The manufacturing geometrical variability in axial compressors is a stochastic source of uncertainty, implying that the real geometry differs from the nominal design. This causes the real geometry to lose the ideal axial symmetry. Considering the aerofoils of a stator vane, the geometrical variability affects the flow traversing it. This impacts the downstream rotor, especially when considering the aeroelastic excitation forces. Optical surface scans coupled with a parametrization method allow for acquiring the information relative to the real aerofoils geometries. The measured data are included in a multi-passage and multi-stage CFD setup to represent the mistuned flow. In particular, low excitation harmonics on the rotor vane are introduced due to the geometrical deviations of the upstream stator. The introduced low engine orders as well as their amplitude depend on the stator geometries and their order. A method is proposed to represent the phenomena in a reduced CFD domain, limiting the size and number of solutions required to probabilistically describe the rotor excitation forces. The resulting rotor excitation forces are reconstructed as a superposition of disturbances due to individual stator aerofoils geometries. This indicates that the problem is linear in in the combination of disturbances from single passages.

Based on a measuring system to determine the statistical heterogeneity of individual polygons we propose a method to use polygonal shape patterns as a source of data in order to determine the Shannon entropy of biological organizations. In this research, the term entropy is a particular amount of data related with levels of spatial heterogeneity in a series of different geometrical meshes and sets of random polygons. We propose that this notion of entropy is important to measure levels of information in units of bits, measuring quantities of heterogeneity in geometrical systems. In fact, one important result is that binarization of heterogeneity frequencies yields a supported metric to determine geometrical information from complex configurations. Thirty-five geometric aggregates are tested; biological and non-biological, in order to obtain experimental results of their spatial heterogeneity which is verified with the Shannon entropy parameter defining low particular levels of geometrical information in biological samples. Geometrical aggregates (meshes) include a spectrum of organizations ranging from cell meshes to ecological patterns. Experimental results show that a particular range (0.08 and 0.27) of information is intrinsically associated with low rates of heterogeneity. We conclude it as an intrinsic feature of geometrical organizations in multi-scaling biological systems.

The Principle of Indifference (“PI”: the simplest non-informative prior in Bayesian probability) has been shown to lead to paradoxes since Bertrand (1889). Von Mises (1928) introduced the “Wine/Water Paradox” as a resonant example of a “Bertrand paradox”, and which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes by a Maximum Entropy (MaxEnt) treatment of the PI that also includes information provided by Benford’s “Law of Anomalous Numbers” (1938). We show that the PI should be understood to represent a family of informationally-identical MaxEnt solutions; each solution being identified with its own explicitly justified boundary condition. In particular, our solution of the Wine/Water Paradox exploits Benford’s Law to construct a non-uniform distribution representing the universal constraint of scale invariance, which is a physical consequence of the Second Law of Thermodynamics.

We investigate the behavior of the solutions of the ray equation in isotropic media. We discuss local and global transversals and wavefronts and give examples of rays without wavefronts.

In this paper, we will describe the general theory of relativity. The special theory of relativity is based on the ideal inertial frames. These ideal inertial frames are assumed to exist in every event in space-time. In contrast, in the general theory of relativity, we do not have to make that assumption. Additionally, the space-time is considered to be part of the solution to the motion. Moreover, interpretation of geometrical quantities is introduced.

Today, mechanical tracking systems have been downsized to allow them to be used in the field of airborne laser communications and in the military domain. Risley systems are used for this purpose, which work by directing a beam of light to a given target point, this procedure is commonly known as the inverse problem. In this paper, an analytical method, the geometric method, has been designed and developed to determine the beam steering in a Risley system and solve the inverse problem. The method focuses on different geometric shapes, like circumference or ellipse, that are described when the beam passes through the second prism. The accuracy and efficiency of the geometric method has been analysed and found to be faster than the two-step method. Furthermore, the geometric method has been implemented in an iterative process and an accuracy of up to 1 pm has been achieved. This high accuracy would allow the geometric method to be applied in fields such as lithography, stereolithography or 3D printers.

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.

