Two ferrocene derivatives, namely, 1,2-(tetramethylene)-ferrocene and 1,2,1’,2’-bis(tetramethylene)-ferrocene, were synthesized in a four-step reaction sequence starting from ferrocene. Friedel-Craft acylation of ferrocene with succinic anhydride gave mono- or bis(3-carboxypropinoyl)ferrocene depending on the stoichiometry of succinic anhydride. The reduction of the keto groups to methylene, followed by ring-closing using trifluoroacetic anhydride gave 1,2-(α-ketotetramethylene)-ferrocene or 1,2,1’,2’-bis(α-ketotetramethylene)ferrocene. The diastereomeric mixture of the latter diketones was separated by column chromatography, characterized by single-crystal X-ray analysis, and assigned their stereochemistry. Reduction of the keto groups to methylene under Clemmensen conditions gave homoannular mono- or bis(tetramethylene) ferrocene derivatives. The molecular structure of 1,2-(tetramethylene)-ferrocene revealed that the ipso carbon atoms of the cyclopentadienyl group are 0.023(3) Å farther away from Fe(II) compared to the remaining three carbon atoms. Both complexes exhibit lower half-wave oxidation potentials than ferrocene, possibly due to the electron-releasing effects of the tetramethylene bridges.

Euler's theorem about the relationship between the numbers of individual elements of planar graphs (vertices, edges, faces) is also satisfied when the graph has zero vertices. The problem of the existence or not of such graphs has so far been the subject of very few studies. Plus, research that didn't prove anything. The paper presents an interpretation of this kind of graphs as mathematical formations lying on piecewise smooth surfaces. The lines of intersection of two or more smooth surfaces (faces of a graph) form edges without a vertex of the graph. The elementary properties of such graphs are discussed, the method of counting non-isomorphic 0-vertex graphs and their importance in physics and chemistry are given.

Cable interference in cable-driven mechanisms (CDM) is a major challenge in their applications. The wrench closure workspace (WCW) in a fully constrained mechanism may not be realized unless cable interference is considered. This paper classifies all possible types of cable interferences for planar cable-driven mechanisms and presents an analytical solution to find the boundaries of the constant-orientation interference region. It is shown that interference between cables starts and ends when one end of a cable coincides with another cable which can be thought as lying a point on a line segment. This geometrical view is developed and applied to provide a framework for the analysis of cable interference. By subtracting the interference space form WCW, the interference-free workspace is obtained in which the robot is able to operate.

A reliable and efficient rail track defect detection system is essential for maintaining rail track integrity and avoiding safety hazards and financial losses. Eddy current (EC) testing is a non-destructive technique that can be employed for this purpose. The trade-off between spatial resolution and lift-off should be carefully considered in practical applications to distinguish closely spaced cracks such as those caused by rolling contact fatigue (RCF). A multi-channel eddy current sensor array has been developed to detect defects on rails. Based on the sensor scanning data, defect reconstruction along the rails is achieved using an inverse algorithm that includes both direct and iterative approaches. In experimental evaluations, the EC system with the developed sensor is used to measure defects on a standard test piece of rail with a probe lift-off of 4-6 mm. The reconstruction results clearly reveal cracks at various depths and spacings on the test piece.

It was recently demonstrated that a co-planar capacitive sensor could be applied to the evaluation of materials without the disadvantages associated with the other techniques. This technique effectively detects changes in the dielectric properties of the materials due to, for instance, imperfections or variations in the internal structure, by moving a set of simple electrodes on the surface of the specimen. An AC voltage is applied to one or more electrodes and signals are detected by others. This is a promising inspection method for imaging the interior structure of the numerous materials, without the necessity to be in contact with the surface of the sample. In this paper, Finite Element (FE) modelling was employed to simulate the electric field distribution from a co-planar capacitive sensor and the way it interacts with a non-conducting sample. Physical experiments with a prototype capacitive sensor were also performed on a Plexiglas sample with sub-surface defects, to assess the imaging performance of the sensor. A good qualitative agreement was observed between the numerical simulation and experimental result.

