The research paper proposes an algorithm to find congruence criteria between two convex polygons in Euclidean Geometry. It begins with a review of triangles, then extends to quadrilaterals and eventually gener- alizes the case to n-sided polygons. It attempts to prove said algorithm using a method of induction and a case-by-case analysis. It also states a corollary to said algorithm

Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The central challenge is to specify how those constraints are themselves updated. In this paper we review and extend the ED framework in several directions. A new version of ED is introduced in which particles follow smooth differentiable Brownian trajectories (as opposed to non-differentiable Brownian paths). To construct the ED we make use of the fact that the space of probabilities and phases has a natural symplectic structure (i.e., it is a phase space with Hamiltonian flows and Poisson brackets). Then, using an argument based on information geometry, a metric structure is introduced. It is shown that the ED that preserves the symplectic and metric structures -- which is a Hamilton-Killing flow in phase space -- is the linear Schrödinger equation. These developments allow us to discuss why wave functions are complex and the connections between the superposition principle, the single-valuedness of wave functions, and the quantization of electric charges. Finally, it is observed that Hilbert spaces are not necessary ingredients in this construction. They are a clever but merely optional trick that turns out to be convenient for practical calculations.

Aim of expressional space is to facilitate the public to explore their hobby and talent in autodidact. The space for expression in Makassar City is necessary for the mural artist and the youngsters who has a hobby to play a skateboard. Their talent is insufficient attention by the government and create negative attitude because there no space to accommodate their activity. Therefore, this paper presents a design idea of expressional space by implementing the architectural geometry. The architectural geometry design principle is based on simple geometry form application as a realization of form from the space. An experience of the space for the user realizes by the relationship between the space and the interaction with the environment through geometry and form processing as the basic of the creation process in the architectural works. Content analysis method providing a landscape layout and transformation of form and building mass in form of the implementation of architectural geometry principles. The principles are pure geometry, composition of geometry, direction wall, and opening for the transformation process of expressional space form to produce and increase aesthetic value of the environment.

Computer extended Descriptive Geometry (CeDG) is a new approach to solve and build computer models of three dimensional (3D) geometrical systems through descriptive geometry procedures that have demonstrated reliability and accuracy. CeDG may calculate a parametric algebraic exact form for the spatial curves generated in the intersection of two surfaces, as well as of the flattened pattern of any developable surface involved in those encounters. This study presents first the theoretical foundations and methodology to calculate those parametric algebraic curves. Secondly, a compound hopper is defined and modelled through CeDG (implemented in GeoGebra) and CAD (Solid Edge 2023) approaches to evaluate the advantages of CeDG against CAD. The results demonstrate the robustness and accuracy of the CeDG technique for surfaces intersection and flattening and the advantages of CeDG against Solid Edge 2023 in solving the hopper case study.

This paper considers the possibility that abstract thinking and advanced synthesis skills might be selection criteria for extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic structures of current physical theories: vector fields, 1-form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission.

Orbit and clock products are used in real-time GNSS precise point positioning without knowing their quality. This study develops a new approach to detect orbit and clock errors through comparing geometry-free and geometry-based wide-lane ambiguities in PPP model. The reparameterization and estimation procedures of the geometry-free and geometry-based ambiguities are described in detail. The effects of orbit and clock errors on ambiguities are given in analytical expressions. The numerical similarity and differences of geometry-free and geometry-based wide-lane ambiguities are analyzed using different orbit and clock products. Furthermore, two types of typical errors in orbit and clock are simulated and their effects on wide-lane ambiguities are numerically produced and analyzed. The contribution discloses that the geometry-free and geometry-based wide-lane ambiguities are equivalent in terms of their formal errors. Although they are very close in terms of their estimates when the used orbit and clock for geometry-based ambiguities are precise enough, they are not the same, in particular, in the case that the used orbit and clock, as a combination, contain significant errors. It is discovered that the discrepancies of geometry-free and geometry-based wide-lane ambiguities are coincided with the actual time-variant errors in the used orbit and clock at the line-of-sight direction. This provides a quality index for real-time users to detect the errors in real-time orbit and clock products, which potentially improves the accuracy of positioning.

As a first step a unification of the gravitational with the electromagnetic interaction within a classical framework is proposed. It is based on a V₅-geometry, with x₅ = q/m. The sole source term is mechanical stress energy, positioned along x₅. The trajectories of test-bodies are placed in V₄ (x₅ = const)-slices. The resulting field equation couples a geometric G-tensor to mechanical stress energy, its momentum with respect to V₅ and the change of this momentum with proper time τ. The following step proceeds to the quantization of space-time to enable the formalization of elementary particles und strong interaction. This is complimented by the inclusion of spin and the weak force, leading finally to a grand total equation, the stationary version of which corresponds to Schroedinger´s equation, the instationary version to the Einstein equation of General Relativity.

Coronary artery disease (CAD) represents a modern pandemic associated with significant morbidity and mortality. The multi-faceted pathogenesis of this entity has long been investigated, highlighting the contribution of systemic factors such as hyperlipidemia and hypertension. Nevertheless, recent research has drawn light to the importance of geometrical features of coronary vasculature on the complexity and vulnerability of coronary atherosclerosis. Various parameters have been investigated so far, including vessel-length, cross-sectional area, curvature, and tortuosity, using primarily invasive angiography and recently non-invasive cardiac computed tomography angiography (CCTA). It is clear that there is correlation between geometrical parameters and both the haemodynamic alterations augmenting the atherosclerosis-prone environment and the extent of plaque burden. The purpose of this review is to discuss the currently available literature regarding this issue and propose a potential non-invasive imaging biomarker, the geometric risk score, which could be of importance to allow early detection of individuals at increased risk of developing CAD.

We employed the Riemannian Face Manifold (RFM) as an alternative approach to the conventional linear Euclidean space for explaining the atypicality bias in face likeliness judgments. The RFM posits that the mental representation of faces is better captured as a manifold of stable states, accounting for the nonlinearity in the physical properties of faces. To examine the relationship between physical and psychological distance of morph and typical/atypical faces, we manipulated the parameter k and incorporated a weight function in the Riemannian metric. Our results indicate that the psychological distance between the morph and typical face was longer than that between the morph and atypical face, consistent with prior research on the atypicality bias in perceptual similarity. The RFM approach provides mathematical support and is a powerful tool for studying face perception and recognition, offering potential implications for explaining the other-race effect. Future research can utilize the RFM to investigate individual differences in face recognition abilities and expertise, and how they impact psychological distance and face processing.

The purpose of the investigation was to determine the relationship between rotary kiln liner loss and steel structure ovality. As measurement apparatus, a terrestrial laser scanner was used. The interior and exterior of the rotary kiln were measured. The primary focus object was inner-lining loss and the geometric characteristics of cylindrical shells. The research uncovered significant disparities in inner lining loss between sections. A correlation was found between ovality and elimination of inner lining. Due to the hypothesis of constant inner linig loss from the middle of the rotary kiln, the investigation found that the loss of brick lining was less than the value reported from the boreholes. The study offers significant information on maintenance and repair strategies for rotary kilns, which have the potential to increase their efficiency and useful life.

This manuscript discusses a novel method to map pressure results from one 3D surface shell mesh onto another. This technique is especially important when transferring results from one numerical analysis to another. This method works independent of the actual pressures, and only focuses on ensuring the surface areas consistently match. By utilizing this approach, the cumulative forces consistently match for all input pressures. This method is demonstrated to work for pressure profiles with precipitous changes in pressures, and with small quadrangular source elements being applied to a mix of large quadrangular and triangular target elements, and the forces at all pressure profiles match remarkably.

The statistical bundle is the set of couples $(Q,W)$ of a probability density Q and a random variable W such that $\mathbb{E}$_Q [W] = 0. On a finite state space, we assume Q to be a probability density with respect to the uniform probability and give an affine atlas of charts such that the resulting manifold is a model for Information Geometry. Velocity and accelleration of a one-dimensional statistical model are computed in this set up. The Euler-Lagrange equations are derived from the Lagrange action integral. An example of Lagrangian using minus the entropy as potential energy is briefly discussed.

In this article, a generalization of the geometry of Grassmannian chain complex of free abelian groups generated by the projective configuration of points and Cathelineau's infinitesimal complex of polylogarithmic groups is proposed. Firstly, homomorphisms for weight n =2 up to weight n=6 will be introduced to connect sub-complexes of Grassmannian and Cathelineau. Lately, generalization of these morphisms will be shown for weight n=N. The associated diagrams will also be proven to be commutative and bi-complex.

This short reading is an extraction from our previous work (Nyambuya 2020), the purpose of which is to make clear that it is very much possible to use Weyl (1918)'s idea of a conformal metric to achieve tensorial affinities. We are of the strong view that this is very important as it is predominantly assumed that this not possible. We want to dispel this myth once and for al-time.

Ampelography, the botanical discipline dedicated to the identification and classification of grapevine cultivars, was grounded on the description of morphological characters and more recently is based on the application of DNA polymorphisms. New methods of image analysis may help to optimize morphological approaches in ampelography. The objective of this study was the classification of representative cultivars of Vitis vinifera conserved in the Spanish collection of IMIDRA according to seed shape. Thirty eight cultivars representing the diversity of this collection were analyzed. A consensus seed silhouette was defined for each cultivar representing the geometric figure that better adjusted to their seed shape. All the cultivars tested were classified in ten morphological groups, each corresponding to a new model. The models are geometric figures defined by equations and similarity to each model is evaluated by quantification of percent of the area shared by the two figures, the seed and the model (J index). The comparison of seed images with geometric models is a rapid and convenient method to classify cultivars. A large proportion of the collection may be classified according to the new models described and the method permits to find new models according to seed shape in other cultivars.

