Search

In this work, an absolute method to calibrate an α-spectrometer is proposed taking into account the Source-to-Detector, and lateral distances due to eccentric source distribution. An analytical formula to calibrate an α-spectrometer is derived and the Simpson's integration method was utilized to solve these equations in its integral form numerically using a written C computer code. The general Monte Carlo N-particle code, MCNP as well as experimental measurements for some standard α-emitters are used to benchmark the proposed method. An agreement was found between the efficiency results calculated by MC and the proposed method with a maximum relative difference of about 0.5%. While, experimental measurement of α-emitters activity employing the proposed method differs by about 1.65% from the certified values. Accounting for the man made error allows to accurately quantify the assayed sample. Therefore, the inaccuracy in the efficiency results due to non-accurate inputs pertained to the source, and detector radii, Source–Detector distances, and eccentric source distribution are investigated in the Source-to-Detector distance range of (4 to 44 mm). The results show that a difference of ±1% in the detector radius, and Source-to-Detector distance than the normal values yields a relative difference of about ±2%, while a difference of ±50% in the source radius or source lateral distance from detector symmetry axis could only yields inaccuracy of less than ±3% in the efficiency results.

Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood. Around 1997, Bárány asked whether for all convex d-polytopes P and all 0 ≤ k ≤ d - 1 , f k (P) ≥ min { f 0 (P) , f d - 1 (P) } . We answer Bárány's question in the affirmative and prove a stronger statement: for all convex d-polytopes P and all 0 ≤ k ≤ d - 1 , f k (P) f 0 (P) ≥ 1 2 [ ⌈ d 2 ⌉ k + ⌊ d 2 ⌋ k ] , f k (P) f d - 1 (P) ≥ 1 2 [ ⌈ d 2 ⌉ d - k - 1 + ⌊ d 2 ⌋ d - k - 1 ]. In the former, equality holds precisely when k = 0 or when k = 1 and P is simple. In the latter, equality holds precisely when k = d - 1 or when k = d - 2 and P is simplicial. [ABSTRACT FROM AUTHOR]

In the plane, the winding number is the number of times a curve wraps around a given point. Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven valuable for geometry processing tasks ranging from surface reconstruction to mesh booleans. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. By processing the derivatives of such functions, we can robustly filter out components of the input that do not bound any region. Ultimately, our algorithm yields (i) a closed, completed version of the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region. The main computational cost is solving a standard Poisson equation, or for surfaces with nontrivial topology, a sparse linear program. We also introduce special basis functions to represent singularities that naturally occur at endpoints of open curves. [ABSTRACT FROM AUTHOR]

Intracardiac electrograms (iEGMs) are time traces of the electrical potential recorded close to the heart muscle. We calculate unipolar and bipolar iEGMs analytically for a myocardial slab with parallel myofibers and validate them against numerical bidomain simulations. The analytical solution obtained via the method of mirrors is an infinite series of arctangents. It goes beyond the solid angle theory and is in good agreement with the simulations, even though bath loading effects were not accounted for in the analytical calculation. At a large distance from the myocardium, iEGMs decay as 1/R (unipolar), 1/R² (bipolar and parallel), and 1/R³ (bipolar and perpendicular to the endocardium). At the endocardial surface, there is a mathematical branch cut. Here, we show how a thicker myocardium generates iEGMs with larger amplitudes and how anisotropy affects the iEGM width and amplitude. If only the leading-order term of our expansion is retained, it can be determined how the conductivities of the bath, torso, myocardium, and myofiber direction together determine the iEGM amplitude. Our results will be useful in the quantitative interpretation of iEGMs, the selection of thresholds to characterize viable tissues, and for future inferences of tissue parameters. [ABSTRACT FROM AUTHOR]

