Degree bounds for separating invariants

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v'). It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree <= |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) <= b(G) <= [G:H] b(H)$, and that b(G) <= b(G/H) b(H)$ in case H is normal. Moreover, we calculate b(G) for some specific finite groups.Comment: 11 page

Fast Convex Decomposition for Truthful Social Welfare Approximation

Approximating the optimal social welfare while preserving truthfulness is a well studied problem in algorithmic mechanism design. Assuming that the social welfare of a given mechanism design problem can be optimized by an integer program whose integrality gap is at most $\alpha$, Lavi and Swamy~\cite{Lavi11} propose a general approach to designing a randomized $\alpha$-approximation mechanism which is truthful in expectation. Their method is based on decomposing an optimal solution for the relaxed linear program into a convex combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition technique relies heavily on the ellipsoid method, which is notorious for its poor practical performance. To overcome this problem, we present an alternative decomposition technique which yields an $\alpha(1 + \epsilon)$ approximation and only requires a quadratic number of calls to an integrality gap verifier

Past changes in riverine input and ocean circulation in the Gulf of Guinea

Large river systems draining the West African Monsoon area deliver sediments and dissolved trace elements into the Gulf of Guinea (GoG) in the easternmost equatorial Atlantic. The different catchment areas of these river systems are characterized by different geological ages and rock types releasing distinct radiogenic neodymium isotope compositions during weathering which are supplied to the GoG. The main rivers discharging into the GoG are the Niger, the Sanaga, the Nyong and the Ntem with present day εNd signatures of -10.5 [1], -12.3, -12.5 and -28.1 [2], respectively. These riverine inputs mix with the tropical Atlantic surface waters. At intermediate water depths Antarctic Intermediate Water (AAIW) prevails whereas the deep basin at this location is mainly filled with NADW. We focus on a marine sediment core that was recovered off the Sanaga and Ntem Rivers and we reconstruct changes in riverine inputs and in mixing of surface and deep water masses over the past 140,000 years. Changes in riverine inputs most likely reflecting latitudinal shifts of the rainfall zones across the different catchment areas were obtained from the Nd isotope signatures of the residual detrital fraction of the sediment. Sediment leachates of several GoG core top samples reflect the riverine input from nearby rivers indicating transport of particles coated in the rivers. Both the sediment leachates and the residual detrital fraction show similar patterns, with shifts towards radiogenic values during the interglacials and least radiogenic values during glacial periods. This shift in εNd values may be attributed to the migration of the rainfall zones towards the north during interglacial times and thus implies the increased influence of the northern rivers, the Sanaga and Nyong. The oxidatively-reductively cleaned planktonic foraminiferal calcite of the core top samples in the GoG reflects surface seawater signatures. Non-reductively cleaned planktonic foraminiferal tests and cleaned shallow endo-benthic and epi-benthic foraminiferal tests were used to acquire information about past bottom waters. Difficulties in cleaning down core foraminiferal samples were experienced and these samples appear to be contaminated by secondary manganese and iron bearing phases, even after cleaning. Those phases may have overprinted the original surface water Nd isotope composition in the planktonic foraminiferal tests. As the planktonic and benthic foraminiferal values are overall similar to the sediment leachates, the foraminiferal isotope signatures are most likely overprinted by isotopic signals originating from the rivers due to remobilization processes in the sediments and formation of secondary phases such as Mn-carbonates, which are attached to the foraminiferal calcites. References: [1] Goldstein et al. (1984) Earth and Planetary Science Letter 70, 221-236. [2] Weldeab et al. (2011) Geophysical Research Letter 38, pp. 5

Localisation of sentinel lymph nodes in patients with melanomas by planar lymphoscintigraphic and hybrid SPECT/CT imaging

