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.     

Infrared reflection-absorption spectra of C2H4 and C2H6 on Cu: effect of surface roughness

Infrared reflection absorption spectroscopy (IRRAS) of the highly symmetric molecules C2H4 and C2H6 adsorbed as mono- and multilayers onto copper films is studied in relation to the type of metal-film roughness. Spectra of C2H4 show Raman lines on cold-deposited Cu films but not on Cu deposited at room temperature. For C2H6, the IR spectra from both types of metal films are similar; the surface infrared selection rule holds and no Raman bands are observed. The Raman lines that appear in the IR spectra already at low exposures are attributed to species adsorbed at special defect sites, identical to the so-called active sites in surface enhanced Raman scattering (SERS). The IR excitation mechanism by transient electron transfer to the adsorbate pi* state can deliver a discrete vibrational band of a Raman-active vibration only under certain circumstances, for example, for adsorbates at the "SERS-active sites". C2H6 at these sites cannot deliver Raman bands in IRRAS, because it has no pi* state. We also discuss IRRAS measurements on Cu(111) and Cu(110) single crystals, where Raman bands of C2H4 have been observed.     

A Deterministic Method for Robust Estimation of Multivariate Location and Shape

Abstract

The existence of outliers in a data set and how to deal with them is an important problem in statistics. The minimum volume ellipsoid (MVE) estimator is a robust estimator of location and covariate structure; however its use has been limited because there are few computationally attractive methods. Determining the MVE consists of two parts—finding the subset of points to be used in the estimate and finding the ellipsoid that covers this set. This article addresses the first problem. Our method will also allow us to compute the minimum covariance determinant (MCD) estimator. The proposed method of subset selection is called the effective independence distribution (EID) method, which chooses the subset by minimizing determinants of matrices containing the data. This method is deterministic, yielding reproducible estimates of location and scatter for a given data set. The EID method of finding the MVE is applied to several regression data sets where the true estimate is known. Results show that the EID method, when applied to these data sets, produces the subset of data more quickly than conventional procedures and that there is less than 6% relative error in the estimates. We also give timing results illustrating the feasibility of our method for larger data sets. For the case of 10,000 points in 10 dimensions, the compute time is under 25 minutes.     

Nanorattles or Yolk–Shell Nanoparticles—What Are They,How Are They Made,and What Are They Good For?

The development of nanotechnology has led to the design of cutting‐edge nanomaterials with increasing levels of complexity. Although “traditional” solid, uniform nanoparticles are still the most frequently reported structures, new generations of nanoparticles have been constantly emerging over the last several decades. The outcome of this nano‐art extends beyond nanomaterials with alternative compositions and/or morphologies. The current state‐of‐the‐art allows for the design of nanostructures composed of different building blocks that exhibit diverse properties. Furthermore, those properties can be a reflection of either individual features, which are characteristic of a particular building block alone, and/or synergistic effects resulting from interactions between building blocks. Therefore, the unique structures as well as the outstanding properties of nanorattles have attracted increasing attention for possible biomedical and industrial applications. Although these nanoparticles resemble core–shell particles, they have a distinctive feature, which is a presence of a void that provides a homogenous environment for the encapsulated core. In this Review, we give a comprehensive insight into the fabrication of nanorattles. A special emphasis is put on the choice of building blocks as well as the choice of preparation method, because those two aspects further influence properties and thus possible future applications, which will also be discussed.     

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.     

MODp‐tests,almost independence and small probability spaces

In this paper, we consider approximations to probability distributions over Z . We present an approach to estimate the quality of approximations to probability distributions towards the construction of small probability spaces. These small spaces are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan, and Veličković [EGLNV], the methods which are used here are simple, and we get smaller sample spaces. Our investigations are motivated by recent work of Azar, Motwani, and Naor [AMN]. They considered the problem to construct in time respective space polynomial in n a good approximation to the joint probability distribution of the mutually independent random variables X₁, X₂,…,X_n. Each X_i has values in {0, 1} and satisfies X_i=0 with probability q and X_i=1 with probability 1−q where q∈[0, 1] is arbitrary. Our considerations improve on results in [EGLNV] and [AMN] for q=1/p and p a prime. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 293–313, 2000