Quantum Kaleidoscopes and Bell's theorem

A quantum kaleidoscope is defined as a set of observables, or states, consisting of many different subsets that provide closely related proofs of the Bell-Kochen-Specker (BKS) and Bell nonlocality theorems. The kaleidoscopes prove the BKS theorem through a simple parity argument, which also doubles as a proof of Bell's nonlocality theorem if use is made of the right sort of entanglement. Three closely related kaleidoscopes are introduced and discussed in this paper: a 15-observable kaleidoscope, a 24-state kaleidoscope and a 60-state kaleidoscope. The close relationship of these kaleidoscopes to a configuration of 12 points and 16 lines known as Reye's configuration is pointed out. The "rotations" needed to make each kaleidoscope yield all its apparitions are laid out. The 60-state kaleidoscope, whose underlying geometrical structure is that of ten interlinked Reye's configurations (together with their duals), possesses a total of 1120 apparitions that provide proofs of the two Bell theorems. Some applications of these kaleidoscopes to problems in quantum tomography and quantum state estimation are discussed.Comment: Two new references (No. 21 and 22) to related work have been adde

Defects and boundary layers in non-Euclidean plates

We investigate the behavior of non-Euclidean plates with constant negative Gaussian curvature using the F\"oppl-von K\'arm\'an reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers -- deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet

Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata

Functions which are equivariant or invariant under the transformations of a compact linear group $G$ acting in an euclidean space $\real^n$, can profitably be studied as functions defined in the orbit space of the group. The orbit space is the union of a finite set of strata, which are semialgebraic manifolds formed by the $G$-orbits with the same orbit-type. In this paper we provide a simple recipe to obtain rational parametrizations of the strata. Our results can be easily exploited, in many physical contexts where the study of equivariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau theory), in equivariant bifurcation theory, in crystal field theory and in most areas where use is made of symmetry adapted functions. A physically significant example of utilization of the recipe is given, related to spontaneous polarization in chiral biaxial liquid crystals, where the advantages with respect to previous heuristic approaches are shown.Comment: Figures generated through texdraw package; revised version appearing in J. Phys. A: Math. Ge

Inter-species variation in colour perception

Inter-species variation in colour perception poses a serious problem for the view that colours are mind-independent properties. Given that colour perception varies so drastically across species, which species perceives colours as they really are? In this paper, I argue that all do. Specifically, I argue that members of different species perceive properties that are determinates of different, mutually compatible, determinables. This is an instance of a general selectionist strategy for dealing with cases of perceptual variation. According to selectionist views, objects simultaneously instantiate a plurality of colours, all of them genuinely mind-independent, and subjects select from amongst this plurality which colours they perceive. I contrast selectionist views with relationalist views that deny the mind-independence of colour, and consider some general objections to this strategy

The oxygen isotopic composition of phytolith assemblages from tropical rainforest soil tops (Queensland, Australia): validation of a new paleoenvironmental tool