BACKGROUND: The aim of the study was to assess the role ofplanar lymphoscintigraphy and fusion imaging of SPECT/CT in sentinellymph node (SLN) identification in patients with melanomas.MATERIAL AND METHODS: Planar and hybrid SPECT/low-doseCT lymphoscintigraphy were performed in 113 consecutive patientswith melanomas (59 men, 54 women, mean age 57.6 withrange 11–87 years, BMI 29.4 ± 12.5). The radiopharmaceuticalwas injected around the tumour (Group A, 59 patients), oraround the scar (Group B, 54 patients). Localisation of melanomas:head and neck 4, trunk 55, upper extremities 28, lowerextremities 26. Planar and SPECT/CT images were interpretedseparately by two nuclear medicine physicians. Abilities of thesetwo techniques to image SLN were compared.RESULTS: SLNs were detected on lymphoscintigraphy comprisingplanar and SPECT-CT images in 108 (95.6%) study patients;there was failure to detect SLNs in the remaining 5 (4.4%) patients. Planar images identified 253 SLNs in 100 (88.5%) pts,with a mean of 2.2 ± 1.7 (range 0–9 nodes) per patient. In theremaining 13 (11.5%) patients no SLNs were detected on planarimages. On SPECT-CT images, 334 hot nodes were detectedin 107 (94.7%) patients with a mean of 3.0 ± 2.1 (range 0–9)nodes per patient. In the remaining 6 (5.3%) patients, SPECT-CTwas negative.SPECT/CT visualised lymphatic drainage in 8 (7.1%) patientswith non-visualisation on planar imaging.CONCLUSIONS: In some patients with melanomas SPECT/CTimproves detection of sentinel lymph nodes. It can image nodesnot visible on planar scintigrams, exclude false positive uptakeand exactly localize SLNs.BACKGROUND: The aim of the study was to assess the role ofplanar lymphoscintigraphy and fusion imaging of SPECT/CT in sentinellymph node (SLN) identification in patients with melanomas.MATERIAL AND METHODS: Planar and hybrid SPECT/low-doseCT lymphoscintigraphy were performed in 113 consecutive patientswith melanomas (59 men, 54 women, mean age 57.6 withrange 11–87 years, BMI 29.4 ± 12.5). The radiopharmaceuticalwas injected around the tumour (Group A, 59 patients), oraround the scar (Group B, 54 patients). Localisation of melanomas:head and neck 4, trunk 55, upper extremities 28, lowerextremities 26. Planar and SPECT/CT images were interpretedseparately by two nuclear medicine physicians. Abilities of thesetwo techniques to image SLN were compared.RESULTS: SLNs were detected on lymphoscintigraphy comprisingplanar and SPECT-CT images in 108 (95.6%) study patients;there was failure to detect SLNs in the remaining 5 (4.4%) patients. Planar images identified 253 SLNs in 100 (88.5%) pts,with a mean of 2.2 ± 1.7 (range 0–9 nodes) per patient. In theremaining 13 (11.5%) patients no SLNs were detected on planarimages. On SPECT-CT images, 334 hot nodes were detectedin 107 (94.7%) patients with a mean of 3.0 ± 2.1 (range 0–9)nodes per patient. In the remaining 6 (5.3%) patients, SPECT-CTwas negative.SPECT/CT visualised lymphatic drainage in 8 (7.1%) patientswith non-visualisation on planar imaging.CONCLUSIONS: In some patients with melanomas SPECT/CTimproves detection of sentinel lymph nodes. It can image nodesnot visible on planar scintigrams, exclude false positive uptakeand exactly localize SLNs

How Does State Fragility Affect Rural Development?

International development agencies are focusing increasingly on fragile and conflict affected states and the challenges they are facing in achieving the Millennium Development Goals (MDGs). Recognizing that fragile states are ‘different’ from more stable and less vulnerable (though not necessarily poorer) developing countries, donors are nowadays paying more attention than before to specific problems of governance, service delivery, the development of core state functions, and – more recently – conflict resolution, peace-building and violence mitigation in fragile and conflict-affected settings. However, despite the fact that many fragile states have large rural populations and (subsistence) agriculture and associated, nonfarm economic activities constitute significant sectors of their economies, the relationship between state fragility and rural development remains underexplored

Vibrant times for mechanical metamaterials

Metamaterials are man-made designer matter that obtains its unusual effective properties by structure rather than chemistry. Building upon the success of electromagnetic and acoustic metamaterials, researchers working on mechanical metamaterials strive at obtaining extraordinary or extreme elasticity tensors and mass-density tensors to thereby mold static stress fields or the flow of longitudinal/transverse elastic vibrations in unprecedented ways. In this prospective paper, we focus on recent advances and remaining challenges in this emerging field. Examples are ultralight-weight, negative mass density, negative modulus, pentamode, anisotropic mass density, Origami, nonlinear, bistable, and reprogrammable mechanical metamaterials

A simple approach for the vertical force of the trampoline bed

Bisherige Arbeiten zur Biomechanik beim Trampolinspringen stellen die vertikale Federwirkung des Trampolinsprungtuches vereinfacht als ideale Feder dar. Bedingt durch die Aufhängung an in Ruhelage horizontal gespannten Federn verhält sich die vertikale Tuchkraft jedoch anders als eine ideale Feder. Ausgehend von einem zweidimensionalen statischen Modell, welches gegenüber der dreidimensionalen Wirklichkeit immer noch vereinfacht ist, wird eine Formel für die vertikale Tuchkraft in Abhängigkeit von der Durchdrücktiefe hergeleitet. Die Parameter werden durch Kraft-/Weg-Messungen an einem konkreten Trampolin bestimmt. Durch Simulationsrechnungen werden die Zeiten des Tuchkontakts, die maximale Durchdrücktiefe und die maximale vertikale Beschleunigung bei verschiedenen Körpermassen und Sprunghöhen berechnet und in Tabellenform dargestellt. Abschließend wird auf Beobachtungen und Messergebnisse anderer Autoren eingegangen, welche mit Hilfe des hier vorgestellten Modells eine bessere Erklärung finden.Some previous papers about biomechanics in trampolining use the ideal spring as a model approach for the vertical force of the trampoline bed. Due to the fixation of the trampoline bed by horizontal springs (while not depressed), the vertical force characteristics differ from an ideal spring. Starting from a two-dimensional static model, which is still simplified from the three-dimensional nature of the trampoline bed, a formula is deduced for the calculation of the vertical force of the trampoline bed as a function of the distance of vertical depression. The parameters are calculated from distance/force measurements with a single existing trampoline. By means of computational simulation the times of bed contact, the maximal bed depression and the maximal vertical acceleration are calculated for different body masses and bouncing heights. The results are presented in tabular form. Finally some previous observations and results of other authors are discussed, which can be explained using the herein presented two-dimensional model