A simple design of an ultrathin six-band polarization-insensitive terahertz perfect metamaterial absorber (PMMA) composed of a metal cross-cave-patch resonator (CCPR) placed over a ground plane was proposed and investigated numerically. The numerical simulation results demonstrate that the average absorption peaks are up to 95% at six resonance frequencies with high quality-factors (>65). In addition, the absorption properties can be kept stability for both normal incident transverse magnetic (TM) and transverse electric (TE) waves. The physical mechanism behind the observed high level absorption is illustrated by the electric and power loss density distributions. The different absorption mainly originates from the higher order multipolar and multipolar plasmon resonance of the structure, which are sharp different to the most previous studies of the PMMAs. Furthermore, the resonance absorption of the PMMA can be tunable by varying the geometric parameters of the unit cell.

Face routing has been adopted in wireless sensor networks (WSNs) where topological changes occur frequently or maintaining full network information is difficult. For message forwarding in networks, a planar graph is used to prevent looping, and because long edges are removed by planarization and the resulting planar graph is composed of short edges, messages are forwarded along multiple nodes connected by them even though they can be forwarded directly. To solve this, face routing using information on all nodes within 2-hop range was adopted to forward messages directly to the farthest node within radio range. However, as the density of the nodes increases, network performance plunges because message transfer nodes receive and process increased node information. To deal with this problem, we propose a new face routing using the planar graphs of neighboring nodes to improve transfer efficiency. It forwards a message directly to the farthest neighbor and reduces loads and processing time by distributing network graph construction and planarization to the neighbors. It also decreases the amount of location information to be transmitted by sending information on the planar graph nodes rather than on all neighboring nodes. Simulation results show that it significantly improves transfer efficiency.

Many studies of hypervelocity impact craters have described the characteristics of quartz grains shock-metamorphosed at high pressures of > 10 GPa. In contrast, few studies have investigated shock metamorphism at lower shock pressures. In this study, we test the hypothesis that low-pressure shock metamorphism occurs in near-surface nuclear airbursts and that this process shares essential characteristics with crater-forming impact events. To investigate low-grade shock microstructures, we compared quartz grains from Meteor Crater, a 1.2-km-wide impact crater, to those from near-surface nuclear airbursts at the Alamogordo Bombing Range, New Mexico in 1945 and Kazakhstan in 1949/1953. This investigation utilized a comprehensive analytical suite of high-resolution techniques, including transmission electron microscopy (TEM) and electron backscatter diffraction (EBSD). Meteor Crater and the nuclear test sites all exhibit quartz grains with closely-spaced, sub-micron-wide fractures that appear to have formed at low shock pressures. Significantly, these micro-fractures are closely associated with Dauphiné twins and are filled with amorphous silica (glass), widely considered a classic indicator of shock metamorphism. Thus, this study confirms that glass-filled shock fractures in quartz form during near-surface nuclear airbursts, as well as crater-forming impact events, and by extension, it suggests that they may form in any near-surface cosmic airbursts in which the shockwave is coupled to Earth’s surface. The robust characterization of such events is crucial because of their potential catastrophic effects on the Earth’s environmental and biotic systems.