This paper presents the light scattering matrices of atmospheric aggregated hexagonal ice particles appearing in cirrus clouds. In this work aggregates consist of the same particles with different spatial orientation and number of these particles. Two types of particle shape were studied: (1) hexagonal column; (2) hexagonal plate. For both shapes we study compact and non-compact cases of arrangement of particles in aggregates. As a result, four sets of aggregates were made: (1) compact columns; (2) non-compact columns; (3) compact plates; (4) non-compact plates. Each set consists of eight aggregates with different number of particles from 2 to 9. For practical reason the bullet-rosette and the aggregate of hexagonal columns with different sizes was also calculated. The light scattering matrices were calculated for the case of arbitrary spatial orientation within the geometrical optics approximation for sets of compact and non-compact aggregates and within the physical optics approximation for two additional aggregates. It was found that light scattering matrix elements for aggregates are depend on arrangement of particles they are consisted.

Based on a measuring system to determine levels of spatial organization in 2D polygons (homogeneous or heterogeneous partition of defined areas) lying on principles of regularity, we propose the entropy term linked to the concept of “information”, from the “information theory field”, in order to obtain an information measurement regarding a quantity of or amount of information in the architecture of complex 2D biological organizations. The term “quantity” does not refer to the amount of data (size), but to the probability of a geometrical basic pattern within a set of possible statistical configurations regarding levels of homogeneity and heterogeneity. It is this notion of information that is important in information theory, and measures of information in units of bits, what we propose to use for measuring quantities of organization in the architecture of complex geometrical systems. Two complex systems are tested, biological and non biological in order to obtain experimental results, which are verified with the evaluation criteria “entropy”. Experimental results show that the lowest levels of information and entropy, in addition with low rates of heterogeneity and high rates of homogeneity are particular features of geometrical organizations in biological systems.

Optinalysis, as a method of symmetry detection, is a new advanced computational algorithm that intrametrically (within elements) or intermetrically (between elements) computes and compares two or more multivariate sequences in an unclustered or clustered manner as a mirror-like reflection of each other (optics-like manner), hence the name is driven. Optinalysis is based by the principles of reflection and moment about a symmetrical line which detects symmetry that reflects a similarity measurement. Optinalysis is suitable for quantitative and qualitative data types, with or without replications, provided it conform the algorithmic requirements there provided. Optinalysis can be organized for geometrical, geostatistical and statistical analysis in one-way, two-way, or three-way approach. A simulation comparisons shows that Optinalysis is a simple alternative approach of multivariate analysis of sociometric, demographic, socio-demographic, psychometric, ecological, experimental, genomic, nanoparticle and shape morphometric data. Optinalysis of these data matrix shows very similar results or conclusions with some multivariate analysis such as skewness measure, one-way ANOVA, paired t-test, one sample t-test, Tukey’s multiple comparisons, BLAST sequence algorithmic analysis (percentages of identity, similarity, gabs, and positives, and the Needleman-Wunsch score), and Riemannian distance.

This study analyzes the modeling and dynamics of a novel passive in Multi-Degree-of-Freedom (MDOF) vibration isolation platform which can achieve significant isolation effect. Symmetrical Scissor-Like structures (SLSs) are utilized in the proposed MDOF isolation platform as the supporting and isolation elastic components. Based on the mathematical modeling and theoretical analysis of the MDOF vibration isolation system with SLSs, the effect of structural parameter and joint friction on stiffness and damping properties is investigated. It is shown that due to geometric relations within the SLSs, the natural frequencies can be reduced via adjusting structural parameters of the SLS for different direction vibration isolation. Theoretical and experimental results show that the SLS isolation platform can achieve much better loading capacity and vibration isolation performance simultaneously by only using linear passive components because of the MDOF adjustable stiffness property. Therefore, with low costing and energy consumption, the proposed novel isolation platform can provide the improvement of vibration suppression in various engineering practices.

The Sun is a frame at relative rest in which sunlight travels at the emitted speed c. Earth travels at the revolving speed v in this frame. The reflection of light as a mechanical phenomenon applies to the modified Michelson interferometer employed by Miller in his experiments with light from the Sun. Unlike the Tomaschek experiments, which use light from stars that may travel in the Universe at velocities different from that of the Sun, the fringe shifts in the Miller experiments are predictable. Based on Michelson's derivation, Miller expected in his experiments at Mount Wilson a 1.12 fringe shift and observed a fringe shift of 0.08 in 1921 and 0.088 in 1925. The reflection of light as a mechanical phenomenon predicts zero fringe shift for Miller's experiment agreeing only with his observations at the Cleveland laboratory in 1924.

