Geometric Paradigm Research Articles

Uncertainty and information in physiological signals: Explicit physical trade-off with log-normal wavelets

Physiological recordings contain a great deal of information about the underlying dynamics of Life. The practical statistical treatment of these single-trial measurements is often hampered by the inadequacy of overly strong assumptions. Heisenberg’s uncertainty principle allows for more parsimony, trading off statistical significance for localization. By decomposing signals into time–frequency atoms and recomposing them into local quadratic estimates, we propose a concise and expressive implementation of these fundamental concepts based on the choice of a geometric paradigm and two physical parameters. Starting from the spectrogram based on two fixed timescales and Gabor’s normal window, we then build its scale-invariant analogue, the scalogram based on two quality factors and Grossmann’s log-normal wavelet. These canonical estimators provide a minimal and flexible framework for single trial time–frequency statistics, which we apply to polysomnographic signals: EEG representations, HRV extraction from ECG, coherence and mutual information between heart rate and respiration.

Exploring secondary school students’ geometrical figure apprehension: cognitive structure and levels of geometrical ability

The present study examines secondary school students’ geometrical figure apprehension based on Duval’s theoretical framework regarding perceptual, operative, and discursive apprehension. The aim is to explore the cognitive structure of the geometrical figure apprehension dimensions (operative, discursive, and perceptual) in three grades of secondary school students. The tasks in the present study were completed by a sample of 881 students attending public secondary education in Cyprus. Confirmatory factor analysis indicated the stability of the structure of the model concerning secondary school students’ geometrical figure apprehension. However, differences were found in the interrelations among the three main aspects of the model in the examined grades (9, 10, and 11). Moreover, it was observed that students find it easier to solve tasks involving perceptual apprehension compared to discursive apprehension tasks, indicating a possible hierarchical structure of figure apprehension. The present study acts as a pilot study of the constructed instrument. Finally, the results are interpreted in relation to the type of geometrical paradigm in which students work at each hierarchical level.

THE RELATIONAL PICTURE OF THE WORLD IN THE PHILOSOPHY OF PHYSICS

The formation of a relational picture of the world is a natural stage in the development of the philosophy of physics. Philosophical reflection on the current state of fundamental theoretical physics is quite diverse. The author proceeds from the fact that the relational picture of the world as a methodological construct is formed in modern physics on the basis of a generalization of the relational paradigm based on three independent fundamental principles: the principle of relational understanding of space-time, the action-at-a-distance principle and Mach's principle. The epistemological role of these principles is extremely significant for revealing the dynamics of the development of physics outside the currently dominant field theoretical paradigm. The geometric paradigm also makes its contribution to the development of modern physics. The principles of the relational paradigm are complemented by a new mathematical apparatus of binary systems of complex relations. The author draws a conclusion about the heuristic value of the theory of binary geometrophysics itself, developed by Yu. S. Vladimirov, and its generalizations in the form of a relational picture of the world for a philosophical understanding of their role in the further development of fundamental theoretical physics.

Assessing the mechanical and static aeroelastic performance of cellular Kirigami wingbox designs

Adaptive wings configurations have been evaluated for morphing airframe applications during the last two decades. Constructions with flexible hinges can be in particular a solution for small top medium-scale air vehicles, while novel Kirigami technologies help to produce flexible and complex structures by enabling novel geometric paradigms. In this study, a cellular Kirigami wingbox with an adaptive hinge is designed and manufactured. The mechanical properties of the wingbox are numerically evaluated, considering the shear modulus of the cellular elements patterning the wingbox. Thus, the equivalent torsional, flexural stiffness, as well as the shear center location of the whole wingbox structure are calculated. The analysis is parametrized against various possible internal cell angles and cell thickness values that define the Kirigami cellular tessellation of the wingbox. The static divergence speed is also evaluated by means of the same parametrization. This study shows the feasibility of using a Kirigami wingbox concept for morphing/adaptive small to medium-scale from a structural and aeroelastic perspective.