We investigate a possible reduction mechanism from (bosonic) Quantum Field Theory (QFT) to Quantum Mechanics (QM), in a way that could explain the apparent loss of degrees of freedom of the original theory in terms of quantum information in the reduced one. This reduction mechanism consists mainly in performing an ansatz on the boson field operator, which takes into account quantum foam and non-commutative geometry. Through the reduction mechanism, QFT reveals its hidden internal structure, which is a quantum network of maximally entangled multipartite states. In the end, a new approach to the quantum simulation of QFT is proposed through the use of QFT's internal quantum network

We reelaborate on the basic properties of PT symmetry from a geometrical Perspective. The transfer matrix associated with these systems induces a Möbius transformation in the complex plane. The trace of this matrix classifies the actions into three types that represent rotations, translations, and parallel displacements. We find that a PT invariant system can be pictured as a complex conjugation followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations and link them with measurable properties of the system.

It is widely acknowledged that the integration of BIM and GIS data is a crucial step forward for future 3D city modelling, but most of the research conducted so far has covered only the semantic aspects of GIS-BIM integration. We present here the results of the GeoBIM project, in which we tackled three integration problems focussing instead on aspects involving geometry processing: (i) the automated processing of complex architectural IFC models, (ii) the integration of existing GIS subsoil data in BIM, and (iii) the georeferencing of BIM models for their use in GIS software. All the problems have been studied using real world models and existing datasets made and used by practitioners in the Netherlands. For each problem, we expose in detail the issues we faced, our proposed solutions, and our recommendations for a more successful integration.

Cosmological models with linearly varying deceleration parameter in the cosmological theory based on Lyra’s geometry have been discussed. Exact solutions have been obtained for a spatially flat FRW model by considering a time dependent displacement field. We have also obtained the time periods during which the universe undergoes decelerated and accelerated expansions for a matter-dominated universe.

We examine here the conductance changes associated with the change in spin state in a variety of different structures, using the example of the spin crossover complex [Fe(H2B(pz)2)2(bipy)] (pz= (pyrazol-1-yl)-borate and bipy = 2,2’-bipyridine) and [Fe(Htrz)2(trz)](BF4)] (Htrz = 1H-1,2,4-triazole) thin films . This conductance change is highly variable depending on the mechanism driving the change in spin state, the substrate and the device geometry. Simply stated, the choice of spin crossover complex used to build a device is not the only factor in determining the change in conductance with the change in spin state.

Snap fit is a common mechanical mechanism. It uses the physical asymmetry that is easy to assemble but difficult to disassemble to provide a simple and fast link between objects. The ingenious combination of geometric shape, bending elasticity and friction of the snap fit is the mechanism behind the easy to assemble but difficult to disassemble disassemble. Yoshida and Wada (2020) has done a groundbreaking work in the analysis of the elastic snap fit. During our study of their paper, while we really enjoyed their research, unfortunately we detected several questioning formulations. After careful checking, we found that those formulations are not typographical, therefore it is necessary to make corrections. This paper reformulates the linear elasticity of a cylindrical snap fit, obtains an exact solution and proposes an accurate relation between the opening angle and bending tangent angle. Under the first order approximation, our formulations can reduced to the results of Yoshida and Wada and hence confirms the scientific correctness of Yoshida and Wada's work. Furthermore, this paper also derives a correct vertical displacement expression, and propose a new way of disassembly by bending for the first time and formulate a scaling law by data fitting. All formulations are validated by finite element simulation and experiment. The research here is helpful to the design of elastic snap fit or adjustable mechanical mechanism and metamaterial cell.

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1+1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1+2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1+2 dimensions.

Currently, various industrial processes are carried out in fluidized bed reactors, such as fermentation, water treatment, and algae growth. Knowing its internal dynamics is fundamental for the intensification of these processes. This work assesses the motion of fluidized calcium alginate spheres under the influence of an upward fluid flow within a 1.2 m high and 0.1 m inner diameter acrylic column. The Radioactive Particle Tracking technique is a proper methodology to study the internal dynamics of these kinds of equipment. Liquid-Solid Fluidized Bed is compared with the Gas-Liquid-Solid Fluidized Bed operation mode in terms of mixing behavior. Data gathered is analyzed in view of Shannon entropy as a dynamic mixing measure. Mixing times are found to be of the same order of magnitude. However, the Liquid-Solid Fluidized Bed achieves a lower homogenization degree. Also, we found good agreement between two criteria for assessing stability for the mixed state: the Shannon Entropy time-series plateau and the Glansdorff-Prigogine criterion based on Fisher information.

A way of systematizing different geometries by geometrical transformations is presented by using Klein’s perspective. Symmetry is considered the basic precedent, and different groups of transformations are introduced by using Erlangen program ideas. So, to focus on the importance of geometric transformations, two purposes are shown: first, the classification of geometries is based on the invariants of geometric transformations; and second, the construction of geometry unlike the theory of deduction can be done by means of geometric transformations.

The asymmetric skew divergence smooths one of the distributions by mixing it, to a degree determined by the parameter $\lambda$, with the other distribution. Such divergence is an approximation of the KL divergence that does not require the target distribution to be absolutely continuous with respect to the source distribution. In this paper, an information geometric generalization of the skew divergence called the $\alpha$-geodesical skew divergence is proposed, and its properties are studied.

Scouring is the most common cause of bridge failure. This study was conducted to evaluate the efficiency of the Artificial Neural Networks (ANN) in determining scour depth around composite bridge piers. The experimental data, attained in different conditions and various pile cap locations, were used to obtain the ANN model and to compare the results of the model with most well-known empirical, HEC-18 and FDOT, methods. The data were divided into training and evaluation sets. The ANN models were trained using the experimental data, and their efficiency was evaluated using statistical test. The results showed that to estimate scour at the composite piers, feedforward propagation network with three neurons in the hidden layer and hyperbolic sigmoid tangent transfer function was with the highest accuracy. The results also indicated a better estimation of the scour depth by the proposed ANN than the empirical methods.

Because only two variables are needed to characterize a simple thermodynamic system in equilibrium, any such system is constrained on a 2D manifold. Of particular interest are the exact 1-forms on the cotangent space of that manifold, since the integral of exact 1-forms is path-independent, a crucial property satisfied by state variables such as internal energy dE and entropy dS. Our prior work [1] shows that given an appropriate language of vector calculus, a machine can re-discover the Maxwell equations and the incompressible Navier-Stokes equations from data. In this paper, We enhance this language by including differential forms and show that machines can re-discover the equation for entropy dS given data. Since entropy appears in various fields of science in different guises, a potential extension of this work is to use the machinery developed in this paper to let machines discover the expressions for entropy from data in fields other than classical thermodynamics.

Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.

Primate vision is an active process that constructs a stable internal representation of the 3D world based on 2D sensory inputs that are inherently unstable due to incessant eye movements. We present here a mathematical framework for processing visual information for a biologically-mediated active vision stereo system with asymmetric conformal cameras. This model utilizes the geometric analysis on the Riemann sphere developed in the group-theoretic framework of the conformal camera, thus far only applicable in modeling monocular vision. The asymmetric conformal camera model constructed here includes the fovea’s asymmetric displacement on the retina and the eye’s natural crystalline lens tilt and decentration, as observed in ophthalmological diagnostics. We extend the group-theoretic framework underlying the conformal camera to the stereo system with asymmetric conformal cameras. Our numerical simulation shows that the 1 theoretical horopter curves in this stereo system are conics that well approximate the empirical longitudinal horopters of the primate vision system.

When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infra-red. Using the octonions as space-time coordinates, we construct a pre-spacetime, pre-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with $SU(2)_R$ chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time.

Ditching-rotary tiller was widely used in tillage and ditching furrow during rapeseed sowing. As its ditching furrow system (consist of four identical ditching plows) would bear great drag resistance from the soil during operation, the ditching plow was optimized based on differential geometry analysis and EDEM simulation. Resistance-velocity simulation indicated the plow with parabolic type guide curve had better reduce-drag characteristics. Differential geometry analysis indicated decay rate of E and growth rate of G could represent the variable regularity of drag resistance with the increase of velocity. The L value could represent the magnitude of drag resistance. The smaller L value, decay rate of E and growth rate of G, the better drag-reduce characteristics. Quadratic rotation orthogonal combination design shown the optimal structural parameter values were: Δθ=15°, θmin=45°, h=250mm, l=100mm. The minimum average drag resistance was 230.1N. The differential geometry analysis of optimal ditching plow shown |dE/dκ| and |dG/dκ| could obtain minimum value, indicating growth rate of drag resistance for optimal plow was the lowest within the experimental level. Field experiment indicated the operation effect of optimized ditching furrow system could met the requirements of rapeseed sowing. The research provided in-depth theoretical guidance for Drag-reduce design of plow.

Fractals offer the ability to generate fascinating geometric shapes with all sorts of unique characteristics (for instance, fractal geometry provides a basis for modelling infinite detail found in nature). While fractals are non-euclidean mathematical objects which possess an assortment of properties (e.g., attractivity and symmetry), they are also able to be scaled down, rotated, skewed and replicated in embedded contexts. Hence, many different types of fractals have come into limelight since their origin discovery. One particularly popular method for generating fractal geometry is using Julia sets. Julia sets provide a straightforward and innovative method for generating fractal geometry using an iterative computational modelling algorithm. In this paper, we present a method that combines Julia sets with dual-quaternion algebra. Dual-quaternions are an alluring principal with a whole range interesting mathematical possibilities. Extending fractal Julia sets to encompass dual-quaternions algebra provides us with a novel visualize solution. We explain the method of fractals using the dual-quaternions in combination with Julia sets. Our prototype implementation demonstrate an efficient methods for rendering fractal geometry using dual-quaternion Julia sets based upon an uncomplicated ray tracing algorithm. We show a number of different experimental isosurface examples to demonstrate the viability of our approach.

