Background: Formation and rupture of cerebral aneurysms may be related to certain types of configuration of the circle of Willis. Analysis of their interdependence can be of great importance. Methods: A group of 114 patients treated operatively for the cerebral aneurysm rupture and a group of 56 autopsied subjects were involved in the study. Four basic types of the circle of Willis configurations were formed–two symmetric types A and C, and two asymmetric types B and D. Results: A statistically significantly higher presence of asymmetry of the circle of Willis in the group of surgically treated subjects (p=0.006) with a significant presence of asymmetric Type B in this group (p=0.017) were determined. The presence of changes in the A1 segment in the group of subjects with solitary aneurysms on the anterior communicating artery showed a statistically significant presence in the group of autopsied subjects (p=0.0004). Analyzing the presence of symmetry of the circle of Willis between the two groups, that is, the total presence of symmetric types A and C indicated their statistically significant presence in the group of autopsied patients (p=0.043). Conclusion: Changes such as hypoplasia or aplasia of A1 and the resulting asymmetry of the circle of Willis directly affect the possibility of the rupture of cerebral aneurysms. Detection of the corresponding types of the circle of Willis after diagnostic examination can be the basis for the development of a protocol for monitoring such patients.

We all have in mind Einstein's famous thought experiment in the elevator where we observe the free fall of a body, and then the trajectory of a light ray. Simply here, in addition to the qualitative aspect, we carry out the exact calculation, and for the first time the worldlines equations are given. We consider a uniformly accelerated reference frame in rectilinear translation, and we show that the trajectories of the particles are semi-ellipses with the center on the event horizon. The frame of reference is non-inertial, the space-time is flat, and the computations are performed within the framework of special relativity. Some experimental consequences are discussed, especially the experiment with the accelerated Michelson-Morley interferometer is solved, and we described an experiment where a new relativistic paradox appears --- a particle of matter seems to go faster than light. The differences, compared to the classical case, are important at large scale and close to the horizon, but they are small in the lift where the interest is above all theoretical. The concepts of metric, coordinated velocity and horizon are discussed, and the analogy with the black hole is made.

This paper aims to introduce the notions of various types of Qc-contractions for which we establish some fixed circle and fixed disc theorems in the setting of metric spaces. Some illustrative examples are also provided to support our results. Moreover, we present some fixed circle and fixed disc results of integral type contractive self-mappings.

An adaptive spatial clustering (ASC) algorithm is proposed in this present study, which employs sweep-circle techniques and a dynamic threshold setting based on the Gestalt theory to detect spatial clusters. The proposed algorithm can automatically discover clusters in one pass, rather than through the modification of the initial model (for example, a minimal spanning tree, Delaunay triangulation or Voronoi diagram). It can quickly identify arbitrarily-shaped clusters while adapting efficiently to non-homogeneous density characteristics of spatial data, without the need of prior knowledge or parameters. The proposed algorithm is also ideal for use in data streaming technology with dynamic characteristics flowing in the form of spatial clustering in large data sets.

Moyamoya disease is also termed a spontaneous obstruction of the arteries in the circle of Willis. It is recognized as the leading cause of ischemic and hemorrhagic stroke. The infrequent cause of stroke is radiologically characterized by the continuous progression of stenosis of the end portion of the bilateral internal carotid artery and compensatory capillaries collaterals. A 1yr old child was admitted to the pediatric I.C.U with a complaint of decreased sensorium, decreased food intake and decreased right upper and lower extremity movements past two days. He had a history of convulsions at seven months of age. He was diagnosed with Moyamoya disease on Magnetic Resonance cerebral angiography. The clinical findings on MR-Angiography show multiple collaterals noted around Circle of Willis, specifically around the posterior communicating artery, anterior communicating artery, posterior cerebral artery, bilateral lenticulostriate, thalamoperforating arteries and medialfrontobasal arteries, which is suggestive of Moyamoya disease. Magnetic resonance imaging of the brain shows Gyral diffusion restriction in a right fronto-parieto-temporal region with mild swelling. The above clinical findings on MR-angiography show the Moyamoya disease in 1year old male child. These clinical findings on MR-Angiography and M.R.I – brain and neck can help the clinician accurately diagnose and manage this rare case.

In the book “Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe”, and further in the Article “Distance Between Two Circles in Any Number of Dimensions is a Vector Ellipse”, it has been mathematically demonstrated that “distance between points on any two different circles in any number of multiple dimensions” is equivalent to “distance of points on a vector ellipse from another fixed or moving point”. Using this mathematical methodology, a method is provided in this Article as a measure for the amount of interaction between two international economies, which are two countries or economic zones, by modelling each economy in terms of a “circle in four dimensions”. Based on this method, the proximity of the two economies at the end of each fiscal period, which is generally a fiscal year, is then measured by the distance between points at the end of the given period on the two circles in four dimensions, associated with these two economies in our model.