Many studies of hypervelocity impact craters have described the characteristics of quartz grains shock-metamorphosed at high pressures of >10 GPa, but in contrast, few studies have investigated shock metamorphism at lower shock pressures. In this study, we test the hypothesis that low-pressure shock metamorphism occurs in near-surface nuclear airbursts and that this process shares important characteristics with impact-cratering events. To investigate low-grade shock microstructures, we compared quartz grains from Meteor Crater, a 1.2-km-wide impact crater, to those from near-surface nuclear airbursts at the Alamogordo Bombing Range, New Mexico in 1945 and Kazakhstan in 1949/1953. This investigation utilized a comprehensive analytical suite of high-resolution techniques, including transmission electron microscopy (TEM) and electron backscatter diffraction (EBSD). Meteor Crater and the nuclear test sites all exhibit metamorphosed quartz grains with closely-spaced, sub-micron-wide fractures that appear to have formed at low shock pressures. Importantly, these micro-fractures are closely associated with Dauphiné twins and are filled with amorphous silica (glass), widely considered to be a classic indicator of shock metamorphism. Thus, this study confirms that glass-filled shock fractures in quartz form during near-surface nuclear airbursts, as well as crater-forming impact events, and by extension, it suggests they also may form in near-surface cosmic airbursts.

Strained layer superlattice (SLS) material system is being considered for large format, small-pitch and low-cost focal plane arrays for different applications. To address difficulties associated to focal plane arrays fabrication arising from mesa-isolation etching or complex surface treatment/ passivation process, planar structures have been considered. In this work, the recent progress on planar SLS photodetector using ion-implantation for device isolation is presented. The devices are presented here are nBn and pBn heterostructure InAs/InAsSb SLS photodetector, where Zn and Si were chosen as the ion implants, respectively. The electrical and optical performance of the devices were compared along with comparison with mesa-etched device, to give deeper view of the device performance.

The research presented in this paper concerns the determination of the attraction basins of the Newton’s iterative method which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The authors considered simple observation systems corresponding to the intersections, or linear and angular resections, used in practice. The main goal was to investigate the properties of the sets of convergent in the initial points of the applied iterative method. An important issue regarding the possibility of automatic and quick selection of such points was also considered. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness or self-similarity were sought. The research also concerned the iterative structures of the basins, i.e. maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method, i.e. the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the essential results of the research is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises and their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points, i.e. located far from the solution points.

The effect of spanwise wing non-planarity, implemented in conjunction with a Gurney flap is presented. Testing was undertaken in a low speed wind tunnel using a rectangular wing with an aspect ratio of 3. The outer 1/3 of the wing was non-planar which took the form of dihedral or a circular arc. A 2% high Gurney flap was implemented such that it could extend over the entire span, or the planar inboard section. Loads were measured using a sting balance. The data shows that non-planarity increases the maximum lift coefficient and the wing’s lift curve slope. Gurney flap lift modulation was enhanced in the presence of non-planarity. The addition of Gurney flaps caused a greater minimum drag coefficient increment for the non-planar wings. The Gurney flaps reduced the lift dependent drag of the wings; however, the minimum drag coefficient was observed to increase. The Gurney flaps reduced the maximum lift to drag ratio (L/D) for the non-planar wings; however, the flat wing showed a small L/D increment with flap addition.

In a software-defined radar (SDR) system, most of the signal processing usually implemented in hardware is implemented by software, thus allowing for higher flexibility and modularity compared to conventional radar systems. However, the majority of SDR demonstrators and proofs-of-concept reported in the open literature so far have been based on simple antenna systems. As a result, the full potentialities of an SDR approach have not been completely exploited yet. In this work, we propose a flexilbe antenna module to be integrated into an active electronically scanning array (AESA) with controlled sidelobe level over a wide angular range, exhibiting polarization reconfigurability with low cross-polarization level and high isolation. For this purpose, analytical and numerically efficient techniques for the synthesis of the aperture distribution and the correct evaluation of the radiating features (e.g., beamwidth, pointing angle, sidelobe levels, etc.) are presented in order to grant a real-time control of the digital beamforming network. A sub-array module demonstrator is fabricated and measured to corroborate the concept.

