A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces

A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a naturel set of spaces generalizing the usual Hilbert cube. In a second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space, with uniform bound and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d.Comment: v2 Largely expanded version, as reflected by the change of title; all part I on generalized Hausdorff dimension is new, as well as the embedding of Hilbert cubes into Wasserstein spaces. v3 modified according to the referee final remarks ; to appear in Journal of Topology and Analysi

Resilience of Supervised Learning Algorithms to Discriminatory Data Perturbations

Discrimination is a focal concern in supervised learning algorithms augmenting human decision-making. These systems are trained using historical data, which may have been tainted by discrimination, and may learn biases against the protected groups. An important question is how to train models without propagating discrimination. In this study, we i) define and model discrimination as perturbations of a data-generating process and show how discrimination can be induced via attributes correlated with the protected attributes; ii) introduce a measure of resilience of a supervised learning algorithm to potentially discriminatory data perturbations, iii) propose a novel supervised learning algorithm that inhibits discrimination, and iv) show that it is more resilient to discriminatory perturbations in synthetic and real-world datasets than state-of-the-art learning algorithms. The proposed method can be used with general supervised learning algorithms and avoids inducement of discrimination, while maximizing model accuracy.Comment: 17 pages, 10 figures, 1 tabl

Proton radiography to improve proton radiotherapy: Simulation study at different proton beam energies

To improve the quality of cancer treatment with protons, a translation of X-ray Computed Tomography (CT) images into a map of the proton stopping powers needs to be more accurate. Proton stopping powers determined from CT images have systematic uncertainties in the calculated proton range in a patient of typically 3-4\% and even up to 10\% in region containing bone~\cite{USchneider1995,USchneider1996,WSchneider2000,GCirrone2007,HPaganetti2012,TPlautz2014,GLandry2013,JSchuemann2014}. As a consequence, part of a tumor may receive no dose, or a very high dose can be delivered in healthy ti\-ssues and organs at risks~(e.g. brain stem)~\cite{ACKnopf2013}. A transmission radiograph of high-energy protons measuring proton stopping powers directly will allow to reduce these uncertainties, and thus improve the quality of treatment. The best way to obtain a sufficiently accurate radiograph is by tracking individual protons traversing the phantom (patient)~\cite{GCirrone2007,TPlautz2014,VSipala2013}. In our simulations we have used an ideal position sensitive detectors measuring a single proton before and after a phantom, while the residual energy of a proton was detected by a BaF$_{2}$ crystal. To obtain transmission radiographs, diffe\-rent phantom materials have been irradiated with a 3x3~cm$^{2}$ scattered proton beam, with various beam energies. The simulations were done using the Geant4 simulation package~\cite{SAgostinelli2003}. In this study we focus on the simulations of the energy loss radiographs for various proton beam energies that are clinically available in proton radiotherapy.Comment: 6 pages, 6 figures, Presented at Jagiellonian Symposium on Fundamental and Applied Subatomic Physics, 7-12 June, 2015, Krak\'ow, Polan

Critical behavior of the metallic triangular-lattice Heisenberg antiferromagnet PdCrO2

We report physical properties of the conductive magnet PdCrO2 consisting of a layered structure with a triangular lattice of Cr3+ ions (S=3/2). We confirmed an antiferromagnetic transition at TN=37.5K by means of specific heat, electrical resistivity, magnetic susceptibility, and neutron scattering measurements. The critical behavior in the specific heat persists in an unusually wide temperature range above TN. This fact implies that spin correlations develop even at much higher temperature than TN. The observed sub-linear temperature dependence of the resistivity above TN is also attributed to the short-range correlations among the frustrated spins. While the critical exponent for the magnetization agrees reasonably with the prediction of the relevant model, that for the specific heat evaluated in the wide temperature range differs substantially from the prediction.Comment: 7 pages, 6 figure

Doenças de hortaliças no Amapá.

Em inspeções feitas nos núcleos rurais do Estado do Amapá e nas bases físicas da UEPAE de Macapá/Embrapa, constataram-se sintomas de doenças fúngicas e bacterianas em hortaliças. Exames microscópicos, cultura pura, testes de patogenicidade e bioquímicos confirmaram a diagnose visual das doenças

Observation of Field-Induced Transverse N\'{e}el Ordering in the Spin Gap System TlCuCl3_3

Neutron elastic scattering experiments have been performed on the spin gap system TlCuCl$_3$ in magnetic fields parallel to the $b$-axis. The magnetic Bragg peaks which indicate the field-induced N\'{e}el ordering were observed for magnetic field higher than the gap field $H_{\rm g}\approx 5.5$ T at $Q=(h, 0, l)$ with odd $l$ in the $a^*-c^*$ plane. The spin structure in the ordered phase was determined. The temperature and field dependence of the Bragg peak intensities and the phase boundary obtained were discussed in connection with a recent theory which describes the field-induced N\'{e}el ordering as a Bose-Einstein condensation of magnons.Comment: 4 pages, 5 eps figures, jpsj styl

Drastic Change of Magnetic Phase Diagram in Doped Quantum Antiferromagnet TlCu1−x_{1-x}Mgx_xCl3_3

TlCuCl$_3$ is a coupled spin dimer system, which has a singlet ground state with an excitation gap of $\Delta/g\mu_{\mathrm B}$ = 5.5 T. TlCu$_{1-x}$Mg$_x$Cl$_3$ doped with nonmagnetic Mg$^{2+}$ ions undergoes impurity-induced magnetic ordering. Because triplet excitation with a finite gap still remains, this doped system can also undergo magnetic-field-induced magnetic ordering. By specific heat measurements and neutron scattering experiments under a magnetic field, we investigated the phase diagram in TlCu$_{1-x}$Mg$_x$Cl$_3$ with $x\sim 0.01$, and found that impurity- and field-induced ordered phases are the same. The gapped spin liquid state observed in pure TlCuCl$_3$ is completely wiped out by the small amount of doping.Comment: 9 pages, 5 figures, jpsj2 class file, to be published in J. Phy. Soc. Jpn. Vol.75 No.3 (2006); layout changed, unrelated figure remove