Crystal structure prediction is demanding in two senses: this is a difficult problem, both experimentally and computationally, yet we want to find new materials with new properties that take unknown crystal structures. Using statistical information on the local environment of already reported stable crystal structures would be helpful when, for instance, finding new crystal structures through visually adding atoms sequentially to a cluster of atoms, which is an intuitive procedure that does not require substantial computational resources. This study conducted hierarchical classification of frequently appearing structure types in the ICSD database by comparing the occurrence ratios of H, B, C, N, O, F, and transition metals (TM) in binary, ternary, and quaternary crystals. The bond angle and solid angle distributions of Delaunay polyhedra with atoms as vertices were considered as descriptors of the local environment for each structure type. Hierarchical clustering of structure types based on bond angle or solid angle distributions revealed that the structure types can be broadly categorized into those favoring TM oxides, having low TM occurrence, and having high TM occurrence. Our results indicate that the anion, or lack of, plays a key role in determining the crystal structure type, and structure types strongly preferred by TM oxides can accommodate a broad range of angles.

Future automotive interior lighting might have the potential to go beyond decorative purposes by influencing alertness, circadian physiology, and sleep. As the available space in the interior of an automobile for lighting applications is limited, understanding the impact of various luminous surface sizes on non-image-forming effects is fundamental in this field. In a laboratory study using a within-subject design, 18 participants were exposed to two bright light conditions with different solid angles and one dim light condition in a balanced, randomized order during the course of the evening. Our results demonstrate that both light conditions significantly increased subjective alertness and reduced salivary melatonin concentration but not cognitive performance compared to dim light. The solid angle of light exposure at constant corneal illuminance only affected visual comfort. While subjective alertness can be increased and melatonin can be attenuated with rather small luminaires, larger solid angles should be considered if visual comfort is a priority.

The random beta polytope is defined as the convex hull of n independent random points with the density proportional to (1 - ‖ x ‖ 2 ) β on the d-dimensional unit ball, where β > - 1 is a parameter. Similarly, the random beta' polytope is defined as the convex hull of n independent random points with the density proportional to (1 + ‖ x ‖ 2 ) - β on R d , where β > d 2 . In a previous work (Kabluchko, Adv. Math. 380:107612, 2021), we established exact and explicit formulae for the expected f-vectors of these random polytopes in terms of certain definite integrals. In the present paper, we use purely algebraic manipulations to derive several identities for these integrals which yield alternative formulae for the expected f-vectors. Similar algebraic manipulations apply to Stirling numbers and yield the following identity: ∑ s = 0 k n - s d - s (d - s) d - s k - s = ∑ s = 0 k (- 1) s n - s d d + 1 k - s = ∑ s = 0 d - k (- 1) s n + 1 d - s d - s k. [ABSTRACT FROM AUTHOR]

Intracardiac electrograms (iEGMs) are time traces of the electrical potential recorded close to the heart muscle. We calculate unipolar and bipolar iEGMs analytically for a myocardial slab with parallel myofibers and validate them against numerical bidomain simulations. The analytical solution obtained via the method of mirrors is an infinite series of arctangents. It goes beyond the solid angle theory and is in good agreement with the simulations, even though bath loading effects were not accounted for in the analytical calculation. At a large distance from the myocardium, iEGMs decay as 1/R (unipolar), 1/R2 (bipolar and parallel), and 1/R3 (bipolar and perpendicular to the endocardium). At the endocardial surface, there is a mathematical branch cut. Here, we show how a thicker myocardium generates iEGMs with larger amplitudes and how anisotropy affects the iEGM width and amplitude. If only the leading-order term of our expansion is retained, it can be determined how the conductivities of the bath, torso, myocardium, and myofiber direction together determine the iEGM amplitude. Our results will be useful in the quantitative interpretation of iEGMs, the selection of thresholds to characterize viable tissues, and for future inferences of tissue parameters.

This paper presents the design of an arc-shaped near-isotropic self-complementary antenna for spectrum sensing application. An arc-shaped dipole with horizontal and vertical arms is used to achieve a near isotropic radiation pattern. The radiation pattern improved by adjusting the horizontal and vertical arm lengths. Simulated and experimental results show that the proposed antenna has an impedance bandwidth of 146% (2.4–18.4 GHz) for VSWR ≤ 2 with a good radiation pattern. In order to quantify the antenna performance, antenna gain variation, bandwidth, efficiency, and size have been compared with previously reported designs. It is shown that the proposed arc-shaped antenna can achieve nearly isotropic radiation patterns with a maximum radiation efficiency of 92%. The isotropic performance of the antenna has been characterized by observing the radiation pattern and solid angle. The FR4 substrate is used as a dielectric with relative permittivity 4.4 and loss tangent of 0.02. ( ε r = 4.4, h = 1.6 mm) The simulated and measured results are in good comparison, and the proposed design is a suitable candidate for spectrum sensing. [ABSTRACT FROM AUTHOR]

