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It is usual that existing material on computer aided geometric design oscillates between over-simplification for programmers and practitioners and over formalism for scientific or academic readers. The first type of publications suppresses the taxonomy and properties of the mathematical concepts discussed when seeking straightforward notation and procedures. The second type of materials in thorough in the mathematical concepts at the expense of increasingly complicated notation and sacrifice of clear procedures for the reader.This book intends to be a compromise between the aforementioned extremes. It recalls basic concepts on functions, relations, transformations, matrices and groups and makes evident their impact in the engineering properties of projections, rotations, translations, perspectives, and so on. The material is of interest for computer scientists and electrical, mechanical, production, mathematical and physics engineers. In particular, it gives valuable insight into robotics, computer vision, design, manufacturing, kinematics and dynamics from a practical point of view while keeping contact whit the underlying decisive mathematical properties of the objects and transformations handled.The exercises and examples discussed in this book were stated, solved, documented and illustrated under the supervision of professors Oscar Ruiz and Carlos Cadavid during the years 2004 to 2007. Scenarios for such a work were the courses introduction to CAD CAM Systems and Geometric Modeling and the activities in the CAD CAM CAE laboratory at EAFIT University.This book intends to be a compromise between the aforementioned extremes. It recalls basic concepts on functions, relations, transformations, matrices and groups and makes evident their impact in the engineering properties of projections, rotations, translations, perspectives, and so on. The material is of interest for computer scientists and electrical, mechanical, production, mathematical and physics engineers. In particular, it gives valuable insight into robotics, computer vision, design, manufacturing, kinematics and dynamics from a practical point of view while keeping contact whit the underlying decisive mathematical properties of the objects and transformations handled.The exercises and examples discussed in this book were stated, solved, documented and illustrated under the supervision of professors Oscar Ruiz and Carlos Cadavid during the years 2004 to 2007. Scenarios for such a work were the courses introduction to CAD CAM Systems and Geometric Modeling and the activities in the CAD CAM CAE laboratory at EAFIT University.The exercises and examples discussed in this book were stated, solved, documented and illustrated under the supervision of professors Oscar Ruiz and Carlos Cadavid during the years 2004 to 2007. Scenarios for such a work were the courses introduction to CAD CAM Systems and Geometric Modeling and the activities in the CAD CAM CAE laboratory at EAFIT University.

Dual medium geometric model is widely used in soil and rock mass environment, which can only reflect the material exchange between cracks and matrix. The dual medium geometric model ignores the hydraulic relationship between the pores and cracks. The paper develops a new method to construct the three-dimensional dual medium geometry model. This method is completely written by Matlab scripting language, can describe the real spatial distribution state of fracture in bedrock. In the method, three-dimensional random fracture network geometric models with different parameters and three-dimensional porous media geometric models with Berlin Noise characteristics are constructed respectively, and then use the three-dimensional direct superposition method to construct the dual medium geometric model. The paper introduces the construction of three-dimensional random fracture network geometry model, three-dimensional porous media geometric model and the dual medium geometric model by direct superposition method. This research is of great significance to the construction of dual medium model and numerical simulation in related fields.

Adaptive evolution ultimately is fuelled by mutations generating novel genetic variation. Non-additivity of fitness effects of mutations (called epistasis) may affect the dynamics and repeatability of adaptation. However, understanding the importance and implications of epistasis is hampered by the observation of substantial variation in patterns of epistasis across empirical studies. Interestingly, some recent studies report increasingly smaller benefits of beneficial mutations once genotypes become better adapted (called diminishing-returns epistasis) in unicellular microbes and single genes. Here, we use Fisher's geometric model (FGM) to generate analytical predictions about the relationship between the effect size of mutations and the extent of epistasis. We then test these predictions using the multicellular fungus Aspergillus nidulans by generating a collection of 108 strains in either a poor or a rich nutrient environment that each carry a beneficial mutation and constructing pairwise combinations using sexual crosses. Our results support the predictions from FGM and indicate negative epistasis among beneficial mutations in both environments, which scale with mutational effect size. Hence, our findings show the importance of diminishing-returns epistasis among beneficial mutations also for a multicellular organism, and suggest that this pattern reflects a generic constraint operating at diverse levels of biological organization.

Little is empirically known about the contribution of mutations to fitness in natural environments. However, Fisher’s Geometric Model (FGM) provides a conceptual foundation to consider the influence of the environment on mutational effects. To quantify mutational properties in the field, we established eight sets of MA lines (7-10 generations) derived from eight founders collected from natural populations of Arabidopsis thaliana from French and Swedish sites, representing the range margins of the species in Europe. We reciprocally planted the MA lines and their founders at French and Swedish sites, allowing us to test predictions of FGM under naturally occurring environmental conditions. The performance of the MA lines relative to each other and to their respective founders confirmed some and contradicted other predictions of the FGM: the contribution of mutation to fitness variance increased when the genotype was in an environment where its fitness was low, that is, in the away environment, but mutations were more likely to be beneficial when the genotype was in its home environment. Consequently, environmental context plays a large role in the contribution of mutations to the evolutionary process and local adaptation does not guarantee that a genotype is at or close to its optimum.