Maximum hardness and hardened depth are the responses of interest in relation to the laser hardening process. These values define heat treatment quality and have a direct impact on mechanical performance. This paper aims to develop models capable of predicting the shape of the hardness profile depending on laser process parameters for controlling laser hardening quality (LHQ), or rather the response values. An experimental study was conducted to highlight hardened profile sensitivity to process input parameters such as laser power (P_L), beam scanning speed (V_S) and initial hardness in the core (H_C). LHQ modeling was conducted by modeling attributes extracted from the hardness profile curve using two effective techniques based on the punctual and geometrical approaches. The process parameters with the most influence on the responses were laser power, beam velocity and initial hardness in the core. The obtained results demonstrate that the geometrical approach is more accurate and credible than the punctual approach according to performance assessment criteria.

Despite the fact that SMA restrainers exhibit a superelastic strain capacity of 7%, this capacity appears inadequate for isolated bridges due to the typically greater than 20cm relative dis-placements between girders during intense seismic events. In order to perform such a stroke, a SMA restrainer of greater than 3 metres in length might be required. In order to reduce the length of restrainers, a novel honeycomb damper constructed from superelastic shape memory alloy (SMA) is proposed. The proposed device, denoted as the superelastic SMA honeycomb damper (SHD), is comprised of steel plates to prevent the SMA plane from collapsing and superelastic SMA honeycomb to provide self-centering capability. By incorporating the large strain capacity of SMA and the geometrically nonlinear deformation of honeycomb structures, SHD has been developed to satisfy the requirements of bridge restrainers with large strokes. It is capable of functioning as a restrainer and energy dissipation device when subjected to dynamic tension and compression loads. The SHD was initially investigated from a theoretical perspective. Following this, a mul-ti-cell SHD specimen was manufactured. The specimen underwent axial tensile and compressive experiments in order to examine the mechanical properties of SHDs. Finally, experimental results were investigated through numerical simulation analyses of the SHDs using a three-dimensional high-fidelity finite element model. Additionally, a method for enhancing SHD was proposed. The findings indicate that SHD is capable of exhibiting superior self-centering capability and sta-ble hysteretic responses when subjected to earthquakes.

GaN nanowires grown on metal substrates attract an increasing interest for a wide range of applications. Herein, we report GaN nanowires grown by plasma-assisted molecular beam epitaxy on thin polycrystalline ZrN buffer layers, sputtered onto Si(111) substrates. The nanowire orientation is studied experimentally and described within a model as a function of the Ga beam angle, nanowire tilt angle and substrate rotation. We show that vertically aligned nanowires grow faster than inclined nanowires, which leads to an interesting effect of geometrical selection of the nanowire orientation in the directional molecular beam epitaxy technique. After a given growth time, this effect depends on the nanowire surface density. At low density, the nanowires continue to grow with random orientations as nucleated. At high density, the effect of preferential growth induced by unidirectional supply of material in MBE starts to dominate. Faster growing nanowires with smaller tilt angles shadow more inclined nanowires that grow slower. This helps to obtain more regular ensembles of vertically oriented GaN nanowires despite their random position induced by the metallic grains at nucleation. The obtained dense ensembles of vertically aligned GaN nanowires on ZrN/Si(111) surfaces are highly relevant for device applications. These results are not specific for GaN nanowires and should be relevant for any nanowires which are epitaxially linked to the randomly oriented surface grains in the directional molecular beam epitaxy.