Let \mathcal Z_d be the zero cell of a d-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k-dimensional faces of \mathcal Z_d, as d\to \infty. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to \sqrt {2\pi /3}\, d^{3/2}, as d\to \infty. We also prove that the expected solid angle of a random cone spanned by d random vectors that are independent and uniformly distributed on the unit upper half-sphere in \mathbb R^{d} is asymptotic to \sqrt 3 \pi ^{-d}, as d\to \infty. [ABSTRACT FROM AUTHOR]

Future automotive interior lighting might have the potential to go beyond decorative purposes by influencing alertness, circadian physiology, and sleep. As the available space in the interior of an automobile for lighting applications is limited, understanding the impact of various luminous surface sizes on non-image-forming effects is fundamental in this field. In a laboratory study using a within-subject design, 18 participants were exposed to two bright light conditions with different solid angles and one dim light condition in a balanced, randomized order during the course of the evening. Our results demonstrate that both light conditions significantly increased subjective alertness and reduced salivary melatonin concentration but not cognitive performance compared to dim light. The solid angle of light exposure at constant corneal illuminance only affected visual comfort. While subjective alertness can be increased and melatonin can be attenuated with rather small luminaires, larger solid angles should be considered if visual comfort is a priority. [ABSTRACT FROM AUTHOR]

Three structurally diverse 5-phenyltetrazolato (Tz) Ti, Zr, and Ta complexes, namely, (C 2 H 8 N)[Ti 2 (C 7 H 5 N 4 ) 5 (C 2 H 6 N) 4 ]·1.45C 6 H 6 or (Me 2 NH 2 )[Ti 2 (NMe 2 ) 4 (2,3-μ-Tz) 3 (2-η 1 -Tz) 2 ]·1.45C 6 H 6 , (1·1.45C 6 H 6 ), [Zr 2 (C 7 H 5 N 4 ) 6 (C 2 H 6 N) 2 (C 2 H 7 N) 2 ]·1.12C 6 H 6 ·0.382CH 2 Cl 2 or [Zr 2 (Me 2 NH) 2 (NMe 2 ) 2 (2,3-μ-Tz) 3 (2-η 1 -Tz) 2 (1,2-η 2 -Tz)]·1.12C 6 H 6 ·0.38CH 2 Cl 2 (2·1.12C 6 H 6 ·0.38CH 2 Cl 2 ), and (C 2 H 8 N) 2 [Ta 2 (C 7 H 5 N 4 ) 8 (C 2 H 6 N) 2 O]·0.25C 7 H 8 or (Me 2 NH 2 ) 2 [Ta 2 (NMe 2 ) 2 (2,3-μ-Tz) 2 (2-η 1 -Tz) 6 O]·0.25C 7 H 8 (3·0.25C 7 H 8 ), where TzH is 5-phenyl-1H-tetrazole, have been synthesized and structurally characterized. All three complexes are dinuclear; the Ti center in 1 is six-coordinate, whereas the Zr and Ta atoms in 2 and 3 are seven-coordinate. The coordination environments of the Ti centers in 1 are similar, and so are the ligations of the Ta centers in 3. In contrast, the two Zr centers in 2 bear a different number of ligands, one of which is a bidentate η 2 -5-phenyltetrazolato ligand that has not been observed previously for d-block elements. The dimethylamido ligand, present in the starting materials, remained unchanged, or was converted to dimethylamine and dimethylammonium during the synthesis. Dimethylamine coordinates as a neutral ligand, whereas dimethylammonium is retained as a hydrogen-bonded entity bridging Tz ligands., (open access.)

