Framework Of Geometry Research Articles

Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity

This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon P, which is the Euclidean closure of the hyperbolic plane H, bounded by n hyperbolic geodesic segments. The polygon P is built by considering the unique geodesic that connects the n+2 vertices z˜,z0,z1,…,zn−1,zn. The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Additionally in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined.

On the Logical Geometry of Geometric Angles

In this paper we provide an analysis of the logical relations within the conceptual or lexical field of angles in 2D geometry. The basic tripartition into acute/right/obtuse angles is extended in two steps: first zero and straight angles are added, and secondly reflex and full angles are added, in both cases extending the logical space of angles. Within the framework of logical geometry, the resulting partitions of these logical spaces yield bitstring semantics of increasing complexity. These bitstring analyses allow a straightforward account of the Aristotelian relations between angular concepts. In addition, also relational concepts such as complementary and supplementary angles receive a natural bitstring analysis.

The ghost fields and the BV extension for finite spectral triples

After arguing why the Batalin–Vilkovisky (BV) formalism is expected to find a natural description within the framework of noncommutative geometry, we explain how this relation takes form for gauge theories induced by finite spectral triples. In particular, we demonstrate how the two extension procedures appearing in the BV formalism, that is, the initial extension via the introduction of ghost/anti-ghost fields and the further extension with auxiliary fields, can be described in the language of noncommutative geometry using the notions of the BV spectral triple and total spectral triple, respectively. The construction is presented in detail for all U(2)-gauge theories induced by spectral triples on the algebra M2(C). Indications are given on how to extend the results to U(n)-gauge theories for n > 2.

Curvature-Based Control for Low-Inertia Systems

This letter proposes a simple and inexpensive control of distributed energy resources aimed at improving the power system dynamic performance. The rationale behind the proposed control relies on a recent interpretation of the frequency in the differential geometry framework. A comparison with well-established controls in terms of eigensensitivity and time-domain performance is carried out to show the effectiveness of the proposed control.

Information Geometrically Generalized Covariate Shift Adaptation.

Many machine learning methods assume that the training and test data follow the same distribution. However, in the real world, this assumption is often violated. In particular, the marginal distribution of the data changes, called covariate shift, is one of the most important research topics in machine learning. We show that the well-known family of covariate shift adaptation methods is unified in the framework of information geometry. Furthermore, we show that parameter search for a geometrically generalized covariate shift adaptation method can be achieved efficiently. Numerical experiments show that our generalization can achieve better performance than the existing methods it encompasses.

Nonlocal Harnack inequalities in the Heisenberg group

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group mathbb {H}^n, whose prototype is the Dirichlet problem for the p-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is p=2, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent s goes to 1.

The symplectic geometrical formulation of quantum hydrodynamics

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

Lorentz-violation-induced arrival time delay of astroparticles in Finsler spacetime

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. We present the arrival time delay of astroparticles subject to Lorentz violation in the framework of Finsler geometry, and the result corresponds to that derived by Jacob and Piran in the standard model of cosmology.

Modeling and Analysis of Priority-Based Distributed Access Control in Vehicular Networks

This paper analyzes priority-based distributed access control in vehicular networks. Characterizing the spatial correlation between vehicles, users, and roads, this paper uses Cox point processes to model vehicles and users, respectively. Then, this paper proposes that each vehicle transmits if it has the highest priority among all of its neighbors. Using the stochastic geometry framework, this paper derives the probability that a typical vehicle is retained to transmit. Then, this paper derives a second-order moment measure of the retained point process to evaluate the signal-to-interference (SIR) distribution of a typical user. The derived SIR distribution is compared to the system-level simulation of the SIR distribution of a typical user.

Spherically symmetric solutions of higher-spin gravity in the IKKT matrix model

We present a systematic study of spherically symmetric vacuum solutions of the IKKT matrix model, within the framework of semi-classical covariant quantum geometries. All asymptotically flat solutions of the equations of motion of the frame are found explicitly. They reproduce the linearized Schwarzschild geometry for large r but deviate from it at the non-linear level, and include contributions from dilaton and axion. They are pertinent to the pre-gravity theory arising on classical brane solutions within the classical matrix model, before taking into account the Einstein-Hilbert term induced by quantum effects. We also address the problem of reconstructing matrix configurations corresponding to some given frame, and show that this problem can always be solved at the geometrical level of the underlying higher spin theory, ignoring possible higher spin modes.

