18,369 research outputs found
Programming matrix optics into Mathematica
The various non-linear transformations incurred by the rays in an optical
system can be modelled by matrix products up to any desired order of
approximation. Mathematica software has been used to find the appropriate
matrix coefficients for the straight path transformation and for the
transformations induced by conical surfaces, both direction change and position
offset. The same software package was programmed to model optical systems in
seventh-order. A Petzval lens was used to exemplify the modelling power of the
program.Comment: 15 page
The hidden geometric character of relativistic quantum mechanics
The presentation makes use of geometric algebra, also known as Clifford
algebra, in 5-dimensional spacetime. The choice of this space is given the
character of first principle, justified solely by the consequences that can be
derived from such choice and their consistency with experimental results. Given
a metric space of any dimension, one can define monogenic functions, the
natural extension of analytic functions to higher dimensions; such functions
have null vector derivative and have previously been shown by other authors to
play a decisive role in lower dimensional spaces. All monogenic functions have
null Laplacian by consequence; in an hyperbolic space this fact leads
inevitably to a wave equation with plane-like solutions. This is also true for
5-dimensional spacetime and we will explore those solutions, establishing a
parallel with the solutions of the Dirac equation. For this purpose we will
invoke the isomorphism between the complex algebra of 4x4 matrices, also known
as Dirac's matrices. There is one problem with this isomorphism, because the
solutions to Dirac's equation are usually known as spinors (column matrices)
that don't belong to the 4x4 matrix algebra and as such are excluded from the
isomorphism. We will show that a solution in terms of Dirac spinors is
equivalent to a plane wave solution. Just as one finds in the standard
formulation, monogenic functions can be naturally split into positive/negative
energy together with left/right ones. This split is provided by geometric
projectors and we will show that there is a second set of projectors providing
an alternate 4-fold split. The possible implications of this alternate split
are not yet fully understood and are presently the subject of profound
research.Comment: 29 pages. Small changes in V3 suggested by refere
A General Method for the Determination of Matrix Coefficients for High Order Optical System Modelling
The non-linear transformations incurred by the rays in an optical system can
be suitably described by matrices to any desired order of approximation. In
systems composed of uniform refractive index elements, each individual ray
refraction or translation has an associated matrix and a succession of
transformations correspond to the product of the respective matrices. This
paper describes a general method to find the matrix coefficients for
translation and surface refraction irrespective of the surface shape or the
order of approximation. The choice of coordinates is unusual as the orientation
of the ray is characterised by the direction cosines, rather than slopes; this
is shown to greatly simplify and generalise coefficient calculation. Two
examples are shown in order to demonstrate the power of the method: The first
is the determination of seventh order coefficients for spherical surfaces and
the second is the determination of third order coefficients for a toroidal
surface.Comment: 12 page
Wavefront and ray-density plots using seventh-order matrices
The optimization of an optical system benefits greatly from a study of its
aberrations and an identification of each of its elements' contribution to the
overall aberration figures. The matrix formalism developed by one of the
authors was the object of a previous paper and allows the expression of
image-space coordinates as high-order polynomials of object-space coordinates.
In this paper we approach the question of aberrations, both through the
evaluation of the wavefront evolution along the system and its departure from
the ideal spherical shape and the use of ray density plots. Using seventh-order
matrix modeling, we can calculate the optical path between any two points of a
ray as it travels along the optical system and we define the wavefront as the
locus of the points with any given optical path; the results are presented on
the form of traces of the wavefront on the tangential plane, although the
formalism would also permit sagital plane plots. Ray density plots are obtained
by actual derivation of the seventh-order polynomials.Comment: 15 pages, 5 figure
Evaluating Hospital Efficiency Adjusting for Quality Indicators: an Application to Portuguese NHS Hospitals
The objective of this paper is to develop a methodology to incorporate measures of hospital quality in efficiency analysis, applied to Portuguese NHS hospitals, in order to assess whether there is a trade-off between efficiency and quality in Portuguese hospitals. We develop and compare two methodologies to compute DEA technical efficiency scores adjusted for output quality, for a sample of Portuguese NHS hospitals in 2009. When DEA efficiency scores are adjusted for output quality, the decision making units that lie on the technical efficiency frontier remain largely unaltered, even if a great weight is given to quality indicators over quantity indicators of output. Nevertheless, we find that outside of the frontier adjusting for quality does have an impact in efficiency scores.We conclude that the empirical evidence is not sufficient to identify a clear trade-off between efficiency and quality in the hospitals under review, implying the possibility that efficiency gains may achieved without a significant sacrifice of service quality. Nevertheless, there is enough evidence to conclude that analyzing hospital efficiency without consideration of differences in quality of service will generate biased results. When perceived quality is brought to the analysis, the gap between efficient and inefficient units tends to widen.Hospital efficiency, Hospital quality, Data Envelopment Analysis
Recognizing pro-R closures of regular languages
Given a regular language L, we effectively construct a unary semigroup that
recognizes the topological closure of L in the free unary semigroup relative to
the variety of unary semigroups generated by the pseudovariety R of all finite
R-trivial semigroups. In particular, we obtain a new effective solution of the
separation problem of regular languages by R-languages
Verified Correctness and Security of mbedTLS HMAC-DRBG
We have formalized the functional specification of HMAC-DRBG (NIST 800-90A),
and we have proved its cryptographic security--that its output is
pseudorandom--using a hybrid game-based proof. We have also proved that the
mbedTLS implementation (C program) correctly implements this functional
specification. That proof composes with an existing C compiler correctness
proof to guarantee, end-to-end, that the machine language program gives strong
pseudorandomness. All proofs (hybrid games, C program verification, compiler,
and their composition) are machine-checked in the Coq proof assistant. Our
proofs are modular: the hybrid game proof holds on any implementation of
HMAC-DRBG that satisfies our functional specification. Therefore, our
functional specification can serve as a high-assurance reference.Comment: Appearing in CCS '1
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