Geometric Derivation Research Articles

Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two Kähler Forms

In this paper, we extend our geometrical derivation of expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two K\"ahler forms. Especially, we take Hirzebruch surfaces F_{0}, F_{3} and Calabi-Yau hypersurface in weighted projective space P(1,1,2,2,2) as examples. We expect that our results can be easily generalized to arbitrary toric manifold.

Automatic Differentiation of the Energy within Self-consistent Tight-Binding Methods.

We present and implement the calculation of analytical n-order geometric derivatives of the energy obtained within the framework of the density functional based tight binding approach. The use of automatic differentiation techniques allows a unique implementation for the calculation of derivatives up to any order providing that the computational facilities are sufficient. As first applications, the derivatives are used to build an analytical potential energy surface around the optimized geometry of acetylene. We also discuss the relevant anharmonic contributions that have to be considered when building such an analytical potential energy surface for acetylene, ethylene, ethane, benzene, and naphtalene.

A dynamic model of the cellular carbon to chlorophyll ratio applied to a batch culture and a continental shelf ecosystem

A novel parameterization of the dynamical relationship between cellular carbon (C) and chlorophyll (Chl) is developed using a Chl synthesis term that includes the physiological status of the cell and the effect of packaging of pigments within cells. The geometric derivation highlights the non‐linear relationship between Chl content and absorption due to the package effect. When parameterized for a generic 3 µm radius phytoplankton cell, the model reproduces the magnitude and daily variations of C : Chl and C : nitrogen ratios of the diatom Skeletonema costatum in published laboratory experiments. The parameterization is then applied in a three‐dimensional biogeochemical model containing three phytoplankton classes in the coastal waters off southeast Tasmania, Australia, which demonstrates the behavior of the dynamic Chl parameterization over a range of light‐ and nutrient‐limiting environments for phytoplankton of different sizes and growth rates. The model produces C : Chl ratios of ∼ 12–20 (weight : weight) and ∼ 60–80 for phytoplankton communities dominated by fast‐growing small and fast‐growing large cells, respectively, close to the ratios of 17 and 76 observed at two sampling stations during periods with diatom‐ and flagellate‐dominated communities. Throughout the simulation, community C : Chl ratios generally vary between 12 and 200, which is similar to the range observed globally. In the new parameterization, C : Chl ratios are most influenced by the package effect for light‐limited, slow‐growing large microalgae, with physiological processes becoming important for smaller, nutrient‐limited, fast‐growing microalgae.

Internal‐to‐Cartesian back transformation of molecular geometry steps using high‐order geometric derivatives

In geometry optimizations and molecular dynamics calculations, it is often necessary to transform a geometry step that has been determined in internal coordinates to Cartesian coordinates. A new method for performing such transformations, the high-order path-expansion (HOPE) method, is here presented. The new method treats the nonlinear relation between internal and Cartesian coordinates by means of automatic differentiation. The method is reliable, applicable to any system of internal coordinates, and computationally more efficient than the traditional method of iterative back transformations. As a bonus, the HOPE method determines not just the Cartesian step vector but also a continuous step path expressed in the form of a polynomial, which is useful for determining reaction coordinates, for integrating trajectories, and for visualization.

A theoretical analysis of interactive coercing effects between urbanization and eco-environment

Objectively, a complex interactive coercing relationship exists between urbanization and eco-environment, and the research of this relationship is primarily divided into three schools, i.e., interactive coercion theory, interactive promotion theory and coupling symbiosis theory. Harmonizing the relationship between urbanization and eco-environment is not only an important proposition for the national development plan but also the only way to promote healthy urbanization. Based on an analysis of urbanization process and its relationship with the eco-environment, this article analyzes interactive coercing effects between urbanization and eco-environment from three perspectives of population urbanization, economic urbanization and spatial urbanization, respectively, and analyzes risk effects of the interactive coercion. Further, it shows six basic laws followed by interactive coercion between urbanization and eco-environment, namely, coupling fission law, dynamic hierarchy law, stochastic fluctuation law, non-linear synergetic law, threshold value law and forewarning law, and divides the interactive coercing process into five stages, namely, low-level coordinate, antagonistic, break-in, ameliorative and high-grade coordinate. Based on the geometric derivation, the interactive coercing relationship between urbanization and eco-environment is judged to be non-linear and it can be explained by a double-exponential function formed by the combination of power and exponential functions. Then, the evolutionary types of the interactive coercing relationship are divided into nine ones: rudimentary coordinating, ecology-dominated, synchronal coordinating, urbanization lagging, stepwise break-in, exorbitant urbanization, fragile ecology, rudimentary break-in and unsustainable types. Finally, based on an interactive coercion model, the degree of interactive coercion can be examined, and then, an evolutionary cycle can be divided into four phases, namely rudimentary symbiosis, harmonious development, utmost increasing and spiral type rising. The study results offer a scientific decision-making of healthy urbanization for achieving the goal of eco-environment protection and promoting urbanization.