Universal alignment in turbulent pair dispersion

Countless processes in nature and industry, from rain droplet nucleation to plankton interaction in the ocean, are intimately related to turbulent fluctuations of local concentrations of advected matter. These fluctuations can be described by considering the change of the separation between particle pairs, known as pair dispersion, which is believed to obey a cubic in time growth according to Richardson’s theory. Our work reveals a universal, scale-invariant alignment between the relative velocity and position vectors of dispersing particles at a mean angle that we show to be a universal constant of turbulence. We connect the value of this mean angle to Richardson’s traditional theory and find agreement with data from a numerical simulation and a laboratory experiment. While the Richardson’s cubic regime has been observed for small initial particle separations only, the constancy of the mean angle manifests throughout the entire inertial range of turbulence. Thus, our work reveals the universal nature of turbulent pair dispersion through a geometrical paradigm whose validity goes beyond the classical theory, and provides a framework for understanding and modeling transport and mixing processes.

Special scattering regimes for conical all-dielectric nanoparticles

All-dielectric nanophotonics opens a venue for a variety of novel phenomena and scattering regimes driven by unique optical effects in semiconductor and dielectric nanoresonators. Their peculiar optical signatures enabled by simultaneous electric and magnetic responses in the visible range pave a way for a plenty of new applications in nano-optics, biology, sensing, etc. In this work, we investigate fabrication-friendly truncated cone resonators and achieve several important scattering regimes due to the inherent property of cones—broken symmetry along the main axis without involving complex geometries or structured beams. We show this symmetry breaking to deliver various kinds of Kerker effects (generalized and transverse Kerker effects), non-scattering hybrid anapole regime (simultaneous anapole conditions for all the multipoles in a particle leading to the nearly full scattering suppression) and, vice versa, superscattering regime. Being governed by the same straightforward geometrical paradigm, discussed effects could greatly simplify the manufacturing process of photonic devices with different functionalities. Moreover, the additional degrees of freedom driven by the conicity open new horizons to tailor light-matter interactions at the nanoscale.

Genesis and progress of virtual power principle

The virtual power principle (VPP) of continuum mechanics states a celebrated variational equality between external and internal virtual powers for any virtual velocity field conforming with linear kinematic constraints. The topic is here addressed to investigate how the original ideas born in the early XIX century are modelled by modern formulations based on Functional Analysis and Differential Geometry. These notions are able to provide an effective mathematical context for proving existence of Lagrange multipliers associated with the constraint of rigidity on velocity fields. The VPP stands as privileged tool for giving to stress fields a consistent definition based on duality with conforming virtual stretching fields. By complementarity, the VPP generates a variational condition for integrability of stretching fields, with self-equilibrated stresses as test fields. Progress is got by the formulation of the rate virtual power principle (RVPP) by time derivation of the VPP along the motion, with internal virtual power integrated per unit mass. The basic distinction between spatial and material fields according to the geometric paradigm is prompted to replace the one previously adopted in the literature. The need for a non-redundant implicit formulation of the rigidity constraint is emphasised to contrast degeneracy. This logical demand avoids proliferation of multipliers, in the spirit of Ockham’s Razor, a celebrated philosophical motto with multiform applications. The shining mathematical theory set out by Leonhard Euler, Jean-Baptiste Le Rond d’Alembert, Joseph Louis Lagrange, and Augustin Cauchy is in this respect a point of optimality. A geometric rate theory of elasticity meets the call for no-dissipation in push-closed elastic cycles, with non need of any finite strain elastic energy functional, thus leading to a proper statement of rate equilibrium problems, basilar for computational formulations and for investigations about instability phenomena and post-critical behaviours.

Arah Kiblat dalam Perspektif Fikih dan Geometri

Facing qibla is prayer mandatory, no debate done. The problem arises when people who are away from mecca had to "face" the Qibla during prayers. The difference of fukaha ijtihad was in syat}r meaning. Linguistically syat}r means direction, intent, and purpose. In addition, syat}r also means al-Nishf and al-Wasath. The two meanings used to assume Earth is spherical, then the syat}r Kakbah means can be conceived in geometric paradigm. This research uses descriptive analysis method with integration-interconnection approach, namely examining the qibla direction concept in the Fiqh point of view by utilizing geometry analysis. The results showed that the fukaha perspective could be integrated with geometry concept using the Syat}r Kakbah concept of which is interpreted as "a vertical semicircular plane through the Kakbah". It means, every place on earth's surface will form a Syat}r Kakbah with that place, this plane that becomes the Qibla direction boundary for Muslims in any hemisphere, this has the same value as 'ain al-Kaaba.