One-loop corrections to the spectral action

We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability in a generalized sense, while staying within the spectral framework of noncommutative geometry. Our result is based on the perturbative expansion of the spectral action in terms of higher Yang-Mills and Chern-Simons forms. In the spirit of random noncommutative geometries, we consider the path integral over matrix fluctuations around a fixed noncommutative gauge background and show that the corresponding one-loop counterterms are of the same form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.

Natural-gradient learning for spiking neurons.

In many normative theories of synaptic plasticity, weight updates implicitly depend on the chosen parametrization of the weights. This problem relates, for example, to neuronal morphology: synapses which are functionally equivalent in terms of their impact on somatic firing can differ substantially in spine size due to their different positions along the dendritic tree. Classical theories based on Euclidean-gradient descent can easily lead to inconsistencies due to such parametrization dependence. The issues are solved in the framework of Riemannian geometry, in which we propose that plasticity instead follows natural-gradient descent. Under this hypothesis, we derive a synaptic learning rule for spiking neurons that couples functional efficiency with the explanation of several well-documented biological phenomena such as dendritic democracy, multiplicative scaling, and heterosynaptic plasticity. We therefore suggest that in its search for functional synaptic plasticity, evolution might have come up with its own version of natural-gradient descent.

Automorphism groupoids in noncommutative projective geometry

We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts.In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras.We analyse this automorphism groupoid, using the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme.We apply our results to two important examples, the two-dimensional quantum projective space and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras.

Particle shape characterizations for energetic materials by computational geometry and stereology method

Particle shapes significantly affect viscosity and flow behavior of energetic materials, and therefore affect their packability and processability. This study presents a computational geometry framework for automatically quantifying two-dimensional (2D) and three-dimensional (3D) particle shapes of energetic materials. A specimen by mixing three typical energetic materials including HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine), RDX (1,3,5-Trinitroperhydro-1,3,5-triazine) and AP (Ammonium Perchlorate) particles is used in this study. This specimen is scanned by high-resolution X-ray computed tomography (X-ray CT), yielding a volumetric image. An improved watershed analysis algorithm is used to process the volumetric image to identify individual 3D particles. The stereology sampling method is used to obtain 2D projections of 3D particles. Computational geometry techniques are developed by this study to analyze 2D particle projections and 3D particle geometries to compute seven commonly used shape descriptors, including convexity, circularity, intercept sphericity, area sphericity, diameter sphericity, circle ratio sphericity, and surface area sphericity. Results show that those different shape descriptors of energetic materials can be divided into three groups based on their numerical ranges. This study also evaluates the effectiveness and accuracy of 2D shape descriptors for quantifying the true 3D shapes. The inconsistent characterization results between 2D and 3D shape descriptors suggest that researchers should be cautious when using 2D images to characterize 3D particle shapes of energetic materials. The computational geometry framework and particle shape analysis results presented in this study can be potentially useful in numerical modeling, experimental analysis, and theoretical investigation for energetic materials.

Stochastic Geometry Framework for Ultrareliable Cooperative Communications With Random Blockages

We study an industry automation scenario where a central controller broadcasts critical messages to the wireless devices (e.g., sensors/actuators). We devise a stochastic geometry framework where the rate coverage probability of devices is modeled by taking into account the density of roaming blockages over the factory floor. To alleviate the loss in the coverage, we adopt a two-phase transmission policy, where in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">broadcast phase</i> , the central controller broadcasts the messages intended for the devices in the network area. The devices in coverage in the broadcast phase act as decode-and-forward relays in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relay phase</i> , so as to reinforce the signal strength at the devices in outage. The total downlink transmission time is, therefore, partitioned into two phases by a tunable factor. Finally, we study the optimal value of the partitioning factor with varying device densities, blockage densities, and file sizes, and we highlight that a longer transmission time should be allotted to the broadcast phase in the case of larger file sizes or lower transmit power of the controller.

Analog Schwarzschild-like geometry in fluids with external pressure

We study acoustic geometry in fluids with external pressure. In particular, we examine the conditions under which the acoustic metric mimics a Schwarzschild-like metric. We demonstrate that it is possible to mimic a Schwarzschild-like geometry in a consistent way only in the framework of relativistic acoustic geometry.