In the field of computer-aided design (CAD) and three-dimensional (3D) modeling, constructive solid geometry (CSG) representations based on primitive 3D shapes and boundary representations (B-Rep) based on geometry and topology are widely used to represent complex shapes. Therefore, it is important to recognize primitive shapes such as cubes, cones, and cylinders and to accurately judge and classify the deformation of primitive shapes. For this purpose, various techniques have been studied, such as a vector-based determination method, a determination method using multiple images from various angles, and a determination method based on positional relationships between points. However, because large datasets are required to classify these shapes and it is difficult to respond to changes in shape due to rotation, the resulting recognition accuracy is not always high. In this work, we propose a method based on solid angles, which do not depend on the positional relationship of vectors, viewpoints, or changes due to rotation, as feature quantities. We demonstrate the effectiveness of primitive 3D figures using features based on solid angles. In addition, we show that the presence or absence of deformation can be determined when part of a primitive 3D figure is deformed. [ABSTRACT FROM AUTHOR]

In this paper we compared two different solid angle formulations to determine the amount of cavitation potential energy that is transfer from a source to a surface. The first formulation used an analytical expression for the solid angle of a planar triangle while the second formulation used a fully continuous form of the solid angle projection approach which represents the surface specific impact power at a point location. The comparison was conducted on a NACA 0018-45 hydrofoil. A block-structured computational mesh was created in ICEM CFD. Transient simulations were performed in Ansys CFX where the SST turbulence model with Reboud's correction was used. For the cavitation modelling the Schnerr-Sauer cavitation model with the recommended values was used. Result from Ansys CFX were imported into Matlab where the potential power applied to the surface was calculated for both formulations. The results of the study show that both formulations work well and predict similar locations where erosion occurred, the difference is in the absolute values which are significantly higher in the formulation that used the fully continuous form of the solid angle. [ABSTRACT FROM AUTHOR]

Virtual reality is increasingly used in medicine for diagnostics, for visualisation of complex structures and for preoperative planning. In interventional radiology, minimally invasive approach could be described with a target point representing the desired needle tip position and an array of all possible trajectories leading to it resembling irregular “cone” or “pyramid”. We present a pilot study of planning a minimally invasive posterior infrazygomatic and suprazygomatic approaches into the pterygopalatine fossa using a solid angle as a measure of size of the approach in five virtually reconstructed heads. The minimally invasive approaches were planned by manually drawing the edges of “pyramids” that described each approach in 3D using virtual reality program Tracer. For each head, a transverse diameter was measured and for each approach a solid angle size, average edge length and estimated area on the skin from where the target point could be reached were calculated. We found that, the solid angle of posterior infrazygomatic approach was significantly larger than suprazygomatic approach (p0.001). Furthermore, the transverse head diameter and solid angle in posterior infrazygomatic approach were negatively correlated (ρ=-0.55, p=0.0002), while transverse head diameter and the estimated area on the skin from where the target point could be reached in the suprazygomatic approach were positively correlated (ρ=0.37, p=0.0206). In conclusion, our findings provide important preliminary evidence on the feasibility of evaluating and comparing different minimally invasive approaches using virtual reality systems, and affirm the validity of solid angle as a measure of the size of the approach.

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements. In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners. Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown. We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions. • Constant solid angle surfaces enable the control of the slope at the boundary. • Points on a constant solid angle surfaces can be found without a mesh. • The solid angle is dimensionless and therefore useful with varying spans. • Enables surface grids in which planar quadrilaterals meet the surface boundaries. • Can be used as Airy stress functions in form finding to direct forces towards corners. [ABSTRACT FROM AUTHOR]

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace. [ABSTRACT FROM AUTHOR]

We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index d - k when also k tends to infinity. [ABSTRACT FROM AUTHOR]