GRAVITATIONAL INTERACTION FROM THE STANDPOINT OF FIELD-THEORETICAL AND GEOMETRIC PARADIGMS

Considered are energy dependences of the dimensionless constants of the electromagnetic, strong and weak interactions. Elementary particle classification depending on their spin is presented as well as dualistic paradigms based on are physical categories of space-time, matter particles and field-carriers of interactions. Gravitational radiation and static gravitational fields are considered both in the framework of the Lagrangian formalism and GR as well as MacCrea-Milne’s Newtonian cosmological model. Both perturbative and nonperbative approaches to gravity quantization are listed. Advantages and disadvantages of gravity description in the framework of field-theoretical and geometric paradigms have been indicated.

GEOMETRIC PARADIGM AND POSSIBLE PHYSICAL PROPERTIES OF THE SPACE-TIME

The article considers and discusses the relational and substantial concepts of the nature of space-time. It is shown that within the framework of the geometric paradigm of physics the substantial concept of space-time is manifested most clearly.

SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions

We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras. We replace the Lie algebra-valued connection one-form A, by a superalgebra-valued polyform tilde{A} mixing exterior-forms of all degrees and satisfying the chiral self-duality condition tilde{A} =^{ast }{tilde{A}}_{chi } , where χ denotes the superalgebra grading operator. This superconnection contains Yang-Mills vectors valued in the even Lie subalgebra, together with scalars and self-dual tensors valued in the odd module, all coupling only to the charge parity CP-positive Fermions. The Fermion quantum loops then induce the usual Yang-Mills-scalar Lagrangian, the self-dual Avdeev-Chizhov propagator of the tensors, plus a new vector-scalar-tensor vertex and several quartic terms which match the geometric definition of the supercurvature. Applied to the SU(2/1) Lie-Kac simple superalgebra, which naturally classifies all the elementary particles, the resulting quantum field theory is anomaly-free and the interactions are governed by the super-Killing metric and by the structure constants of the superalgebra.

On forms of geometric work: a study with pre-service teachers based on the theory of Mathematical Working Spaces

In this paper, we identify various forms of geometric work carried out by student teachers who were asked to perform a geometric task for the estimation of a land area. The theory of Mathematical Working Spaces is used to analyze and characterize the work produced. This study provides evidence that students developed forms of geometric work that are compliant with at least two distinct geometric paradigms, one characterized by the utilization of measuring and drawing tools and the other by a property-based discourse on proof. Significantly, a sizable number of students also developed work forms that do not correspond to any geometrical paradigm. A broader purpose of this paper is to highlight three criteria born by the theory and shown to be useful for the description and evaluation of geometric work: compliance, completeness, and correctness.

Toeplitz Structured Covariance Matrix Estimation for Radar Applications

Following a geometric paradigm, the estimation of a Toeplitz structured covariance matrix is considered. The estimator minimizes the distance from the Sample Covariance Matrix (SCM) while complying with some specific constraints modeling the covariance structure. The resulting constrained optimization problem is solved globally resorting to the Dykstra’ projection framework. Each step of the procedure involves the solution of two convex sub-problems, whose minimizers are available in closed form. Simulation results related to typical radar environments highlight the effectiveness of the devised method.

Scaling Organic Electrochemical Transistors Down to Nanosized Channels.

The perspective of downscaling organic electrochemical transistors (OECTs) in the nanorange is approached by depositing poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) on electrodes with a nanogap designed and fabricated by electromigration induced break junction (EIBJ) technique. The electrical response of the fabricated devices is obtained by acquiring transfer characteristics in order to clarify the specific main characteristics of OECTs with sub-micrometer-sized active channels (nanogap-OECTs). On the basis of their electrical response to different scan times, the nanogap-OECT shows a maximum transconductance unaffected upon changing scan times in the time window from 1 s to 100 µs, meaning that fast varying signals can be easily acquired with unchanged amplifying performance. Hence, the scaling down of the channel size to the nanometer scale leads to a geometrical paradigm that minimizes effects on device response due to the cationic diffusion into the polymeric channel. A comprehensive study of these features is carried out by an electrochemical impedance spectroscopy (EIS) study, complemented by a quantitative analysis made by equivalent circuits. The propagation of a redox front into the polymer bulk due to ionic diffusion also known as the "intercalation pseudocapacitance" is identified as a limiting factor for the transduction dynamics.

