Helly-Type Theorems in Property Testing

Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If $S$ is a set of $n$ points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be partitioned into $k$ clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object $G$. In this paper, as an application of Helly's theorem, by taking a constant size sample from $S$, we present a testing algorithm for $(k,G)$-clustering, i.e., to distinguish between two cases: when $S$ is $(k,G)$-clusterable, and when it is $\epsilon$-far from being $(k,G)$-clusterable. A set $S$ is $\epsilon$-far $(0<\epsilon\leq1)$ from being $(k,G)$-clusterable if at least $\epsilon n$ points need to be removed from $S$ to make it $(k,G)$-clusterable. We solve this problem for $k=1$ and when $G$ is a symmetric convex object. For $k>1$, we solve a weaker version of this problem. Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability

Bleeding Meckel's diverticulum diagnosis: an unusual indication for computed tomography

Despite the wide use of modern investigation techniques, the diagnosis of complications related to Meckel's diverticulum (MD) remains difficult. Arteriography is commonly indicated for acute bleeding, and radionuclide scans may help in identifying the site of intestinal hemorrhage. In contrast, computed tomography (CT) is usually considered little use in the diagnosis of bleeding MD. We present the case of a young patient with massive gastrointestinal hemorrhage, in whom the diagnosis of MD bleeding was preoperatively made with contrast-enhanced CT after two negatives arteriographie

Trapped by the Entrapment.

Popliteal entrapment syndrome results from extrinsic compression of the popliteal artery by the surrounding musculotendinous structures and is a rare cause of limb ischaemia. The purpose of this report is to highlight potential mistakes in the management of popliteal entrapment. In 2000, a 23 year old man underwent a popliteal to popliteal artery bypass surgery for what was initially diagnosed as a traumatic popliteal artery thrombosis. After being initially lost to follow up for 13 years, this "unspecified traumatic" thrombosis led to several inappropriate endovascular and open procedures misinterpreted as being caused by late graft failure. These included thrombectomy, aneurysmorrhaphy, polytetrafluoroethylene covered stent graft, a redo femoropopliteal bypass, and bypass thrombolysis. The diagnosis was reached 19 years after the initial surgery, when the patient underwent a redo bypass using a retrogeniculate approach. An abnormal lateral insertion of the gastrocnemius muscle medial head, and its accessory slip, constricted the artery, and also involved the popliteal vein (Type V), thus explaining previous revascularisation failures. Surgery consisted of resecting the accessory slip and the aneurysmal bypass. The artery was reconstructed with the cephalic vein. The patient was discharged on clopidogrel 75 mg, with no further complication, and a patent bypass at six months. Based on post-operative imaging (duplex ultrasound and magnetic resonance imaging), with forced plantarflexion and dorsiflexion, asymptomatic popliteal entrapment was also present on the contralateral side. The finding of an isolated popliteal artery lesion in a young individual should be considered to be caused by popliteal artery entrapment, unless proven otherwise. Definitive surgical release of the popliteal artery should be favoured over other strategies

SCD Patterns Have Singular Diffraction

Among the many families of nonperiodic tilings known so far, SCD tilings are still a bit mysterious. Here, we determine the diffraction spectra of point sets derived from SCD tilings and show that they have no absolutely continuous part, that they have a uniformly discrete pure point part on the z-axis, and that they are otherwise supported on a set of concentric cylinder surfaces around this axis. For SCD tilings with additional properties, more detailed results are given.Comment: 11 pages, 2 figures; Accepted for Journal of Mathematical Physic

Liquid‐Liquid Equilibrium and Interfacial Properties of the System Water + Hexylacetate + 1‐Hexanol

Experimental and theoretical investigations of the phase diagram and the interfacial tension are presented. The theoretical framework was able to predict the phase behavior and the interfacial tension with a high accuracy, where only binary experimental data enter the model parameter. The theory permits the calculation of the concentration profiles across the interface. The profiles show that 1-hexanol will be enriched, which was expected. In same circumstances a competition between hexylacetate and 1-hexanol was figured out leading to slight minima in the profile of 1-hexanol

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder, A. Deza, and A. Ivic Weiss (eds), Springe