Synergistic Antibacterial Activity of New Isomeric Carborane Derivatives Through Combination with Nanoscaled Titania

As a novel strategy to combat the nosocomial infection caused by multidrug-resistant (MDR) bacteria like Acinetobacter baumannii, the antibacterial effect of a pair of geometrical isomers ferrocene-carborane derivatives (designated as FcSB1 and FcSB2) combining with nanoscaled titania (nano TiO2) against five MDR A. baumannii strains were explored in this contribution. The drug interaction assay and time-kill studies demonstrate the considerable enhancement effect of the TiO2 nanoparticles on the antibacterial activity of FcSB1/or FcSB2 against clinically MDR-resistant A. baumannii. Meanwhile, the relevant antibacterial efficiency assessed by a nonparametric approach (fractional inhibitory concentration [FIC] index model) illustrates that the FIC index values ranged from 0.375 to 0.106 for the combination of FcSB1 and nano TiO2, and 0.250 to 0.083 for that of FcSB2 with nano TiO2, indicating the synergistic antibacterial activity of FcSB1/or FcSB2 with nano TiO2 on target bacteria. This might shed light on the promising bio-application of these new carborane derivatives in combination with nano TiO2 to combat the nosocomial infection caused by multidrug-resistant (MDR) bacteria including A. baumannii.

Geometrical Derivation of Differential Kinematics to Calibrate Model Parameters of Flexible Manipulator

In this paper differential kinematics was geometrically derived to be utilized in a calibration algorithm that improves the accuracy of the manipulation of a robot. Even though the mechanical compo...

On radial stationary solutions to a model of non-equilibrium growth

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.

On the Optimal Control of Impulsive Hybrid Systems on Riemannian Manifolds

This paper provides a geometrical derivation of the hybrid minimum principle (HMP) for autonomous impulsive hybrid systems on Riemannian manifolds, i.e., systems where the manifold valued component of the hybrid state trajectory may have a jump discontinuity when the discrete component changes value. The analysis is expressed in terms of extremal trajectories on the cotangent bundle of the manifold state space. In the case of autonomous hybrid systems, switching manifolds are defined as smooth embedded submanifolds of the state manifold and the jump function is defined as a smooth map on the switching manifold. The HMP results are obtained in the case of time invariant switching manifolds and state jumps on Riemannian manifolds.

Concentration Ellipsoids, Their Planes of Support, and the Linear Regression Model

The relationship between the concentration ellipsoid of a random vector and its planes of support is exploited to provide a geometric derivation and interpretation of existing results for a general form of the linear regression model. In particular, the planes of support whose points of tangency to the ellipsoid are contained in the range (or column space) of the design matrix are the source of all linear unbiased minimum variance estimators. The connection between this idea and estimators based on projections is explored, as is also its use in obtaining and interpreting some existing relative efficiency results.

Time-Dependent Approach to Resonance Raman Spectra Including Duschinsky Rotation and Herzberg-Teller Effects: Formalism and Its Realistic Applications.

