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.

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.

Apple-like Shape of Freezing Paraffin Wax Droplets and Its Origin.

Paraffin wax stores energy in the form of latent heat at a nearly constant temperature during melting and releases this energy during solidification. This effect is used in industrial energy storage. At the same time, the possible deformation of even small volumes of material as a result of phase change is insufficiently studied. In this paper, the physical nature of such deformation, probably for the first time, is studied on the example of a droplet of paraffin wax. An unusual change in the shape of a melted droplet of paraffin wax placed on a relatively cold glass plate was observed in the laboratory experiments. As the droplet solidifies, its upper surface becomes nearly flat, and a dimple is formed in the center of this surface, making the droplet look like a fruit (pumpkins are more commonly shaped like this, but the authors prefer apples). A series of experiments, as well as physical and numerical modeling of the droplet's thermal state, taking into account the formation of a mushy zone between liquidus and solidus, made it possible to understand the role of gravity and gradual increase in viscosity and density of paraffin wax on changing the droplet shape and, in particular, to clarify the mechanism of formation of the dimple on its upper. It was shown that the mushy zone between the liquidus and solidus of the paraffin wax is responsible for the dimple formation.

Universality of Scaling Laws Governing Contact and Spreading Time Spans of Bouncing Liquid Marbles and its Physical Origin.

The impact of liquid marbles coated with a diversity of hydrophobic powders with various solid substrates, including hydrophobic, hydrophilic, and superhydrophobic ones, was investigated. The contact time of the bouncing marbles was studied. Universal scaling behavior of the contact time tc as a function of the Weber number (We) was established; the scaling law tc = tc(We) was independent of the kind of powder and the type of solid substrate. The total contact time consists of spreading time and retraction time. It is weakly dependent on We and this is true for all kinds of studied powders and substrates. This observation hints to the surface tension/inertia spring model governing the impact. By contrast, the spreading time ts scales as [Formula: see text], n = 0.28 - 0.30 ± 0.002. We relate the origin of this scaling law to the viscous dissipation occurring within the spreading marbles. The retraction time tr grows weakly with the Weber number. The scaling law was changed at threshold values of We ≅ 15-20. It is reasonable to explain this change with the breaking of the Leidenfrost regime of spreading under high values of We.

Mapping Grip Force Characteristics in the Measurement of Stress in Driving.

Reducing drivers' stress can potentially increase road safety. However, state-of-the-art physiological stress indices are intrusive and limited by long time lags. Grip force is an innovative index of stress that is transparent to the user and, according to our previous findings, requires a two- to five-second time window. The aim of this study was to map the various parameters affecting the relationship between grip force and stress during driving tasks. Two stressors were used: the driving mode and the distance from the vehicle to a crossing pedestrian. Thirty-nine participants performed a driving task during either remote driving or simulated driving. A pedestrian dummy crossed the road without warning at two distances. The grip force on the steering wheel and the skin conductance response were both measured. Various model parameters were explored, including time window parameters, calculation types, and steering wheel surfaces for the grip force measurements. The significant and most powerful models were identified. These findings may aid in the development of car safety systems that incorporate continuous measurements of stress.

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.

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.

Interfacial crystallization in the polyhedral liquid marbles.

HYPOTHESIS: We hypothesized that interfacial crystallization occurring within evaporated polyhedral liquid marbles may be controlled by hydrophilization of the polymer plates coating the marbles. The hypothesis was tested with polyhedral marbles coated with hydrophobized and cold plasma-hydrophilized PET (Polyethylene terephthalate) plates. EXPERIMENTS: Interfacial crystallization within polyhedral liquid marbles was investigated experimentally. Two types of polyhedral marbles filled with saturated saline were prepared: i) liquid marbles coated with hydrophobized PET plates (Marbles A); ii) liquid marbles coated with Janus PET plates, one facet of which was plasma hydrophilized and the other hydrophobized (Marbles B). The hydrophobized side of the PET plate was in contact with the saline solution, whereas, the hydrophilized facet contacted air. Crystallization occurring within the marbles under their evaporation was monitored in situ. FINDINGS: It was established that for both kinds of marbles, NaCl crystallization was initiated at the edges of the plates. NaCl crystallization on the hydrophobized PET surfaces was not registered. When Marbles B were evaporated, the outer hydrophilic side of the PET plates was coated by the saline creep process. For both kinds of marbles the process resulted in the formation of hollow shells built of PET plates and NaCl crystals. The thermodynamic explanation of the observed phenomena is suggested.