The Daurian Redstart Phoenicurus auroreus, formerly considered a winter resident in Japan, is expanding its breeding area. Although this species was originally a cavity-nesting bird, it has been selecting nesting sites in exhaust fan rain covers, with wider openings than birdhouses and mailboxes. To understand the reasons, we compared two types of exhaust fan rain covers, one that birds used and one that they did not, focusing on the view from the nest. As a result, the species was found to use exhaust fan rain covers with significantly smaller vertical viewing angles or solid angles, regardless of the size of the opening. In addition, exhaust fan rain covers had the characteristically narrow fields of view from the nest, making it difficult for potential predators to enter the field of view. Therefore, we suggest that the Daurian Redstart used exhaust fan rain covers as nesting sites to avoid detection by predators. Investigating the field of view from the nest using a solid angle is useful for evaluating the nesting site selection patterns. [ABSTRACT FROM AUTHOR]

The geometrical efficiency of a source-to-detector configuration is considered to be necessary in the calculation of the full energy peak efficiency, especially for NaI(Tl) and HPGe gamma-ray spectroscopy detectors. The geometrical efficiency depends on the solid angle subtended by the radioactive sources and the detector surfaces. The present work is basically concerned to establish a new mathematical approach for calculating the solid angle and geometrical efficiency, based on conversion of the geometrical solid angle of a non-axial radioactive point source with respect to a circular surface of the detector to a new equivalent geometry. The equivalent geometry consists of an axial radioactive point source with respect to an arbitrary elliptical surface that lies between the radioactive point source and the circular surface of the detector. This expression was extended to include coaxial radioactive circular disk source. The results were compared with a number of published data to explain how significant this work is in the efficiency calibration procedure for the γ-ray detection systems, especially in case of using isotropic radiating γ-ray sources in the form of point and disk shapes.

In this paper, we present a straightforward analytical method for calculating the efficiencies of an ovoid shape (elliptical cylindrical) detector by using a randomly located point source. To determine the activity of an unknown radioactive source, absolute efficiency is required and we should take into account the attenuation of the gamma-ray photons by the source container and the detector housing materials. The soundness of the resultant straightforward analytical method was successfully confirmed by comparison with some published data.

This paper presents the design of an arc-shaped near-isotropic self-complementary antenna for spectrum sensing application. An arc-shaped dipole with horizontal and vertical arms is used to achieve a near isotropic radiation pattern. The radiation pattern improved by adjusting the horizontal and vertical arm lengths. Simulated and experimental results show that the proposed antenna has an impedance bandwidth of 146% (2.4–18.4 GHz) for VSWR ≤ 2 with a good radiation pattern. In order to quantify the antenna performance, antenna gain variation, bandwidth, efficiency, and size have been compared with previously reported designs. It is shown that the proposed arc-shaped antenna can achieve nearly isotropic radiation patterns with a maximum radiation efficiency of 92%. The isotropic performance of the antenna has been characterized by observing the radiation pattern and solid angle. The FR4 substrate is used as a dielectric with relative permittivity 4.4 and loss tangent of 0.02. (εr = 4.4, h = 1.6 mm) The simulated and measured results are in good comparison, and the proposed design is a suitable candidate for spectrum sensing.

Consider a random simplex [ X 1 , ... , X n ] defined as the convex hull of independent identically distributed (i.i.d.) random points X 1 , ... , X n in R n - 1 with the following beta density: Let J n , k (β) be the expected internal angle of the simplex [ X 1 , ... , X n ] at its face [ X 1 , ... , X k ] . Define J ~ n , k (β) analogously for i.i.d. random points distributed according to the beta ′ density f ~ n - 1 , β (x) ∝ (1 + ‖ x ‖ 2 ) - β , x ∈ R n - 1 , β > (n - 1) / 2. We derive formulae for J n , k (β) and J ~ n , k (β) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of β . For J n , 1 (± 1 / 2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K n , d : = [ U 1 , ... , U n ] where U 1 , ... , U n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K n , d : lim n → ∞ n - (d - 1) / (d + 1) E f (K n , d) = c d · Ω (K) , where Ω (K) is the affine surface area of K, and c d is an unknown vector not depending on K. We compute c d explicitly in dimensions up to d = 10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere. [ABSTRACT FROM AUTHOR]