Efficient quantum dynamical and electronic structure approaches are presented to calculate resonance Raman spectroscopy (RRS) with inclusion of Herzberg-Teller (HT) contribution and mode-mixing (Duschinsky) effect. In the dynamical method, an analytical expression for RRS in the time domain is proposed to avoid summation over the large number of intermediate vibrational states. In the electronic structure calculations, the analytic energy-derivative approaches for the excited states within the time-dependent density functional theory (TDDFT), developed by us, are adopted to overcome the computational bottleneck of excited-state gradient and Hessian calculations. In addition, an analytic calculation to the geometrical derivatives of the transition dipole moment, entering the HT term, is also adopted. The proposed approaches are implemented to calculate RR spectra of a few of conjugated systems, phenoxyl radical, 2-thiopyridone in water solution, and free-base porphyrin. The calculated RR spectra show the evident HT effect in those π-conjugated systems, and their relative intensities are consistent with experimental measurements.

The unified orthogonal architecture of industrial serial manipulators

The well-known inverse kinematics problem of six-degree-of-freedom serial manipulators has been solved with several approaches, but no attention has been given to provide an explicit solution involving the generality of industrial manipulators architecture. With the aim of embracing a large spectrum of industrial manipulators in a unified platform, an anthropomorphic classification is introduced. This classification considers sixteen different architectures, whose inverse kinematics is solved with a single approach, due to the geometric derivation introduced here as well. It is remarked that with this derivation, the user has the control of the eight possible solutions for a given pose, as they are identified by three Boolean variables defined in the pseudo-code presented. To emphasize this feature, an example is given to show the advantage of being able to visualize alternative configurations. A complementary formulation is introduced to solve the inverse kinematics of five-degree-of-freedom manipulators. Furthermore, it is presented a case study in which a survey of industrial manipulators is tagged according to the classification defined here. Highlights? An orthogonal classification embracing industrial serial manipulators is introduced. ? Deriving the inverse kinematics closed form solution of anthropomorphic manipulators. ? Optional configurations obtained with the designed skeleton of a given manipulator. ? A survey of industrial manipulators is embedded in the orthogonal classification.

Geometry of the Welch bounds

A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular, a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames of symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.

Comment on “Geometric derivation of the quantum speed limit”

Recently, Jones and Kok [P. J. Jones and P. Kok, Phys. Rev. A 82, 022107 (2010)] presented alternative geometric derivations of the Mandelstam-Tamm [L. Mandelstam and I. Tamm, J. Phys. (USSR) 9, 249 (1945)] and Margolus-Levitin [N. Margolus and L. B. Levitin, Phys. D 120, 188 (1998)] inequalities for the quantum speed of dynamical evolution. The Margolus-Levitin inequality followed from an upper bound on the rate of change of the statistical distance between two arbitrary pure quantum states. We show that the derivation of this bound is incorrect. Subsequently, we provide two upper bounds on the rate of change of the statistical distance, expressed in terms of the standard deviation of the generator K and its expectation value above the ground state. The bounds lead to the Mandelstam-Tamm inequality and a quantum speed limit which is only slightly weaker than the Margolus-Levitin inequality.

Image-Based System for Measuring Objects on an Oblique Plane and Its Applications in 2-D Localization

This paper presents an image-based framework for measuring target objects on an oblique plane by using a single charge-coupled device camera and two laser projectors mounted in parallel beside the camera. Because of the alignment of the laser beams, which form in parallel with the optical axis of the camera, laser-projected spots in the image can be processed to establish relationships between distance and pixel counts of the projected spots in the image. Based on simple geometrical derivations without complex image processing, the proposed approach can successfully measure the photographic distance, the distance between two arbitrary points on the oblique surface, and the incline angle of the oblique surface. Thanks to its ranging capability, the proposed image-based measuring system is further applied to localize objects on a ground surface in addition to depth measurement. To demonstrate the feasibility of the proposed approach for practical applications, we propose a surveillance framework under which a pan-tilt-zoom camera tracks objects in an environment according to the 2-D localization results obtained via the proposed method. Experimental results have demonstrated the effectiveness of the proposed approach in distance measurement, as well as localization of objects on an oblique plane.