This article investigates on the radial and non-radial geodesic structures of the generalized K-essence Vaidya spacetime. Within the framework of K-essence geometry, it is important to note that the metric does not possess conformal equivalence to the conventional gravitational metric. This study employs a non-canonical action of the Dirac-Born-Infeld kind. In this work, we categorize the generalized K-essence Vaidya mass function into two distinct forms. Both the forms of the mass functions have been extensively utilized to analyze the radial and non-radial time-like or null geodesics in great details inside the comoving plane. Indications of the existence of wormhole can be noted during the extreme phases of spacetime, particularly in relation to black holes and white holes, which resemble the Einstein-Rosen bridge. In addition, we have also detected the distinctive indication of the quantum tunneling phenomenon around the central singularity.

Deep neural networks with powerful auto-optimization tools have been widely applied in various research fields, e.g., NLP and computer vision. However, existing neural network architectures typically are constructed using different inductive biases, e.g., preconceptions, expecting to decrease parameter search space during training, reduce computational cost or introduce expert knowledge in the neural network design. As an alternative, Multilayer Perceptron (MLP) provides much better freedom for exploration, has a lower inductive bias than convolutional neural networks (CNNs), and offers good flexibility in learning complex patterns. Even though, such neural architectures are commonly built in a flat Euclidean space, which is not necessarily the optimal space for any data, and is especially not good for modeling hierarchical correlations. Hyperbolic neural networks (HNNs) have gained attention for their ability to capture hierarchical structures present in complex data types like graphs. Recently, there has been an increasing interest to extend HNNs to computer vision tasks, motivated by the observations that images possess rich hierarchical relations. However, this is generally applied by employing a Euclidean backbone for learning higher-level semantic representations and only incorporating a hyperbolic classifier for classification, which, we argue, does not make full use of the advantage of hyperbolic space. Considering the recovery of the attention-free Multilayer Perceptron(MLP), in this paper, we extend it to non-Euclidean space and propose a novel architecture, named Hyperbolic Res-MLP (HR-MLP), that leverages fully hyperbolic layers to learn feature embeddings and perform image classification in an end-to-end fashion. With the help of the proposed Lorentz cross-patch and cross-channel layers, we can directly perform operations in the hyperbolic domain with fewer parameters, making it faster to train and providing comparatively better performance than its Euclidean counterpart. Experiments on CIFAR10, CIFAR100, and MiniImageNet demonstrate a comparable and superior performance when compared to Euclidean baselines. Our code is available at (https://github.com/Ahmad-Omar-Ahsan/HR-MLP)

As a radiant light source within the dynamic range of most spacecraft payloads, the moon pro-vides an excellent reference for on-orbit radiometric calibration. This research hinges on the pre-cise simulation of lunar spectral irradiances and the Earth-based Moon observation geometry. The paper leverages the Hapke model to simulate the temporal changes in lunar spectral irradi-ances, utilizing datasets obtained from Lunar Reconnaissance Orbiter Camera (LROC). The re-search also details the transformation process from the lunar geographic coordinate system to the instantaneous projection coordinate system, thereby delineating the necessary observational geometry. The insights offered by this study have the potential to enhance future in-orbit space-craft calibration procedures, thereby boosting the fidelity of data gathered from satellite obser-vations.

The present paper provides a new definition for dual spline curves in a geometric intuitive way based on adaptive curve refinement techniques. The dual spline is an implementation of the interpolatory subdivision scheme for curve modeling, which is comprised of polynomial segments of different degrees. Specially, the dual spline curves are mainly aims to solve the difficult geometry defeaturing problems in the existing computer-aided technology. Dual spline curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, continuous and discontinuous boundary approximation. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual spline is flexible and convenient for approximating a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of approximating the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method.

In this paper, we introduce the notions of nearly Sasakian and nearly Kähler statistical structures with a non-trivial example. The conditions for a real hypersurface in a nearly Kähler statistical manifold to admit a nearly Sasakian statistical structure are given. We also study Invariant and anti-invariant statistical submanifolds of nearly Sasakian statistical manifolds. At the end, some conditions under which such a submanifold of a nearly Sasakian statistical manifold is itself a nearly Sasakian statistical manifold are given.

During few last years, climate change including global warming which is attributed to human activities and also its long-term adverse effects on the planet’s functions have been identified as the most challenging discussion topics which have arisen many concerns and efforts to find the possible solutions. Since the warmth arising from Earth’s landscapes affects the world’s weather and climate patterns, we decided to study the changes in the Land Surface Temperature (LST) patterns in different seasons through non-linear methods. Here, we particularly want to estimate the non-integer dimension and fractal structure of the land surface temperature. For this study, the (LST) data has been obtained during the daytime by the Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Terra satellite. Depending on what time of the year data has been collected, temperatures change in different ranges. Since equatorial regions remain warm, and Antarctica and Greenland remain cold, and also because altitude affects temperature, we selected Riley County in the U.S. state of Kansas, which does not belong to any of this type locations and we are interested to observe the seasonal changes in temperature in this county. The results of the present study show that the Land Surface Temperature (LST) belongs to the class of fractal process with non-integer dimension.

Acid fracturing simulation is used widely to optimize carbonate reservoirs and improve acid fracturing treatment performance. In this study, a method was used to minimize the risk of the acid fracturing treatment. First, optimal fracture geometry parameters with UFD methods are calculated. After that, design components change as long as fracture geometry parameters reach their optimal values. The results showed a high flow rate needed to achieve optimal fracture geometry parameters with increasing acid volume. Sensitivity analysis was performed on controllable and reservoir parameters. It observed that a high flow rate should be applied for a low fluid viscosity to achieve the optimization goals. Straight acid reaches optimal conditions at a high flow rate and low volume. These conditions for retarded acids appear only at a low flow rate and high volume. The study of the acid concentration for gelled acid showed that as it increased, the flow rate and volume increased. Besides, for low permeability formation, a large fracture half-length and small fracture width are desirable. In this case, a higher flow rate will be required. The sensitivity analysis showed that the optimum flow rate and acid volume increase and decrease for the high Young's modulus. The effect of closure stress was also investigated and observed for a sample with high closure stress, low flow rate, and high acid volume are required.

The present paper investigates the influence of the specimen size of EN-AW6082 wrought aluminium alloy subjected to very high cycle fatigue (VHCF) tests. The hourglass specimens were tested under fully reversed loading condition, up to 10^9 cycles, by means of the ultrasonic fatigue testing machine developed by Italsigma® (Italy). Three specimens groups were considered, with a diameter in the middle cross-section ranging from 3 mm up to 12 mm. The stress field in the specimens was determined numerically and by strain gauge measurements in correspondence of the cross-section surface. The dispersion of experimental results has been accounted for, and data are reported in P-S-N diagrams. The decrease in fatigue resistance with increasing specimen size is evident. Theoretical explanation for the observed specimen-size effect is provided, based on Fractal Geometry concepts, allowing to obtain scale independent P-S*-N curves. The fatigue life expectation in the VHCF regime of the EN-AW6082 aluminium alloy full-scale components is rather overestimated if it is assessed only from standard small specimens of 3 mm in diameter. Experimental tests carried out on larger specimen, and a proper extrapolation, are required to assure safe structural design.

New class of conserved quantities is constructed. These quantities find direct application in mechanics of dissipative (generally non-conservative) dynamical systems. Approach demands formulation in the language of geometric mechanics, providing theoretical framework for situations with energy flow in and out of the system. As a by product, we suggest possibility of existence of Hamiltonian form for every autonomous ODE system, evolution of which is governed by non-potential generator of motion. Various examples are provided, ranging from physics and mathematics, to chemical kinetics and population dynamics in biology. Applications of these ideas in geometric integration techniques (GNI) of numerical analysis are discussed, and as an example of such, new discrete gradient-based numerical method is introduced.

Physical measurements might display range values extending towards infinite. The occurrence of infinity in physical equations, such as the black hole singularities, is a troublesome issue that causes many theories to break down when assessing extreme events. Different methods, such as re-normalization, have been proposed to avoid detrimental infinity. Here a novel technique is proposed, based on physical geometrical considerations and the Alexander Horned sphere, that permits to undermine infinity in physical equations. In this unconventional approach, a continuous monodimensional line becomes an assembly of countless bidimensional lines that superimpose in quantifiable knots and bifurcations.

The Heisenberg Lie algebra (HA) plays an important role in mathematics with Fourier transforms, as well as for the foundations of quantum theory where it expresses the operators of space-time, X, and their commutation rules with the momentum operators, D, that execute infinitesimal translations in X. Yet it is known that space-time is curved and thus the D operators must interfere thus giving “structure constants” that vary with location which suggests a mathematical generalization of the concept of a Lie algebra to allow for “structure constants” that are functions of X. We here investigate the mathematics of such a “generalized Heisenberg algebra” (GHA) which has “structure constants” that are functions of X and thus are in the enveloping algebra rather than constants. As expected, the Jacobi identity no longer holds globally but only in small regions of space-time where the [D, X] commutator can be considered locally constant and thus where one has a true Lie algebra. We show that one is able to reframe Riemannian geometry in this GHA. As an example, it is then shown that one can express the Einstein equations of general relativity as commutation rules. If one requires that the GHA commutator reduces to the HA of quantum theory in the limit of no curvature, then there are observable effects for quantum theory in this curved space time.