This article deals with special curves similar to those known in mathematics as Lamé curves. Their equations have a certain similarity with the equations of Lamé curves. But these curves are more like ellipses in appearance. Like ellipses, they have major and minor axes. These are such curves that their major and minor semi-axis coincide with an ellipse but are different than an ellipse. The main feature of these curves is that for any triangle whose base coincides with the axis of a given curve and the vertex is on a given curve, the condition aⁿ+bⁿ=cⁿ is obtained. The number “n” is called the degree or rank of this curve. To distinguish this group of curves, we called them Elbe curves. The equations of these lines are also provided in the paper.

The conceptualisation of happiness varies across different cultures. In Chinese culture, happiness is not only inclusive of oneself, but also hinges on the people is related to and society. Chinese social development have influenced psychological traditionality and modernity and individual's happiness. In Experienment1, 450 participants were recruited and surveyed to examine the levels of happiness amongst Chinese undergraduates with diverse cultural values. The results showed that individuals with both traditional and modern psychological orientations had high levels of happiness. Those with higher psychological traditionality showed more satisfaction with family relationships and social happiness, while those with higher psychological modernity demonstrated a higher level of relational happiness. In Experienment2, 280 participants were recruited, and a "happiness token distribution" and "happiness importance survey" were employed to draw a 'happiness circle' for Chinese undergraduates. The results revealed that those with a higher degree of psychological traditionality assigned a greater proportion of their happiness to ascribed relationships. In contrast, those with a higher degree of psychological modernity allocated a larger share of their happiness to interactive relationships and social relationships. Individuals with lower values of both traditionality and modernity perceived themselves as the most important component in their happiness circle.

Iterations of continuous maps are the simplest models of generic dynamical systems. In particular, circle maps display several key properties of complex dynamics, such as phase-locking and the quasi-periodicity route to chaos. Our work points out that Planck’s constant may be derived from the scaling behavior of circle maps in the asymptotic limit.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

In the paper, the author finds bounds of the sine and cosine along a circle on the complex plane in terms of two double inequalities for the norms of the sine and cosine along a circle on the complex plane.

The application of 3D LiDAR technology has become increasingly extensive in tunnel monitoring due to the large density and high accuracy of the acquired spatial data. The proposed processing method aims at circle tunnels and provides a clear workflow to automatically process raw point data and easily interpretable results to analyze tunnel health state. The proposed automatic processing method employs a series of algorithms to extract point cloud of a single tunnel segment without obvious noise from entire raw tunnel point cloud mainly by three steps: axis acquisition, segments extraction and denoising. Tunnel axis is extracted by fitting boundaries of the tunnel point cloud rejection in plane with RANSAC algorithm. With guidance of axis, the entire preprocessed tunnel point cloud is segmented by equal division to get a section of tunnel point cloud which corresponds to a single tunnel segment. Then the noise in every single point cloud segment is removed by clustering algorithm twice, based on distance and intensity. Finally, clean point clouds of tunnel segments are processed by effective deformation extraction processor to get ovality and three-dimensional deformation nephogram.

During tensile testing of materials, strain measurement is conducted using either contact or non-contact methods. Contact methods offer high accuracy and precision but are limited by the specimen's thickness and dimensions, whereas non-contact methods minimize damage to thin specimens and allow measurements in various environments, though they require longer prep-aration and calculation times. This paper proposes a circular grid marking pattern and a strain prediction algorithm using artificial intelligence (AI), which simplifies the preparation process and allows strain prediction without additional equipment. The circular grid pattern can be ar-ranged in various configurations from 1×5 to 5×7, and a laser marker, which requires minimal time, was used to engrave the pattern on the specimen to shorten the preparation time. The AI model, trained on image-based data, enables strain calculation regardless of the specimen's gauge length and size, and allows measurement of local strain as well as gauge-length strain. The reliability of this concept was verified by applying it to tensile testing.

