Integral formulae on hypersurfaces in Riemannian manifolds

The coefficients of the complete set of n fundamental forms of a hypersuface V_n−1 imbedded in an n-dimensional Riemannian space V_n, as recently introduced[(5)], are used to construct certain tensor fields over V_n−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of V_n, the coefficients of the n fundamental forms, and the r^th curvatures of V_n−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on V_n−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of V_n−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever V_n is a space of constant curvature. This research was supported by the National Research Council of Canada. Entrata in Redazione il 21 agosto 1970.     

The reduction of certain boundary value problems to variational problems by means of transversality conditions

It is shown by means of the classical theory of the transversality conditions of the calculus of variations that certain boundary value problems are equivalent to necessary conditions for the attainment of extreme values of a fundamental integral of a variational problem with variable boundaries. Systems of second order ordinary, as well as partial, differential equations are considered. The method is illustrated for both cases by means of examples, from which well-known theorems of exceptional practical importance emerge effortlessly.     

Canonical formalism for parameter-invariant integrals in the Calculus of Variations whose Lagrange functions involve second order derivatives

Summary In a recent paper [4] a general theory of parameter-invariant integrals in the Calculus of Variations whose Lagrangians involve higher derivatives was developed, and in particular a certain canonical formalism for such problems was discussed. From the point of view of applications it was found that this formalism proved inadequate inas-much as the suggested Hamiltonian function did not depend explicitly on the first derivatives of the positional coordinates. In the present note an alternative Hamiltonian function is defined, which gives rise to a new canonical formalism. The latter is less complicated than the formalism suggested in [4] and is more readily applicable to special problems. A brief discussion of the resulting Hamilton-Jacobi theory is given, and in conclusion the method is illustrated explicitly by means of an example of fairly general character.     

Local imbedding of Einstein spaces

Summary A special class of hypersurfaces of a Riemannian space is examined, this class being defined by the stipulation that the coefficients of the third fundamental form be expressible as linear combinations of the coefficients of the first and second fundamental forms. It is jound that these so-called C-hypersurfaces are umbilical provided that certain conditions (which may depend on dimension) are satisfied. An (n-1)-dimensional Einstein space imbedded in an n-dimensional space of constant curvature is such a C-hypersurface; accordingly the theory may be applied to the problem of the local imbedding of such spaces. Entrata in Redazione il 23 giugno 1971.     

The theory of problems in the calculus of variations whose Lagrangian function involves second order derivatives: A new approach

Summary Variational principles whose Lagrangian functions involve higher order derivatives have, in the past, been applied to certain aspects of the theory of elementary particles. The corresponding Lagrangian functions must satisfy certain conditions if consistency with the classical electromagnetic interaction terms is sought, and it is found that these conditions are closely related to the requirement that the action integral be invariant under a parameter transformation. If, however, the latter condition is accepted, the usual expression for the Hamiltonian function vanishes identically, resulting in a complete break-down of the canonical equations. Thus an alternative approach to the theory of parameter-invariant problems in the calculus of variations whose Lagrangians depend on second order derivatives is developed. A general Finsler metric is introduced in a natural manner, which provides a geometrical background to the theory as well as useful analytical techniques. It is possible to define an alternative Hamiltonian function corresponding to which a canonical formalism is developed. The method of equivalent integrals is generalised, giving rise to a new and rigorous derivation of theEuler-Lagrange equations, which in turn leads to a generalisation of the so-called excess-function and the analogue of the well-known condition of Weierstrass in the calculus of variations. To Enrico Bompiani on his scientific Jubilee.     

An extension of Noether's theorem to transformations involving position-dependent parameters and their derivatives

Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only the independent and the dependent field variables, but also a set of position-dependent parameters together with their first derivatives. The stipulation that an action integral associated with the field variables be invariant under such transformations gives rise to a set of three conditions involving the Lagrangian and its derivatives, together with derivatives of the functions that define the transformations. These invariance identities constitute an extension of the classical theorem of Noether to general transformations of this kind. An application to the case of gauge fields demonstrates the existence of two distinct types of conservation laws for such fields.     

Tutorial review on validation of liquid chromatography–mass spectrometry methods: Part II

This is the part II of a tutorial review intending to give an overview of the state of the art of method validation in liquid chromatography mass spectrometry (LC–MS) and discuss specific issues that arise with MS (and MS–MS) detection in LC (as opposed to the “conventional” detectors). The Part II starts with briefly introducing the main quantitation methods and then addresses the performance related to quantification: linearity of signal, sensitivity, precision, trueness, accuracy, stability and measurement uncertainty. The last section is devoted to practical considerations in validation. With every performance characteristic its essence and terminology are addressed, the current status of treating it is reviewed and recommendations are given, how to handle it, specifically in the case of LC–MS methods.     

Chern-Simons expectation values and quantum horizons from loop quantum gravity and the Duflo map

We report on a new approach to the calculation of Chern-Simons theory expectation values, using the mathematical underpinnings of loop quantum gravity, as well as the Duflo map, a quantization map for functions on Lie algebras. These new developments can be used in the quantum theory for certain types of black hole horizons, and they may offer new insights for loop quantum gravity, Chern-Simons theory and the theory of quantum groups.     

Passivity and Microlocal Spectrum Condition

In the setting of vector-valued quantum fields obeying a linear wave-equation in a globally hyperbolic, stationary spacetime, it is shown that the two-point functions of passive quantum states (mixtures of ground- or KMS-states) fulfill the microlocal spectrum condition (which in the case of the canonically quantized scalar field is equivalent to saying that the two-pnt function is of Hadamard form). The fields can be of bosonic or fermionic character. We also give an abstract version of this result by showing that passive states of a topological *-dynamical system have an asymptotic pair correlation spectrum of a specific type. Received: 9 February 2000 / Accepted: 7 June 2000