scholarly journals Prolongations of G-Structures to Tangent Bundles of Higher Order

1970 ◽  
Vol 38 ◽  
pp. 153-179 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Tangent Bundle ◽  
Higher Order ◽  
First Order ◽  
First Contact ◽  
Tangent Bundles ◽  
Derivatives Of ◽  
Special Case

In the previous paper [4] we have studied the prolongations of G-structures to tangent bundles. The purpose of the present paper is to generalize the previous prolongations and to look at them from a wide view as a special case by considering the tangent bundles of higher order. In fact, in some places, the arguments and calculations in [4] are more or less simplified. Since the usual tangent bundle T(M) of a manifold M considers only the first derivatives or first contact elements of M, the previous paper contains, in most parts, only the calculation of derivatives of first order.

scholarly journals Causality and passivity in elastodynamics

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150256 ◽  
Author(s):  
Ankit Srivastava
Keyword(s):  
Fourier Transforms ◽  
Constitutive Relations ◽  
Higher Order ◽  
Causal Effects ◽  
Passive System ◽  
One Dimensional ◽  
The Real ◽  
Analogous Question ◽  
Derivatives Of ◽  
Special Case

What are the constraints placed on the constitutive tensors of elastodynamics by the requirements that the linear elastodynamic system under consideration be both causal (effects succeed causes) and passive (system does not produce energy)? The analogous question has been tackled in other areas but in the case of elastodynamics its treatment is complicated by the higher order tensorial nature of its constitutive relations. In this paper, we clarify the effect of these constraints on highly general forms of the elastodynamic constitutive relations. We show that the satisfaction of passivity (and causality) directly requires that the hermitian parts of the transforms (Fourier and Laplace) of the time derivatives of the constitutive tensors be positive semi-definite. Additionally, the conditions require that the non-hermitian parts of the Fourier transforms of the constitutive tensors be positive semi-definite for positive values of frequency. When major symmetries are assumed these definiteness relations apply simply to the real and imaginary parts of the relevant tensors. For diagonal and one-dimensional problems, these positive semi-definiteness relationships reduce to simple inequality relations over the real and imaginary parts, as they should. Finally, we extend the results to highly general constitutive relations which include the Willis inhomogeneous relations as a special case.

On the Decomposition of Spinor Fields which satisfy a Nonlinear Higher Order Equation

1983 ◽  
Vol 38 (12) ◽  
pp. 1293-1295
Author(s):  
D. Großer
Keyword(s):  
Field Theory ◽  
Higher Order ◽  
Order Equation ◽  
Field Theories ◽  
First Order ◽  
Spinor Fields ◽  
Fermion Fields ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
Higher Order Equation

Abstract A field theory which is based entirely on fermion fields is non-renormalizable if the kinetic energy contains only derivatives of first order and therefore higher derivatives have to be included. Such field theories may be useful for describing preons and their interaction. In this note we show that a spinor field which satisfies a higher order field equation with an arbitrary nonlinear selfinteraction can be written as a sum of fields which satisfy first order equations.

Generalized systems of linear differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
Rainer Pfaff
Keyword(s):  
Differential Equations ◽  
Bounded Variation ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Generalized Systems ◽  
Linear Differential ◽  
Derivatives Of

SynopsisWe consider ordinary linear differential systems of first order with distributional coefficients and distributional nonhomogeneous terms. Firstly the coefficients are assumed to be functions, secondly to be first order distributions (i.e. first derivatives of functions which are integrable or of bounded variation), and thirdly to be distributions of higher order.

scholarly journals Prolongations of Connections to Tangential Fibre Bundles of Higher Order

1970 ◽  
Vol 40 ◽  
pp. 85-97 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Tangent Bundle ◽  
Higher Order ◽  
Fibre Bundles ◽  
Affine Connections ◽  
Tangent Bundles

In the previous paper [3] we have studied the prolongations of G-structures to tangent bundles of higher order. The purpose of the present paper is to study the prolongations of connections to tangential fibre bundles of higher order, and to generalize the results due to S. Kobayashi [1] for the case of usual tangent bundle —— in fact, the arguments in [1] will be, in a sense, more or less simplified and clarified by using the notion of tangent bundles of higher order. In addition, as a consequence of our results, we shall obtain the prolongations of linear (affine) connections to tangent bundles of higher order.

scholarly journals Rotatory vibration of sphere of higher order viscoelastic solid

1994 ◽  
Vol 17 (4) ◽  
pp. 799-806 ◽  
Author(s):  
P. R. Sengupta ◽  
Nibedita De (Das) ◽  
Manidipa Kar ◽  
Lokenath Debnath
Keyword(s):  
Strain Rate ◽  
Elastic Solid ◽  
Frequency Equation ◽  
Higher Order ◽  
Stress Rate ◽  
Viscoelastic Solid ◽  
Viscoelastic Solids ◽  
First Order ◽  
General Frequency ◽  
Special Case

