On the reconstruction of planar lattice-convex sets from the covariogram

A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\subseteq \mathbb{Z}^d$ is the function associating to each u \in \integer^d the cardinality of $K\cap (K+u)$. Daurat, G\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.Comment: accepted in Discrete and Computational Geometr

Bayesian history matching for structural dynamics applications

Computer models provide useful tools in understanding and predicting quantities of interest for structural dynamics. Although computer models (simulators) are useful for a specific context, each will contain some level of model-form error. These model-form errors arise for several reasons e.g., numerical approximations to a solution, simplifications of known physics, an inability to model all relevant physics etc. These errors form part of model discrepancy; the difference between observational data and simulator outputs, given the ‘true’ parameters are known. If model discrepancy is not considered during calibration, any inferred parameters will be biased and predictive performance may be poor. Bayesian history matching (BHM) is a technique for calibrating simulators under the assumption that additive model discrepancy exists. This ‘likelihood-free’ approach iteratively assesses the input space using emulators of the simulator and identifies parameters that could have ‘plausibly’ produced target outputs given prior uncertainties. This paper presents, for the first time, the application of BHM in a structural dynamics context. Furthermore, a novel method is provided that utilises Gaussian Process (GP) regression in order to infer the missing model discrepancy functionally from the outputs of BHM. Finally, a demonstration of the effectiveness of the approach is provided for an experimental representative five storey building structure

Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets $\varLambda\subset\R^d$ with long-range order by $X$-rays in prescribed $\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\varLambda$. Here, an $X$-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. For our main result, we introduce the notion of algebraic Delone sets $\varLambda\subset\R^2$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\varLambda$-directions.Comment: 15 pages, 2 figures; condensed and revised versio

First-Digit Law in Nonextensive Statistics

Nonextensive statistics, characterized by a nonextensive parameter $q$, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore the unevenness of the first digit distribution of nonextensive statistics analytically and numerically. We find that the first-digit distribution follows Benford's law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature. The fluctuation decreases when $q$ increases, and the result converges to Benford's law exactly as $q$ approaches 2. The relevant regularities between nonextensive statistics and Benford's law are also presented and discussed.Comment: 11 pages, 3 figures, published in Phys. Rev.

The HADES facility for high activity decommissioning engineering & science: part of the UK national nuclear user facility

Research and innovation is key to delivering UK Government's civil nuclear energy policy, in particular to accelerate reduction in the hazard, timescale and cost of legacy decommissioning and geological disposal of radioactive wastes. To address this challenge, a national centre of excellence, the HADES Facility, has been established to support research and innovation in High Activity Decommissioning Engineering & Science, as part of the wider network of UK National Nuclear User Facilities. Herein, we describe the development of this user facility, the current status of its capability, and functional equipment specifications. The unique capabilities of the HADES Facility, in the UK academic landscape, are emphasised, including: handling of weighable quantities of 99Tc and transuranics; quantitative electron probe microanalysis of radioactive materials; hot isostatic pressing of radioactive materials; and laboratory-based X-ray absorption and emission spectroscopy. An example case study of the application of the HADES capability is described, involving thermal treatment of a real radioactive ion exchange resin waste to produce a conceptual vitrified waste form

Detection and Quantification of Grapevine Bunch Rot Using Functional Data Analysis and Canonical Variate Analysis Biplots of Infrared Spectral Data

Grapevine bunch rot assessment has economic significance to wineries. Industrial working conditionsrequire rapid assessment methods to meet the time constraints typically associated with grape intakeat large wineries. Naturally rot-affected and healthy white wine grape bunches were collected overfive vintages (2013 to 2016, 2020). Spectral data of 382 grape must samples were acquired using threedifferent, but same-type attenuated total reflection mid-infrared (ATR-MIR) ALPHA spectrometers. Thepractical industrial problem of wavenumber shifts collected with different spectrometers was overcome byapplying functional data analysis (FDA). FDA improved the data quality and boosted data mining effortsin the sample set. Canonical variate analysis (CVA) biplots were employed to visualise the detection andquantification of rot. When adding 90 % alpha-bags to CVA biplots minimal overlap between rot-affected(Yes) and healthy (No) samples was observed. Several bands were observed in the region 1734 cm-1 to 1722cm-1 which correlated with the separation between rot-affected and healthy grape musts. These bandsconnect to the C=O stretching of the functional groups of carboxylic acids. In addition, wavenumber 1041cm-1, presenting the functional group of ethanol, contributed to the separation between categories (severity% range). ATR-MIR could provide a sustainable alternative for rapid and automated rot assessment.However, qualitative severity quantification of rot was limited to only discriminating between healthy andsevere rot (> 40 %). This study is novel in applying FDA to correct wavenumber shifts in ATR-MIR spectraldata. Furthermore, visualisation of the viticultural data set using CVA biplots is a novel application of thistechnique

Stability for Borell-Brascamp-Lieb inequalities

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.Comment: to appear in GAFA Seminar Note

Estimates for measures of sections of convex bodies

A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these inequalities from stability in comparison problems for different generalizations of intersection bodies