We survey the main results emerging in the rapidly developing field that considers the geometry of thermodynamics and statistical mechanics. Our ideas of stability depend on variational principles – Maximum Entropy together with Least Action (and the isomorphic Least Exertion) – and this stability (invariance in time) indicates the centrality of the geometries of the systems. We explore a variety of examples of the application of thermodynamics in both macro- and micro-systems, and propose that irreversibility is an intrinsic and fundamental property of the physical realm (instead of being merely emergent). Using the concept of complex time, we examine the underlying mathematical unity of quantum mechanics, statistical mechanics, and geometrical thermodynamics. We show how the Planck and Boltzmann constants provide the fundamental quantum units for the phenomenological characteristics of the two Noether-conserved quantities controlling overall thermodynamic behaviour: respectively, energy and entropy production. We discuss the far-reaching isomorphisms (between all of complex time, temperature and the system symmetry) that always exist for Maximum Entropy systems, even as they exhibit either irreversible (dissipative, yet still dynamically stable) or reversible (including stable and static geometries) behaviours.

Double-spiral galaxies are common in the Universe. It is known that the logarithmic double spiral is a Maximum Entropy geometry and represents spiral galaxies well. It is also known that the virial mass of such a galaxy can be approximately determined from the entropy of its central supermassive black hole. But over time the black hole must accrete mass, and therefore the overall galactic entropy must increase. From the associated entropic Euler-Lagrange equation (forming the basis of the Principle of Least Exertion, and also enabling the application of Nöther’s theorem) we show that the galactic entropy production is a conserved quantity, and we derive an appropriate expression for the relativistic entropic Hamiltonian of an idealised spiral galaxy. We generalise Onsager’s celebrated expression for entropy production and demonstrate that galactic entropy production has two parts, one many orders of magnitude larger than the other, and where the smaller is comparable to the Hawking radiation of the central supermassive black hole. We conclude that galaxies cannot be isolated, since even idealised spiral galaxies have non-zero entropy production.

The building blocks of a scientific facility based on particle beams comprise magnets and electro-magnetic devices. The design of an overall accelerator usually imposes a demanding accuracy on their exact position and orientation. Their non-geometrical functional features are frequently either not clearly defined – what is the « magnetic axis » of a magnet –, or not explicitly used along their lifecycle. Improving the process to establish and use these functional features would contribute to meeting demanding challenges. The European Spallation Source (ESS) is aiming at providing a powerful proton linear accelerator and a target system to produce pulsed neutrons. The challenging complex design and integration yielded to introducing a tool for all stakeholders along the lifecycle: the "situation features", as defined in ISO GPS (Geometrical Product Specifications) standards. This paper deals with the beyond-mechanics non-geometrical use cases and generalizes the use of the association operation to theoretically exact features. The examples of the functional specification, fiducialization (measure of the real features with respect to external marks) and installation phases of quadrupole magnets is developed. Perspectives of generic use are highlighted.

Electrical Resistance Tomography (ERT) has been used in the literature to monitor the gas-liquid separation. However, the image reconstruction algorithms used in the studies take a considerable amount of time to generate the tomograms, which is far above the time scales of the flow inside the inline separator and, as a consequence, the technique is not fast enough to capture all the rele-vant dynamics of the process, vital for control applications. This article proposes a new strategy based on the physics behind the measurement and simple logics to monitor the separation with a high temporal resolution by minimizing both the amount of data and the calculations required to reconstruct one frame of the flow. To demonstrate its potential, the electronics of an ERT system are used together with a high-speed camera to measure the flow inside an inline swirl separator. For the 16-electrode system used in this study, only 12 measurements are required to reconstruct the whole flow distribution with the proposed algorithm, 10x less than the minimum number of measurements of ERT (120). In terms of computational effort, the technique was shown to be 1000x faster than solving the inverse problem non-iteratively via the Gauss-Newton approach, one of the computationally cheapest techniques available. Therefore, this novel algorithm has the potential to achieve measurement speeds in the order of 104 times the ERT speed in the context of inline swirl separation, pointing to flow measurements at around 10kHz while keeping the aver-age estimation error below 6 mm in the worst case scenario.

Stochastic modelling of geometrically imperfect steel frame structures requires inputs of statistical values of the imperfection parameters, in some cases also with suitably selected mutual correlations between these input parameters. The paper presents a verification study of two stochastic methods to directly model the initial global geometrical imperfections of the steel frame structures. The stochastic input values of geometrical imperfections are derived from the tolerance criteria of the corresponding European Standard. Two advanced stochastic methods have been developed, referred as #RSS (random story sway) and #RSP (random story position). These methods are verified using random sampling simulations. This study provides useful provisions for the advanced numerical analyses of multi-story steel frames of various geometries. The proposed methods are verified for equidistant story heights and for structures of up to 24 stories, hence might be used for the most types of the standard steel frame structures.

