Shannon Entropy of Ramsey Graphs with up to Six Vertices.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and ß-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic ß-colored convex polygons with m ß-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sß are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph; Sß is, in turn, an average uncertainty to find the red ß-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings.

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns.

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.

Clustering and self-organization in small-scale natural and artificial systems.

Physical properties of clusters, i.e. systems composed of a 'small' number of particles, are qualitatively different from those of infinite systems. The general approach to the problem of clustering is suggested. Clusters, as they are seen in the graphs theory, are discussed. Various physical mechanisms of clustering are reviewed. Dimensional properties of clusters are addressed. The dimensionality of clusters governs to a great extent their properties. Weakly and strongly coupled clusters are discussed. Hydrodynamic and capillary interactions giving rise to clusters formation are surveyed. Levitating droplet clusters, turbulent clusters and droplet clusters responsible for the breath-figures self-assembly are considered. Entropy factors influencing clustering are considered. Clustering in biological systems results in non-equilibrium multi-scale assembly, where at each scale, self-driven components come together by consuming energy in order to form the hierarchical structure. This article is part of the theme issue 'Bioinspired materials and surfaces for green science and technology (part 3)'.

Self-Arranged Levitating Droplet Clusters: A Reversible Transition from Hexagonal to Chain Structure.

Water microdroplets condense over locally heated water-vapor interfaces and levitate in an ascending vapor-air flow forming self-assembled ordered monolayer clusters. The droplets do not coalesce due to complex aerodynamic interactions between them. The droplet cluster formation is governed by the condensation/evaporation balance and by coupling of heat flux and vapor flow with aerodynamic forces. Here, we report the observations of a reversible structural transition from the ordered hexagonal-structure cluster to the chain-like structure and provide an explanation of its mechanism and conditions under which the transition occurs. The phenomenon provides new insights on the fundamental physical and chemical processes with microdroplets including their role in reaction catalysis in nature and their potential for aerosol and microfluidic applications.

Is the Voronoi Entropy a True Entropy? Comments on "Entropy, Shannon's Measure of Information and Boltzmann's H-Theorem", Entropy 2017, 19, 48.

The goal of this comment note is to express our considerations about the recent paper by A. Ben Naim (Entropy 2017, 19, 48). We strongly support the distinguishing between the Shannon measure of information and the thermodynamic entropy, suggested in the paper. We demonstrate that the Voronoi entropy should also be clearly distinguished from the entropy of a two-dimensional gas. Actually, the Voronoi entropy being an intensive value is the averaged Shannon measure of ordering for a given pattern.

Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation.

The Voronoi entropy for random patterns and patterns demonstrating various elements of symmetry was calculated. The symmetric patterns were characterized by the values of the Voronoi entropy being very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points constituting the pattern. Extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables the elements of symmetry of the patterns to be revealed.

Toward an Understanding of Magnetic Displacement of Floating Diamagnetic Bodies, I: Experimental Findings.

Diamagnetic objects (polymer and metallic plates and spheres, ceramic beads, and liquid marbles), floating on water, and a variety of organic liquids may be driven by a steady magnetic field of 0.1 T, registered at the water-vapor surface. Diamagnetic bodies are attracted to the magnet, when the apparent contact angle at the solid/liquid interface is obtuse and repelled from the magnet, when the angle is acute. Cold plasma-treated polyolefin rafts and spheres, demonstrating underwater floating, are repelled by a permanent magnet. Addition of a surfactant to the water, as well as cold plasma treatment of the polyolefin bodies, can turn the attraction into the repulsion. We conjecture that the observed effects are caused by the interplay of two main phenomena. The first is the gravity, which induces sliding of the particle on the deformed liquid/vapor interface (the Moses effect). The second cause is the hysteresis of the contact angle at the bodies' boundaries.

Characterization of Self-Assembled 2D Patterns with Voronoi Entropy.