Sigma map dynamics and bifurcations

Some interesting variants of walking droplet based discrete dynamical bifurcations arising from diffeomorphisms are analyzed in detail. A notable feature of these new bifurcations is that, like Smale horseshoes, they can be represented by simple geometric paradigms, which markedly simplify their analysis. The two-dimensional diffeomorphisms that produce these bifurcations are called sigma maps or double sigma maps for reasons that are made manifest in this investigation. Several examples are presented along with their dynamical simulations.

A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

A new class of disturbance covariance matrix estimators for radar signal processing applications is introduced following a geometric paradigm. Each estimator is associated with a given unitary invariant norm and performs the sample covariance matrix projection into a specific set of structured covariance matrices. Regardless of the considered norm, an efficient solution technique to handle the resulting constrained optimization problem is developed. Specifically, it is shown that the new family of distribution-free estimators shares a shrinkagetype form; besides, the eigenvalues estimate just requires the solution of a one-dimensional convex problem whose objective function depends on the considered unitary norm. For the two most common norm instances, i.e., Frobenius and spectral, very efficient algorithms are developed to solve the aforementioned one-dimensional optimization leading to almost closed form covariance estimates. At the analysis stage, the performance of the new estimators is assessed in terms of achievable Signal to Interference plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming different data statistical characterizations. The results show that interesting SINR improvements with respect to some counterparts available in the open literature can be achieved especially in training starved regimes.

Unification of gravity and quantum field theory from extended noncommutative geometry

We make biframe and quaternion extensions on the noncommutative geometry, and construct the biframe spacetime for the unification of gravity and quantum field theory (QFT). The extended geometry distinguishes between the ordinary spacetime based on the frame bundle and an extra non-coordinate spacetime based on the biframe bundle constructed by our extensions. The ordinary spacetime frame is globally flat and plays the role as the spacetime frame in which the fields of the Standard Model are defined. The non-coordinate frame is locally flat and is the gravity spacetime frame. The field defined in both frames of such “flat” biframe spacetime can be quantized and plays the role as the gravity field which couples with all the fields to connect the gravity effect with the Standard Model. Thus, we provide a geometric paradigm in which gravity and QFT can be unified.

On clustering shape data

ABSTRACTAmong the statistical methods to model stochastic behaviours of objects, clustering is a preliminary technique to recognize similar patterns within a group of observations in a data set. Various distances to measure differences among objects could be invoked to cluster data through numerous clustering methods. When variables in hand contain geometrical information of objects, such metrics should be adequately adapted. In fact, statistical methods for these typical data are endowed with a geometrical paradigm in a multivariate sense. In this paper, a procedure for clustering shape data is suggested employing appropriate metrics. Then, the best shape distance candidate as well as a suitable agglomerative method for clustering the simulated shape data are provided by considering cluster validation measures. The results are implemented in a real life application.

Logical and set calculations in the framework of geometrical informatics paradigm

Possibilities of using a computational model of geometrical informatics for solving problems involving set-theoretic and logical calculations are discussed. New programming methods for solving these problems are proposed. These methods are characterized by high level of concurrency when using modern graphical processors.

Étude d’une situation de reproduction de figures par pliage en cycle 2 : le pliox

This work presents the study of a situation of reproduction of figure by folding which is called a PLIOX (Guille-Biel Winder, 2013). We put this situation as a spatial problem into the 3D micro-space (Berthelot & Salin, 1992). If we consider it as a geometrical problem, we identify that G1 is the corresponding geometric paradigm (Houdement & Kuzniak, 2000 & 2006), we analyze the function of this situation for the development of the geometrical thought (Van Hiele, 1986), we use a cognitive and semiotic point of view (Duval, 1995) to see that the situation uses shape modifications (decompositions and reconfigurations), as well as positional modifications. We finally identify various didactic variables and mathematical knowledges at stake. This work leads us to the presentation and analysis of different phases of PLIOX situation.