A novel elasto-geometric model is introduced that simultaneously estimates joint compliances and geometric error parameters by employing the loaded double ball bar apparatus. The model parameters are estimated from tests at different force levels by distinguishing between errors that change with the applied force (compliance effect) from those that do not (geometric effects). At lower forces, the geometric errors are dominant while at higher forces compliance errors dominate. Using all data to build a single global geometry and compliance set of parameters (global constant compliance model), the radial volumetric variations due to geometric errors and compliance are estimated at 0.019 mm and 0.046 mm, respectively, making compliance dominant by more than three times. The impact of dominant and non-dominant equivalent global compliance C XXX , C YYY , C XYX , C CXY , C CYY , and C CCY on the loaded circular test readings at the highest force level of 742 N are predicted to be around 0.045, 0.034, 0.00058, 0.0022, 0.0014, and 0.0045 mm peak-to-peak, respectively. The impact of loaded geometric parameters E XX1 , E YY1 , E YX2 , E XY2 , E C(0Y)X , E Xt0 , and E Yt0 on the loaded circular test readings is predicted to be around 0.019, 0.014, 0.0074, 0.012, 0.00017, 0.0076, and 0.0012 mm peak-to-peak, respectively. The dominant global compliances are C XXX and C YYY at 0.0619 and 0.0461 μ m / N , respectively.

Spaceborne multistatic synthetic aperture radar (SAR) tomography (SMS-TomoSAR) systems take full advantage of the flexible configuration of multistatic SAR in the space, time, phase, and frequency dimensions, and simultaneously achieve high-precision height resolution and low-deformation measurement of three-dimensional ground scenes. SMS-TomoSAR currently poses a series of key issues to solve, such as baseline optimization, spatial transmission error estimation and compensation, and the choice of imaging algorithm, which directly affects the performance of height-dimensional imaging and surface deformation measurement. This paper explores the impact of baseline distribution on height-dimensional imaging performance for the baseline optimization issue, and proposes a feasible baseline optimization method. Firstly, the multi-base multi-pass baselines of an SMS-TomoSAR system are considered equivalent to a group of multi-pass baselines from monostatic SAR. Secondly, we establish the equivalent baselines as a symmetric-geometric model to characterize the non-uniform characteristic of baseline distribution. Through experimental simulation and model analysis, an approximately uniform baseline distribution is shown to have better SMS-TomoSAR imaging performance in the height direction. Further, a baseline design method under uniform-perturbation sampling with Gaussian distribution error is proposed. Finally, the imaging performance of different levels of perturbation is compared, and the maximum baseline perturbation allowed by the system is given.

This study presents a novel approach to metric spaces through the lens of geometric calculus, redefining traditional structures with new operations and properties derived from non-Newtonian measures. Specifically, we develop and prove geometric versions of the Hölder and Minkowski inequalities, which provide foundational support for applying these spaces in analysis. Additionally, we establish key relationships between geometric and classical metric spaces, examining concepts such as openness, closedness, and separability within this geometric framework. By exploring topological characteristics and separability conditions in geometric metric spaces, this work enhances the understanding of metric spaces’ structural properties, offering potential applications in fields that require flexible metric adaptations, such as data science, physics, and computational geometry. This framework’s adaptability makes it relevant for scenarios where non-Euclidean or high-dimensional spaces are needed, allowing for versatile applications and extending classical metric concepts into broader analytical contexts.

Substitution Box (S-Box) has had been a cardinal component of various cryptographic systems. In this paper, we introduce a novel S-Box design that merges octagonal geometry with chaotic dynamics to enhance the security effects of the cryptographic systems. In particular, the proposed method leverages the geometric properties of octagons and the unpredictability of chaotic maps to construct a novel S-Box with improved security features. The mathematical construct octagon carries out the necessary operation of confusion in the proposed S-Box. The centres of these octagons are hypothetically created within the confines of the 16 × 16 matrix of numbers. Further, these octagons have different radii, locations, and the amounts with which the numbers lying on their boundaries have to be circularly shifted clockwise or anti-clockwise to create the confusion effects. In case, a portion of octagon goes past the edges of the matrix, the numbers lying on its boundary have been wrapped out. This process has been repeated numerous times to come up with a reliable and a secured S-Box. The comprehensive security analyses validate that the proposed S-Box is furnished with nice security effects and has the requisite resilience to defy the varied cryptanalytic threats. The results of non-linearity and differential probability are 105.625 and 0.0391 respectively which signals towards the inherent robustness of the suggested S-Box.

We propose the χ-index as a bibliometric indicator that generalises the h-index. While the h-index is determined by the maximum square that fits under the citation curve of an author when plotting the number of citations in decreasing order, the χ-index is determined by the maximum area rectangle that fits under the curve. The height of the maximum rectangle is the number of citations ck to the kth most-cited publication, where k is the width of the rectangle. The χ-index is then defined as [Formula: see text], for convenience of comparison with the h-index and other similar indices. We present a comprehensive empirical comparison between the χ-index and other bibliometric indices, focusing on a comparison with the h-index, by analysing two datasets-a large set of Google Scholar profiles and a small set of Nobel prize winners. Our results show that, although the χ and h indices are strongly correlated, they do exhibit significant differences. In particular, we show that, for these data sets, there are a substantial number of profiles for which χ is significantly larger than h. Furthermore, restricting these profiles to the cases when ck > k or ck < k corresponds to, respectively, classifying researchers as either tending to influential, i.e. having many more than h citations, or tending to prolific, i.e. having many more than h publications.