The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav-Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters.

Fungal Contamination of Methylprednisolone Causing Recurrent Lumbosacral Intradural Abscess.

Fungal meningitis transmitted through injections of methylprednisolone contaminated with Exserohilum rostratum affected 753 persons and caused 61 deaths in the United States in 2012. We report a case of infection recurrence after 24-months with the unique manifestation of an intradural fungal abscess. Fungal disease should remain on the differential diagnosis list for previously exposed patients.

Superposition of Translational and Rotational Motions under Self-Propulsion of Liquid Marbles Filled with Aqueous Solutions of Camphor.

Self-locomotion of liquid marbles, coated with lycopodium or fumed fluorosilica powder, filled with a saturated aqueous solution of camphor and placed on a water/vapor interface is reported. Self-propelled marbles demonstrated a complicated motion, representing a superposition of translational and rotational motions. Oscillations of the velocity of the center of mass and the angular velocity of marbles, occurring in the antiphase, were registered and explained qualitatively. Self-propulsion occurs because of the Marangoni solutocapillary flow inspired by the adsorption of camphor (evaporated from the liquid marble) by the water surface. Scaling laws describing translational and rotational motions are proposed and checked. The rotational motion of marbles arises from the asymmetry of the field of the Marangoni stresses because of the adsorption of camphor evaporated from marbles.

Under-Liquid Self-Assembly of Submerged Buoyant Polymer Particles.

The self-assembly of submerged cold-plasma-treated polyethylene beads (PBs) is reported. The plasma-treated immersed millimetrically sized PBs formed well-ordered 2D quasicrystalline structures. The submerged floating of "light" (buoyant) PBs is possible because of the energy gain achieved by the wetting of the high-energy plasma-treated polymer surface prevailing over the energy loss due to the upward climb of the liquid over the beads. The capillary "immersion" attraction force is responsible for the observed self-assembly. The observed 2D quasicrystalline structures demonstrate "dislocations" and "point defects". The mechanical vibration of self-assembled rafts built of PBs leads to the healing of point defects. The immersion capillary lateral force governs the self-assembly, whereas the elastic force is responsible for the repulsion of polymer beads.

Fermat Principle, Ramsey Theory and Metamaterials.

Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the "inverse graph" is introduced and discussed. The total number of triangles in the "direct" (source) and "inverse" Ramsey optical graphs is the same. The applications of "Ramsey optics" are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested.

Ramsey theory and thermodynamics.

Re-shaping of thermodynamics with the graph theory and Ramsey theory is suggested. Maps built of thermodynamic states are addressed. Thermodynamic states may be attainable and non-attainable by the thermodynamic process in the system of constant mass. We address the following question how large should be a graph describing connections between discrete thermodynamic states to guarantee the appearance of thermodynamic cycles? The Ramsey theory supplies the answer to this question. Direct graphs emerging from the chains of irreversible thermodynamic processes are considered. In any complete directed graph, representing the thermodynamic states of the system the Hamiltonian path is found. Transitive thermodynamic tournaments are addressed. The entire transitive thermodynamic tournament built of irreversible processes does not contain a cycle of length 3, or in other words, the transitive thermodynamic tournament is acyclic and contains no directed thermodynamic cycles.

Branched droplet clusters and the Kramers theorem.

Scaling laws inherent for polymer molecules are checked for the linear and branched chains constituting two-dimensional (2D) levitating microdroplet clusters condensed above the locally heated layer of water. We demonstrate that the dimensionless averaged end-to-end distance of the droplet chain r[over ¯] normalized by the averaged distance between centers of the adjacent droplets l[over ¯] scales as r[over ¯]/l[over ¯]∼n^{0.76}, where n is the number of links in the chain, which is close to the power exponent ¾, predicted for 2D polymer chains with excluded volume in the dilution limit. The values of the dimensionless Kuhn length b[over ̃]≅2.12±0.015 and of the averaged absolute value of the bond angle of the droplet chains |θ|[over ¯]=22.0±0.5^{0} are determined. Using these values we demonstrate that the predictions of the Kramers theorem for the gyration radius of branched polymers are valid also for the branched droplets' chains. We discuss physical interactions that explain both the high value of the power exponent and the applicability of the Kramers theorem including the effects of the excluded volume, surrounding droplet monomers, and the prohibition of extreme values of the bond angle.