Phytoliths are micrometric particles of amorphous silica that form inside or between the cells of higher plant tissues throughout the life of a plant. With plant decay, phytoliths are either incorporated into soils or exported to sediments via regional watersheds. Phytolith morphological assemblages are increasingly used as proxy of grassland diversity and tree cover density in inter-tropical areas. Here, we investigate whether, along altitudinal gradients in northeast Queensland (Australia), changes in the δ<sup>18</sup>O signature of soil top phytolith assemblages reflect changes in mean annual temperature (MAT) and in the oxygen isotopic composition of precipitation (δ<sup>18</sup>O<sub>precipitation</sub>), as predicted by equilibrium temperature coefficients previously published for silica. Oxygen isotopic analyses were performed on 16 phytolith samples, after controlled isotopic exchange (CIE), using the IR Laser-Heating Fluorination Technique. Long-term mean annual precipitation (MAP) and MAT values at the sampled sites were calculated by the ANUCLIM software. δ<sup>18</sup>O<sub>precipitation</sub> estimates were calculated using the Bowen and Wilkinson (2002) model, slightly modified. An empirical temperature-dependant relationship was obtained: δ<sup>18</sup>O<sub>wood phytolith-precipitation</sub> (‰ vs. VSMOW) = −0.4 (±0.2) <i>t</i> (°C) + 46 (±3) (<i>R</i><sup>2</sup> = 0.4, <i>p</i> < 0.05; <i>n</i> = 12). Despite the various unknowns introduced when estimating δ<sup>18</sup>O<sub>precipitation</sub> values and the large uncertainties on δ<sup>18</sup>O<sub>wood phytolith</sub> values, the temperature coefficient (−0.4 ± 0.2‰ °C<sup>−1</sup>) is in the range of values previously obtained for natural quartz, fresh and sedimentary diatoms and harvested grass phytoliths (from −0.2 to −0.5‰ °C<sup>−1</sup>). The consistency supports the reliability of δ<sup>18</sup>O<sub>wood phytolith</sub> signatures for recording relative changes in mean annual δ<sup>18</sup>O<sub>soil water</sub> values (which are assumed to be equivalent to the weighted annual δ<sup>18</sup>O<sub> precipitation</sub> values in rainforests environments) and MAT, provided these changes were several ‰ and/or several °C in magnitude

The Maxwell Lagrangian in purely affine gravity

The purely affine Lagrangian for linear electrodynamics, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the Einstein-Maxwell equations in the metric-affine and metric formulation. We show that this equivalence is related to the invariance of the Maxwell Lagrangian under conformal transformations of the metric tensor. We also apply to a purely affine Lagrangian the Legendre transformation with respect to the tensor of homothetic curvature to show that the corresponding Legendre term and the new Hamiltonian density are related to the Maxwell-Palatini Lagrangian for the electromagnetic field. Therefore the purely affine picture, in addition to generating the gravitational Lagrangian that is linear in the curvature, justifies why the electromagnetic Lagrangian is quadratic in the electromagnetic field.Comment: 9 pages; published versio

A Kind of Affine Weighted Moment Invariants

A new kind of geometric invariants is proposed in this paper, which is called affine weighted moment invariant (AWMI). By combination of local affine differential invariants and a framework of global integral, they can more effectively extract features of images and help to increase the number of low-order invariants and to decrease the calculating cost. The experimental results show that AWMIs have good stability and distinguishability and achieve better results in image retrieval than traditional moment invariants. An extension to 3D is straightforward

Action Principle for the Generalized Harmonic Formulation of General Relativity

An action principle for the generalized harmonic formulation of general relativity is presented. The action is a functional of the spacetime metric and the gauge source vector. An action principle for the Z4 formulation of general relativity has been proposed recently by Bona, Bona--Casas and Palenzuela (BBP). The relationship between the generalized harmonic action and the BBP action is discussed in detail.Comment: This version is contains more thorough presentations and discussions of the key results. To be published in PRD. (8 pages, no figures

Univalent Foundations and the UniMath Library

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

Does \u2018bigger\u2019mean \u2018better\u2019? Pitfalls and shortcuts associated with big data for social research

\u2018Big data is here to stay.\u2019 This key statement has a double value: is an assumption as well as the reason why a theoretical reflection is needed. Furthermore, Big data is something that is gaining visibility and success in social sciences even, overcoming the division between humanities and computer sciences. In this contribution some considerations on the presence and the certain persistence of Big data as a socio-technical assemblage will be outlined. Therefore, the intriguing opportunities for social research linked to such interaction between practices and technological development will be developed. However, despite a promissory rhetoric, fostered by several scholars since the birth of Big data as a labelled concept, some risks are just around the corner. The claims for the methodological power of bigger and bigger datasets, as well as increasing speed in analysis and data collection, are creating a real hype in social research. Peculiar attention is needed in order to avoid some pitfalls. These risks will be analysed for what concerns the validity of the research results \u2018obtained through Big data. After a pars distruens, this contribution will conclude with a pars construens; assuming the previous critiques, a mixed methods research design approach will be described as a general proposal with the objective of stimulating a debate on the integration of Big data in complex research projecting