Virtual reality is increasingly used in medicine for diagnostics, for visualisation of complex structures and for preoperative planning. In interventional radiology, minimally invasive approach could be described with a target point representing the desired needle tip position and an array of all possible trajectories leading to it resembling irregular "cone" or "pyramid". We present a pilot study of planning a minimally invasive posterior infrazygomatic and suprazygomatic approaches into the pterygopalatine fossa using a solid angle as a measure of size of the approach in five virtually reconstructed heads. The minimally invasive approaches were planned by manually drawing the edges of "pyramids" that described each approach in 3D using virtual reality program Tracer. For each head, a transverse diameter was measured and for each approach a solid angle size, average edge length and estimated area on the skin from where the target point could be reached were calculated. We found that, the solid angle of posterior infrazygomatic approach was significantly larger than suprazygomatic approach (p<0.001). Furthermore, the transverse head diameter and solid angle in posterior infrazygomatic approach were negatively correlated (p=-0.55, p=0.0002), while transverse head diameter and the estimated area on the skin from where the target point could be reached in the suprazygomatic approach were positively correlated (p=0.37, p=0.0206). In conclusion, our findings provide important preliminary evidence on the feasibility of evaluating and comparing different minimally invasive approaches using virtual reality systems, and affirm the validity of solid angle as a measure of the size of the approach. [ABSTRACT FROM AUTHOR]

Purpose: The concept of 4πc radiotherapy is a radiotherapy planning technique receiving much attention in recent times. The aim of this article is to disprove the feasibility of the 4π radiotherapy using a cantilever-type linear accelerator or any other external-beam delivery machines. Materials and Methods: A surface integral-based mathematical derivation for the maximum achievable solid angle for a linear accelerator was carried out respecting the rotational boundary conditions for gantry and couch in three-dimensional Euclidean space. The allowed movements include a gantry rotation of 0–2πc and a table rotation of πc/2 - 0 - 3πc/2) . Results: Total achievable solid angle by cantilever-type linear accelerator (or any teletherapy machine employing a cantilever design) is 1/r2 π ∫ - θ=0 π ∫ φ=0 (rSinθdφ). (rdθ) = 2πc, which is applicable only for the foot and brain radiotherapy where the allowed table rotation is 90°–0°–270°. For other sites such as pelvis, thorax, or abdomen, achievable solid angle as the couch rotation comes down significantly. Practically, only suitable couch angle is 0° by avoiding gantry–couch–patient collision. Conclusions: Present cantilever design of linear accelerator prevents achieving a 4π radian solid angle at any point in the patient. Even the most modern therapy machines like CyberKnife which has a robotic arm also cannot achieve 4π geometry. Maximum achievable solid angle under the highest allowable boundary condition(s) cannot exceed 2πc, which is restricted for only extremities such as foot and brain radiotherapy. For other parts of the body such as pelvis, thorax, and abdomen, the solid angle is reduced to 1/5th (maximum value) of the 4πc. To obtain a 4πc solid angle in a three-dimensional Euclidean space, the patient has to be a zero-dimensional point and X-ray head of the linear accelerator has a freedom to rotate in every point of a hypothetical sphere of radius 1 m. This article establishes geometrically why it is not possible to achieve a 4πc solid angle.

The situation was analyzed that has been established in the International System of Units (SI) as a result of adoption of the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities. It was shown that such decision was based on a misunderstanding of the mathematical formula, relating the arc length of a circle with its radius and corresponding central angle, as well as series expansions of trigonometric functions. As follows from the conducted analysis, the plane angle value does not depend on any of the SI quantities and should be treated as a base quantity, while its unit (radian) should be added to the population of base SI quantities. Therefore, the value of a solid angle becomes a derived quantity from a plane angle, and its unit (steradian) represents a coherent derived unit equal to a square radian. [ABSTRACT FROM AUTHOR]

