Coordinate-free Approach Research Articles

A Direct Approach to the Polar Representation of Plane Tensors

We show we can derive the so called polar representation of 2D symmetric second and fourth order real tensors essentially just relying on the spectral theorem for unitary tensors in the complex field. The use of a coordinate-free approach allows us to clearly and methodically detect rotation-invariant quantities and to readily (and directly) deduce the representation theorems.

A Coordinate-Free Approach to the Design of Generalized Griffis-Duffy Platforms

Architectural singularity belongs to the Type II singularity, in which a parallel manipulator (PM) gains one or more degrees of freedom and becomes uncontrollable. PMs remaining permanently in a singularity are beneficial for linear-to-rotary motion conversion. Griffis-Duffy (GD) platform is a mobile structure admitting a Bricard motion. In this paper, we present a coordinate-free approach to the design of generalized GD platforms, which consists in determining the shape and attachment of both the moving platform and the fixed base. The generalized GD platform is treated as a combination of six coaxial single-loop mechanisms under the same constraints. Owing to the inversion, hidden in the geometric structure of these single-loop mechanisms, the mapping from a line to a circle establishes the geometric transformation between the fixed base and the moving platform based on the center of inversion, and describes the shape and attachment of the generalized GD platform. Moreover, the center of inversion not only identifies the location of rotation axis, but also affects the shape of the platform mechanism. A graphical construction of generalized GD platforms using inversion, proposed in the paper, provides geometrically feasible solutions of the manipulator design for the requirement of the location of rotation axis.

THE BEST LINEAR UNBIASED ESTIMATORS OF REGRESSION COEFFICIENTS IN A MULTIVARIATE GROWTH-CURVE MODEL. A COORDINATE-FREE APPROACH

The problems of the existence of the best linear unbiased estimators (BLUE) and of their equality to the ordinary least squares estimators (OLSE) of the expected value of the observations are treated in a coordinate-free approach using a multivariate growth-curve model. It will be proved that the alternative forms of the necessary and sufficient conditions used in solving these problems are independent of the between-individuals design matrix of the model.

Local orthogonal maps and rigidity of holomorphic mappings between real hyperquadrics

We propose a coordinate-free approach to study the holomorphic maps between the real hyperquadrics in complex projective spaces. It is based on a notion of orthogonality on the projective spaces induced by the Hermitian structures that define the hyperquadrics. There are various kinds of special linear subspaces associated to this orthogonality which are well respected by the relevant holomorphic maps and we obtain rigidity theorems by analyzing the images of these linear subspaces, together with techniques in projective geometry. Our method allows us to recover and generalize a number of well-known results in the field with simpler arguments.

Friedman-reflexivity

In the present paper, we explore an idea of Harvey Friedman to obtain a coordinate-free presentation of consistency. For some range of theories, Friedman's idea delivers actual consistency statements (modulo provable equivalence). For a wider range, it delivers consistency-like statements.We say that a sentence C is an interpreter of a finitely axiomatised A over U iff it is the weakest statement C over U, with respect to U-provability, such that U+C interprets A. A theory U is Friedman-reflexive iff every finitely axiomatised A has an interpreter over U. Friedman shows that Peano Arithmetic, PA, is Friedman-reflexive.We study the question which theories are Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free, or Herbrand consistency statements. We illustrate that Peano Corto as a base theory has additional desirable properties.We prove a characterisation theorem for the Friedman-reflexivity of sequential theories. We provide an example of a Friedman-reflexive sequential theory that substantially differs from the paradigm cases of Peano Arithmetic and Peano Corto.Interpreters over a Friedman-reflexive U can be used to define a provability-like notion for any finitely axiomatised A that interprets U. We explore what modal logics this idea gives rise to. We call such logics interpreter logics. We show that, generally, these logics satisfy the Löb Conditions, aka K4. We provide conditions for when interpreter logics extend S4, K45, and Löb's Logic. We show that, if either U or A is sequential, then the condition for extending Löb's Logic is fulfilled. Moreover, if our base theory U is sequential and if, in addition, its interpreters can be effectively found, we prove Solovay's Theorem. This holds even if the provability-like operator is not necessarily representable by a predicate of Gödel numbers.At the end of the paper, we briefly discuss how successful the coordinate-free approach is.

Sobolev spaces and $$\nabla $$-differential operators on manifolds I: basic properties and weighted spaces

We study covariant Sobolev spaces and $$\nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted $$\nabla $$ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the $$\nabla $$ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection $$\nabla $$ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our $$\nabla $$ -differential operators. We also introduce and study the notion of a $$\nabla $$ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for $$\nabla $$ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

Differential operators mod p : analytic continuation and consequences

This paper concerns certain $\mod p$ differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the $\mod p$ reduction of the $p$-adic theta operators previously studied by some of the authors. In the characteristic $0$, $p$-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our $\mod p$ operators extend ("analytically continue", in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre's conjecture, we discuss some effects of these operators on Galois representations. Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with $q$-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases.

