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We develop and approach to unsupervised semantic medical image segmentation that extends previous work with generative adversarial networks. We use existing edge detection methods to construct simple edge diagrams, train a generative model to convert them into synthetic medical images, and construct a dataset of synthetic images with known segmentations using variations on extracted edge diagrams. This synthetic dataset is then used to train a supervised image segmentation model. We test our approach on a clinical dataset of kidney ultrasound images and the benchmark ISIC 2018 skin lesion dataset. We show that our unsupervised approach is more accurate than previous unsupervised methods, and performs reasonably compared to supervised image segmentation models. All code and trained models are available at https://github.com/kiretd/Unsupervised-MIseg., Comment: 16 pages, 8 figures

The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$. Other consequences of such a factorization are new results about odd perfect polynomials.

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity averse decision maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a general characterization of the value of the optimal timing policy and the worst case measure in terms of a family of an explicitly identified excessive functions generating an appropriate class of supermartingales. In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting. Somewhat surprisingly, we find that ambiguity may result into stationarity in models which typically do not possess stationary behavior. In this way, our results indicate that ambiguity may act as a stabilizing mechanism.

We consider the impact of ambiguity on the optimal timing of a class of two-dimensional integral option contracts when the exercise payoff is a positively homogeneous measurable function. Hence, the considered class of exercise payoffs includes discontinuous functions as well. We identify a parameterized family of excessive functions generating an appropriate class of supermartingales for the considered problems and then express the value of the optimal policy as well as the worst case measure in terms of these processes. The advantage of our approach is that it reduces the analysis of the multidimensional problem to the analysis of an ordinary one-dimensional static optimization problem. In that way it simplifies earlier treatments of the problem without ambiguity considerably. We also illustrate our findings in explicitly parameterized examples., Comment: 23 pages, 3 figures

According to conventional wisdom, ambiguity accelerates optimal timing by decreasing the value of waiting in comparison with the unambiguous benchmark case. We study this mechanism in a multidimensional setting and show that in a multifactor model ambiguity does not only influence the rate at which the underlying processes are expected to grow, it also affects the rate at which the problem is discounted. This mechanism where nature also selects the rate at which the problem is discounted cannot appear in a one-dimensional setting and as such we identify an indirect way of how ambiguity affects optimal timing.

Fast magnetic reconnection events can play an important role in accretion disk systems. A potential model to explain the non-thermal very-high-energy (VHE) emission (from GeV to TeV) observed in black-hole binaries (BHBs) and Active Galatic Nuclei (AGNs) can be attributed to fast magnetic reconnection induced in the turbulent corona of accretion disks and/or jets. In this work, we will discuss the results of global general relativistic MHD (GRMHD) simulations of accretion disks around black holes, whose turbulence is naturally driven by MHD instabilities, such as the magnetorotational instability (MRI). We will also present studies of magnetic reconnection driven by kink instabilities inside jets employing special relativistic MHD (SRMHD) simulations. As we expect, our simulations reveal the development of a nearly steady-state turbulence driven by these instabilities. We have performed a detailed statistical analysis to identify the presence of current sheets in the turbulent regions of both the accretion flow and jet. We then determined the magnetic reconnection rates in these locations obtaining average reconnection velocities in Alfv\'{e}n speed units of the order of $0.01-0.7$, which are consistent with the predictions of the theory of turbulence-induced fast reconnection., Comment: Proceeding at the International Conference on Black Holes as Cosmic Batteries: UHECRs and Multimessenger Astronomy - BHCB2018, 12-15 September 2018, Foz du Iguazu, Brasil, in press in Procs. of Science

We prove that there is no circulant Hadamard matrix $H$ with first row $[h_{1},\ldots,h_{n}]$ of order $n>4$, under a condition about a sum of scalar products of rows of two other circulant matrices of size $n/2$ associated to $H.$, Comment: Submitted to AMEN

