Geometric Definition Research Articles

Algebraic and geometric definitions of the cross product: the link

Given two vectors $\overrightarrow u = {({u_1},\,\,{u_2},\,{u_3})^t}$ and $\overrightarrow y = {({v_1},\,{v_2},\,{v_3})^t}$ in ${{\mathcal{R}}^3}$ , the cross product $\overrightarrow u \times \overrightarrow v $ is defined as follows (see [1] or [2]): $$Algebraic{\rm{ }}definition{\rm{:}}\,\,\overrightarrow u \times \overrightarrow v \, = \,\left[ {\matrix{ {{u_2}{v_3}} \hfill & - \hfill & {{u_3}{v_2}} \hfill \cr {{u_3}{v_1}} \hfill & - \hfill & {{u_1}{v_3}} \hfill \cr {{u_1}{v_2}} \hfill & - \hfill & {{u_2}{v_1}} \hfill \cr } } \right].$$

Large synthesis of in situ field measurements of the size distribution of mineral dust aerosols across their life cycles

Abstract. Mineral dust aerosol is important in the Earth system, and the correct representation of its size distribution is fundamental for shaping the current state and evolution of the climate. Despite many observational dust size data that are available in the literature, using this body of information to properly guide the development and validation of climate models and remote sensing retrievals remains challenging. In this study we collect, evaluate, harmonize, and synthesize 58 size distribution data from the past 50 years of in situ field observations with the aim of providing a consistent dataset to the community for use in constraining the representation of dust size across its life cycle. Four levels (LEVs) of data treatment are defined, going from original data (LEV0), data interpolated and normalized on a standardized diameter grid (LEV1), and data in which original particle diameters are converted to a common geometrical definition under both spherical (LEV2a) and aspherical (LEV2b) assumptions. Size distributions are classified as emission or source (SOURCE, <1 d from emission; number of datasets in this category N=12), mid-range transport (MRT, 1–4 d of transport; N=36), and long-range transport (LRT, >4 d of transport; N=10). The harmonized dataset shows consistent features suggesting the conservation of airborne particles with time and a decrease in the main coarse-mode diameter from a value on the order of 10 µm (in volume) for SOURCE dust to a value on the order of 1–2 µm for LRT conditions. An additional mode becomes evident below 0.4 µm for MRT and LRT dust. Data for the three levels (LEV1, LEV2a, and LEV2b) and the three categories (SOURCE, MRT, and LRT), together with statistical metrics (mean, median, 25th and 75th percentiles, and standard deviation), are available as follows: SOURCE (https://doi.org/10.57932/58dbe908-9394-4504-9099-74a3e77140e9, Formenti and Di Biagio, 2023a); MRT (https://doi.org/10.57932/31f2adf7-74fb-48e8-a3ef-059f663c47f1, Formenti and Di Biagio, 2023b); LRT (https://doi.org/10.57932/17dc781c-3e9d-4908-85b5-5c99e68e8f79, Formenti and Di Biagio, 2023c).

Short Cyclic Algorithm for Turbo Block Construction

This paper presents perhaps the shortest possible algorithm for drafting a turbo block in a standard CAD environment. Though not defining the complete turbo roundabout geometry (the legs are defined by their centerlines only), turbo block construction stays at the very core of its geometrical definition. Relevant national policies elaborate turbo block construction at various levels of detail. Some policies skip certain definitions of spirals’ centers and some even allow for slight discontinuities in spiraling lanes’ geometries. But, as our computer program had to be based on precisely defined geometrical features, no question, be it a location of a particular center or relation between the interconnecting arcs, could be left unanswered. Consequently, while explaining the basics on which the program relies, all the elements of the turbo block are discussed. Even the problems not addressed in renowned national policies (such as alternatively matching marking and curb lines) had to be resolved before being incorporated into the program. After resolving and elaborating turbo block roundabout technics in detail, a new cyclic algorithm for turbo block construction is presented. The program is very short and could be easily understood and modified by the average programmer.

Physiological definition for region of interest selection in electrical impedance tomography data: description and validation of a novel method

Objective. Geometrical region of interest (ROI) selection in electrical impedance tomography (EIT) monitoring may lack sensitivity to subtle changes in ventilation distribution. Therefore, we demonstrate a new physiological method for ROI definition. This is relevant when using ROIs to compute subsequent EIT-parameters, such as the ventral-to-dorsal ratio during a positive end-expiratory pressure (PEEP) trial.Approach.Our physiological approach divides an EIT image to ensure exactly 50% tidal impedance variation in the ventral and dorsal region. To demonstrate the effects of our new method, EIT measurements during a decremental PEEP trial in 49 mechanically ventilated ICU-patients were used. We compared the center of ventilation (CoV), a robust parameter for changes in ventro-dorsal ventilation distribution, to our physiological ROI selection method and different commonly used ROI selection methods. Moreover, we determined the impact of different ROI selection methods on the PEEP level corresponding to a ventral-to-dorsal ratio closest to 1.Main results.The division line separating the ventral and dorsal ROI was closer to the CoV for our new physiological method for ROI selection compared to geometrical ROI definition. Moreover, the PEEP level corresponding to a ventral-to-dorsal ratio of 1 is strongly influenced by the chosen ROI selection method, which could have a profound clinical impact; the within-subject range of PEEP level was 6.2 cmH2O depending on the chosen ROI selection method.Significance.Our novel physiological method for ROI definition is sensitive to subtle ventilation-induced changes in regional impedance (i.e. due to (de)recruitment) during mechanical ventilation, similar to the CoV.