Exploring Energy Landscapes of Intrinsically Disordered Proteins: Insights into Functional Mechanisms.

Intrinsically disordered proteins (IDPs) lack a rigid three-dimensional structure and populate a polymorphic ensemble of conformations. Because of the lack of a reference conformation, their energy landscape representation in terms of reaction coordinates presents a daunting challenge. Here, our newly developed energy landscape visualization method (ELViM), a reaction coordinate-free approach, shows its prime application to explore frustrated energy landscapes of an intrinsically disordered protein, prostate-associated gene 4 (PAGE4). PAGE4 is a transcriptional coactivator that potentiates the oncogene c-Jun. Two kinases, namely, HIPK1 and CLK2, phosphorylate PAGE4, generating variants phosphorylated at different serine/threonine residues (HIPK1-PAGE4 and CLK2-PAGE4, respectively) with opposing functions. While HIPK1-PAGE4 predominantly phosphorylates Thr51 and potentiates c-Jun, CLK2-PAGE4 hyperphosphorylates PAGE4 and attenuates transactivation. To understand the underlying mechanisms of conformational diversity among different phosphoforms, we have analyzed their atomistic trajectories simulated using AWSEM forcefield, and the energy landscapes were elucidated using ELViM. This method allows us to identify and compare the population distributions of different conformational ensembles of PAGE4 phosphoforms using the same effective phase space. The results reveal a predominant conformational ensemble with an extended C-terminal segment of WT PAGE4, which exposes a functional residue Thr51, implying its potential of undertaking a fly-casting mechanism while binding to its cognate partner. In contrast, for HIPK1-PAGE4, a compact conformational ensemble enhances its population sequestering phosphorylated-Thr51. This clearly explains the experimentally observed weaker affinity of HIPK1-PAGE4 for c-Jun. ELViM appears as a powerful tool, especially to analyze the highly frustrated energy landscape representation of IDPs where appropriate reaction coordinates are hard to apprehend.

Dirac Equation on the Kerr–Newman Spacetime and Heun Functions

By employing a pseudoorthonormal coordinate-free approach, the Dirac equation for particles in the Kerr–Newman spacetime is separated into its radial and angular parts. In the massless case to which a special attention is given, the general Heun-type equations turn into their confluent form. We show how one recovers some results previously obtained in literature, by other means.

On a fifth-order nonselfadjoint boundary value problem

In this paper we aim to share a way to impose some nonselfadjoint boundary conditions for the solutions of a formally symmetric fifth-order differential equation. Constructing a dissipative operator related with the problem we obtain some informations on spectral properties of the problem. In particular, using coordinate-free approach we construct characteristic matrix-function related with the contraction which is obtained with the aid of the dissipative operator.

Thermal buckling of thin injection-molded FRP plates

The occurrence of buckling during cooling of an injection-molded fiber-reinforced plastic (FRP) object is a complex phenomenon which can result in unacceptably large deformations. The present work focuses on the thermal buckling of injection-molded thin disks, as they are representative for the local behavior in parts of a more complex object. An analytical approach based on the Föppl-von Kármán theory for thin elastic plates is presented and used to analyze its buckling stability and initial post-buckling behavior. The system of equations is formulated in a coordinate-free approach for a thin anisotropic linear thermo-elastic body with residual thermal stresses. Large out-of-plane displacements are assumed. The particular case of a free disk with cylindrical orthotropy and uniform thermo-elastic properties through its thickness is considered. A general technique to study the bifurcation in the solutions (i.e., buckling) based on a perturbation approach and the Fourier-Galerkin method to determine post-buckling deflections is proposed. Experimental and numerical results available in the literature are used as validation cases. The results of the present model show qualitative agreement with the experimental data. It is found that the periodicity of the first buckling mode for orthotropic disks is fully determined by the ratio between the elastic moduli in the tangential and radial directions. The predictions are also in excellent quantitative agreement with the numerical results of FEM models for the same cases. The effect of the disk thickness on the buckling temperature and warpage magnitude is correctly captured.

Massless fermions on static general prolate metrics and their Heun solutions

Employing a pseudo-orthonormal coordinate-free approach, we write down the Dirac equation in spacetimes with static general prolate metrics. As examples, we consider the electrically charged C-metric, the vacuum C-metric, the Reissner–Nordström and Schwarzschild spacetimes and the BBMB black hole and show that the solutions to the Dirac equations for particles in these spacetimes can be derived in terms of Heun’s general functions and their confluent and double confluent forms. By imposing boundary conditions to the radial solutions, the so-called resonant frequencies are obtained.