We study the optimal sustainable harvesting of a population that lives in a random environment. The novelty of our setting is that we maximize the asymptotic harvesting yield, both in an expected value and almost sure sense, for a large class of harvesting strategies and unstructured population models. We prove under relatively weak assumptions that there exists a unique optimal harvesting strategy characterized by an optimal threshold below which the population is maintained at all times by utilizing a local time push-type policy. We also discuss, through Abelian limits, how our results are related to the optimal harvesting strategies when one maximizes the expected cumulative present value of the harvesting yield and establish a simple connection and ordering between the values and optimal boundaries. Finally, we explicitly characterize the optimal harvesting strategies in two different cases, one of which is the celebrated stochastic Verhulst Pearl logistic model of population growth., Comment: 20 pages, 2 figures

This paper addresses the gain of global fractional regularity in Nikolskii spaces for solutions of a class of quasilinear degenerate equations with $(p,q)$-growth. Indeed, we investigate the effects of the datum on the derivatives of order greater than one of the solutions of the $(p,q)$-Laplacian operator, under Dirichlet's boundary conditions. As it turns out, even in the absence of the so-called Lavrentiev phenomenon and without variations on the order of ellipticity of the equations, the fractional regularity of these solutions ramifies depending on the interplay between the growth parameters $p$, $q$ and the data. Indeed, we are going to exploit the absence of this phenomenon in order to prove the validity up to the boundary of some regularity results, which are known to hold locally, and as well provide new fractional regularity for the associated solutions. In turn, there are obtained certain global regularity results by means of the combination between new a priori estimates and approximations of the differential operators, whereas the nonstandard boundary terms are handled by means of a careful choice for the local frame.

Given a convex $n$-gon, we can draw $n$ disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the $n$ disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and P\'erez-Lantero in 2016, proved to be planar and Hamiltonian. In this paper we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an $O(n)$-time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in $O(n)$ time. We further study the independence number of this graph, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex $n$-gon we can select at least $\lceil n/4 \rceil$ pairwise disjoint disks, and we prove that for every $n\ge 3$ there exist convex $n$-gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs.

Magnetohydrodynamic instabilities play an important role in accretion disks systems. Besides the well-known effects of the magnetorotational instability (MRI), the Parker-Rayleigh-Taylor instability (PRTI) also arises as an important mechanism to help in the formation of the coronal region around an accretion disk and in the production of magnetic reconnection events similar to those occurring in the solar corona. In this work, we have performed three-dimensional magnetohydrodynamical (3D-MHD) shearing-box numerical simulations of accretion disks with an initial stratified density distribution and a strong azimuthal magnetic field with a ratio between the thermal and magnetic pressures of the order of unity. This study aimed at verifying the role of these instabilities in driving fast magnetic reconnection in turbulent accretion disk/corona systems. As we expected, the simulations showed an initial formation of large-scale magnetic loops due to the PRTI followed by the development of a nearly steady-state turbulence driven by both instabilities. In this turbulent environment, we have employed an algorithm to identify the presence of current sheets produced by the encounter of magnetic flux ropes of opposite polarity in the turbulent regions of both the corona and the disk. We computed the magnetic reconnection rates in these locations obtaining average reconnection velocities in Alfv\'en speed units of the order of $0.13 \pm 0.09$ in the accretion disk and $0.17 \pm 0.10$ in the coronal region (with mean peak values of order of $0.2$), which are consistent with the predictions of the theory of turbulence-induced fast reconnection., Comment: 23 pages, 14 figures. Accepted for publication in ApJ

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the $p$-Laplacian, in general, the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness. Moreover, despite its own interest, fractional regularity methods may be used as a tool for the investigation of some Partial Differential Equations which are not usually addressed in this manner. Thus, the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the $p$-Laplacian. This is done by means of explicit a priori estimates regarding Lebesgue and Nikolskii spaces, which are part of the present contribution. As a consequence, this approach allows a relaxation on some of the standard conditions employed in this class of problems.

We consider the determination of the optimal stationary singular stochastic control of a linear diffusion for a class of average cumulative cost minimization problems arising in various financial and economic applications of stochastic control theory. We present a set of conditions under which the optimal policy is of the standard local time reflection type and state the first order conditions from which the boundaries can be determined. Since the conditions do not require symmetry or convexity of the costs, our results cover also the cases where costs are asymmetric and non-convex. We also investigate the comparative static properties of the optimal policy and delineate circumstances under which higher volatility expands the continuation region where utilizing the control is suboptimal., Comment: 23 pages, 3 figures