Design, Implementation, Utilization, and Sustainability of a Fast Healthcare Interoperability Resources-Based Inpatient Rounding List.

OBJECTIVE: We designed and implemented an application programming interface (API)-based electronic health record (EHR)-integrated rounding list and evaluated acceptability, clinician satisfaction, information accuracy, and efficiency related to the application. METHODS: We developed and integrated an application, employing iterative design techniques with user feedback. EHR and application user action logs, as well as hospital safety reports, were evaluated. Rounding preparation characteristics were obtained through surveys before and after application integration. To evaluate usability, inpatient providers, including residents, fellows, and attendings were surveyed 2 weeks prior to and 6 months after enterprise-wide EHR application integration. Our primary outcome was provider time savings measured by user action logs; secondary outcomes include provider satisfaction. RESULTS: The application was widely adopted by inpatient providers, with more than 69% of all inpatients queried by the application within 6 months of deployment. Application utilization was sustained throughout the study period with 79% (interquartile range [IQR]: 76, 82) of enterprise-wide unique patients accessed per weekday. EHR action logs showed application users spent -3.24 minutes per day (95% confidence interval [CI]: -6.8, 0.33), p = 0.07 within the EHR compared with nonusers. Median self-reported chart review time for attendings decreased from 30 minutes (IQR: 15, 60) to 20 minutes (IQR: 10, 45) after application integration (p = 0.04). Self-reported sign-out preparation time decreased by a median of 5 minutes (p < 0.01), and providers were better prepared for hand-offs (p = 0.02). There were no increased safety reports during the study period. CONCLUSION: This study demonstrates successful integration of a rounding application within a commercial EHR using APIs. We demonstrate increasing both provider-reported satisfaction and time savings. Rounding lists provided more accurate and timely information for rounds. Application usage was sustained across multiple specialties at 42 months. Other application designers should consider data density, optimization of provider workflows, and using real-time data transfer using novel tools when designing an application.

Quantification of ordering in active light driven colloids.

HYPOTHESIS: Light driven diffusioosmosis allows for the controlled self-assembly of colloidal particles. Illuminating of colloidal suspensions built of nanoporous silica microspheres dispersed in aqueous solution containing photosensitive azobenzene cationic surfactant enables manufacturing self-assembled well-ordered 2D colloidal patterns. We conjectured that ordering in this patterns may be quantified with the Voronoi entropy. EXPERIMENTS: Depending on the isomerization state the surfactant either tends to absorb (trans-state) into negatively charged pores or diffuse out (cis-isomer) of the particles generating an excess concentration near the colloids outer surface and thus resulting in the initiation of diffusioosmotic flow. The direction of the flow can be controlled by the wavelength and intensity of irradiation. Under irradiations with blue light the colloids separate within a few seconds forming equidistant particle ensemble where long range diffusioosmotic repulsion acts over distances exceeding several times the particle diameter. Hierarchy of ordering in the studied colloidal systems is distinguished, namely: i) ordering of individual separated colloidal particles; ii) ordering of clusters built of colloidal particles; iii) ordering within clusters of individual colloidal particles. FINDINGS: The study of the temporal change in the Voronoi entropy for the light illuminated colloidal dispersions allowed quantification of ordering evolution on different lateral scales and under different irradiation conditions. Fourier analysis of the time evolution of the Voronoi entropy is presented. Fourier spectrum of the "small-area" (100 × 100 µm) reveals the pronounced peak at f = 1.125 Hz reflecting the oscillations of individual particles at this frequency. Ordering in hierarchical colloidal system emerging on different lateral scales is addressed. The minimal Voronoi entropy is intrinsic for the close packed 2D clusters.