An attempt is made to investigate the rotatory vibration of a sphere of higher order viscoelastic solid considering higher order strain rate and stress rate. The general frequency equation is obtained for this type of vibration of a sphere. As a special case of this analysis, the frequency equations for the first order and the second order viscoelastic solids are derived. It is shown that the classical frequency equation for an isotropic elastic solid follows from this analysis.

scholarly journals On the relativistic invariance of quantized field theories

1948 ◽  
Vol 195 (1042) ◽  
pp. 365-376 ◽  
Keyword(s):  
Present Author ◽  
Higher Order ◽  
Field Theories ◽  
Relativistic Invariance ◽  
Field Equations ◽  
First Order ◽  
Generalized Poisson ◽  
General Relativistic ◽  
Quantized Field ◽  
Derivatives Of

Heisenberg & Pauli (1929) have shown how to quantize field theories derived from Lagrangians containing first-order derivatives of the field quantities. They showed their quantization to be Lorentz invariant. Fuchs (1939) subsequently showed that the quantized theory was in fact invariant under general transformations of co-ordinates. The present author in another paper has shown how the theory of Heisenberg & Pauli can be extended to field equations derived from higher order Lagrangians, i. e. Lagrangians containing higher deri­vatives than the first of the field quantities. In the present paper the general relativistic invariance of the higher order quantized theories is established, making use of the generalized Poisson brackets introduced by Weiss.

Higher-order differential equations and higher-order lagrangian mechanics

1986 ◽  
Vol 99 (3) ◽  
pp. 565-587 ◽  
Author(s):  
M. Crampin ◽  
W. Sarlet ◽  
F. Cantrijn
Keyword(s):  
Order Differential Equation ◽  
Higher Order ◽  
Point Of View ◽  
Lagrangian Mechanics ◽  
First Order ◽  
The Subject ◽  
Tangent Bundles ◽  
Order Tangent ◽  
Higher Order Differential Equations ◽  
Number Of Authors

The study of higher-order mechanics, by various geometrical methods, in the framework of the theory of higher-order tangent bundles or jet spaces, has been undertaken by a number of authors recently: for example, Tulczyjew [16, 17], Rodrigues [14, 15] de León [8], Krupka and Musilova [11, and references therein]. In this article we wish to complement these studies by approaching the subject from a new point of view, one which we developed for second-order differential equation fields and first-order Lagrangian mechanics in [19]. In particular, our aim is to show that many of the results we obtained there may be extended to the higher-order case.

scholarly journals Eichler cohomology and zeros of polynomials associated to derivatives of L-functions

2021 ◽  
Vol 2021 (770) ◽  
pp. 1-25
Author(s):  
Nikolaos Diamantis ◽  
Larry Rolen
Keyword(s):  
Modular Forms ◽  
Critical Role ◽  
Cusp Forms ◽  
Zeros Of Polynomials ◽  
First Order ◽  
Higher Derivative ◽  
Eichler Cohomology ◽  
Derivatives Of ◽  
Special Case ◽  
Function Field Case

Abstract In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section 5.3) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series. Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L-derivatives in the function field case. A critical role in their work was played by a notion of “super-positivity”, which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang.

Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities

2018 ◽  
Vol 18 (3) ◽  
pp. 543-553 ◽  
Author(s):  
Mengxia Dong
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Higher Order ◽  
Extensive Study ◽  
Extremal Functions ◽  
Compact Embedding ◽  
First Order ◽  
Nirenberg Inequality ◽  
Higher Order Derivative ◽  
Derivatives Of

Abstract Though there has been an extensive study on first-order Caffarelli–Kohn–Nirenberg inequalities, not much is known for the existence of extremal functions for higher-order ones. The higher-order derivative of the Caffarelli–Kohn–Nirenberg inequality established by Lin [14] states \bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{j}u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{% \frac{1}{r}}\leq C\bigg{(}\int_{\mathbb{R}^{N}}\lvert D^{m}u|^{p}\frac{dx}{|x|% ^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{(}\int_{\mathbb{R}^{N}}\lvert u|^{q}\frac{% dx}{|x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}, where {C=C(p,q,r,\mu,\sigma,s,m,j)} and {p,q,r,\mu,\sigma,s,m,j} are parameters satisfying some balanced conditions. The main purpose of this paper is to establish the existence of extremal functions for a family of this higher-order derivatives of Caffarelli–Kohn–Nirenberg inequalities under numerous circumstances of parameters. Moreover, we study the compactness of the weighted Sobolev space for higher-order derivatives and prove that {\dot{H}^{m,p}_{\mu}(\Omega)\cap L^{q}_{\sigma}(\Omega)\hookrightarrow\dot{H}^% {j,r}_{s}(\Omega)} is a compact embedding within some range of the parameters.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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