Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $\gamma \in [0,1)$ and satisfy $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely, for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$

Determination of 2´-5´-Oligoadenylate Synthetase in Serum and Peripheral Blood Mononuclear Cells before and after Subcutaneous Application of Recombinant Interferon beta and gamma

Peer Reviewe

Multiple Instance Learning for Heterogeneous Images: Training a CNN for Histopathology

Multiple instance (MI) learning with a convolutional neural network enables end-to-end training in the presence of weak image-level labels. We propose a new method for aggregating predictions from smaller regions of the image into an image-level classification by using the quantile function. The quantile function provides a more complete description of the heterogeneity within each image, improving image-level classification. We also adapt image augmentation to the MI framework by randomly selecting cropped regions on which to apply MI aggregation during each epoch of training. This provides a mechanism to study the importance of MI learning. We validate our method on five different classification tasks for breast tumor histology and provide a visualization method for interpreting local image classifications that could lead to future insights into tumor heterogeneity

Quantum properties of dichroic silicon vacancies in silicon carbide

The controlled generation and manipulation of atom-like defects in solids has a wide range of applications in quantum technology. Although various defect centres have displayed promise as either quantum sensors, single photon emitters or light-matter interfaces, the search for an ideal defect with multi-functional ability remains open. In this spirit, we investigate here the optical and spin properties of the V1 defect centre, one of the silicon vacancy defects in the 4H polytype of silicon carbide (SiC). The V1 centre in 4H-SiC features two well-distinguishable sharp optical transitions and a unique S=3/2 electronic spin, which holds promise to implement a robust spin-photon interface. Here, we investigate the V1 defect at low temperatures using optical excitation and magnetic resonance techniques. The measurements, which are performed on ensemble, as well as on single centres, prove that this centre combines coherent optical emission, with up to 40% of the radiation emitted into the zero-phonon line (ZPL), a strong optical spin signal and long spin coherence time. These results single out the V1 defect in SiC as a promising system for spin-based quantum technologies

Asymptotics of self-similar solutions to coagulation equations with product kernel

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(\xi,\eta)= (\xi \eta)^{\lambda}$ with $\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda} g(x)$ in the limit $\lambda \to 0$. It turns out that $h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference

Deep Modeling of Growth Trajectories for Longitudinal Prediction of Missing Infant Cortical Surfaces

Charting cortical growth trajectories is of paramount importance for understanding brain development. However, such analysis necessitates the collection of longitudinal data, which can be challenging due to subject dropouts and failed scans. In this paper, we will introduce a method for longitudinal prediction of cortical surfaces using a spatial graph convolutional neural network (GCNN), which extends conventional CNNs from Euclidean to curved manifolds. The proposed method is designed to model the cortical growth trajectories and jointly predict inner and outer cortical surfaces at multiple time points. Adopting a binary flag in loss calculation to deal with missing data, we fully utilize all available cortical surfaces for training our deep learning model, without requiring a complete collection of longitudinal data. Predicting the surfaces directly allows cortical attributes such as cortical thickness, curvature, and convexity to be computed for subsequent analysis. We will demonstrate with experimental results that our method is capable of capturing the nonlinearity of spatiotemporal cortical growth patterns and can predict cortical surfaces with improved accuracy.Comment: Accepted as oral presentation at IPMI 201

Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension

The qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling is studied. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise below, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria is studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.Comment: 56 pages. Expanded introduction, added references. This revised version is published in Arch. Ration. Mech. Anal. (207) (2013), 611-66

Scalable quantum photonics with single color centers in silicon carbide

Silicon carbide is a promising platform for single photon sources, quantum bits (qubits), and nanoscale sensors based on individual color centers. Toward this goal, we develop a scalable array of nanopillars incorporating single silicon vacancy centers in 4H-SiC, readily available for efficient interfacing with free-space objective and lensed-fibers. A commercially obtained substrate is irradiated with 2 MeV electron beams to create vacancies. Subsequent lithographic process forms 800 nm tall nanopillars with 400–1400 nm diameters. We obtain high collection efficiency of up to 22 kcounts/s optical saturation rates from a single silicon vacancy center while preserving the single photon emission and the optically induced electron-spin polarization properties. Our study demonstrates silicon carbide as a readily available platform for scalable quantum photonics architecture relying on single photon sources and qubits

Pedestrians moving in dark: Balancing measures and playing games on lattices

We present two conceptually new modeling approaches aimed at describing the motion of pedestrians in obscured corridors: * a Becker-D\"{o}ring-type dynamics * a probabilistic cellular automaton model. In both models the group formation is affected by a threshold. The pedestrians are supposed to have very limited knowledge about their current position and their neighborhood; they can form groups up to a certain size and they can leave them. Their main goal is to find the exit of the corridor. Although being of mathematically different character, the discussion of both models shows that it seems to be a disadvantage for the individual to adhere to larger groups. We illustrate this effect numerically by solving both model systems. Finally we list some of our main open questions and conjectures