Haemodynamic simulations are increasingly used to study vascular diseases like Intracranial Aneurysms (IA) and to further develope treatment options. However, due to limited data, some aspects must rely on heuristics, especially at the simulation’s distal ends. In literature, Murray’s Law is often used to model the outflow split based on vessel cross-section area, but this poses challenges for the communicating arteries in the Circle of Willis (CoW). In this study, we contribute by assessing the impact of Murray’s Law in patient-specific geometries featuring IA at the posterior communication. We simulate different domain extensions, representing common modelling choices. We establish Full CoW simulations as a baseline to evaluate the effect of these modelling assumptions on haemodynamic indicators, focusing on IA growth and rupture-related factors like Wall Shear Stress (WSS) and Oscillatory Shear Index (OSI). Our findings reveal qualitative alterations in haemodynamics when not modeling posterior communication. Comparisons between computing the anterior circulation and computing the whole Circle of Willis reveal quantitative changes in WSS may reach up to 80%, highlighting the significance of modelling choices in assessing IA risks and treatment strategies.

Taiwan is the world’s leading country regarding tuna longline fisheries; the combined catches with those from Japan account for more than half of the global total. However, the bycatch of sea turtles, seabirds, and dolphins in longline fisheries has attracted international concern. Regarding sea turtle conservation, the use of circle hooks in longline fisheries will inevitably become a trend. Meanwhile, Taiwanese fishermen are concerned whether the use of circle hooks will have a negative impact on the catch efficiency of target species. This study was conducted to compare the catch efficiency of circle and tuna hooks by the Taiwanese commercial tuna longline fishing vessels in the eastern Pacific Ocean. The results indicated that if the effects of immersion time were considered, the catch rates of all groups were the same between 4.2-sun circle and tuna hooks. However, the catch rate of 4.0-sun circle hooks was significantly higher than the two types of 4.2-sun circle and tuna hooks. Regarding survival rates, fish caught with circle hooks showed higher survival rates than tuna hooks for total commercial fish, tunas and yellowfin tuna (Thunnus albacares). However, the survival rate was only determined at haulback; the relationship between survival rate and immersion time still needs further investigation. Regarding catch sizes, there were no significant differences in the sizes of bigeye (Thunnus obesus) and yellowfin tuna between circle and tuna hooks. But based solely on mean weight, the value of yellowfin tuna caught using circle hooks may be higher than those using tuna hooks. In summary, the circle hooks used in this study did not show a negative impact on the catch efficiency of Taiwanese longline fisheries, and could show a superior performance than that of tuna hooks. These results could be actively promoted in the industry through the education and training of fishermen; thus enabling Taiwan to comply with international sea turtle conservation trends.

Students have historically fared poorly on the West African Examinations Council’s (WAEC) mathematics exam. This study investigated senior high school students’ difficulties with studying Geometry in the Cape Coast Metropolis, Central Region, Ghana. Using the descriptive survey design, data were collected from 300 Senior High School Two (SHS 2) students using simple random sampling. Students responded to the questionnaire and a 30-item multiple-choice achievement test on geometry. Four geometry concepts students perceived difficult to study are Circle theorems, Perpendicularity of Tangent and radius of a circle, Angle between Tangent and a chord and Tangents from an external point. The reasons for the difficulty in studying geometry topics are the unavailability of instructional materials, the geometry topics being complex, and teachers not teaching geometry concepts to the students’ understanding. Students’ gender did not influence the learning of concepts in geometry at a 0.05 level of significance. There was an insignificantly weak positive correlation between the students’ performance and their perceived difficult geometry concepts. It was concluded that the Ministry of Education does not provide relevant instructional materials to the various schools to teach geometry concepts. Schools do not organise workshops and seminars for teachers on effectively teaching geometry concepts in mathematics.

Green-space resources, in the context of urbanisation, cannot meet the actual needs of residents well, and the study of the balance of green-space resource allocation from the relationship between supply and demand is an urgent problem to be solved. This study quantitatively evaluates the green-space supply level from four dimensions by constructing a framework for assessment in the community life circle. It also evaluates the matching of green space supply and demand, resource distribution fairness, and distribution equilibrium under group differentiation through the supply–demand coupling matrix, Gini coefficient, and the Kruskal–Wallis H non-parametric rank-sum test, respectively. This study shows that (1) a significant spatial imbalance exists in green-space resource allocation in community life circles in different dimensions. (2) The comprehensive green-space supply level in the community life circle matched the total demand of residents to a low degree. (3) There was significant inequality in green-space resource allocation within the community life circle (the Gini coefficient of each evaluation perspective was > 0.4). Based on the study results, we were able to identify community life circles with spatial mismatches, different supply and demand, and other green-space resource allocation problems, which is of great significance to urban green-space research and planning practice under the "green justice" framework.