Stress-Adaptive Training: An Adaptive Psychomotor Training According to Stress Measured by Grip Force.

Current training methods show advances in simulation technologies; however, most of them fail to account for changes in the physical or mental state of the trainee. An innovative training method, adaptive to the trainee's stress levels as measured by grip force, is described and inspected. It is compared with two standard training methods that ignore the trainee's state, either leaving the task's level of difficulty constant or increasing it over time. Fifty-two participants, divided into three test groups, performed a psychomotor training task. The performance level of the stress-adaptive group was higher than for both control groups, with a main effect of t = -2.12 (p = 0.039), while the training time was shorter than both control groups, with a main effect of t = 3.27 (p = 0.002). These results indicate that stress-adaptive training has the potential to improve training outcomes. Moreover, these results imply that grip force measurement has practical applications. Future studies may aid in the development of this training method and its outcomes.

Hierarchical liquid marbles formed using floating hydrophobic powder and levitating water droplets.

HYPOTHESIS: Liquid marbles i.e. droplets coated by hydrophobic particles may be formed not only on the solid substrates but also on the floating layers of hydrophobic powders such as fluorinated fumed silica or polytetrafluoroethylene. EXPERIMENTS: Formation and growth of liquid marbles on fluorinated fumed silica or polytetrafluoroethylene powder floating on a heated water-vapor interface is reported. Marbles emerge from condensation of water droplets levitating above the powder layer. FINDINGS: The kinetics of the growth of droplets is reported. Growth of droplets results from three main mechanisms: water condensation, absorption of small droplets and merging of droplets with neighboring ones. Growing droplets are coated with the hydrophobic powder, eventually giving rise to the formation of stable liquid marbles. Formation of hierarchical liquid marbles is reported. Growth of liquid marbles emerging from water condensation follows the linear temporal dependence. A phenomenological model of the liquid marble growth is suggested.

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.

Effect of asymmetric cooling of sessile droplets on orientation of the freezing tip.

HYPOTHESIS: The shape of the "freezing tip" formed by the crystallization of water droplets demonstrated remarkable universality - no dependence on the cooling rate and physico-chemical properties of the substrate has been observed. At the same time, the spatial orientation of the freezing cone may be varied. We hypothesized that the orientation of the freezing tip is determined by the direction of heat flux at the base of the sessile droplet. This direction is expected to be changed when the substrate with a low thermal diffusivity is not cooled uniformly. EXPERIMENTS: We studied the freezing of water droplets placed on the inclined surface of wedges made from a variety of materials (polymers: Polymethylmethacrylate, Polytetrafluoroethylene, Polyurethane and metal: Titanium), which were cooled from below. The shape of the frozen droplets was controlled in situ. COMPUTATIONS: The computational model was suggested for the transient temperature field in the polymer wedge to determine a time variation of the local heat flux under the droplets. A comparison of numerical results and the measurements enabled us to confirm the aforementioned hypothesis relating the orientation of the freezing tips to the direction of the heat flux. FINDINGS: It was established that the orientation of the freezing cone axis depends on the location of the frozen droplet on the inclined surface of the wedge. Calculations of the transient temperature field of the wedge confirmed our hypothesis about the physical reason of the various spatial orientations of the freezing cones.

Grip Force on Steering Wheel as a Measure of Stress.