On sequences of convex records in the plane

Abstract Convex records have an appealing purely geometric definition. In a sequence of d-dimensional data points, the nth point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the bivariate (i.e. two-dimensional) setting. For iid (independent and identically distributed) points, we establish an identity relating the mean number ⟨ R n ⟩ of convex records up to time n to the mean number ⟨ N n ⟩ of vertices in the convex hull of the first n points. By combining this identity with extensive numerical simulations, we provide a comprehensive overview of the statistics of convex records for various examples of iid data points in the plane: uniform points in the square and in the disk, Gaussian points and points with an isotropic power-law distribution. In all these cases, the mean values and variances of Nn and Rn grow proportionally to each other, resulting in the finite limit Fano factors FN and FR . We also consider planar random walks, i.e. sequences of points with iid increments. For both the Pearson walk in the continuum and the Pólya walk on a lattice, we characterise the growth of the mean number ⟨ R n ⟩ of convex records and demonstrate that the ratio R n / ⟨ R n ⟩ keeps fluctuating with a universal limit distribution.

Simplified Procedure for the Design and Optimisation of Oval Tensile Spoke Wheels for Roof Structures

This paper presents a simplified procedure for the design and shape optimisation of oval tensile spoke wheels to be used in roof structures for grandstands in sports stadiums. This procedure combines graphical methods with simple numerical methods to obtain the geometrical definition of an oval wheel and to estimate the prestressing forces. It also allows for the compression forces in the outer ring to be estimated and compared with those in a circular wheel. This makes it easy to make small geometrical adjustments for optimisation. Application examples are given for each step of the procedure. Several wheel designs are defined, showing how the shape of the inscribing oval determines the magnitude of the forces in the outer ring.

Development of a coupled process - design numerical model for an automotive assembly

Abstract The product development process is a very challenging one, especially in the case of the automotive industry. The actual constructions require performance and quality while focusing on mechanical performance, lightweight and cost-effective. Thus, the virtual development stage is critical. Once the product is validated according to its geometrical definition and features, it is necessary to evaluate its mechanical performance. This process is usually performed using numerical simulation methods. In this paper, the numerical process performed for the validation of a safety part is discussed. As a specific feature, the assembly is finished using a riveting process. The manufacturing process’s influence on the assembly’s mechanical performance is further investigated. A discussion of the methods available for the design validation summarizes the finding of this specific work.

Weak solutions of anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials

We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in RN with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.

An index theorem for loop spaces

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of KK-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to specify an “appropriate form” of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact S1-manifolds with a compact fixed-point set. In order to formulate it, we use a ring of formal power series.

The Generic Failure of Lower-Semicontinuity for the Linear Distortion Functional

AbstractWe consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, $$n\ge 3$$ n ≥ 3 . The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if $$ \{ f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion $$H({f_{n}})$$ H ( f n ) ), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence $$ \{f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ with $$ {f_{n}}\rightarrow {f}$$ f n → f locally uniformly and with $$\limsup _{n\rightarrow \infty } H( {f_{n}})<H( {f})$$ lim sup n → ∞ H ( f n ) < H ( f ) . Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each $$\alpha <\sqrt{2}$$ α < 2 there is $${f_{n}}\rightarrow {f}$$ f n → f locally uniformly with f affine and $$\begin{aligned} \alpha \; \limsup _{n\rightarrow \infty } H( {f_{n}}) < H( {f}) \end{aligned}$$ α lim sup n → ∞ H ( f n ) < H ( f ) We conjecture $$\sqrt{2}$$ 2 to be best possible.

Geometric morphometrics analysis of forensically important Sarcophaga species

Abstract Geometric morphometrics is a method that offers a mathematical description of biological forms based on geometric definitions of their size and shape, enabling the discrimination between species. In this study, we aimed to utilize geometric morphometrics as a valuable tool for identifying Sarcophagidae species, a family of flies that are challenging to distinguish morphologically. We collected a total of 80 wings (both left and right) from male individuals representing nine species (Sarcophaga africa, S. argyrostoma, S. hirticrus, S. jacobsoni, S. lehmanni, S. melanura, S. pandellei, S. portschinskyi, and S. teretirostris). These wings were removed, mounted on microscope slides, photographed, and digitized using 15 landmarks. The results demonstrated effective differentiation among seven species, representing significant progress in the expedited identification of Sarcophaga species. Due to its speed, affordability, and user-friendly nature, the wing landmark-based geometric morphometrics, when combined with taxonomy, can enhance the robustness of Sarcophagidae analyses and prove useful for discriminating flesh flies in forensic contexts.

