Linearly continuous maps discontinuous on the graphs of twice differentiable functions

A function g : R n → R is linearly continuous provided its restriction g ` to every straight line ` ⊂ R n is continuous. It is known that the set D(g) of points of discontinuity of any linearly continuous g : R n → R is a countable union of isometric copies of (the graphs of) f P, where f : R n−1 → R is Lipschitz and P ⊂ R n−1 is compact nowhere dense. On the other hand, for every twice continuously differentiable function f : R → R and every nowhere dense perfect P ⊂ R there is a linearly continuous g : R 2 → R with D(g) = f P. The goal of this paper is to show that this last statement fails, if we do not assume that f 00 is continuous. More specifically, we show that this failure occurs for every continuously differentiable function f : R → R with nowhere monotone derivative, which includes twice differentiable functions f with such property. This generalizes a recent result of professor Ludek Zajicek and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasFALSEMinisterio de Ciencia e Innovación (MICINN)/FEDERunpu

A framework for adaptive MCMC targeting multimodal distributions

We propose a new Monte Carlo method for sampling from multimodal distributions. The idea of this technique is based on splitting the task into two: finding the modes of a target distribution π and sampling, given the knowledge of the locations of the modes. The sampling algorithm relies on steps of two types: local ones, preserving the mode; and jumps to regions associated with different modes. Besides, the method learns the optimal parameters of the algorithm while it runs, without requiring user intervention. Our technique should be considered as a flexible framework, in which the design of moves can follow various strategies known from the broad MCMC literature. In order to design an adaptive scheme that facilitates both local and jump moves, we introduce an auxiliary variable representing each mode and we define a new target distribution π~ on an augmented state space X × I, where X is the original state space of π and I is the set of the modes. As the algorithm runs and updates its parameters, the target distribution π~ also keeps being modified. This motivates a new class of algorithms, Auxiliary Variable Adaptive MCMC. We prove general ergodic results for the whole class before specialising to the case of our algorithm

The Formation and Gravitational-Wave Detection of Massive Stellar Black-Hole Binaries

If binaries consisting of two 100 Msun black holes exist they would serve as extraordinarily powerful gravitational-wave sources, detectable to redshifts of z=2 with the advanced LIGO/Virgo ground-based detectors. Large uncertainties about the evolution of massive stars preclude definitive rate predictions for mergers of these massive black holes. We show that rates as high as hundreds of detections per year, or as low as no detections whatsoever, are both possible. It was thought that the only way to produce these massive binaries was via dynamical interactions in dense stellar systems. This view has been challenged by the recent discovery of several stars with mass above 150 Msun in the R136 region of the Large Magellanic Cloud. Current models predict that when stars of this mass leave the main sequence, their expansion is insufficient to allow common envelope evolution to efficiently reduce the orbital separation. The resulting black-hole--black-hole binary remains too wide to be able to coalesce within a Hubble time. If this assessment is correct, isolated very massive binaries do not evolve to be gravitational-wave sources. However, other formation channels exist. For example, the high multiplicity of massive stars, and their common formation in relatively dense stellar associations, opens up dynamical channels for massive black hole mergers (e.g., via Kozai cycles or repeated binary-single interactions). We identify key physical factors that shape the population of very massive black-hole--black-hole binaries. Advanced gravitational-wave detectors will provide important constraints on the formation and evolution of very massive stars.Comment: ApJ accepted, extended description of modelin

The role of supernova convection for the lower mass gap and the isolated binary formation of gravitational wave sources

Understanding astrophysical phenomena involving compact objects requires an insight about the engine behind core-collapse supernovae (SNe) and the fate of the stellar collapse of massive stars. In particular, this insight is crucial in developing an understanding of the origin and formation channels of detected population of BH-BH, BH-NS and NS-NS mergers. To gain this understanding, we must tie our current knowledge of pre-SN stars properties and their potential explosions to the final NS or BH mass distribution. The timescale of convection growth may have a large effect on the strength of SN explosion and therefore also on the mass distribution of stellar remnants. In this study we adopt the new formulas for the relation between the pre-SN star properties and its remnant from Fryer et al. 2022 in prep. into StarTrack population synthesis code and check how they impact double compact object (DCO) mergers formed via isolated binary evolution. The new formulas give one ability to test a wide spectrum of assumptions on the convection growth time. In particular, different variants allow for a smooth transition between having a deep lower mass gap and a remnant mass distribution filled by massive NSs and low mass BHs. In this paper we present distribution of masses, mass ratios and the local merger rate densities of DCO mergers for different variants of new remnant mass formulas. We test them together with different approaches to other highly uncertain processes. We find that mass distribution of DCO mergers up to m_1+m_2 < 35 Msun is sensitive to adopted assumption on SN convection growth timescale. Between the two extreme tested variants the probability of compact object formation within the lower mass gap may differ up to 2 orders of magnitude. The mass ratio distribution of DCO mergers is significantly influenced by SN model only for our standard mass transfer stability criteria.Comment: 20 pages, submitted to MNRAS, comments welcom

Captions in 360 Video : Rapid Prototyping for User Testing

Extended reality is reinventing our approach to work, learning, culture, and social interaction. Nevertheless, the integration of accessible services within immersive environments is still in progress. This presentation will introduce new prototyping for immersive captioning and discuss how to achieve an optimal and fully inclusive viewing experience