Although Cable driven rehabilitation devices (CDRD) have advantages over traditional link-driven devices, including lightweight, ease of reconfiguration, remote actuation, etc, the majority of existing lower-limb CDRD is limited to rehabilitation in the sagittal plane. In this work, we extend our previous sagittal plane model (2DOF) to accommodate hip motion in the frontal plane (abduction/adduction) toward studying the feasibility of tracking bi-planar motion (combined frontal and sagittal plane motion) via intra-planar cable routing. Two optimization problems have been formulated to estimate an optimal location of the hip cuffs, first to estimate the optimal cuff location at each time step to identify a single ‘averaged’ optimal cuff location and second to calculate a single optimal cuff location for the entire gait cycle. The optimization solutions identified for the 3DOF model revealed that optimization of the location of cuffs on the anterior and posterior side of the hip joint for 4 cable configuration is not sufficient to generate the desired bi-planar motion. For simultaneous tracking of the bi-planar motion, 2 additional cables have been added at hip joints and are routed in an intra-planar manner. The simulation result with the 3DOF model confirmed successful bi-planar trajectory tracking. The various number of cables and cable routings for tracking bi-planar motion will be studied in future work.

We investigated fabrication of neodymium doped thin film optical waveguide-based devices as potential active sources for planar integrated optics. Liquid-phase epitaxial growth was used to fabricate neodymium-doped yttrium aluminum borate films on compatible lattice-matched un-doped yttrium aluminum borate substrates. We observed the refractive index contrast of the doped and un-doped crystal layers by differential interference contrast microscopy. In addition, characterization by X-ray powder diffraction, optical absorption and luminescence spectra demonstrated the crystal quality and uniformity and optical guiding of the resulting thin films.

Dynamic behavior of coaxial axisymmetric planar cracks in the transversely isotropic magneto-electro-elastic (MEE) material in transient in-plane magneto-electro-mechanical loading is studied. Magneto-electrically impermeable as well as permeable cracks are assumed for crack surface. In the first step, considering prismatic and radial dynamic dislocations, electric and magnetic jumps are obtained through Laplace and Hankel transforms. These solutions are utilized to derive singular integral equations in the Laplace domain for the axisymmetric penny-shaped and annular cracks. Derived Cauchy singular type integral equations are solved to obtain the density of dislocation on the crack surfaces. Dislocation densities are utilized in computation of the dynamic stress intensity factors, electric displacement and magnetic induction in the vicinity tips of crack tips. Finally, some numerical case studies of a single and multiple cracks are presented. The effect of system parameters on the results is then discussed.

The feasibility of replacing the toxic chlorobenzene antisolvent with the environmentally friendly anisole in the fabrication of planar triple cation perovskite solar cells was explored. The successful integration of anisole not only ensures comparable device performance but also contributes to the development of more sustainable and green fabrication processes for next-generation photovoltaic technologies. Nevertheless, to ensure the possibility to achieve well-functioning unencapsulated devices, at any time of the year, with changed temperature and relative humidity, several batches with different cesium concentration were realized. We found that 10% of CsI in the precursor solution makes devices with low hysteresis index and sustained performance stability over a 90-day period both with cholorobenzene and anisole antisolvent. However the use of anisole as a green antisolvent ensures enhanced performance.

Goal of this article is to provide the numerical and experimental assessments of an effective near-field to far-field transformation (NF–FF T) technique with planar spiral scanning for flat antennas under test (AUTs), which requires a non-redundant, i.e. minimum, amount of NF measurements. This technique has its roots in the theory of non-redundant sampling representations of electromagnetic fields and has been devised by suitably applying the unified theory of spiral scans for non-volumetric antennas to the case in which the considered AUT is modeled by a circular disk having its radius equal to one half the AUT maximum dimension. It makes use of a 2-D optimal sampling interpolation (OSI) formula to accurately determine the massive NF data required by the classical plane-rectangular NF–FF T technique from the non-redundant ones gathered along the spiral. It must be emphasized that, when considering flat AUTs, the developed transformation allows to further and significantly save measurement time as compared to that needed by the previously developed NF–FF T techniques with planar spiral scans based on the quasi-planar antennas modelings, because the number of turns of the spiral and that of NF data to be acquired depend somehow on the area of the modeling surface. The reported numerical simulations assess the accuracy of the proposed NF–FF T technique, whereas the experimental tests prove its practical feasibility.