On the Local Form and Transitions of Pre-symmetry Sets

Symmetry is an important cue in shape analysis. It has lead to the definition of popular shape descriptors like the medial axis. Its properties have been analyzed with a superset, called the symmetry set which represents the midpoints of circles that are at least bitangent to a shape. In this work we investigate the pre-symmetry set. This set considers the pairs of points at which the bitangent contact occurs. One thus obtains pairwise symmetric points of a 2D shape. A closed 2D shape has a parameterization P with finite length. Its pre-symmetry can therefore be represented by a symmetric diagram curves formed from the pairs of points (p i ,p j )?S 1×S 1. We discuss the properties of the pre-symmetry set visualized by this diagram. We firstly give the so-called transitions, changes caused by a perturbation of the shape and show the changes of the curves in the pre-symmetry set diagram. Secondly, we investigate curves that are spanned by all points on the shape. We name the curves essential loops and discuss their properties and transitions. As one important result we show that their are either zero or two essential loops. In the latter case a part of the medial axis is spanned by an essential loop and can therefore be considered as the main axis of the medial axis. As application of pre-symmetry sets, we discuss two possibilities for shape matching based on representations of the pre-symmetry set. The first shape descriptor we present is given by a circular diagram representing the shape, with a set of points representing the extrema of the curvature in the order they appear on the shape. They are pairwise connected and endowed with a length measure. This descriptor is directly related to the curves and their lengths in the pre-symmetry set diagram. The second descriptor is given by a binary array representing the areas enclosed by the curves in the pre-symmetry set diagram. It is area based and relates to the geometric derivation of the symmetry set.

The Catenary as Roulette

SummaryThe focus of a parabola rolling without sliding along a straight line traces a catenary. We give some historical notes on this classical kinematical construction, observe that it was rediscovered in a recent article, and give a simpler and more geometric alternative derivation.

Quantized response and topology of magnetic insulators with inversion symmetry

We study three-dimensional insulators with inversion symmetry in which other point group symmetries, such as time reversal, are generically absent. We find that certain information about such materials' behavior is determined by just the eigenvalues under inversion symmetry of occupied states at time reversal invariant momenta (TRIM parities). In particular, if the total number of $\ensuremath{-}1$ eigenvalues at all TRIMs is odd then the material cannot be an insulator. A likely possibility is that it is then a ``Weyl'' semimetal. Additionally if the material is an insulator and has vanishing Hall conductivity, then a magnetoelectric response, parameterized by $\ensuremath{\theta}$, can be defined, and is quantized to $\ensuremath{\theta}=0,\phantom{\rule{0.16em}{0ex}}\ensuremath{\pi}$. The value is $\ensuremath{\pi}$ if the total number of TRIM parities equal to $\ensuremath{-}1$ is twice an odd number. This generalizes the rule of Fu and Kane that applies to materials in which time reversal is unbroken. This result may be useful in the search for magnetic insulators with large $\ensuremath{\theta}$. These two results are obtained as part of a classification of the band topology of inversion-symmetric insulators. Such band structures can be classified by two sets of numbers: the TRIM parities and three Chern numbers. The TRIM parities have the physical implications just described, and additionally they constrain the values of the Chern numbers modulo 2. An alternate geometrical derivation of our results is obtained by using the entanglement spectrum of the ground-state wave function.

Geometrically Derived Ray-Theory Results and Direct Verification of the Pekeris Solution for Unbounded Constant-Gradient Media

This paper directly verifies the Pekeris solution to the point-source Helmholtz equation for an unbounded constant-gradient medium using only elementary vector calculus. Self-contained geometrical derivations of ray-theory results for such a medium are presented: (1) ray-path location and travel time as a function of source location, ray start angle, and ray angle; (2) the wavefront equation as a function of source location and travel time; (3) the wavefront location and ray angle along a ray as a function of source location, ray start angle, and travel time; and (4) source angle and receiver angle as a function of source location and receiver location. A short mathematical derivation gives the travel time between two points for a given source location and a given receiver location. In some cases, the form of the results seems to be simpler than that of the equivalent results previously given in the literature.

The Exponential Gradient HMP Algorithm for the Optimization of Hybrid Systems On Lie Groups

This paper provides an algorithm for the optimization of autonomous and controlled hybrid systems on Lie groups. First, a geometrical derivation of the Hybrid Minimum Principle (HMP) for hybrid systems whose state manifolds constitute a Lie group (G, ⋆) which is left invariant under the controlled dynamics of the system is presented. Second, a geometrical algorithm is developed by employing the notion of exponential curves on Lie groups. The convergence analysis for the proposed algorithm is based on Lasalle Theory. Simulation results are provided at the end of the paper.