Quite a number of new metered parking systems have been introduced in recent years by the local authorities of various spa towns in Poland in connection with home zone conversion projects. The traffic signs posted in these locations were limited to the beginning and end of the demarcated parking area. Traffic circle (TC) is an example of traffic calming measures (TCM) used in home zones to slow down the traffic (case study – home zone in a small spa village). This article presents the results of a study investigating the speed reductions obtained with a home zone and traffic circle used as traffic calming measures. The indispensable speed surveys were carried out in relation to this study in two periods: in summer when the streets are crowded with tourists and in September with little pedestrian traffic. Two research hypotheses were formulated as part of the speed data analysis to verify the slowing effect of traffic circle and the relevance of traffic circles design parameters and location, road function and the surrounding streetscape. For each hypothesis statistical analyses were carried out using two nonparametric tests: two-sample Kolmogorov–Smirnov test and median test. The third research hypothesis formulated in this study was related to sustainable development factors related to fuel consumption and traffic-related air pollution including carbon dioxide, carbon monoxide, nitrogen oxide and hydrocarbons. This hypothesis was verified by estimating the amount of air pollution in the home zone under analysis in three different situations (scenarios): in summer with the travel speed reduced by pedestrian traffic to ca. 8–10 km/h and in September with a small number of pedestrians and 20–25 km/h resulting speed between traffic circles, reduced at traffic circle, and in a theoretical 30 km/h zone with 25–30 km/h assumed speed between traffic circles, dropping at traffic circle. These analyses confirmed the appropriateness of traffic circle as a home zone traffic calming measure, as long as its design is based on a detailed analysis of the relevant factors, including location, road function and the surrounding streetscape.

In the extended malfatti problem, the numbers of circles in the triangle are triangle numbers, because the tangency properties among the internal circles and 3 sides of the triangle have special type of structure. That is, the corner circle tangent to two sides of the triangle and two boundary circles, the boundary circles tangent to one side of the triangle and 4 other circles, and the inner circles always tangent to 6 other circles. The circles we find in extended general Malfatti’s problem has the property that the smallest radius and the largest radius for the circle has big difference. In this paper, we proposed algorithms to solve the problem that the tangency properties among circles and sides of triangle is not fixed, so that the number of circles in the triangle is not necessary a triangle numbers. The purpose of this change is try to find the radii of the circles in the triangle is within a small range.

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other. According to Kepler’s 1^st Law, “orbit of a planet with respect to the Sun is an ellipse, with the Sun at one of the two foci.” Orbit of the Moon with respect to Earth is also distinctly elliptical, but this ellipse has a varying eccentricity as the Moon comes closer to and goes farther away from the Earth in a harmonic style along a full cycle of this ellipse. In this paper, our research results are summarized, where it is first mathematically shown that the “distance between points around any two different circles in three dimensional space” is equivalent to the “distance of points around a vector ellipse to another fixed or moving point, as in two dimensional space”. What is done is equivalent to showing that bodies moving on two different circular orbits in space vector wise behave as if moving on an elliptical path with respect to each other, and virtually seeing each other as positioned at an instantaneously stationary point in space on their relative ecliptic plane, whether they are moving with the same angular velocity, or different but fixed angular velocities, or even with different and changing angular velocities with respect to their own centers of revolution. This mathematical revelation has the potential to lead to far reaching discoveries in physics, enabling more insight into forces of nature, with a formulation of a new fundamental model regarding the motions of bodies in the Universe, including the Sun, Planets, and Satellites in the Solar System and elsewhere, as well as at particle and subatomic level. Based on the demonstrated mathematical analysis, as they exhibit almost fixed elliptic orbits relative to one another over time, the assertion is made that the Sun, the Earth, and the Moon must each be revolving in their individual circular orbits of revolution in space. With this expectation, individual orbital parameters of the Sun, the Earth, and the Moon are calculated based on observed Earth to Sun and Earth to Moon distance data, also using analytical methods developed as part of this research to an approximation. This calculation and analysis process have revealed additional results aligned with observation, and this also supports our assertion that the Sun, the Earth, and the Moon must actually be revolving in individual circular orbits.

In this paper, we present a complete, flexible and safe convex-optimization-based method to solve speed planning problems over a fixed path for autonomous driving in both static and dynamic environments. Our contributions are five fold. First, we summarize the most common constraints raised in various autonomous driving scenarios as the requirements for speed planner developments and metrics to measure the capacity of existing speed planners roughly for autonomous driving. Second, we introduce a more general, flexible and complete speed planning mathematical model including all the summarized constraints compared to the state-of-the-art speed planners, which addresses limitations of existing methods and is able to provide smooth, safety-guaranteed, dynamic-feasible, and time-efficient speed profiles. Third, we emphasize comfort while guaranteeing fundamental motion safety without sacrificing the mobility of cars by treating the comfort box constraint as a semi-hard constraint in optimization via slack variables and penalty functions, which distinguishes our method from existing ones. Fourth, we demonstrate that our problem preserves convexity with the added constraints, thus global optimality of solutions is guaranteed. Fifth, we showcase how our formulation can be used in various autonomous driving scenarios by providing several challenging case studies in both static and dynamic environments. A range of numerical experiments and challenging realistic speed planning case studies have depicted that the proposed method outperforms existing speed planners for autonomous driving in terms of constraint type covered, optimality, safety, mobility and flexibility.

