Graphic Representation of a Dimensional Expansion of Triangular Fuzzy Number

We calculate Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers. We prove that if the 3-dimensional result is limited to 2 dimensions, it is the same as the 2-dimensional result, which is shown as a graph. Since a 3-dimensional graph cannot be drawn, the value of the membership function is expressed with color density. We cut a 3-dimensional triangular fuzzy number by a perpendicular plane passing a vertex, and consider the cut plane as a domain. The value of the membership function for each point on the cut plane is also expressed with color density. The graph expressing the value of the membership function, defined in the plane as a 3-dimensional graph using the z -axis value instead of expressing with color density, is consistent with the results in the 2-dimensional case.

POSITIVE AND NEGATIVE ROUGHNESS ACCORDING TO LOCAL DIFFERENCES BETWEEN DEM SURFACE AND 3D REFERENCE PLANES

The irregularities of the earth’s surface are quantified by means of roughness measurements using Digital Elevation Models (DEM’s). This article presents a roughness measurement method that is based on the calculation of the difference of altitude existing between a plane passing through the centre of a moving window and the altitude of the DEM surface inside this window. This method differs from the measure of the standard deviation and best fit plane, in the sense that it considers all difference values, positives or negatives. The measurement is done in a 3 × 3 or a 5 × 5 moving window and contemplates inside this window the plane which passes through the centre of the window and the highest pixel located in the border or perimeter of this window. According to the 3D configuration of the DEM surface inside the moving window, the sum of all the differences is positive or negative, allowing to discriminate the local morphology independently of the global roughness. The roughness variable which distinguishes negative and positive values allows to classify accurately landscape units such as watersheds, riverbeds, volcanic assemblages as well as landforms associated with tectonic structures.

Anatomic Variations of Ethmoid Roof and Risk of Skull Base Injury in Endoscopic Sinus Surgery: Statistical Correlations

Background Recent developments in endoscopic sinus surgery (ESS) have increased the need to investigate the complex anatomic variations in the ethmoid roof and skull base, to inform the surgeon about the risk of damaging these crucial areas during ESS. Objective To offer a detailed description of sinus anatomy focusing on the key surgical landmarks in ESS and frontal recess surgery to standardize a systematic approach during the preoperative sinuses imaging evaluation. Methodology: A total of 220 computed tomography (CT) scans were reviewed to obtain six sets of measurements: the depth of the cribriform plate (CP); the length of the lateral lamella of the cribriform plate (LLCP); the angle formed by the LLCP and the continuation of the horizontal plane passing through the CP; the position of the anterior ethmoidal artery (AEA) at the skull base; the extent of frontal sinus pneumatization (FSP); the type of superior attachment of the uncinate process (SAUP). Results The length of the LLCP was statistically significantly correlated with the different Keros classification types, the angle formed by the LLCP with the continuation of the horizontal plane passing through the CP, and with the AEA position at the skull base. The depth of the olfactory fossa was correlated with FSP. Conclusions According to the Keros and Gera classifications, the data obtained from these evaluations allow the assessment of anatomic-radiological risk profiles and can help identify those patients who are high risk for ethmoid roof injury.

On topological properties of weakly m-convex sets

The topological properties of classes of generally convex sets in multidimensional real Euclidean space $\mathbb{R}^n$, $n\ge 2$, known as $m$-convex and weakly $m$-convex, $1\le m<n$, are studied in the present work. A set of the space $\mathbb{R}^n$ is called \textbf{\emph{$m$-convex}} if for any point of the complement of the set to the whole space there is an $m$-dimensional plane passing through this point and not intersecting the set. An open set of the space is called \textbf{\emph{weakly $m$-convex}}, if for any point of the boundary of the set there exists an $m$-dimensional plane passing through this point and not intersecting the given set. A closed set of the space is called \textbf{\emph{weakly $m$-convex}} if it is approximated from the outside by a family of open weakly $m$-convex sets. These notions were proposed by Professor Yuri Zelinskii. It is known the topological classification of (weakly) $(n-1)$-convex sets in the space $\mathbb{R}^n$ with smooth boundary. Each such a set is convex, or consists of no more than two unbounded connected components, or is given by the Cartesian product $E^1\times \mathbb{R}^{n-1}$, where $E^1$ is a subset of $\mathbb{R}$. Any open $m$-convex set is obviously weakly $m$-convex. The opposite statement is wrong in general. It is established that there exist open sets in $\mathbb{R}^n$ that are weakly $(n-1)$-convex but not $(n-1)$-convex, and that such sets consist of not less than three connected components. The main results of the work are two theorems. The first of them establishes the fact that for compact weakly $(n-1)$-convex and not $(n-1)$-convex sets in the space $\mathbb{R}^n$, the same lower bound for the number of their connected components is true as in the case of open sets. In particular, the examples of open and closed weakly $(n-1)$-convex and not $(n-1)$-convex sets with three and more connected components are constructed for this purpose. And it is also proved that any compact weakly $m$-convex and not $m$-convex set of the space $\mathbb{R}^n$, $n\ge 2$, $1\le m<n$, can be approximated from the outside by a family of open weakly $m$-convex and not $m$-convex sets with the same number of connected components as the closed set has. The second theorem establishes the existence of weakly $m$-convex and not $m$-convex domains, $1\le m<n-1$, $n\ge 3$, in the spaces $\mathbb{R}^n$. First, examples of weakly $1$-convex and not $1$-convex domains $E^p\subset\mathbb{R}^p$ for any $p\ge3$, are constructed. Then, it is proved that the domain $E^p\times\mathbb{R}^{m-1}\subset\mathbb{R}^n$, $n\ge 3$, $1\le m<n-1$, is weakly $m$-convex and not $m$-convex.