The conventional rolled-up model for carbon nanocones assumes that the cone is constructed from a rolled-up graphene sheet joined seamlessly, which predicts five distinct vertex angles. This model completely ignores any effects due to the changing curvature and all bond lengths and bond angles are assumed to be those for the planar graphene sheet. Clearly curvature effects will become more important closest to the cone vertex, and especially so for the cones with the smaller apex angles. Here we construct carbon nanocones which in the assembled cone are assumed to comprise bond lengths and bond angles which are, as far as possible, equal throughout the structure at the same distance from the conical apex. Predicted bond angles and bond lengths are shown to agree well with those obtained by relaxing the conventional rolled-up model using the LAMMPS software. The major objective here is not simply to model physically realisable carbon nanocones for which numerical procedures are far superior, but rather to produce an improved model that takes into account curvature effects close to the vertex, and from which we may determine an analytical formula which represents an improvement on that for the conventional rolled-up model.

The concept of smart morphing blades, which can control themselves to reduce or eliminate the need for active control systems, is a highly attractive solution in blade technology. In this paper an innovative passive control system based on Shape Memory Alloys (SMAs) is proposed. On the basis of previous thermal and shape characterization of a single morphing blade for a heavy-duty automotive cooling axial fan, this study deals with the numerical analysis of the aerodynamic loads acting on the fan. By coupling CFD and FEM approaches it is possible to analyze the actual blade shape resulting from both the aerodynamic and centrifugal loads. The numerical results indicate that the polymeric blade structure ensures proper resistance and enables shape variation due to the action of the SMA strips.

Aluminum alloy Al 2024-T3 were successfully joined using friction stir spot jwelding joining (FSSW). Satisfactory joint strengths were obtained at different welding parameters (tool rotational speed, tool plunge depth, dwell time) and tool pin profile (straight cylindrical, triangular and tapered cylindrical). Resulting joints were welded with welded zone. The different tools significantly influenced the evolution on the stir zone in the welds. Lap-shear tests were carried out to find the weld strength. Weld cross section appearance observations were also done. The macrostructure shows that the welding parameters have a determinant effect on the weld strength (x: the nugget thickness, y: the thickness of the upper sheet and SZ: stir zone). The main fracture mode was pull out fracture modes in the tensile shear test of joints. The results of tensile shear tests showed that the tensile-shear load increased with increasing rotational speed in the shoulder penetration depth of 0.2 mm. Failure joints were obrerved in the weld high shoulder penetration depth and insufficient tool rotation.

The real systems are often complex, nonlinear, and noisy in various areas including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of the complex system by analyzing the given time series. The SymEn estimation is a kind of entropy based on symplectic principal component analysis (SPCA) which represent organized but unpredictable behaviors of systems. The key to SPCA is to preserve the global submanifold geometrical properties of the systems through symplectic transform in the phase space, which is a kind of the measure-preserving transforms. The capability of preserving the global geometrical characteristics makes the SymEn a test statistic to detect the nonlinear characteristics in several typical chaotic time series and the stochastic characteristic in the Gaussian white noise. The results are in agreement with findings in the approximate entropy (ApEn), the sample entropy (SampEn) and the fuzzy entropy (FuzzyEn). Moreover, the SymEn method is also used to analyze the nonlinearities of the real signals (including the EEG signals for ASD and healthy subjects, and the sound and vibration signals for the mechanical systems). The results indicate that the SymEn estimation can be taken as a measure for describing the nonlinear characteristics in the data collected from the natural complex systems.

Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.

When the minimal length approach emerging from noncommutative Heisenberg algebra, generalized uncertainty principle (GUP), and thereby integrating gravitational fields to this fundamental theory of quantum mechanics (QM) is thoughtfully extended to Einstein field equations, the possible deformation of the metric tensor could be suggested. This is a complementary term combining the effects of QM and general relativity (GR) and comprising noncommutative algebra together with maximal spacelike four–acceleration. This deformation compiles with GR as curvature in relativistic eight–dimensional spacetime tangent bundle, generalization of Riemannian spacetime, is the recipe applied to derive the deformed metric tensor. This dictates how the affine connection on Riemannian manifold is straightforwardly deformed. We have discussed the symmetric property of deformed metric tensor and affine connection. Also, we have evaluated the dependence of a parallel transported tangent vector on the spacelike four–acceleration given in units of L, where L = rℏcG3 is a universal constant, c is speed of light, and ℏ is Planck constant, and G is Newton’s gravitational constant.

Hydraulic relationships are important for water resource management, hazard prediction, and modelling. Since Leopold first identified power law expressions that could relate streamflow to top-width, depth, and velocity, hydrologists have been estimating ‘At-a-station Hydraulic Geometries’ (AHG) to describe average flow hydraulics. As the amount of data, data sources, and application needs increase, the ability to apply, integrate and compare disparate and often noisy data is critical for applications ranging from reach to continental scales. However, even with quality data, the standard practice of solving each AHG relationship independently can lead to solutions that fail to conserve mass. The challenge addressed here is how to extend the physical properties of the AHG relations, while improving the way they are hydrologically addressed and fit. We present a framework for minimizing error while ensuring mass conservation at reach - or hydrologic Feature - scale geometries’(FHG) that complies with current state-of-the-practice conceptual and logical models. Through this framework, FHG relations are fit for the United States Geological Survey’s (USGS) Rating Curve database, the USGS HYDRoacoustic dataset in support of the Surface Water Oceanographic Topography satellite mission (HYDRoSWOT), and the hydraulic property tables produced as part of the NOAA/Oakridge Continental Flood Inundation Mapping framework. The paper describes and demonstrates the accuracy, interoperability, and application of these relationships to flood modelling and presents this framework in an R package.

Radiation Fireballs are singular phenomena which involve severe thermal radiation and consequently, they need to be duly assessed and prevented. Although the radiative heat transfer produced by a sphere is relatively well known, the shadowing measures implemented to control the fireball’s devastating effects have frequently posed a difficult analytical instance, mainly due to its specific configuration. In this article, since the usual solving equations for the said cases are impractical, the authors propose a novel graphic-algorithm method that sorts the problem efficiently for different kinds of obstructions and relative positions of the fireball and the defenses. Adequate application of this method may improve the safety of a significant number of facilities exposed to such risks.

This hypothesis is the result of my conviction that science will oneday prove everything in space and time is part of a unification. In "A Brief History of Time", Stephen Hawking wrote, "If everything in the universe depends on everything else in a fundamental way, it might be impossible to get close to a full solution (of the universe's puzzles) by investigating parts of the problem (such as general relativity and quantum mechanics) in isolation." The goal: to establish a “proof of concept” to which equations can be added. It’s concluded the Steady State, Big Bang, Inflation and Multiverse theories all ultimately fail and a topological model including bits (binary digits), Mobius strips, figure-8 Klein bottles and Wick rotation works better. The failed cosmologies have impressive points leading to the idea that they’re all necessary stepping-stones. For example, the Big Bang is seen here as violation of the 1st Law of Thermodynamics but its supposed origin from quantum fluctuations is reminiscent of bits switching between 1 and 0. The topological hypothesis has potential to explain dark matter, dark energy, and electromagnetic-gravitational union. Finally, the article introduces what is called vector-tensor-scalar geometry - and extensions of Einstein's Gravity and Maxwell's Electromagnetism.

The sTetro is a stair cleaning robot which can climb the staircase with its shapeshifting capabilities. As this robot is intended to traverse multi-floor environment autonomously, hence its localization/positioning information is an essential component of the overall system. Usually, the indoor mobile robots rely on some external system for localization information, e.g., WiFi, UWB, vision, RFID signals, or indoor Global Positioning System (GPS). This requires the installation of additional hardware and/or modification of the working environment for precise positioning information of the mobile platform. As the dimensions of the staircase are known a priori, this knowledge can be used to localize the sTetro robot on the stairs. In this article, the geometry of the staircase has been exploited to localize the robot in 3D space with measurements from the onboard time-of-flight (ToF) range sensors only. The heading angle of the robot is also estimated with two ToF sensors installed in front of the sTetro robot. Results achieved by conducting experiments on real robot prove the efficacy of the proposed approach.

This paper presents an extension for principles of Differential Geometry on Surfaces (re-hashed through the budding field of CMS, the Calculus of Moving Surfaces). It analyzes mostly 2D Hypersurfaces with Riemannian Geometry and proposes the construction of a 3D Static Frame combining the Surface Basis Vectors with the Orthogonal Normal Field as a 3D Orthogonal Vector Frame. The paper introduces conventions for manipulating Tensors defined using this 3D Orthogonal Vector Frame as well as Curvature Connections associated with this Vector Frame. It then finally introduces Symbols and Tensors to describe Inner Products and Variance within the 3D Vector Frame and then extends all the above concepts to a surface which is Dynamic utilizing principles from CMS. This formulation has potential to extend identities and concepts from CMS and from Differential Geometry in a compact Tensorial Framework, which agrees with the new Framework proposed by CMS.

Despite the successful reconciliation of special relativity and quantum mechanics (QM), a good century ago, general relativity (GR) in its current classical formulation still does not compile with QM. While GR gives precise sensical predictions at large scales, QM is obviously dominated at small scales! We think that the everlasting battle for exploring and understanding the Universe and for privileging a consistent perception of reality is therefore awaiting a consolidator rather than a conqueror. This should be capable of either unifying the two different benchmarks or at least bringing one closer to the other! The latter describes the consolidating approach, which is based on extending QM to relativistic energies and gravitational fields and generalizing Riemann to discretized Finsler geometry. Some properties of the additional geometric structures and curvatures which are disclosed by the proposed quantization and apparently overlooked in Einstein's GR, are analytically and numerically characterized. The analytical analyses introduce dynamics with possible nonlinear connections. We conclude that i) additional geometric structures and curvatures are intrinsic essential and ii) the spacetime at the quantum scale is no longer smooth or continuous. A maximal proper force is predicted, which maximally gravitationally accelerates a test particle and importantly manifests the quantum geometric nature of the generalized spacetime curvature.