Discrete Double Hilbert Exponential sums along polynomials. This is preprint version (not peer reviewed) And we now check the proof of (ii) of Lemma 3.2 whether it is really true.

In the Brazilian Amazon, rural settlements are increasingly isolated by large-scale production farms, jeopardizing their sustainability and the good living of family farmers. Works were carried out in settlements to measure sustainability. However, the majority does not consider the participation and the collectively of those involved. In this way, we propose to evaluate, in a collective and participatory way, the sustainability and good living of the SDP São Paulo Rural Settlement, of the northern Amazon of Mato Grosso. We used the didactic-pedagogical method Circle of Sustainability, developed from five points: 1st - circle of investigation of generating themes; 2nd - circle of the history of the subject world; 3rd - circle of diagnosis of rural settlements; 4th - circle of exchange of knowledge; and 5th - circle of sustainable perceptions and narratives. The historical, socioeconomic and cultural characterization of the settlement allowed us to understand how sustainability and good living are being built in the settlement history process. Sustainability and good living are dialectical processes, are under construction, in movement.

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other, whereas objects follow parabolic escape orbits while moving away from Earth and bodies asserting a gravitational pull, and some comets move in near-hyperbolic orbits when they approach the Sun. In this article, it is first mathematically proven that the “distance between points on any two different circles in three-dimensional space” is equivalent to the “distance of points on a vector ellipse from another fixed or moving point, as in two-dimensional space.” Then, it is further mathematically demonstrated that “distance between points on any two different circles in any number of multiple dimensions” is equivalent to “distance of points on a vector ellipse from another fixed or moving point”. Finally, two special cases when the “distance between points on two different circles in multi-dimensional space” become mathematically equivalent to distances in “parabolic” or “near-hyperbolic” trajectories are investigated. Concepts of “vector ellipse”, “vector hyperbola”, and “vector parabola” are also mathematically defined. The mathematical basis derived in this Article is utilized in the book “Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe” in asserting a new Circular Orbital Model for moving bodies in the Universe, leading to further insights in Astrophysics.

The astro-structures with the shape of comet and semi-circle, are found in the electronegative inductively coupled plasmas (ICPs), e.g., Ar/O2 and Ar/SF6. They are formed by means of the self-coagulation theory. This theory is built and based on two-dimensional fluid model simulation dynamic data of the above ICP sources. Concretely, quasi-Helmholtz equation is constructed through the free diffusion transport component and a novel negative chemical source term. The absolute value of recombination rate is higher than the attachment rate and so the net source of simulated anion is negative. This is quite a new chemical phenomenon in the electronegative and gaseous discharging plasmas when compared to the previously reported normal positive chemical source that will generate ambi-polar diffusion potential. In electronegative plasma, the self-coagulation is ambi-polar to sustain the electrical neutrality. Besides, when the anions are accelerated by ambi-polar potential before self-coagulated, the self-coagulation is again advective which leads to the formation of blue sheath. At the ambi-polar self-coagulation of high electronegativity, the electrons dynamics are relatively decoupled from ions and meanwhile they have their own relatively weak, spontaneous, and monopolar self-coagulation in the periphery of astro-structure. At the ambi-polar self-coagulation, both the anion and cation are individually treated as mass point models that carry charge (point charge models, more precisely). When the charge amounts of two individual astr-structures, e.g., semi-circle SF3+ and Ar+, are equal, the coagulated profile of heavy species SF3+ collapses by the expelling effect of Coulomb force that exists between the two point-charges. This is an anti-collective behavior of astro-structure, instead of the collective interaction of plasma, since the collapse of coagulated heavy species creates spatially dispersed charge density that cannot be shielded. The simulation shows that the lighter the species, the easier it self-coagulates, determined by the inertial effect of density quantity. The discharge picture of electronegative plasma is combined by a dynamic continuum transport (e.g., forming the ellipse background) and the mechanical balance contained in the static self-coagulation process (proven by dimensional analysis of continuity equation). The popular astro-structure of electronegative ICP source awaits urgently for the validation of experiment.