Driver performance is crucial for road safety. There is a relationship between performance and stress such that too high or too low stress levels (usually characterized by stressful or careless driving, respectively) impair driving quality. Therefore, monitoring stress levels can improve the overall performance of drivers by providing either an alert or intervention when stress levels are sub-optimal. Commonly used stress measures suffer from several shortcomings, such as time delays in indication and invasiveness of sensors. Grip force is a relatively new measure that shows promising results in measuring stress during psychomotor tasks. In driving, grip force sensor is non-invasive and transparent to the end user as drivers must continuously grip the steering wheel. The aim of the current research is to examine whether grip force can be used as a useful measure of stress in driving tasks. Twenty-one participants took part in a field experiment in which they were required to brake the vehicle in various intensities. The effects of the braking intensity on grip force, heart rate, and heart rate variability were analyzed. The results indicate a significant correlation between these three parameters. These results provide initial evidence that grip force can be used to measure stress in driving tasks. These findings may have several applications in the field of stress and driving research as well as in the vehicle safety domain.

Osmotic evolution of composite liquid marbles.

HYPOTHESIS: We hypothesized that the reported evolution (growth) of composite water marbles filled with saline water and coated with lycopodium dispersed in a thin layer of silicone oil is due to the osmotic mass transfer. The hypothesis is supported by the semi-empirical model of osmotic growth of small liquid marbles floating on distilled water. EXPERIMENTS: Saline composite, silicone oil-coated marbles floating on distilled water grew with time; whereas, composite marbles filled with distilled water floating on aqueous solutions of NaCl lost mass with time and shrunk. However, composite liquid marbles filled with saline water and floating on aqueous solutions of NaCl remained stable during 25 h of the laboratory experiment. FINDINGS: The reported findings are reasonably attributed to osmotic mass transport through the thin silicon layer filled with lycopodium particles coating the marbles, acting as an osmotic membrane. This is supported by the suggested model for the osmotic growth of marbles.

Study of wetting of the animal retinas by Water and organic liquids and its Implications for ophthalmology.

Interfacial properties of the animal retinas are reported. Wetting of the retina-retinal pigment epithelium-choroid-sclera tissues of cow, sheep, and pig eyes by water, silicone and castor oil was explored experimentally. Both water and silicone oils demonstrated complete wetting of the retina, regardless of the viscosity of the silicone oil, whereas the castor oil demonstrated a partial wetting regime. Similar wetting regimes were observed for sheep, cow and pig retinas. The intact surface of animal retina was found to be both hydrophilic and oleophilic. Wetting experiments with double sandwich oil/water layers were performed. Water demonstrated stronger affinity to the retina than silicone and castor oils, and eventually replaced the oils at the liquid/retina interface. We conclude that aqueous solutions continuously secreted in the living eye may displace silicone oil from the retinal surface and contribute to retinal re-detachment. Study of dynamics of wetting of the animal retina by water and organic oils is reported. The exponent describing the dynamics of spreading of the castor oil is lower than that predicted by the Tanner law. Castor oil may provide more effective tamponade than silicone oil.

Composite Liquid Marbles as a Macroscopic Model System Representing Shedding of Enveloped Viruses.

A model macroscopic system imitating the entry of viruses into living cells is suggested. The system represents the contact of a composite (core-shell) liquid marble with hydrophobic/hydrophilic particles. Composite liquid marbles are water droplets coated with silicone oil armored with nanometer-sized hydrophobic particles serving as an interfacial model of a living cell. Composite marbles absorbed hydrophilic polymer particles but prevented hydrophobic particles from entering their core. Swallowing of hydrophilic particles by composite marbles resembles the penetration of viruses into living cells. The interfacial mechanism of absorption is suggested.

Manufacture and properties of composite liquid marbles.

HYPOTHESIS: Liquid marbles are non-stick droplets coated with colloidal usually hydrophobic particles. We suggest that "composite" liquid marbles, i.e. bi-liquid droplets, may be prepared with water droplets coated by a thin silicone oil layer containing hydrophobic, colloidal particles. EXPERIMENTS: The process enabling manufacturing water marbles coated with silicone oil containing fumed fluorosilica particles is reported. The marbles remained stable when placed on solid and liquid supports. Bouncing and coalescence of the composite marbles was explored. FINDINGS: Non-coalescence prolonged (ca. 20 min) jumping of composite marbles above a vibrating water bath was observed. Composite marbles withstand coalescence better than colloidal particle-stabilized liquid marbles. The effective surface tension of the composite marbles is markedly lower than that of water marbles coated with fumed fluorosilica particles. The coefficient of restitution of the composite marbles bouncing on a hydrophobic solid substrate is lower than that established for water marbles. This observation is related to the viscous dissipation occurring within the silicone layer making up the composite marbles.