This study scrutinizes the complexities of designing and exploring the potential energy surface (PES) of systems containing more than twenty atoms with planar tetracoordinate carbons (ptCs). To tackle this issue, we utilized an established design rule to design a Naphtho [1,2-b:4,3-b′:5,6-b′′:8,7-b′′′]tetrathiophene derivative computationally. This process began with substituting S atoms with CH– units, then replacing three sequential protons with two Si2+ units in the resultant polycyclic aromatic hydrocarbon polyanion. Despite not representing the global minimum, the newly designed Si8C22 system with four ptCs provided valuable insights into strategic design and PES exploration. Our results underscore the importance of employing adequate methodologies to confirm the stability of newly designed molecular structures containing planar hypercoordinate carbons.

This paper focuses on the non-destructive dielectric measurement for low-loss planar materials with a thickness of less than 3 mm using a large coaxial probe with an outer diameter of 48 mm. The aperture probe calibration procedure required only to make a measurement of the half-space air and three offset shorts. The reflection coefficient for the thin material is measured using a Keysight E5071C network analyzer from 0.3 MHz to 650 MHz and then converted to a relative dielectric constant and tangent loss via closed form capacitance model and lift-off calibration process. Measurement error of dielectric constant, Δε_r is less than 2.5 % from 1 MHz to 400 MHz and the resolution of loss tangent, tan δ measurement is capable of achieving 3×10^-3.

This paper proposes a compact planar Wi-Fi antenna with optimized radiation patterns for small uncrewed aerial vehicle (UAV) applications in both urban and open areas. It is suitable for mounting on the outermost side of the non-metallic wing of small UAVs. It has small dimensions of 16.5 mm (L) by 30.3 mm (W) by 1.6 mm (h), and the measured results of its prototype are in agreement with simulated data. Its impedance bandwidths over the two frequency ranges are 2.11 to 2.58 GHz and 5.06 to 7.5 GHz ( ). At 5.8 GHz, it has stronger radiation below the small UAV to reduce interference from rare-use directions. Its maximum radiations, the directions of the maximum radiation in each elevation plane, are below the UAV and between 14° and 29° from the horizontal plane. At 2.4 GHz, it has quasi-omnidirectional radiation to ensure a stable link in all directions, and its maximum radiations are near the horizontal plane. The optimized radiation patterns at 5.8 and 2.4 GHz can provide more antenna gain when the small UAV flies farther in urban and open areas, respectively. In addition, it has good vertically polarized radiation for long-distance applications.

The symmetry properties of an ornament contain information about its civilisation and its interactions with other cultural sources. Two-dimensional periodic ornaments can be strictly classified into a limited set of 17 mathematical symmetry groups, also known as wallpaper groups. The collection of ornaments thus classified for a civilisation is characteristic of the cultural group and serves as a fingerprint to identify that group. If the distribution of wallpaper groups is available for several societies, mathematical methods can be applied to determine similarities and differences between the art practices of these communities. This method allows a systematic approach to the general ornamental practices within a culture and their interactions in the form of similarity of fingerprints. We test the feasibility of the method on examples of medieval Armenians, Byzantium, Seljuks first in Persia and then in Anatolia and among Arabs in the Middle East. For this purpose we present the distribution of the planar ornaments and calculate the Euclidean distances in pairs. We tested to what extend geographical and religious factors could account for the observed similarity of ornamental groups between cultures. The results suggest an intensive interaction between the Seljuk Turks and Arab craftsmen who produced the ornaments. Therefore the cultural interactions are religiously motivated.