Dickinsonia: mobile and adhered

The classical genus of Ediacaran macroorganisms,Dickinsonia, was part of an extensive benthic marine community inhabiting the fields of microbial mats. The remains ofDickinsoniaare commonly preserved in the position of adhesion to the habitat substrate. However, these were mobile organisms. In addition to the already known feeding traces ofDickinsonia, structures described as traces of motor activity are reported. Long parallel furrows, extending from the posterior end of the body imprint, are interpreted as imprints of ridges left by an organism moving along the surface of the substrate. Groups of differently shaped grooves laying in the depression that enhalo theDickinsoniabody imprints or accompany their individual areas are interpreted as imprints of ridges and cords of mucous material. They are considered to represent structures of self-determined stretching and lift-off of the body margins from the substrate. The rings and arcs of silt- and sand-sized mineral particles bordering the body imprints are composed of material that was supposedly brushed off from the surface of the microbial mat byDickinsonia. They are considered traces of the adhesion of these organisms to the substrate. Accumulations of multidirectional pulling and tear-off structures, lacking the body imprint but accompanied by the joint plane passing into the overlying sediment and cutting through the bedding, are interpreted as escape traces. The dual modality of the behaviour (attachment and mobility) could indicate the adaptability ofDickinsoniato life in extremely shallow-water environments.

Effects of unilateral posterior missing-teeth on the temporomandibular joint and the alignment of cervical atlas

Cervical atlas alignment changes are associated with craniofacial development. Disturbance of craniofacial development may be associated with temporal mandibular joint function. Therefore, we examined the possibility of a correlation between unilateral missing teeth and morphologic changes of the spine and posture. We collected eighty-nine patients (38 men and 51 women) with unilateral posterior missing teeth and twenty patients without previous orthodontic treatment or missing posterior teeth by tracing and analyzing their panoramic and cephalometric film. We measured the angulations of articular eminence, cranio-cervical angle, and the percentage of the occlusal plane passing through the first and second cervical vertebrae with other morphologic geometric data. The angle of articular eminence inclination was higher in the non-missing teeth group than the missing teeth group (46.66° and 42.28°, respectively). The cranio-cervical angle was smaller in the missing posterior teeth group than the non-missing posterior teeth group (99.81° and 103.27°, respectively). The missing teeth group also showed fewer occlusal planes passing through the intersection of the first and second cervical vertebrae compared to the non-missing teeth group (28.9% and 65%, respectively). Individuals with unilateral missing teeth had lower articular eminence inclination, smaller cranio-cervical angle, and a lower percentage of the occlusal plane passing through the intersection of the first and second cervical vertebrae.

Measurement of the Flow Field Generated by Multicopter Propellers

This paper presents the results of research on the airflow around a multirotor aircraft. The research consisted of the analysis of the velocity field using particle image velocimetry. Based on the tests carried out in a wind tunnel, the distribution of the velocity and its components in the vertical plane passing through the propeller axis were determined for several values of the angle of attack of the tested object for two values of airflow velocity inside the tunnel, i.e., vwt = 0 m/s and vwt = 12.5 m/s. Determining the velocity value as a function of the coordinates of the adopted reference system allowed for defining the range of impact of the horizontal propellers and the fuselage of the research object itself. The tests allowed for quantitative and qualitative analyses of the airflow through the horizontal rotor. Particular attention was paid to the impact of the airflow and the angle of attack on the obtained velocity field distributions.