The paradox of pedestrian's risk aversion.

Traffic accidents are becoming a significant cause for unnatural deaths around the world, with more than 1.25 million fatalities in road accidents each year, and over 20 million people severely injured. A large portion of accidents that result in fatalities involve interaction between vehicles and pedestrians. In the literature, researchers speculate on a wide range of reasons for these figures. This paper focuses on the relationship between pedestrians' urgency to cross a busy road and the resulting level of risk for an accident. The probability for an accident is determined by a prediction model for a collision between drivers and pedestrians at congested conflict spots. The model is based on a motion planner called the Probabilistic Navigation Function (PNF), initially designed for robot navigation in dynamic cluttered and uncertain environments. The model predicts pedestrians' trajectories when crossing a busy road in a sub-meter accuracy, based on the risk they are willing to take (a reflection of the level of urgency to cross the road). The paper describes an unexpected and surprising pedestrian behavior in simple road crossings scenarios. When the model is given a loose risk boundary (that reflects a high level of pedestrian urgency to cross), the resulting trajectory exposes the pedestrian to a lower risk compared with a trajectory constructed with a strict risk boundary (that reflects a more conservative pedestrian). This is equivalent to claiming that, paradoxically, pedestrians in some scenarios who are willing to take higher levels of risk, face a decreased probability for an accident while crossing a congested road. The paper introduces the PNF model for crossing pedestrians, analyses their performance in a set of simulations, and explains its rationale. Next, an analytic estimation for the risk level as a function of the crossing angle of the selected trajectory is provided. A series of experiments conclude the paper and support the claim that this phenomenon is frequent among crossing pedestrians. The experimental results suggest that in some common scenarios, more cautious pedestrians may lower the initial risk for an accident at the expense of a total higher risk for an accident during the entire road crossing process, compared with a pedestrian who takes an initial higher level of risk that results in, overall, a decreased probability for an accident. A statistical analysis implies that there are significant differences in this occurrence between adults and children.

Mini-Generator of Electrical Power Exploiting the Marangoni Flow Inspired Self-Propulsion.

The mini-generator of electrical energy exploiting Marangoni soluto-capillary flows is reported. The interfacial flows are created by molecules of camphor emitted by the "camphor engines" placed on floating polymer rotors bearing permanent magnets. Camphor molecules adsorbed by the water/vapor interface decrease its surface tension and create the stresses resulting in the rotation of the system. The alternative magnetic flux in turn creates the current in the stationary coil. The long-lasting nature of rotation (approximately 10-20 h) should be emphasized. The brake-specific fuel consumption of the reported generator is better than that reported for the best reported electrical generators. Various engineering implementations of the mini-generator are reported.

Traffic accident risk assessment with dynamic microsimulation model using range-range rate graphs.

Analysis of accidents that involve vehicles and pedestrians requires accurate reproduction of the dynamics of the vehicles and pedestrians immediately prior to and during the accident. In many cases, only centimeters and milliseconds separate survival from disaster, particularly when high-speed aggressive drivers and careless pedestrians are involved. In this paper we present a methodology for analyzing the dynamic interaction between drivers in conflict scenarios with pedestrians. We assess the safety of a traffic location's environment with a high-resolution, spatially explicit, dynamic agent-based simulation model - SAFEPED. Based on the resulting data, Range-Range Rate (R-RR) graphs are generated. These graphs provide compact, simple, and objective presentation of the dynamic interaction between vehicles and pedestrians. Significant traffic risk indicators such as Time-To-Collision, acceleration/deceleration rates, and minimal distances between vehicles and pedestrians are easily extracted from the R-RR graphs. These indicators can provide insights on particular traffic scenarios and can assist road planners and developers of traffic safety measures in understanding the dynamic behavior of drivers and pedestrians before and during a conflict scenario.