Negative Effective Mass in Plasmonic Systems II: Elucidating the Optical and Acoustical Branches of Vibrations and the Possibility of Anti-Resonance Propagation.

We report the negative effective mass metamaterials based on the electro-mechanical coupling exploiting plasma oscillations of free electron gas. The negative mass appears as a result of the vibration of a metallic particle with a frequency ω which is close to the frequency of the plasma oscillations of the electron gas m2, relative to the ionic lattice m1. The plasma oscillations are represented with the elastic spring constant k2=ωp2m2, where ωp is the plasma frequency. Thus, the metallic particle vibrating with the external frequency ω is described by the effective mass meff=m1+m2ωp2ωp2-ω2, which is negative when the frequency ω approaches ωp from above. The idea is exemplified with two conducting metals, namely Au and Li embedded in various matrices. We treated a one-dimensional lattice built from the metallic micro-elements meff connected by ideal springs with the elastic constant k1 representing various media such as polydimethylsiloxane and soda-lime glass. The optical and acoustical branches of longitudinal modes propagating through the lattice are elucidated for various ratios ω1ωp, where ω12=k1m1 and k1 represents the elastic properties of the medium. The 1D lattice, built from the thin metallic wires giving rise to low frequency plasmons, is treated. The possibility of the anti-resonant propagation, strengthening the effect of the negative mass occurring under ω = ωp = ω1, is addressed.

Cherenkov-Like Surface Thermal Waves Emerging from Self-Propulsion of a Liquid Marble.

We explore the thermal field related to the self-propulsion of floating liquid marbles filled with aqueous ethanol. Cherenkov-like thermal waves arising from self-propulsion are registered. The opening angle of the thermal field Cherenkov triangle is governed by the inter-relation between the velocity of self-propulsion and the phase velocity of the capillary waves. The self-propulsion is driven by soluto-capillarity accompanied by thermo-capillarity. A semiquantitative analysis of the effect is presented. The empirical selection rule for capillary waves responsible for the mass, momentum, and heat transfer is outlined. The soluto-capillarity leads to much stronger spatial variations of the surface tension than the thermo-capillarity.

Attenuating hypoxia driven malignant behavior in glioblastoma with a novel hypoxia-inducible factor 2 alpha inhibitor.

Hypoxia inducible factor (HIFs) signaling contributes to malignant cell behavior in glioblastoma (GBM). We investigated a novel HIF2α inhibitor, PT2385, both in vitro, with low-passage patient-derived cell lines, and in vivo, using orthotopic models of glioblastoma. We focused on analysis of HIF2α expression in situ, cell survival/proliferation, and survival in brain tumor-bearing mice treated with PT2385 alone and in combination with standard of care chemoradiotherapy. HIF2α expression increased with glioma grade, with over half of GBM specimens HIF2α positive. Staining clustered in perivascular and perinecrotic tumor regions. Cellular phenotype including proliferation, viability, migration/invasion, and also gene expression were not altered after PT2385 treatment. In the animal model, PT2385 single-agent treatment did improve median overall survival compared to placebo (p = 0.04, n = 21) without a bioluminescence correlate (t = 0.67, p = 0.52). No difference in animal survival was seen in combination treatment with radiation (RT)/temozolomide (TMZ)/PT2385 (p = 0.44, n = 10) or mean tumor bioluminescence (t 1.13, p = 0.32). We conclude that HIF2α is a reasonable novel therapeutic target as expressed in the majority of glioblastomas in our cohort. PT2385 as a single-agent was efficacious in vivo, however, an increase in animal survival was not seen with PT2385 in combination with RT/TMZ. Further study for targeting HIF2α as a therapeutic approach in GBM is warranted.