A planar microwave sensor devoted to the detection of humidity in underwear and clothes in general is proposed. The ultimate goal of the sensor is to detect the presence of liquids in fabrics, of interest to aid patients that suffer from certain pathologies, such as hyperhidrosis and enuresis. The main target in the design of the sensor, considering the envisaged application, is simplicity. Thus, the sensor operates at a single frequency, and the working principle is the variation in the mag-nitude of the transmission coefficient of a matched line loaded with an open-ended quar-ter-wavelength sensing stub resonator. The stub, which must be in contact with the so-called fabric under test (FUT), generates a notch in the transmission coefficient with a resonance frequency that depends on the humidity level of the fabric. By designing the stub with a moderately high quality factor, the variation in the resonance frequency causes a significant change in the magnitude level at the operating frequency, the resonance frequency when the sensing stub is loaded with the dry fabric, and the presence of liquid can be detected by means of an amplitude detector. A prototype device is proposed and experimentally validated.

An increasing number of aircraft is equipped with wing tip devices, which either are installed by the aircraft manufacturer at the production line or are retrofitted after the delivery of the aircraft to its operator. Installation of wing tip devices has not been a popular choice for regional turboprop aircraft and the novelty of the current study is to investigate the feasibility of retrofitting the British Aerospace (BAe) Jetstream 31 with an appropriate wing tip device (or winglet) to increase its cruise range performance, taking also into account the aerodynamic and structural impact of the implementation. To the best of the knowledge of the authors, no previous study exists which has attempted to assess the winglet retrofit of an existing aircraft type of similar size and operating profile. The optimal winglet design achieved a 2.38% increase of the maximum range by reducing the total drag by 1.19% at a mass penalty of 3.25%, as compared with the baseline aircraft configuration. Other designs were found to be more effective in reducing the total drag, but the structural reinforcement required for their implementation outweighed the achieved performance improvements. Since successful winglet retrofit programs for typical short to medium range narrow body aircraft report even more than 3% of block fuel improvements, undertaking the project of installing an optimal winglet design to the BAe Jetstream 31, should also consider a Direct Operating Cost (DOC) assessment on top of the aerodynamic and structural aspects of the retrofit.

This paper presents a demonstration of a linearly-polarized transmit-arrays antenna with high-efficiency and wideband operating in X-band. The topology of the unit cell consists of two identical patch antennas loaded by C-loop slot and interconnected by a metalized via through a ground plane sandwiched between two identical substrates [18]. A waveguide measurement system WGMS has been used for single cell characterization (reflection/transmission coefficients). Good results have been demonstrated in terms of matching performances; the reflection coefficient S11 is under 10-dB in the 9.4-10.4 GHz band (i.e. 10.1% around 9.9 GHz) and with only 0.35 dB-insertion loss at 10 GHz. The transmit-arrays prototype permitted to achieve 22.4 dBi of directivity and 20.1 dBi of gain. The aperture and radiation efficiencies are about 28.2% and 58.3%, respectively. With a focal ratio (F/D) of 1/2 the spill-over loss is less than 2 dB. Beam-scanning up to ± 35° has been achieved with the same phase quantization (passive) and less than 3 dB of gain losses.

The planar cell polarity (PCP) of epithelial ciliated cells is essential for effective mucociliary clearance (MCC) in the sinonasal mucosa. We hypothesize that MCC coordination is impaired in nasal polyp (NP) mucosae due to suppressed expression of a series of CPLANE (ciliogenesis and planar cell polarity effector) complex proteins in chronic rhinosinusitis (CRS) patients. To inves-tigate this hypothesis, we subjected sinonasal mucosal samples to live video recording to measure mucociliary transport velocity (MCTV) and scanning electron microscopy to evaluate surface morphology. The expression and distribution of a panel of PCP proteins, e.g. WDPCP and FUZ, were investigated in relation to inflammatory cytokine levels and clinical features. The mean MCTV of NP mucosae was significantly lower than that of the inferior turbinate mucosae. The CRS group with NPs (CRSwNP group) (n=28) showed increased expression of IL-13 and CCL26 mRNA compared to CRS patients without NPs (n=25) and controls (n=30). WDPCP and FUZ mRNA levels were significantly decreased in NP mucosae compared to ethmoid sinus mucosae in CRSwNP patients. WDPCP protein distribution was reduced in the cytoplasmic region of ciliated cells in CRSwNP patients. We conclude that suppression of WDPCP in ciliated cells is responsible for the impaired MCC of nasal polyps with type-2 inflammation. This mechanism might explain the decreased clearance and the potential for worsening symptoms of CRSwNP.