In this work, an absolute method to calibrate an α -spectrometer is proposed taking into account the Source-to-Detector, and lateral distances due to eccentric source distribution. An analytical formula to calibrate an α -spectrometer is derived and the Simpson's integration method was utilized to solve these equations in its integral form numerically using a written C computer code. The general Monte Carlo N-particle code, MCNP as well as experimental measurements for some standard α -emitters are used to benchmark the proposed method. An agreement was found between the efficiency results calculated by MC and the proposed method with a maximum relative difference of about 0.5%. While, experimental measurement of α -emitters activity employing the proposed method differs by about 1.65% from the certified values. Accounting for the man made error allows to accurately quantify the assayed sample. Therefore, the inaccuracy in the efficiency results due to non-accurate inputs pertained to the source, and detector radii, Source-Detector distances, and eccentric source distribution are investigated in the Source-to-Detector distance range of (4 to 44 mm). The results show that a difference of ± 1 % in the detector radius, and Source-to-Detector distance than the normal values yields a relative difference of about ± 2 % , while a difference of ± 50 % in the source radius or source lateral distance from detector symmetry axis could only yields inaccuracy of less than ± 3 % in the efficiency results., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

Light not only enables humans to perceive their surroundings, but also influences their sleep–wake cycle, mood, concentration and performance. Targeted use of these so called nonvisual effects could also have a positive contribution in automobiles by keeping passengers alert, minimizing error rates or bootsting attention in general. Since construction space in vehicle interios is scarce, this study compared the influence of differently-sized light panels and thus solid angles on nonvisual effects. In a counterbalanced order, 32 volunteers were exposed to three lighting conditions in the morning: baseline (12 lx, 2200 K), small (200 lx, 6500 K, 0.05 sr) and large (200 lx, 6500 K, 0.44 sr). During each session of 60 min, alertness, concentration and working memory were assessed before and during light exposure. After data analysis no significant main effects of light, measurement point or interaction between light and measurement point could be seen. [ABSTRACT FROM AUTHOR]

Ultra-widefield fluorescein angiography (UWFA) is an emerging imaging modality used to characterise pathologies in the retinal vasculature, such as microaneurysms (MAs) and vascular leakages. Despite its potential value for diagnosis and disease screening, objective quantitative assessment of retinal pathologies by UWFA is currently limited because laborious manual processing is required. In this report, we describe a geometrical method for uneven brightness compensation inherent to UWFA imaging technique. The correction function is based on the geometrical eyeball shape, therefore it is fully automated and depends only on pixel distance from the center of the imaged retina. The method’s performance was assessed on a database containing 256 UWFA images with the use of several image quality measures that show the correction method improves image quality. The method is also compared to the commonly used CLAHE approach and was also employed in a pilot study for vascular segmentation, giving a noticeable improvement in segmentation results. Therefore, the method can be used as an image preprocessing step in retinal UWFA image analysis.

• Two new theorems in attitude kinematics were developed using the slewing motion of a single axis and rotation about the axis. • Derived Ishlinskii's theorem and the Goodman-Roberson theorem based on the newly developed kinematic theorems. • Found that a general attitude transformation consists of a slewing transformation of an axis followed by a rotational transformation about the axis, where the two transformations commute. Attitude kinematics using a single axis was developed to establish the relationships among the axis trajectory, enclosed solid angle and the associated attitude transformation. An arbitrary transformation due to a sequence of finite rotations or due to propagation consisting of an infinite number of infinitesimal rotations is equivalent to the product of a slewing transformation along a great circle arc linking the initial and final orientations of the axis followed by a rotational transformation about the axis. A key finding of this work is that the slewing transformation commutes with the rotational transformation, if the trajectory of the slewing transformation is fixed in space and the axis orientation is accounted for in the rotational transformation. The results were used to establish two new kinematic theorems. The new theorems provided simple derivations of Ishlinskii's theorem, the Goodman-Roberson theorem and a result found by Montgomery. The findings in this work were applied to attitude propagation by separating attitude rotational increments that are normal to the axis with those that are parallel to the axis. The normal and parallel angular increments were integrated separately to form the slewing transformation and rotational transformation, respectively. The product of the two commuting transformations formed the total propagation transformation. The findings were validated by using them to create and implement a propagation algorithm. Numerical testing showed that the new algorithm has accuracy comparable to traditional attitude propagation algorithms when stress tested with pure coning motion. [ABSTRACT FROM AUTHOR]