Effects of Heat Input and Intertrack Overlap on the Microstructure and Properties of Inconel 686 Weld Overlays.

The objective of this study was to investigate how weld overlays with nickel superalloys are important for the integrity, due the high temperatures and corrosive environments that can be experienced in mineral processing environments, of mining and processing equipment. The Ni-Cr-Mo superalloy Inconel 686 overlays are fabricated through automatic gas metal arc welding with variations in arc voltage and travel speed (i.e., heat input), and they have overlap between adjacent weld tracks for applications in the mining and minerals sector. The impact of variations in the process parameters and the size of the weld overlapping on the dilution, solidification morphology, microsegregation, and microhardness were investigated. Both geometric and chemical composition definitions were used to quantify the extent of the weld dilution. Subsequently, the weld geometry and dilution were correlated with the solidification microstructure and phase transformations. The maximum dilutions were measured to be 13.63% (1/2 overlap, 5.96 kJ·cm-1) and 15.39% (1/3 overlap, 4.77 kJ·cm-1), which shows that less of an overlap increases the dilution level. Scanning electron microscopy and chemical composition analysis revealed that an increase in weld heat input and dilution level led to higher levels of microsegregation for Mo and Cr, as well as the volume fraction of Mo- and Cr-rich phases in the interdendritic/intercellular regions in the overlay layer. Analysis of the weld overlays in the current study revealed strong and unprecedented connections between the weld overlay process conditions, the resultant metallurgy (i.e., dendrite arm spacing, microsegregation, and phase formation), and the hardness of the overlay. It was concluded that the optimal weld overlays in the processing window studied in this investigation were fabricated at mid-level heat inputs (i.e., 4-5 kJ·cm-1) and a 1/2 track overlap.

Inverse Geometric Reconstruction Based on MW-NURBS Curves

Currently, rational curves such as the Non-Uniform Rational B-Spline (NURBS) play a significant role in both shape representation and shape reconstruction. NURBS weights are often real in nature and are referred to as challenging to assign, with the exception of conics. ‘Matrix Weighted Rational Curves’ are the expanded form of rational curves that result from replacing these real weights with matrices, or matrix weights. The only difference between these curves and conventional curves is the geometric definition of the matrix weights. In this paper, MW-NURBS curves are used to reconstruct space curves from their stereo perspectives. In particular, MW-NURBS fitting is carried out in stereo views, and the weight matrices for the MW-NURBS curves are produced using the normal vectors provided at the control points. Instead of needing to solve a complicated system, the MW-NURBS model can reconstruct curves by choosing control points and control normals from the input data. The efficacy of the proposed strategy is verified by using many examples based on both synthetic and real images. The various error types are compared to those of conventional methods like point-based and NURBS-based approaches. The results demonstrate that the errors acquired from the proposed approach are much fewer than those obtained from the point-based method and the NURBS-based method.

Statistical investigation of interdependence of trabecular microarchitectural parameters from micro-computed tomography

Standard microarchitectural analysis of bone using micro-computed tomography produces a large number of parameters that quantify the structure of the trabecular network. Analyses that perform statistical tests on many parameters are at elevated risk of making Type I errors. However, when multiple testing correction procedures are applied, the risk of Type II errors is elevated if the parameters being tested are strongly correlated. In this article, we argue that four commonly used trabecular microarchitectural parameters (thickness, separation, number, and bone volume fraction) are interdependent and describe only two independent properties of the trabecular network. We first derive theoretical relationships between the parameters based on their geometric definitions. Then, we analyze these relationships with an aggregated in vivo dataset with 2987 images from 1434 participants and a synthetically generated dataset with 144 images using principal component analysis (PCA) and linear regression analysis. With PCA, when trabecular thickness, separation, number, and bone volume fraction are combined, we find that 92 % to 97 % of the total variance in the data is explained by the first two principal components. With linear regressions, we find high coefficients of determination (0.827–0.994) and fitted coefficients within expected ranges. These findings suggest that to maximize statistical power in future studies, only two of trabecular thickness, separation, number and bone volume fraction should be used for statistical testing.