GEODYNAMIC PROCESSES DURING ASCENT OF A PLUME WITH INTERMEDIATE THERMAL POWER THROUGH THE CONTINENTAL LITHOSPHERE AND DURING ITS ERUPTION ON THE SURFACE

The study is focused on thermochemical mantle plumes with intermediate thermal power (1.15 < Ka < 1.9). Previously we have shown that these plumes are diamondiferous. Based on the laboratory modeling data, the flow structure of a melt in a plume conduit is represented. A plume melts out and ascends from the core – mantle boundary to the bottom of the continental lithosphere. The plume roof moves upwards in the lithosphere because of melting of the lithospheric matter at the plume roof and due to the effect of superlithostatic pressure on the roof, which causes motion in the lithosphere block above the plume roof. The latter manifests itself by uplifting of the ground surface above the plume. As the plume ascends through the lithosphere, the elevation of the surface increases until the plume ascends to critical level xкр, where an eruption conduit is formed. In our model, plume ascent velocity uпл is the rate of melting at the plume roof. Values of uпл and the ascent velocity of a spherical plume roof due to superlithostatic pressure U are calculated. Relationships are found between these velocities and the plume roof depth. The dependence of the velocity of the surface’s rise on the dynamic viscosity of the lithosphere block above the plume is obtained. A relationship is determined between the maximum surface elevation and the lithosphere viscosity. The elevation values are determined for different times and different lithosphere viscosities.The results of laboratory modeling of flow structure at the plume conduit/eruption conduit interface are presented. The flow was photographed (1) in the plane passing through the axes of the plume conduit and the eruption conduit; and (2) in case of the line-focus beam perpendicular to the axial plane. The photographs were used for measuring the flow velocities in the plume conduit and the eruption conduit. Corresponding Reynolds numbers and flow regimes are determined. The relation of dynamic pressure in the eruption conduit to that in the plume conduit is found for intermediate-power plumes. The melt flow velocity in the eruption conduit depends on superlithostatic pressure on the plume roof, plume diameter and kinematic viscosity of the melt. Its values are determined for different kinematic viscosities of melt.

PARAMETERS OF THE GRAIN TRAJECTORY AFTER TOUCHING THE SIEVE OF A SEMI-AUTOMATIC GRAIN CLEANING MACHINE

The main disadvantage of grain cleaning machines with rectangular sieves is limited throughput, which takes place due to a logical contradiction. It consists in the fact that as it passes through the sieve, the amount of material to be cleaned on the sieve decreases, and the width of the sieve remains unchanged. To overcome this contradiction, a grain cleaning machine with a sieve representing an inverted truncated cone that performs vertical vibrations is proposed. First, the parameters of the grain trajectory are determined after the first touch of the semiautomatic grain cleaning machine sieve when the sieve is in the lower position, and then when the sieve is in the upper position. Two variants of the grain trajectory are considered: when the grain trajectory after its collision with the sieve is in the vertical plane passing through the velocity vector, and when the grain trajectory after its collision with the sieve is in the vertical plane passing through the total force vector. The parameters of this interaction were determined by analyzing the grain path after the first sieving of a semi-automatic grain cleaning machine. Specific values of parameters are revealed, in particular, aggregate speed of the grain at the moment of its collision with the sieve, the angle between the grain velocity vector after reflection from the sieve and the horizontal, time of the grain's take-off over the sieve after falling on it, the range of the grain's flight over the sieve after falling. Based on the analysis, the conclusion is made: in order for the grain separation to be intensive, the sieve must move downwards with acceleration close to the acceleration of free fall.

On circular sections of a quadric surface

A brief overview of the history of conic sections is given. Circular sections of ellipsoids and hyperboloids with planes passing through the center of the surface are considered. In general, there are two such secant planes. Generalizing the concept that arose in rigid-body mechanics, a straight line passing through the center of an ellipsoid is called the Galois axis if the orthogonal plane intersects this ellipsoid along a circle. Let us consider the pencil of planes passing through the intermediate principal axis of a triaxial ellipsoid. Each section of an ellipsoid with such a plane is an ellipse, one of the axes of which coincides with the intermediate principal axis of the ellipsoid. When the secant plane rotates around the intermediate principal axis of the ellipsoid, the length of the other axis of the ellipse continuously changes, taking values between the lengths of the minor and major axes of the ellipsoid. Therefore, some such section is a circle whose diameter is the intermediate principal axis of the ellipsoid. A triaxial ellipsoid has two such sections. They transform into each other when mirrored relative to the plane passing through the intermediate and other principal axes of the ellipsoid. Both Galois axes are orthogonal to the intermediate principal axis of the triaxial ellipsoid, and for a non-sphere ellipsoid of rotation, both Galois axes coincide with one axis and are orthogonal to the other principal axes of the ellipsoid. A method for constructing Galois axes from the known principal axes of an ellipsoid is proposed. This construction serves as one of the natural examples of geometric problems. In addition, the Galois axis can be correctly defined not only for the ellipsoid (for which it was originally introduced), but also for some other classes of centrally symmetric surfaces, including hyperboloids.