Star block-copolymers (SBCs) are macromolecules formed by a number of diblock copolymers anchored to a common central core, being the internal monomers solvophilic and the end monomers solvophobic. Recent studies have demonstrated that SBCs constitute a self-assembling building blocks with specific softness, functionalization, shape, and flexibility. Depending on different physical and chemical parameters the SBCs can behave as flexible patchy particles. In this paper, we study the rotational dynamics of isolated SBCs using a hybrid mesoscale simulation technique. We compare three different approaches to analyse the dynamics: the laboratory frame, the non-inertial Eckart's frame, and a geometrical approximation relating the conformation of the SBC to the velocity profile of the solvent. We find that the geometrical approach is adequate when dealing with very soft systems while in the opposite extreme, the dynamics is best explained using the laboratory frame. On the other hand, the Eckart frame is found to be very general and to reproduced well both extreme cases. We also compare the rotational frequency and the kinetic energy with the definitions of the angular momentum and inertia tensor different from recent publications.

A Berry geometrical phase is identified in a strongly metastable system containing dynamically responsive clathrate hydrate structures within a crystal-fluid material. High energy degeneracy in the associated chemistry produces local stability and false vacuum conditions that lead to non-extensive and non-additive contributions in the fundamental thermodynamic relation. Application of Ginzburg-Landau theory and the scaling laws reveals a coherence length and a penetration depth for a macro-scale dual superconductor. The coherence length determines non-extensive volume changes whilst its inverse gives the Higgs mass. The penetration depth determines the extent of QCD vacuum suppression whilst its inverse gives the vector boson mass together with indirect manifestation as non-additive hyperbolic curvature. Simultaneous emergence of the Ginzburg-Landau superconducting phase transition is consistent with gauge-invariant coupling of the scalar field to the Yang-Mills action in QCD. The discovery of an energy gap in the gradient energy term of the system Lagrangian is associated with a critical correlation length revealed in the transition from a gapped to a gapless superconducting state. Together with the emergence and absorption of the Higgs-like scalar field, a mechanism for describing a renormalized QCD mass gap arises.

A new pathway leading to the n-10 fatty acid series has been recently evidenced, starting from sapienic acid - a monounsaturated fatty acid (MUFA) resulting from the transformation of palmitic acid by delta-6 desaturase. Sapienic acid attracts attention as novel marker of cancer cell plasticity. Here, we analyzed fatty acids including the n-10 fatty acid contents, and compared for the first time cell membranes and the corresponding extracellular vesicles (EV) of two human prostatic adenocarcinoma cell lines of different aggressiveness (PC3 and LNCaP). The n-10 components were 9-13% of the total fatty acids in both cancer cell lines and EVs, with total MUFA levels significantly higher in EVs of the most aggressive cell type (PC3). High sapienic/palmitoleic ratios indicated the preference for delta-6 vs. delta-9 desaturase enzymatic activity in these cell lines. The expressions analysis of enzymes involved in desaturation and elongation by qRT-PCR showed a higher desaturase activity in PC3 and a higher elongase activity toward polyunsaturated fatty acids than toward saturated fatty acids, compared to LNCaP cells. Our results improve the present knowledge in cancer fatty acid metabolism and lipid phenotypes, highlighting EV lipidomics to monitor positional fatty acid isomer profiles and MUFA levels in cancer.

Emission, propagation, and reflection of light as mechanical phenomena in inertial frames are based on the behavior of balls at the limit when their mass is zero. The kinematics of massless balls is like that of balls with mass. Light as a wave or particle is a massless entity. Therefore, it is natural to apply the kinematics of the massless balls to light. Consequently, the kinematics of light depends on its kinetics of electromagnetic nature and its kinematics of mechanical nature in its interactions of emission and reflection with the matter. The study of the physics phenomena in the frame at absolute rest includes those in the inertial frames. Considering this and applying the emission, propagation, and reflection of light as mechanical phenomena in the vacuum of the frame at absolute rest, this study derives formulas for the speed of the wavefront of a ray of light reflected by a fixed and moving mirror when the light comes from a fixed and moving source. The derived formulas apply to the modified Michelson interferometer, employed independently by R. Tomaschek and D. C. Miller in their experiments.