Submersion and Homogeneous Spray Geometry

We introduce the submersion between two spray structures and propose the submersion technique in spray geometry. Using this technique, as well as global invariant frames on a Lie group, we setup the general theoretical framework for homogeneous spray geometry. We define the spray vector field \(\eta \) and the connection operator N for a homogeneous spray manifold \((G/H,{\mathbf {G}})\) with a linear decomposition \({\mathfrak {g}}={\mathfrak {h}}+{\mathfrak {m}}\). These notions generalize their counter parts in homogeneous Finsler geometry. We prove the correspondence between \({\mathbf {G}}\) and \(\eta \) when the given decomposition is reductive, and that between geodesics on \((G/H,{\mathbf {G}})\) and integral curves of \(-\eta \). We find the ordinary differential equations on \({\mathfrak {m}}\) describing parallel translations on \((G/H,{\mathbf {G}})\), and we calculate the S-curvature and Riemann curvature of \((G/H,{\mathbf {G}})\), generalizing L. Huang’s curvature formulae for homogeneous Finsler manifolds.

An analytical framework of the kinematic geometry for general point-contact gears from contact path

Based on the idea that the relative curvature relationships along total contact path will better reflect the contact characteristics of gears, this study attempts to develop an analytical framework for the kinematic geometry of point-contact gears from the contact paths. First, a general method for obtaining the contact path of a gear from a pinion is proposed. Second, a normal curvature and geodesic torsion calculation method in the contact path direction is developed. Then, based on the linearly dependent condition of the tangent vectors of the two contact paths, two relative normal curvature analysis methods are proposed by applying the fundamental form theory. Finally, a sample case is presented to show the application and complete the verification of the kinematic geometry methods. The sample case agrees with the analytical model.

Stochastic Geometry Analysis of IRS-Assisted Downlink Cellular Networks

Using stochastic geometry tools, we develop a comprehensive framework to analyze the downlink performance of various types of users (e.g., users served by direct base station (BS) transmissions and indirect intelligent reflecting surface (IRS)-assisted transmissions) in a cellular network with multiple BSs and IRSs. For the proposed users, we provide the approximate expressions for the performance in terms of coverage probability, ergodic capacity, and energy efficiency (EE). The proposed stochastic geometry framework can capture the impact of channel fading, locations of BSs and IRSs, arbitrary phase-shifts and interference experienced by a typical user supported by direct transmission and/or IRS-assisted transmission. For IRS-assisted transmissions, we first model approximate the desired signal power from the nearest IRS as a sum of scaled generalized gamma (GG) random variables whose parameters are functions of the IRS phase shifts. Then, we derive the Laplace Transform (LT) of the received signal power in a closed form. Also, we approximate the aggregate interference from multiple IRSs as the sum of normal random variables. Then, we derive the LT of the aggregate interference from all IRSs and BSs. The derived LT expressions are used to calculate coverage probability, ergodic capacity, and EE for users served by direct BS transmissions as well as users served by IRS-assisted transmissions. Finally, we derive the overall network coverage probability, ergodic capacity, and EE based on the fraction of direct and IRS-assisted users, which is defined as a function of the deployment density of IRSs, as well as blockage probability of direct transmission links. Numerical results validate the derived analytical expressions and extract useful insights related to the number of IRS elements, large-scale deployment of IRSs and BSs, and the impact of IRS interference on direct transmissions.

A Stochastic Geometry Analysis for Energy-Harvesting-Based Device-to-Device Communication

The rapidly developing energy harvesting (EH) technology is a promising solution to the durability issue in the battery-powered Internet of Things (IoT) systems. In this article, underlaid device-to-device (D2D) transmission powered by radio signals harvested from cellular systems is studied. By considering the dilemmas among EH, D2D transmission opportunity, and interference management, we propose two transmission policies: 1) Policy 1 requires that the available power in the battery should be no less than the D2D transmission power and 2) Policy 2 not only sets the constraint on available power but also introduces the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">guard zone</i> rule to protect D2D transmissions from severe interference. The employment of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">guard zone</i> in this situation is technically challenging since the original distribution of energy arrival will thereby be changed. We derive expressions in closed or semiclosed forms for the considered D2D transmission performance metrics with the stochastic geometry framework and Poisson hole process. With numerical simulation results, the influences of varying network parameters on D2D performances are illustrated. The results show that by introducing a guard zone, the D2D successful transmission rate can be increased by 41.2%. All the developed D2D frameworks and the summarized useful remarks are used to provide meaningful design insights and guidelines for the deployment strategies of EH-based D2D wireless networks.