Coordinate-Free Approach for the Characteristic Function of a Fourth-Order Dissipative Operator

In this article, we investigate some spectral properties of a singular dissipative fourth-order dissipative operator in case at the singular point. For this purpose we construct the characteristic function of both maximal simple dissipative operator and completely non-unitary contraction which is the Cayley transform of the dissipative operator. Using the properties of the characteristic operator-function we obtain the related results of the boundary value problem. Moreover we obtain the selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing eigenfunctions by using coordinate-free approach.

Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator

In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential equation and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal nonselfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.

TESTING HYPOTHESES ABOUT STRUCTURE OF PARAMETERS IN MODELS WITH BLOCK COMPOUND SYMMETRIC COVARIANCE STRUCTURE

Abstract In this article we deal with testing the hypotheses of the so-called structured mean vector and the structure of a covariance matrix. For testing the above mentioned hypotheses Jordan algebra properties are used and tests based on best quadratic unbiased estimators (BQUE) are constructed. For convenience coordinate-free approach (see Kruskal (1968) and Drygas (1970)) is used as a tool for characterization of best unbiased estimators and testing hypotheses. To obtain the test for mean vector, linear function of mean vector with the standard inner product in null hypothesis is changed into equivalent hypothesis about some quadratic function of mean parameters (it is shown that both hypotheses are equivalent and testable). In both tests the idea of the positive and negative part of quadratic estimators is applied to get the test, statistics which have F distribution under the null hypothesis. Finally, power functions of the obtained tests are compared with other known tests like LRT or Roy test. For some set for parameters in the model the presented tests have greater power than the above mentioned tests. In the article we present new results of coordinate-free approach and an overview of existing results for estimation and testing hypotheses about BCS models.

The mass in terms of Einstein and Newton

It is shown that the mass of an asymptotically flat manifold with a noncompact boundary can be computed in terms of limiting surface integrals involving the Einstein tensor of the interior metric and the Newton tensor attached to the second fundamental form of the boundary. This extends to this setting previous results by several authors in the boundaryless case. The method outlined below, which is based on a coordinate-free approach due to Herzlich, also applies to asymptotically hyperbolic manifolds, again with a noncompact boundary, for which a similar notion of mass has been recently considered by Almaraz and the first named author, and both cases will be discussed here.

Heun Functions Describing Bosons and Fermions on Melvin’s Spacetime

Employing a pseudo-orthonormal coordinate-free approach, the solutions to the Klein–Gordon and Dirac equations for particles in Melvin spacetime are derived in terms of Heun’s biconfluent functions.

Decentralised attitude synchronisation of multiple rigid bodies on lie group SO (3)

Here, the global and coordinate-free approach has been applied to study the leaderless attitude synchronisation problem of multiple rigid bodies with kinematic evolving on Lie group SO (3). A decentralised consensus law based on the gradient of a smooth cost function which depends on the relative attitude of neighbouring agents is proposed. The advantage of using this cost function lies in the fact that there is a relation between the Hessian matrix of this function in a consensus configuration and the Laplacian matrix of a graph. In addition, a new approach for analysing the attitude synchronisation problem was developed and proved that convergence to synchronisation can be achieved for any connected and undirected graph if the initial attitudes of neighbouring agents are sufficiently close. Using the critical points of the cost function, the effects of graph topology on synchronisation problem were investigated. The initial attitudes sets ensuring the convergence to synchronisation are obtained for the undirected tree, complete, and ring graphs. The initial set is more restrictive for ring graph owing to the existence of other critical points induced by graph topology. Eventually, using the appropriate Lyapunov function, sufficient condition that guarantees convergence to synchronisation for general connected graph was obtained, and numerical simulation carried out to illustrate the theoretical results.

Stress-driven integral elastic theory for torsion of nano-beams

Functionally graded elastic annular nano-beams subjected to torsion are studied by a coordinate-free approach. Strain- and stress-driven integral elasticity models are formulated for investigation of structural behavior of beam-like components of nano-electro-mechanical systems (NEMS). The analysis reveals that the Eringen strain-driven fully nonlocal model cannot be used in Structural Mechanics. The stress-driven theory is instead mathematically and mechanically appropriate for nanotechnological applications. Exact solutions of elastic torsional rotations of nano-beams of technical interest are established by adopting the new stress-driven integral relation equipped with error and bi-exponential kernels. Effectiveness of the new nonlocal strategy is tested by comparing the contributed results, with the ones corresponding to the first-gradient elasticity theory and to the Eringen special differential law.

Exact Fourier inversion formula over manifolds

We show an exact (i.e. no smooth error terms) Fourier inversion type formula for differential operators over Riemannian manifolds. This provides a coordinate free approach for the theory of pseudo-differential operators.