Derivation of light paths in the Michelson interferometer is based on the hypothesis that the speed of light does not change after reflection by a mirror in motion. The Michelson-Morley experiment predicts a fringe shift of 0.40. The same fringe shift is predicted for a particular Michelson interferometer in which the beam splitter of the interferometer makes an angle of 45° with the direction of light from the source. Light behaves like a wave and also as a particle. Thus, it is reasonable to consider the reflection of light as a mechanical phenomenon. With this hypothesis, the speed of light changes after reflection, and the predicted fringe shift for the particular Michelson interferometer is zero which is in accordance with the result of the Michelson-Morley experiment. Apparently, light travels in any inertial frame as if this particular interferometer belongs to a fixed frame. The velocity of light is considered independent of the velocity of its source, which is in accordance with astronomers’ observations of the binary stars, and the experiment performed at CERN, Geneva, in 1964.

The study of the emission, propagation, and reflection of balls in Newtonian mechanics, applying the laws of conservation of momentum and energy when the mass of the balls converges to zero, concludes that the kinematics of light are like those of massless balls. The kinematics of light explain why the speed of light is the constant c in each inertial frame in which the light source and reflective mirror are at rest, why the laws of physics have the same form in each inertial frame, and why any light experiment in an inertial frame cannot prove the motion of that inertial frame. The following law of mechanics governs the kinematics of balls and electromagnetic radiation: Each inertial frame drags at its velocity the balls and electromagnetic radiation emitted by bodies at rest in that inertial frame. The theory of special relativity misapplies the symmetry observed in phenomena to two inertial frames; therefore, it duplicates a physical phenomenon from one inertial frame, which is considered stationary, into another. There are multiple issues rooted in Lorentz's and Einstein's transformations. For example, transformations offer the speed of light at the constant c in the moving and opposite directions of the inertial frame. At the same time, it is variable in any other direction, converging to infinity. Time contraction in the moving direction of the inertial frame is different from time dilation in the opposite direction. It requires a ruler and time synchronization in each direction. There are no length contractions in Lorentz's and Einstein's transformations to support the fundamental concept of length contraction in special relativity. With these unacceptable conclusions, the theory of special relativity is self-negating.

The ocean of the planet Earth, which is as thin as a cicada's wings, play a role of catalyst to the evolving of the ancient plates. The ocean and the ancient plates of the planet Earth formed a magic Whirlpool of the Cross “十” shape with double whirl arms. The central core of the Cross “十” shape Ocean-Plate Whirlpool is the Hint Black Hole (HBH) located at the Northwestern Pacific Ocean floor which is decorated with invisible geological age distribution of clam shell. The double whirl arms of ancient plates are distinguished to be left and right at Bering Strait and just formed the ring of fire of the Pacific Ocean.The Cross “十” shape Ocean-Plate Whirlpool of the planet Earth was born at the equatorial area about 66 Ma ago and is rotating anticlockwise and drifting northward by a mini footprint of spiral at s speed of as slow as a snail moves. Obviously, it would come to an end at the North Pole about 100 Ma in the future. So, the lifespan of the Cross “十” shape Ocean-Plate Whirlpool would be about 166 Ma, from birth to death on the planet Earth.What is hidden behind of the evolving of the oceans and the ancient plates is the golden law of geometrical progression of ladder pagoda of attributes evolving of them, which is “1➨2➨4➨2➨1”, including upgrading and downgrading process. It seems to hint that what is happening on the planet Earth, including the top-level species of human race, is actually unique in the Milky Way galaxy in essence.The property roles of expanding and shrinking are existing in the Cross “十” shape Ocean-Plate Whirlpool by face to face and the exchange of property roles took place tens of millions of years ago. And that situation would happen again tens of millions of years in the future. It seems a giant circle of evolving of the exchange of property roles would be drawn by the Cross “十” shape Ocean-Plate Whirlpool in her lifetime.