A mixture of nonlabeled (14N2H4) and 15N labeled hydrazine (15N2H4) in an aqueous solution is oxidized to 15N2, 14N2, and 14N15N molecules, indicating the intermediate existence of the 14NH2-14NH-15NH-15NH2 with subsequent hydrogen transfers and splitting of side N-N bonds. Structures, thermodynamics, and electron characteristics of various N4H6 molecules in aqueous solutions are investigated using theoretical treatment at the CCSD/cc-pVTZ level of theory to explain the crucial part of the hydrazine oxidation reaction. Most N4H6 structures in aqueous solutions are decomposed during geometry optimization. Splitting the bond between central nitrogen atoms is the most frequent, but the breakaway of the side nitrogen is energetically the most preferred. The N-N fissions are enabled by suitable hydrogen rearrangements. Gibbs free energy data indicate the dominant abundance of NH3... N2... NH3 species. The side N atoms have very high negative charges, which should support hydrogen transfers in aqueous solutions. The only stable cyclo-(NH)4…H2 structure has a too high Gibbs energy and breaks the H2 molecule. The remaining initial cyclic structures are split into hydrazine and HN≡NH or H2N≡N species, and their relative abundance in aqueous solutions is vanishing.

We propose a promising model of discrete spacetime based on nonassociative geometry and complex networks. Our approach treats space as a simplicial 3-complex (or complex network), built from "atoms" of spacetime and entangled states forming n-dimensional simplices (n=1,2,3). At large scales, a highly connected network is a coarse, discrete representation of a smooth spacetime. We show that for high temperatures, the network describes disconnected discrete space. At the Planck temperature, the system experiences the phase transition, and for low temperatures, the space becomes a triangulated discrete space. We show that the cosmological constant depends on the universe’s topology. The "foamy" structure, analogous to Wheeler’s "spacetime foam," significantly contributes to the effective cosmological constant, which is determined by the Euler characteristic of the universe.

Central to an understanding of the physical nature of biosystems is an apprehension of their ability to control entropy dynamics in their environment.  To achieve ongoing stability and survival, living systems must adaptively respond to incoming information signals concerning matter and energy perturbations in their biological continuum (biocontinuum).  Entropy dynamics for the living system are then determined by the natural drive for reconciliation of these information divergences in the context of the constraints formed by the geometry of the biocontinuum information space.  The configuration of this information geometry is determined by the inherent biological structure, processes and adaptive controls that are necessary for the stable functioning of the organism.  The trajectory of this adaptive reconciliation process can be described by an information-theoretic formulation of the living system’s procedure for actionable knowledge acquisition that incorporates the axiomatic inference of the Kullback Principle of Minimum Information Discrimination (a derivative of Jaynes’ principle of maximal entropy). Utilizing relative information for entropic inference provides for the incorporation of a background of the adaptive constraints in biosystems within the operations of Fisher biologic replicator dynamics.  This mathematical expression for entropic dynamics within the biocontinuum may then serve as a theoretical framework for the general analysis of biological phenomena.

Eric Lord has observed: “In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant.” This paper proves Lord’s Observation. Constant-time algorithms are given for the segmented helix generated from the intrinsic properties of a stacked object and its conjoining rule. Standard results from screw theory and previous work are combined with corollaries of Lord’s observation to allow calculations of segmented helices from either transformation matrices or four known consecutive points. The construction of these from the intrinsic properties of the rule for conjoining repeated subunits of arbitrary shape is provided, allowing the complete parameters describing the unique segmented helix generated by arbitrary stackings to be easily calculated. Free-libre open-source interactive software and a website is provided which performs this computation for arbitrary prisms along with interactive 3D visualization . We prove that any subunit can produce a toroid-like helix or a maximally-extended helix, forming a continuous spectrum based on joint-face normal twist. This software, website and paper, taken together, compute, render, and catalog an exhaustive “zoo” of 28 uniquely-shaped platonic helices, such as the Boerdijk-Coxeter tetrahelix and various species of helices formed from dodecahedra.

A theoretical description of quantum jumps at the level of elementary particles is proposed, based on a micro-cosmological interpretation of their de Broglie phase. The third quantization formalism proposed in current literature for the description of baby universes in quantum cosmology is used here to describe the breakdown of unitarity in the transition from the pre-jump to the post-jump wave function. The corpuscular aspect manifested by the particle in the micro-interaction that originates the jump is represented by a pair of evanescent "micro-universes", respectively pre- and post-jump, connected by a wormhole. The latter represents the actual implementation of the interaction that leads to the projection on the outgoing state; this interaction is always local, even when the selected outgoing state is entangled. Therefore, the decoherence which leads to the emergence of classicality is originated by the same fundamental interactions of the Standard Model involved in the unitary evolution of the wave function. The objective nature of the reduction process admits implications on the possibility of using the formalism in the cosmological context, which are briefly discussed.

In this work, generalized geometry of second-order tangent groups and affine configuration complexes is proposed. Initially, geometry for higher weights n=4 and weights n=5 is presented through some interesting and suitable homomorphisms, finally, this geometry is extended and generalized for any weight n.

In our recently proposed theory of quantum gravity, a black hole arises from the spontaneous localisation of an entangled state of a large number of atoms of space-time-matter [STM]. Prior to localisation, the non-commutative curvature of an STM atom is described by the spectral action of non-commutative geometry. By using the techniques of statistical thermodynamics from trace dynamics, we show that the gravitational entropy of a Schwarzschild black hole results from the microstates of the entangled STM atoms and is given (subject to certain assumptions) by the classical Euclidean gravitational action. This action, in turn, equals the Bekenstein-Hawking entropy (Area/$4{L_P}^2$) of the black hole. We argue that spontaneous localisation is related to black-hole evaporation through the fluctuation-dissipation theorem.

Temperature field is an essential attribute of metal additive manufacturing in view of its bearings on the prediction, control, and optimization of residual stress, part distortion, fatigue, balling effect, etc. This work provides an analytical physics-based approach to investigate the effect of scan strategy parameters including time delay between two irradiations and hatching space on thermal material properties and melt pool geometry. This approach is performed through the analysis of the distribution of material properties and temperature profile in three-dimensional space. The moving point heat source approach is used to predict the temperature field. To predict the temperature field during the additive manufacturing process some important phenomena are considered. 1) Due to the high magnitude of temperature in the presence of the laser, the temperature gradient is usually high which has a crucial influence on thermal material properties. Consequently, the thermal material properties of stainless steel grade 316L are considered to be temperature-dependent. 2) Due to the repeated heating and cooling, part usually undergoes several melting and solidification cycles. This physical phenomenon is considered by modifying the heat capacity using the latent heat of melting. 3) The multi-layer aspect of metal AM process is considered by incorporating the temperature history from the previous layer since the interaction of the successive layers has an impact on heat transfer mechanisms. 4) Effect of heat affected zone on thermal material properties is considered by the superposition of material properties in regions where the temperature fields of two consecutive irradiations have an overlap since the consecutive irradiations change the behavior of the material properties. The goals are to 1) investigate the effects of temperature-sensitive material properties and constant material properties on the temperature field. 2) Study the behavior of thermal material properties under different scan strategies. 3) Study the importance of considering the effect of heat affected zone on thermal material through the prediction of melt pool geometry. 4) Investigate the effect of hatching space on melt pool geometry. This work is purely employed physics-based analytical models to predict the behavior of material properties and temperature field under different process conditions, and no finite element modeling is used.

In this paper we review a proposal to represent the geometric degrees of freedom of the gravitational field as a branched covering space, and introduce a new application of this in which the branch loci are 1- or 2-knots. This allows one to construct arbitrary smooth, closed 3- and 4-manifolds with enough geometric and topological information to write down a partition function and calculate statistical quantities in the thermodynamic limit. Further, we find clear evidence for a dimensional reduction of the spacetime geometry from four to two. As an example, we choose a family of smooth 4-manifolds presented in this way, and calculate the entropy of the system.

In this paper, we generalize the problems of finding simple polygons with the minimum area, maximum perimeter and maximum number of vertices so that they contain a given set of points and their angles are bounded by $\alpha+\pi$ where $\alpha$ ($0\leq\alpha\leq \pi$) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with the minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate the three generalized problems as nonlinear programming models, and then present a Genetic Algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.

Abstract: Characterization the 3-D structure of clouds is needed for a more complete understanding of the Earth's radiative and latent heat fluxes. Here we develop and explore a “ray casting” algorithm applied to the Multi-angle Imaging SpectroRadiometer (MISR) on board the Terra satellite, to reconstruct 3-D cloud volumes for observed clouds. The ray casting algorithm is first applied to geometrically simple synthetic clouds to show that, under the assumption of perfect, clear-conservative cloud masks, the reconstruction method yields overestimation whose magnitude depends on the cloud geometry and the resolution of the reconstruction grid relative to the image pixel resolution. The method is then applied to two select MISR scenes, fully accounting for MISR’s viewing geometry for reconstructions over the Earth’s ellipsoidal surface. The MISR Radiometric Camera-by-camera Cloud Masks at 1.1 km resolution and custom cloud masks at 275 m resolution independently derived from MISR RGB channels are used as input cloud masks. A wind correction method, termed “cloud spreading”, is devised and applied to the cloud masks to offset potential cloud movements over short time intervals (around 7 minutes at maximum) between the cameras. The MISR cloud top height product is used as a constraint to reduce the overestimation at the cloud top. The reconstruction results show that their uncertainty is significant when the wind correction is applied, and that they have more refined structures when the input cloud mask has a higher resolution. Recommendations for improving the presented cloud volume reconstructions as well as for future passive remote sensing satellite missions are discussed.