The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

Classification of wines by means of multivariate data analysis using the SPME/CGC-chromatograms of volatile aroma compounds

The solid phase microextraction (SPME) is an effective solvent-free sample preparation technique for the capillary gas chromatographic (CGC) analysis of volatile aroma compounds of wines. Using discriminant analysis based upon only two terpene compounds, it was possible to analytically discern between the varieties Riesling, Muller-Thurgau and Silvaner grown in the same region. The discrimination of these varieties was unsuccessful for wines of different vintages (1988-1995). In order to obtain a highly significant classification, it was necessary to consider further aroma components described in wine literature. The differentiation between these wines by a similar high classification rate was obtained using a set of variables selected by mathematical methods. Wines prepared from known grape varieties were qualitatively recognized by factor- and cluster-analyses as well as the relative peak intensities of the terpene compounds in the SPME-CGC chromatograms. The composition of wine blends was quantitatively determined

The local atomic quasicrystal structure of the icosahedral Mg25Y11Zn64 alloy

A local and medium range atomic structure model for the face centred icosahedral (fci) Mg25Y11Zn64 alloy has been established in a sphere of r = 27 A. The model was refined by least squares techniques using the atomic pair distribution (PDF) function obtained from synchrotron powder diffraction. Three hierarchies of the atomic arrangement can be found: (i) five types of local coordination polyhedra for the single atoms, four of which are of Frank-Kasper type. In turn, they (ii) form a three-shell (Bergman) cluster containing 104 atoms, which is condensed sharing its outer shell with its neighbouring clusters and (iii) a cluster connecting scheme corresponding to a three-dimensional tiling leaving space for few glue atoms. Inside adjacent clusters, Y8-cubes are tilted with respect to each other and thus allow for overall icosahedral symmetry. It is shown that the title compound is essentially isomorphic to its holmium analogue. Therefore fci-Mg-Y-Zn can be seen as the representative structure type for the other rare earth analogues fci-Mg-Zn-RE (RE = Dy, Er, Ho, Tb) reported in the literature.Comment: 12 pages, 8 figures, 2 table

Multimodal imaging of pancreatic beta cells in vivo by targeting transmembrane protein 27 (TMEM27)

Aims/hypothesis: Non-invasive diagnostic tools specific for pancreatic beta cells will have a profound impact on our understanding of the pathophysiology of metabolic diseases such as diabetes. The objective of this study was to use molecular imaging probes specifically targeting beta cells on human samples and animal models using state-of-the-art imaging modalities (fluorescence and PET) with preclinical and clinical perspective. Methods: We generated a monoclonal antibody, 8/9-mAb, targeting transmembrane protein 27 (TMEM27; a surface N-glycoprotein that is highly expressed on beta cells), compared its expression in human and mouse pancreas, and demonstrated beta cell-specific binding in both. In vivo imaging was performed in mice with subcutaneous insulinomas overexpressing the human TMEM27 gene, or transgenic mice with beta cell-specific hTMEM27 expression under the control of rat insulin promoter (RIP-hTMEM27-tg), using fluorescence and radioactively labelled antibody, followed by tissue ex vivo analysis and fluorescence microscopy. Results: Fluorescently labelled 8/9-mAb showed beta cell-specific staining on human and mouse pancreatic sections. Real-time PCR on islet cDNA indicated about tenfold higher expression of hTMEM27 in RIP-hTMEM27-tg mice than in humans. In vivo fluorescence and PET imaging in nude mice with insulinoma xenografts expressing hTMEM27 showed high 8/9-mAb uptake in tumours after 72h. Antibody homing was also observed in beta cells of RIP-hTMEM27-tg mice by in vivo fluorescence imaging. Ex vivo analysis of intact pancreas and fluorescence microscopy in beta cells confirmed these findings. Conclusions/interpretation: hTMEM27 constitutes an attractive target for in vivo visualisation of pancreatic beta cells. Studies in mouse insulinoma models and mice expressing hTMEM27 demonstrate the feasibility of beta cell-targeted in vivo imaging, which is attractive for preclinical investigations and holds potential in clinical diagnostic