Parametric Analysis of “Zollinger Type” Ribbed Timber Shells

The following article presents the study and further development of the roof system developed and patented by Friedrich Zollinger in 1920. The so-called Zollinger construction system is based on short timber components ‐ lamellae, connected by a bolt and a form of decking only for covering purpose. By developing the micro step joint, an increase in joint stiffness can be achieved. In addition, by applying and activating a load-bearing roof covering, a further increase in system stiffness can be achieved and demonstrated. Especially with regard to large spans, the increase of the stiffness and reduction of the deflection becomes inevitable against the background of an economic feasibility. Based on the analysis of the historic Zollinger system, it is possible to evaluate potential for improvements in terms of system stiffness as well as contemporary design, development and manufacturing. Due to the large number of different influencing factors, the system was modeled as a parametric master model. In this, a closed process cycle can be developed from the geometric definition, to the static analysis and to the derivation of the computer-aided manufacturing data. In the present study especially the exact recording, description and modeling of the stiffness and feasibility of the connection details “lamella‐lamella” and “lamella‐planking” was investigated. A parametric finite element model can be used to determine the stiffness parameters as a function of the fasteners used and the spacing of the fasteners. Staple connections are used as the main means of connection between the lamella and the planking. The implementation of the staple connections in the finite element analysis was carried out using spring parameters, which are determined in the course of the development. The analysis and optimization of the connection stiffnesses is presented against the background of an economical structure development. By using digital planning methods and interfaces, a contemporary way of working is presented. In addition, a contribution is made to digitalization in the construction industry.

Geometric Definition of Transformable Surfaces with Homogeneous ABC Reciprocal Meshes

Geometric Definition of Transformable Surfaces with Homogeneous ABC Reciprocal Meshes

Limits and Thresholds

Abstract of Keynote Presentation: ‘Between Excess and Austerity’, IDEA Conference, Sydney, August 8–11, 2003. Abstract: Before we can approach the tension between austerity and excess – or any other productive tension in a design practice – we must first enter the metaphysical space of ‘interior’ itself. The notion of interior is always defined by at least three ‘working parts’: inside, outside, and (most important yet least noticed) the threshold setting one off from the other. Properly speaking, this threshold is neither outside nor inside; rather, in setting the limit between them, it partakes of both. Like the skin of a body or the cladding of a building, indeed like any surface, the threshold comes into contact with what lies on both sides of it, linking the two environments in the act of separating them. ‘A surface separates from out and belongs no less to one than to the other’ (Don DeLillo). The threshold is an ontological anomaly, a space outside of space, existing only in its vanishing. In theory, a threshold, as with any line in geometrical definition, has extension but no dimension. It is literally zero degrees thick. The function of the threshold, therefore, is not to be wide but to be crossed. Every limit is also its own negation. Drawing the limit-line is coeval with the possibility, the suggestion, we might even say the demand that the line be crossed. Once established, boundaries ‘ask for’ breaching – traditionally, a task for heroes. This crossing is no simple matter, though we may blithely perform it dozens or even hundreds of times a day. What is involved here? What relationships of time and space, of consciousness and identity, of necessity and freedom, are created by the move from outside to in, and back again? In this paper, I approach the thought of limits (and maybe the limits of thought) by takingthe interior back to its moment of geometric inception, its philosophical ground zero: the drawing of lines and circles. Once inside, but only then, we may begin to ask what it is we seek in this crossing over, this transgression, of the threshold. Is it comfort? Security? Control? Or perhaps something deeper, and more challenging: the act of thinking itself.

Local Head Losses And Drag Coefficients Characterization In Coastal Infrastructures

Some coastal infrastructures (i.e. intakes, outfalls, gates) incorporate singular elements (meshes, nets or grids) which, through their interaction with the water flow, can produce large changes in the water level (upstream and downstream) and forces on them. Therefore, an accurate definition of local head losses (to characterize water level changes) and drag coefficient (to characterize drag forces) allows to optimize their designs (configuration and geometric definition).

A geometric property of quadrilaterals

Quadrilaterals in the complex plane play a significant part in the theory of planar quasiconformal mappings. Motivated by the geometric definition of quasiconformality, we prove that every quadrilateral with modulus in an interval \([1/K, K]\), where \(K>1\), contains a disk lying in its interior, of radius depending only on the internal distances between the pairs of opposite sides of the quadrilateral and on \(K\).

The geometrization of Maxwell's equations and the emergence of gravity and antimatter

Coupling the Maxwell tensor to the Riemann-Christoffel curvature tensor is shown to lead to a geometricized theory of electrodynamics. While this geometricized theory leads directly to the classical Maxwell equations, it also extends their physical interpretation by giving charge density and the four-velocity that describe its motion geometric definitions. Introducing mass using a conserved energy-momentum tensor, all solutions to the geometricized theory of electrodynamics developed here are shown to be consistent with the emergence of gravity obeying the General Relativity field equation augmented by a term that mimics the properties of dark matter and/or dark energy. Finally, due to the symmetries of the theory, the properties and phenomenology of antimatter emerge in solutions.