In a wireless network, locations of base stations (BSs)/access points (APs)/sensor nodes can be modeled based on stochastic processes, e.g., a Poisson point process (PPP) or a deterministic pattern planned ahead by providers. While deterministic deployment does not provide tractable interference analysis in general, PPP yields tractable analysis for interference. However, PPP allows APs to be deployed very close to each other and gives pessimistic results compared to the field measurements. In this study, in order to address this issue, Lloyd’s algorithm, which functions as a bridge between random and structural APs deployments, is investigated for analyzing coverage probability in a network. The link distance distribution is modeled as a mixture of Weibull distributions and its parameters are obtained by using the expectation-maximization (EM) algorithm for each iteration of Lloyd’s algorithm. The link distance distribution is further utilized for calculating the coverage probability approximately by exploiting the tractability of PPP.

Recently, federated learning (FL) has been receiving great attention as an effective machine learning method to avoid the security issue in raw data collection as well as to distribute the computing load to edge devices. However, even though wireless communications is a essential part for implementing FL in edge networks, there have been a few works that analyze the effect of wireless networks on FL. In this paper, we investigate FL in small-cell networks where multiple base stations (BSs) and users are located according to homogeneous Poisson point process (PPP) with different densities. We comprehensively analyze the effects of geographic node deployment on the model aggregation in FL on the basis of stochastic geometry-based analysis. We derive the closed-form expressions of coverage probability with tractable approximations and discuss the minimum required BS density for achieving a target model aggregation rate in small-cell networks. Our analysis and simulation results provide insightful information for understanding the behaviors of FL in small-cell networks, and it can be exploited as a guideline for designing the network facilitating the wireless FL.

There must exist a reformulation of quantum field theory which does not refer to classical time. We propose a pre-quantum, pre-spacetime theory, which is a matrix-valued Lagrangian dynamics for gravity, Yang-Mills fields, and fermions. The definition of spin in this theory leads us to an eight dimensional octonionic space-time. The algebra of the octonions reveals the standard model; model parameters are determined by roots of the cubic characteristic equation of the exceptional Jordan algebra. We derive the asymptotic low energy value 1/137 of the fine structure constant, and predict the existence of universally interacting spin one Lorentz bosons, which replace the hypothesised graviton. Gravity is not to be quantized, but is an emergent four-dimensional classical phenomenon, precipitated by the spontaneous localisation of highly entangled fermions.

In the world industry plant, solid and solid metals are always in contact even their motions not independent. Modeling of rolling die contact with slab primarily needs to describe the Tribology of contact phenomena. Consideration of continuum theory of rolling contact how a contact region is formed between rolling die and slab, and how the tangential force is distributed over the contact area with coefficient of friction is important. The central concern of numerical model is used in this work to indicate a set of equations, derived from the contact principle, that transfer the physical event into the mathematical equations including the laws of nature, such as newton’s laws, boundary conditions, state of stresses and their derivatives at particular time and locations. In this paper the elasticity stress behavior of rolling die contact with slab for number of cyclic loads is modeled. The model is including new proposed constitutive equations for discontinuity of the velocity, pressure distribution in rolling contact from the enter side to exit side of the neutral point. To verify the model, finite element simulation and experimental data from the literature are considered. The results show good agreement with finite element simulation and experimental data.

We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustrations of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairaut-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

The objective of this research is to conduct a comparative analysis between rectangular-shaped ASTM D3039 specimens with different angles (0°, 15°, and 90°) and various tensile test specimens based on ASTM and ISO standards, all 3D-printed using polyethylene terephthalate glycol (PETG) material through FDM. The study aimed to assesses the weak spot commonly found at the neck of the specimens. Two different printing orientations, namely flat and on-edge, were investigated, and a comprehensive examination and comparison were made regarding the variations in tensile strength, E-modulus, strain, and elongation at break among the tested samples. Additionally, the broken areas was evaluated, and a numerical study utilizing the finite element method (FEM) was conducted to identify stress risers' locations in each specimen type. The experimental results demonstrated that the ASTM D3039-0° specimen exhibited the best tensile properties when printed in the on-edge orientation, while the flat orientation yielded the best results in terms of the broken area. On the other hand, the ISO 527-2 specimens displayed the lowest tensile properties, regardless of the printing orientation. The study highlighted the enhanced tensile properties achieved with the rectangular shape. Specifically, the tensile strength of ASTM D3039-0° was 17.87% and 21% higher than that of the ISO 527 geometry shape for the flat and on-edge orientations, respectively. The numerical analysis indicated that the ISO 527-2 specimen had either no or minimal stress raisers, and the higher stresses observed in the narrow section were isolated from the gripping location.

When generalized noncommutative Heisenberg algebra accommodating impacts of finite gravitational fields as specified by loop quantum gravity, doubly–special relativity, and string theory, for instance, is thoughtfully applied to the eight-dimensional manifold (M8), the generalization of the Riemannian manifold becomes imminent. By constructing the deformed affine connections on a four-dimensional Riemannian manifold, we have determined the minimal length deformation of Riemann curvature tensor and its contractions, the Ricci curvature tensor, and Ricci scalar. Consequently, we have been able to construct the deformed Einstein tensor. As in Einstein’s classical theory of general relativity, we have proved that the covariant derivative of the deformed Einstein tensor vanishes, as well. We conclude that the minimal length correction suggests a correction to the spacetime curvature which is manifest in the additional curvature terms of the corrected Riemann tensor and its contractions. Accordingly, the spacetime curvature endows additional curvature and geometrical structure.

Physics-informed neural networks are a promising method to yield surrogate models of flow fields. We present a metamodeling technique for variable geometries based on physics-informed neural networks. The method was applied to the DU99W350 airfoil at a Reynolds number of 1×10⁵. The model predicted the Reynolds-averaged velocity and pressure field around the airfoil for arbitrary angles of attack between 10.0° and 17.5°. The model was trained with data from CFD simulations for a limited set of angles of attack. Additionally, satisfaction of the a priori known boundary conditions as well as the Reynolds-averaged Navier-Stokes equations were trained. A sensitivity analysis concerning the Reynolds number, the amount and distribution of training data, and the turbulence model was conducted showing the superiority of the pseudo-Reynolds stress method and the demand of labeled training data in the domain. The trained network was capable of predicting the developing flow separation on the suction surface and exhibited excellent agreement with CFD results even in the proximity to the wall for interpolations as well as extrapolations from the labeled data set.

There must exist a reformulation of quantum field theory, even at low energies, which does not depend on classical time. The octonionic theory proposes such a reformulation, leading to a pre-quantum pre-spacetime theory. The ingredients for constructing such a theory, which is also a unification of the standard model with gravitation, are : (i) the pre-quantum theory of trace dynamics – a matrix-valued Lagrangian dynamics, (ii) the spectral action principle of non-commutative geometry, (iii) the number system known as the octonions, for constructing a non-commutative manifold and for defining elementary particles via Clifford algebras, (iv) a Lagrangian with $E_8 \times E_8$ symmetry. The split bioctonions define a sixteen dimensional space (with left-right symmetry) whose geometry (evolving in Connes time) relates to the four known fundamental forces, while predicting two new forces, $SU(3)_{grav}$ and $U(1)_{grav}$. This latter interaction is possibly the theoretical origin of MOND. Coupling constants of the standard model result from left-right symmetry breaking, and their values are theoretically determined by the characteristic equation of the exceptional Jordan algebra of the octonions. The quantum-to-classical transition, precipitated by the entanglement of a critical number of fermions, is responsible for the emergence of classical spacetime, and also for the familiar formulation of quantum theory on a spacetime background.

We developed a novel approach in the field of spatiotemporal modelling, based on the spatialisation of time: the Timescape algorithm. It is especially aimed at sparsely distributed datasets in ecological research, whose spatial and temporal variability is strongly entangled. The algorithm is based on the definition of a spatiotemporal distance that incorporates a causality constraint and that is capable of accommodating the seasonal behaviour of the modelled variable as well. The actual modelling is conducted exploiting any established spatial interpolation technique, substituting the ordinary spatial distance with our Timescape distance, thus sorting, from the same input set of observations, those causally related to each estimated value at a given site and time. The notion of causality is expressed topologically and it has to be tuned for each particular case. The Timescape algorithm originates from the field of stable isotopes spatial modelling (isoscapes), but in principle it can be used to model any real scalar random field distribution.

A universally applicable method for the prediction of the isobaric heat capacities of the liquid and solid phase of molecules at 298.15 K is presented, derived from their “true” volume. The molecules’ “true” volume in A³ is calculated on the basis of their geometry-optimized structure and the Van-der-Waals radii of their constituting atoms by means of a fast numerical algorithm. Good linear correlations of the “true” volume of a large number of compounds encompassing all classes and sizes with their experimental liquid and solid heat capacities over a large range have been found, although noticeably distorted by intermolecular hydrogen-bond effects. To take account of these effects, the total amount of 1303 compounds with known experimental liquid heat capacities has been subdivided into three subsets consisting of 1102 hydroxy-group-free compounds, 164 monoalcohols/monoacids, and 36 polyalcohols/polyacids. The standard deviations for Cp(liq,298) were 20.7 J/mol/K for the OH-free compunds, 22.91 J/mol/K for the monoalcohols/monoacids and 16.03 J/mol/K for the polyols/polyacids. Analogously, 797 compounds with known solid heat capacities have been separated into a subset of 555 OH-free compounds, 123 monoalcohols/monoacids and 119 polyols/polyacids. The standard deviations for Cp(sol,298) were calculated to 23.14 J/mol/K for the first, 21.62 J/mol/K for the second, and 19.75 J/mol/K for the last subset. A discussion of structural and intermolecular effects influencing the heat capacities as well as of some special classes, in particular hydrocarbons, ionic liquids, siloxanes and metallocenes, has been given. In addition, the present method has successfully been extended to enable the prediction of the temperature dependence of the solid and liquid heat capacities in the range between 250 and 350 K.

