases: the parallel alignment and the arbitrary, continuous orientation of the shapes within the packing. The packings were generated numerically. Apart from determining the kinetics of packing growth in low- and high-density regimes, we analyzed the results in terms of packing density and probed the microstructural properties using the density autocorrelation function.
Journal Article
Conformal Graph Directed Markov Systems on Carnot Groups
We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped
with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit
set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples
of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include
the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the
non-real classical rank one hyperbolic spaces.
eBook
Perfectly packing a cube by cubes of nearly harmonic sidelength
Let d be an integer greater than
$1$
, and let t be fixed such that
$\\frac {1}{d} < t < \\frac {1}{d-1}$
. We prove that for any
$n_0$
chosen sufficiently large depending on t, the d-dimensional cubes of sidelength
$n^{-t}$
for
$n \\geq n_0$
can perfectly pack a cube of volume
$\\sum _{n=n_0}^{\\infty } \\frac {1}{n^{dt}}$
. Our work improves upon a previously known result in the three-dimensional case for when
$\\frac {1}{3} < t \\leq \\frac {4}{11} $
and
$n_0 = 1$
and builds upon recent work of Terence Tao in the two-dimensional case.
Journal Article