The minimum energy structures of the Si3C5 and Si4C8 clusters are planar and contain planar tetracoordinate carbons (ptCs). These species have been classified, qualitatively, as global () and local () aromatics according to the adaptive natural density partitioning (AdNDP) method, which is an orbital localization method. This work evaluates these species' aromaticity, focusing on confirming and quantifying their global and local aromatic character. For this purpose, we use an orbital localization method based on the partitioning of the molecular space according to the topology of the electronic localization function (LOC-ELF). In addition, the magnetically induced current density is analyzed. The LOC-ELF-based analysis coincides with the AdNDP study (double aromaticity, global and local). Besides, the current density analysis detects global and local ring currents. The strength of the global and local current circuit is significant, involving 4n+2 - and -electrons, respectively. The latter implicates the Si-ptC-Si fragment, which would be related to the 3c-2e -bond detected by the orbital localization methods in this fragment.

Motivated by a common Mathematical Finance topic, this paper surveys several results concerning windings of 2-dimensional processes, including planar Brownian motion, complex-valued Ornstein-Uhlenbeck processes and planar stable processes. In particular, we present Spitzer's asymptotic Theorem for each case. We also relate this study to the pricing of Asian options.

Leaky-wave antennas (LWAs) represent a recent class of radiating systems, first appeared in the second half of the past century. Initially treated with a certain dose of skepticism due to the apparent non-physical nature of supported wave solutions, LWAs have so far deeply evolved from the original configuration of slotted metallic lossy waveguide and adapted to a huge number of practical application fields, from microwave to millimeter waves and even up to the terahertz frequency range. Due to physics of relevant working principle, LWAs are often found in planar and low-profile configurations, being critically required in many cutting-edge applications, where device miniaturization, scaling and integration are required. Here we propose a review of some of the most recent and perspective array-fed 2-D LWAs, selecting two of most widespread categories in applications, Fabry–Perot Cavity Antennas (FPCAs) and Bull’s-Eye Antennas (BEAs), discussing and illustrating some advanced features recently proposed.

Large values and gradients of stresses and deformations, triggering concentrations of stresses and deformations, arise in the corner areas of a structure. The action of forced deformations, leading to the finite rupture of the contact between the elements of a structure, also triggers the concentration of stresses, while the rupture reaches an irregular point, a line on the area boundary. The theoretical analysis of the stress-strain state (SSS) of areas with angular cutouts in the boundary under the action of discontinuous forced deformations is reduced to the study of singular solutions to the homogeneous problem of the elasticity theory that has power-related features. The calculation of stress concentration coefficients in the domain of a singular solution to the elastic problem makes no sense. It is experimentally proven that the zone, that is close to the vertex of the angular cutout in the area boundary, has substantial deformations, rotations, and it corresponds to rising values of the first and second derivatives of displacements along the radius in cases of sufficiently small radii in the neighbourhood of the irregular point of the boundary. For such areas, it is necessary to consider the plane problem of the elasticity theory, taking into account the geometric nonlinearity under the action of forced deformations. This will allow analyzing the effect of relations between orders of values of deformations, rotations, and forced deformations on the form of the equation of equilibrium. The purpose of this work is to analyze the effect of relations of deformation orders, rotations, forced deformations on the form of the equilibrium equation in the polar coordinate system for a V-shaped area under the action of forced temperature-induced deformations with regard for the geometrical non-linearity and physical linearity.

This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.