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes, this enables a comparison between curves’ shapes and allows then the classification of processes’ measure.

Robot workpiece machining is interesting in industry since it offers some advantages, such as higher flexibility in comparison with the conventional approach based on the CNC technology. However, in recent years we have been facing a strong progressive shift to custom based manufacturing and low volume/high mix production, which require a novel approach to automation by the employment of collaborative robotics. However, collaborative robots feature only limited motion capability, to provide safety in cooperation with human workers. Thus, it is highly necessary to perform more detailed robot task planning to ensure its feasibility and optimal performance. In this paper, we deal with the problem of studying kinematic robot performance in the case of such manufacturing tasks, where the robot tool is constrained to follow the machining path embedded on the workpiece surface at a prescribed orientation. The presented approach is based on the well-known concept of manipulability, although the latter suffers from physical inconsistency due to mixing different units of linear and angular velocity in a general 6 DOF task case. Therefore, we introduce the characteristics of the workpiece surface constraint in the robot kinematic analysis, that enables evaluation of its available velocity capability in a reduced dimension space. Such constrained robot kinematics transform the robot`s task space to a two-dimensional surface tangent plane, and the manipulability analysis may be limited to the space of linear velocity only. Thus, the problem of physical inconsistency is avoided effectively. We show the theoretical derivation of the proposed method, which was verified by numerical experiments.

We review the main features of the non-local gauge, named the contour gauge. The contour gauge belongs to the axial type of gauges and extends the local gauge used in the most of approaches. The geometry of gluon fields and the path-dependent formalism are the essential tools for the description of non-local gauges. The principle feature of the contour gauge is that there are no the residual gauges which are left in the finite domain of space. In the review, we present the useful correspondence between the contour gauge conception and the Hamiltonian (Lagrangian) formalism. The Hamiltonian formalism is turned out to be a very convenient framework for the understanding of contour gauges. The comprehensive comparison analysis of the local and non-local gauges advocates the advantage of the contour gauge use. As an example of practical worth, we consider the Drell-Yan process and discuss the gauge invariance of the corresponding hadron tensor. We show that the appropriate use the contour gauge leads to the existence of extra diagram contributions. These additional contributions, first, restore the gauge invariance of the hadron tensor and, second, give the important terms for the observable quantities. We also demonstrate the significant role of the additional diagrams to form the relevant contour in the Wilson path-ordered exponential. Ultimately, it leads to the spurious singularity fixing. Moreover, in the present review, we discuss in detail the problem of spin and orbital angular momentum separation. We show that in SU(3) gauge theories the gluon decomposition on the physical and pure gauge components has a strong mathematical evidence provided the contour gauge conception has been used. In addition, we prove that the contour gauge possesses the special kind of residual gauge that manifests at the boundary of space. Besides, the boundary field configurations can be associated with the pure gauge fields.

Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s Theorem from analytic Euclidean geometry into the Poincar´e ball model of analytic hyperbolic geometry, which is based on M¨obius addition. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of the hyperbolic trigonometry, called gyrotrigonometry, to which the Poincar´e ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.

A high-voltage transmission tower consists of structures to avoid the risk of electric shock and to prevent the risk of collapse. Hence, towers are generally designed to be high-rise for efficiency. The main posts of the tower are primary structural members that resist loads under various load conditions. Therefore, the contribution of the secondary member to securing the stiffness and strength of the main posts by reducing the effective buckling length is important. However, there are no detailed design criteria for secondary members. In this study, the structural effects of horizontal members and braces on the torsional stiffness, elastic buckling strength, and load-carrying capacity of transmission towers were observed through various structural analysis methods including linear-elastic, eigenvalue, and geometric nonlinear and inelastic analyses under governing load combinations. According to the analytical study, rather than the horizontal members, the brace spacing significantly affects the structural performance. Therefore, the number of horizontal members can be minimized if sufficient brace members are erected. If the brace spacing is wide, it is recommended that the horizontal members be erected to create K bracing, and the buckling resistance of the main posts can be thus enhanced.

The novel coronavirus has presented specimen-access challenges to geoscientific researchers, including paleobiologists interested in fossil ammonoids. Ammonoid sutures are geometric patterns formed by the intersection of the septa and the shell wall, and have long been a diagnostic tool for ammonite researchers for such applications as species identification, taxonomic relationships, ontogenetic change, functional and evolutionary morphology, and other aspects of ammonoid paleobiology. Without access to specimens in museum and institutional collections, researchers must rely on previously published illustrations and photographs of ammonoid sutures. However, many of these illustrations were published decades ago without an index of scale. Suture tracings lacking a scale bar are not usable by researchers interested in applying the quantitativeness of fractal geometry to the interpretation of septal complexity. Additonally, distortion of the marginal elements of suture geometry occurs due to shell curvature near the venter and umbilicus. The revised approach described here eliminates the problem of missing scale information in fractal analysis of ammonite sutures, making use of just the lateral lobe and adjacent saddle. Our revised method’s non-requirement of a full hemisuture also facilitates comparisons among sutures within an ontogenetic sequence, or sutures from multiple ammonite taxa.

We explore several ways, in order to include fermionic structures naturally in a physical theory in C⁴. We begin with the standard Dirac formalism and we proceed by using Cartan's property of triality as a second option. Afterwards, we suggest a new approach (in a preliminary basis), by introducing an 1-linear form, as the "square root of the geometry" derived by the usual 2-linear forms (quadratic forms). Keeping this way, we introduce n-linear forms, in order to formulate a new geometric structure, which could be suitable for the formulation of a pure geometric unied theory.

In the fifth generation communication system, millimeter wave (mmWave) networks will coexist with traditional micro wave (μWave) networks, which allows for higher data transmission rate and better user experience. In this paper, we give a comprehensive framework of mathematical models and analytical methods for multi-tier mmWave and μWave hybrid caching networks based on stochastic geometry. We propose an association strategy by assuming average biased-received power association and Rayleigh fading. Accordingly, by using a D-ball approximating blockage model of mmWave networks, expressions of the cell association probability and the coverage probability of the hybrid networks are derived. Also, we use the average successful delivery probability as the performance metric to analyze the existing caching placement strategies. Simulation results validate the accuracy of our analytical conclusions.

We present a summary of the main results within the Scale Invariant Vacuum (SIV) paradigm based on the Weyl Integrable Geometry (WIG) as an extension to the standard Einstein General Relativity (EGR). After a brief review of the mathematical framework, where we also highlight the connection between the weak-field SIV equations and the notion of un-proper time parametrization within the reparametrization paradigm, we continue with the main results related to early Universe; that is, applications to inflation, Big Bang Nucleosynthesis, and the growth of the density fluctuations within the SIV. In the late time Universe the applications of the SIV paradigm are related to scale-invariant dynamics of galaxies, MOND, dark matter, and the dwarf spheroidals where one can find MOND to be a peculiar case of the SIV theory. Finally, within the recent time epoch, we highlight that some of the change in the length-of-the-day (LOD), about 0.92 cm/yr, can be accounted for by SIV effects in the Earth-Moon system.

We present a summary of the main results within the Scale Invariant Vacuum (SIV) paradigm as related to the Weyl Integrable Geometry. After a brief review of the mathematical framework, we will highlight the main results related to inflation within the SIV [9], the growth of the density fluctuations [8], and the application of the SIV to scale-invariant dynamics of Galaxies, MOND, Dark Matter, and the Dwarf Spheroidals [7]. The connection of the weak-field SIV results to the un-proper time parametrization within the reparametrization paradigm is also discussed [14].

Here we studied Bianchi type-III string cosmological models generated by means of a cloud of strings with particles connected to them in the framework of Lyra geometry considering five-dimensional space-time. To obtain the exact solutions of field equations we consider that the shear scalar and the scalar expansion are proportional, σ∞θ which leads to D=cn and secondly we adopt the assumption considering the Reddy String Condition, ρ+λ=0. From the two different cases obtained here, first case leads to the Bianchi type-III string cosmological model in Lyra geometry in five-dimensional space-time and the second case leads to the five-dimensional vacuum universe in general relativity. The Geometrical and physical properties of the model universe are studied comparing with the present day’s observational findings. The model universe obtained here starts with the big bang and as time progresses both particle density ρp and energy density ρ decreases with expansion of our Universe.

The nuclear lamina consists of a dense fibrous meshwork made of nuclear lamins, Type V intermediate filaments, and is ~14 nm thick according to recent cryo-electron tomography studies. Recent advances in light microscopy have extended the resolution to a scale allowing for the fine structure of the lamina to be imaged in the context of the whole nucleus. We review quantitative approaches to analyze the imaging data of the nuclear lamina as acquired by structured illumination microscopy (SIM) and single molecule localization microscopy (SMLM), as well as the requisite cell preparation techniques. In particular, we discuss the application of steerable filters and graph based methods to segment the structure of the four mammalian lamin isoforms (A, C, B1, and B2) and extract quantitative information.