In recent years, measuring the speed of vehicles by a single camera has been done using several methods including the road geometric information, the difference in the number of image pixels of moving objects, and the homographic mapping of the motion vector from the image plane to the global coordinate plane. In this paper, we present a new method based on the geometry of the imaging system and the definition of solid angle. Our method does not require 3D modeling; It has a lower computational cost than the existing methods; And it has an accuracy comparable to the other approaches. In this method, we use the video images taken with a single camera on the road to extract the license plate in the image. Then, using the geometric information of the system and the distance travelled by vehicles the speed is computed. The average relative error of the proposed method is 3.23% and the mean absolute error is 1.32 km/h, which is comparable to the available algorithms. Furthermore, the computational cost of our method is less than the existing ones, which makes it suitable for implementing on embedded systems. [ABSTRACT FROM AUTHOR]

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$. We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$. As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$. The main tool used in the proofs is the conic version of the Crofton formula. [ABSTRACT FROM AUTHOR]

Given a smooth closed oriented manifold M of dimension n embedded in R n + 2 , we study properties of the 'solid angle' function Φ : R n + 2 \ M → S 1 . It turns out that a non-critical level set of Φ is an explicit Seifert hypersurface for M. This gives an explicit analytic construction of a Seifert surface in higher dimensions. [ABSTRACT FROM AUTHOR]

A new Rococo 2 X‐ray fluorescence detector was implemented into the cryogenic sample environment at the Hard X‐ray Micro/Nano‐Probe beamline P06 at PETRA III, DESY, Hamburg, Germany. A four sensor‐field cloverleaf design is optimized for the investigation of planar samples and operates in a backscattering geometry resulting in a large solid angle of up to 1.1 steradian. The detector, coupled with the Xspress 3 pulse processor, enables measurements at high count rates of up to 106 counts per second per sensor. The measured energy resolution of ∼129 eV (Mn Kα at 10000 counts s−1) is only minimally impaired at the highest count rates. The resulting high detection sensitivity allows for an accurate determination of trace element distributions such as in thin frozen hydrated biological specimens. First proof‐of‐principle measurements using continuous‐movement 2D scans of frozen hydrated HeLa cells as a model system are reported to demonstrate the potential of the new detection system. [ABSTRACT FROM AUTHOR]

ST elevation on an electrocardiogram is a hallmark of acute transmural ischemia. However, the underlying mechanism remains unclear. We hypothesized that high ischemic sensitivities of epicardial adenosine triphosphate-sensitive potassium (IKATP) and sodium (INa) currents play key roles in the genesis of ST elevation. Using a multi-scale heart simulation under moderately ischemic conditions, transmural heterogeneities of IKATP and INa created a transmural gradient, opposite to that observed in subendocardial injury, leading to ST elevation. These heterogeneities also contributed to the genesis of hyper-acute T waves under mildly ischemic conditions. By contrast, under severely ischemic conditions, although action potentials were suppressed transmurally, the potential gradient at the boundary between the ischemic and normal regions caused ST elevation without a contribution from transmural heterogeneity. Thus, transmural heterogeneities of ion channel properties may contribute to the genesis of ST-T changes during mild or moderate transmural ischemia, while ST elevation may be induced without the contribution of heterogeneity under severe ischemic conditions. [ABSTRACT FROM AUTHOR]

In the present work, the design of a high-precision luminosity sensor (HPLS) is shown. On sunny and clear days, the design allows the sun position to be determined with 99.7% accuracy; in the presence of clouds (partly sunny or cloudy conditions), the most clear area, which represents the area with the greatest amount of direct solar radiation at the moment, is determined. In both cases (clear and cloudy sky conditions), the above is achieved through the location in real time of the maximum luminosity point (MLP) in the sky through the construction of luminosity gradients, which are obtained under the acquisition and processing of the output signals of a set of sensors constituting the HPLS. The distribution of the sensors is determined through optical and geometric analyses. The HPLS is validated through a set of experiments in which the search and signaling of the MLP were carried out on a sunny day and on days where partly sunny and cloudy conditions were present. Finally, a technical comparative analysis between the HPLS and other sensors is presented; the cost of the HPLS is $293.50 USD (obtained in a calculation performed in this work), and a cost comparison with some commercial sensors is presented as well. [ABSTRACT FROM AUTHOR]