As noted in the introductory quotation, an ideal map was long ago seen as the map of the map, the map of the map, of the map, and so on endlessly. This recursive perspective on maps, however, has received little attention in cartography. Cartography, as a scientific discipline, is essentially founded on Euclidean geometry and Gaussian statistics, which deal with respectively regular shapes, and more or less similar things. It is commonly accepted that geographic features – such as rivers, cities, streets and building – are not regular and that the Earth’s surface is full of fractal or scaling or living phenomena with far more small things than large ones at different levels of scale. This paper argues for a new paradigm in mapping, based on fractal or living geometry and Paretian statistics, and – more critically – on the new conception of space, conceived and developed by Christopher Alexander, that space is neither lifeless nor neutral, but a living structure capable of being more living or less living. The fractal geometry is not limited to Benoit Mandelbrot’s framework, but is extended towards Christopher Alexander’s living geometry and based upon the third definition of fractal: A set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times. Paretian statistics deals with far more small things than large ones, so it differs fundamentally from Gaussian statistics, which deals with more or less similar things. Under the new paradigm, I make several claims about maps and mapping: (1) Topology of geometrically coherent things – in addition to that of geometric primitives – enables us to see a scaling or fractal or living structure; (2) Under the third definition, all geographic features are fractal or living, given the right perspective and scope; (3) Exactitude is not truth – to paraphrase Henri Matisse – but the living structure is; and (4) Töpfer’s law is not universal, but scaling law is. All these assertions are supported by evidence, drawn from a series of previous studies. This paper demands a monumental shift in perspective and thinking from what we are used to on the legacy of cartography and GIS.

The crack geometry and associated strain field around Berkovich and Vickers indents on silicon have been studied by X-ray diffraction imaging and micro-Raman spectroscopy scanning. The techniques are complementary, the Raman data coming from within a few micrometers of the indentation, whereas the X-ray image probes the strain field at a distance of typically tens of micrometers. For example, Raman data provides an explanation for the central contrast feature in the X-ray images of an indent. Strain relaxation from breakout and high temperature annealing are examined and it is demonstrated that millimeter length cracks, similar to those produced by mechanical damage from misaligned handling tools, can be generated in a controlled fashion by indentation within 75 micrometers of the bevel edge of 200mm diameter wafers.

We exploit the properties of complex time to obtain an analytical relationship based on considerations of causality between the two conserved quantities of a system’s Hamiltonian and its Entropy Production. In natural units, when complexified, the one is simply the Wick-rotated complex-conjugate of the other. A Hilbert transform relation is constructed in the formalism of Quantitative Geometrical Thermodynamics which enables system irreversibility to be handled analytically within a framework that unifies both the microscopic and macroscopic scales, and which also unifies the treatment of both reversibility and irreversibility as complementary parts of a single physical description. In particular the thermodynamics of two unitary entities are considered: the alpha particle which is absolutely stable (that is, trivially reversible with zero entropy production), and a black hole whose absolute disequilibrium (unconditional irreversibility) is characterized by a non-zero entropy production for which we show an alternate derivation confirming our previous one (Universe 7, 2021, 325). The thermodynamics of a canonical decaying harmonic oscillator are also considered. In this treatment the complexification of time also enables a meaningful physical interpretation of both “imaginary time” and “imaginary energy”.

We have shown that according to the predictions of the standard quantum theory, signaling is possible using a postponed compensation Hong-Ou-Mandel interferometer, although it violates Lorentz invariance. We argue that new physics is required to deal with such a paradox. Either quantum principles or relativity needs reassessment. We propose an experiment that can decide whether relativity rules over quantum or vice versa.

Purpose – Extend the Blade Element Momentum Theory (BEMT) such that rotors with pronounced cone and axis angle (tilt or yaw) can be calculated. Derive an equation for the speed ratio (lambda) as a function of Tip Speed Ratio (TSR), radius, blade, cone and axis angle. This converts the BEMT into an Unsteady BEMT or UBEMT. Present the Wagner rotor as one such rotor geometry. --- Methodology – Literature review and calculations. --- Findings – The UBEMT can be used to calculate highly unconventional rotor geometries. --- Research Limitations – Although the aerodynamic coefficients used in the UBEMT are from measurements in steady flow conditions, they can be used with success. --- Practical Implications – Also conventional Horizontal Axis Wind Turbines (HAWT) with noticeable cone and axis angle should be calculated with the UBEMT. The accuracy of power calculations of these HAWTs can be slightly improved. --- Originality – Analytic equations for rotors with cone and axis angle have barely been discussed.

The goal of this work is to show that the underlying symmetries of effective field theory can be traced to the onset of self-similarity. In particular, we argue that the scale-free structure of fractal geometry lies at the heart of invariance principles in classical and Quantum Field Theory.

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group $G$ with relations. A valid subgroup $H$ of index $d$ in $G$ leads to a \lq magic' state $\left|\psi\right\rangle$ in $d$-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite \lq contextual' geometry. In the present work, we choose $G$ as the fundamental group $\pi_1(V)$ of an exotic $4$-manifold $V$, more precisely a \lq small exotic' (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $\mathbb{R}^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (a) it shows the occurence of standard contextual geometries such as the Fano plane (at index $7$), Mermin's pentagram (at index $10$), the two-qubit commutation picture $GQ(2,2)$ (at index $15$) as well as the combinatorial Grassmannian Gr$(2,8)$ (at index $28$) , (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$'s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of \lq quantum gravity'.

In this work, we study the weak gravitational lensing in the background of Kerr-Newman black hole with quintessential dark energy. Initially, we compute the deflection angle of light by charged black hole with quintessential dark energy by utilizing the Gauss-Bonnet theorem. Firstly, we suppose the light rays on the equatorial plane in the axisymmetric spacetime. In doing so, we first find the corresponding optical metrics and then calculate the Gaussian optical curvature to utilize in Gauss-Bonnet theorem. Consequently, we calculate the deflection angle of light for rotating charged black hole with quintessence. Additionally, we also find the deflection angle of light for Kerr-Newman black hole with quintessential dark energy. In order to verify our results, we derive deflection angle by using null geodesic equations which reduces to the deflection angle of Kerr solution with the reduction of specific parameters. Furthermore, we analyze the graphical behavior of deflection angle $\Theta$ w.r.t to quintessence parameter $\alpha$, impact parameter $b$, BH charge $Q$ and rotation parameter $a$. Our graphical analysis retrieve various results regarding to the deflection angle by the Kerr-Newman black hole with quintessential dark energy.

This paper presents a novel real time control strategy on four-wheel independent steering system for electric vehicle with stepper motors using neural networks combined with fuzzy logic. The control of the four wheels is based on a variation of Hopfield Neural Network (VHNN) method, in which the input is the error of each steering motor and the output is processed by a hyperbolic tangent function (HTF) feeding the fuzzy logic controller (FLC), which ultimately drives the stepper motor. The whole system is made up of the four aforementioned blocks where they work in sync and inseparable from each other with the common goal of driving all the steering stepper motors at the same time. Simulation results demonstrate that the proposed control method has strong robustness and practicability under different angles and speeds.

This research endeavors to illuminate the path toward efficient and optimized propeller design by delving into the realm of aerodynamics. Propellers, as vital mechanical marvels, constitute the core of numerous propulsion systems, spanning from aircraft and marine vessels to wind turbines. The pursuit of enhanced propeller efficiency and performance holds paramount importance in industries where propulsion assumes a critical role, underscoring the indispensable need for a profound understanding of aerodynamics. Through meticulously crafted theoretical insights and their practical applications, this study embarks on a journey into the fascinating realm of propeller aerodynamics. While the paper primarily addresses the aerodynamic aspects of propeller design and optimization, it serves as a foundational guide for those seeking to comprehend the intricacies of propeller engineering.The research journey commences with an in-depth exploration of the fundamental principles governing propeller performance. It unveils the intricate dynamics of airflow surrounding propeller blades, shedding light on their proficiency in generating thrust, lift, and torque. The bedrock of the theoretical insights is a comprehensive analysis encompassing airfoil profiles, blade geometry, and the underlying principles of blade element theory. In navigating the ensuing pages, the study embarks on the task of demystifying the intricate methods and calculations that propel the design and optimization of propellers. Leveraging cutting-edge methodologies, the research systematically assesses the aerodynamic characteristics of propellers across various operational conditions. These invaluable insights empower the tailoring of propeller designs with precision, aligning them seamlessly with specific performance objectives. The aim is not only to impart a comprehensive understanding of propeller aerodynamics but also to provide actionable insights for engineers and researchers striving to enhance propeller efficiency. This research endeavor serves as a guiding light for both propeller enthusiasts and professionals alike. It not only enriches the broader discourse on propeller technology but also equips readers with the requisite knowledge and tools to embark on their own quests for optimized propeller design. As propulsion technologies continue their evolution, the steadfast principles elucidated herein will persist, guiding future innovations and advancements in the realm of propeller engineering. This paper serves as a profound resource, offering new researchers a comprehensive understanding of propellers and their intricate design processes.

Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi Polygons; Delaunay Triangulations; Perpendicular Bisectors; Circumcenters; Convex Hulls; Minimal Spanning Trees; Ulam Trees; Pitteway Violations; Circularity; Clark-Evans spatial statistics; Variance to Mean Ratios; Gabriel Graphs; and, Minimal Spanning Trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis’s Law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch’s Law (the number of edges of convex polygons are positively correlated with the perimeters of these polygons); and Errara’s Law (daughter cell areas should be roughly half that of their parent cells’ areas). We introduce a new Pitteway Law that the number of sides of the convex polygons in a Voronoi tessellation of biological epithelia is proportional to the minimal interior angle of the convex polygons as angles less than 90 degrees result in Pitteway violations of the Delaunay dual of the Voronoi tessellation.