Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(\Omega)$ and $\widetilde{H}^s(\Omega)$, for $s\in \mathbb{R}$ and an open $\Omega\subset \mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $\Omega$ for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if $\Omega$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large

The makewaves tsunami tests and their relevance to tsunami engineering and risk management

MAKEWAVES is an international multi-partner collaborative project bringing together nine academic institutions and two commercial consultancies. The objective of the collaboration is to develop experimental data and associated numerical modelling on tsunami inundation and interaction with boulders, buildings, natural and engineered barriers, towards the development of new internationally accepted guidance for structural codes and standards. Using a pneumatic tsunami simulator (TS) developed jointly by HR Wallingford and UCL the team conducted experiments between November 2022 and April 2023 within a highly instrumented 100m long flume. The TS is capable of simulating realistic trough and crest-led tsunami waves at 1:50, including traces from the The TS is capable of generating very long trough and crest-led waves, and can reproduce at 1:50 scale waves from real life events such as the Mercator trace from the 2004 Indian Ocean event and the and 2011 Tohoku tsunamis. The TS capability has been further extended to include bore-waves. The characteristics of the waves are controlled by adjusting the flow rate and total volume of water drawn in and discharged by the TS. The experimental campaign is was subdivided into discrete research areas, each aimed at furthering knowledge on how different tsunami wave characteristics affect their interaction with manmade and natural structures environments. These include tests aimed at understanding: (1) how roughness representative of coastal forests and mangroves affects tsunami inundation characteristics, (2) how tsunami interact with boulders (3) the effectiveness of offshore breakwaters as tsunami barriers (4) how structural loads and foundation scour are affected by building permeability. This paper presents an overview of the tests conducted and some of the important early observations made that are relevant to future engineering standards and to tsunami disaster management

Sobolev spaces on non-Lipschitz subsets of Rn with application to boundary integral equations on fractal screens

We study properties of the classical fractional Sobolev spaces on non-Lipschitz subsets of Rn. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschitz, including cases where the screen is fractal or has fractal boundary

The DECam Ecliptic Exploration Project (DEEP) IV: Constraints on the shape distribution of bright TNOs

We present the methods and results from the discovery and photometric measurement of 26 bright (VR $>$ 24 trans-Neptunian objects (TNOs) during the first year (2019-20) of the DECam Ecliptic Exploration Project (DEEP). The DEEP survey is an observational TNO survey with wide sky coverage, high sensitivity, and a fast photometric cadence. We apply a computer vision technique known as a progressive probabilistic Hough transform to identify linearly-moving transient sources within DEEP photometric catalogs. After subsequent visual vetting, we provide a photometric and astrometric catalog of our TNOs. By modeling the partial lightcurve amplitude distribution of the DEEP TNOs using Monte Carlo techniques, we find our data to be most consistent with an average TNO axis ratio b/a $<$ 0.5, implying a population dominated by non-spherical objects. Based on ellipsoidal gravitational stability arguments, we find our data to be consistent with a TNO population containing a high fraction of contact binaries or other extremely non-spherical objects. We also discuss our data as evidence that the expected binarity fraction of TNOs may be size-dependent

A Triple Protostar System Formed via Fragmentation of a Gravitationally Unstable Disk

Binary and multiple star systems are a frequent outcome of the star formation process, and as a result, almost half of all sun-like stars have at least one companion star. Theoretical studies indicate that there are two main pathways that can operate concurrently to form binary/multiple star systems: large scale fragmentation of turbulent gas cores and filaments or smaller scale fragmentation of a massive protostellar disk due to gravitational instability. Observational evidence for turbulent fragmentation on scales of $>$1000~AU has recently emerged. Previous evidence for disk fragmentation was limited to inferences based on the separations of more-evolved pre-main sequence and protostellar multiple systems. The triple protostar system L1448 IRS3B is an ideal candidate to search for evidence of disk fragmentation. L1448 IRS3B is in an early phase of the star formation process, likely less than 150,000 years in age, and all protostars in the system are separated by $<$200~AU. Here we report observations of dust and molecular gas emission that reveal a disk with spiral structure surrounding the three protostars. Two protostars near the center of the disk are separated by 61 AU, and a tertiary protostar is coincident with a spiral arm in the outer disk at a 183 AU separation. The inferred mass of the central pair of protostellar objects is $\sim$1 M$_{sun}$, while the disk surrounding the three protostars has a total mass of $\sim$0.30 M_{\sun}. The tertiary protostar itself has a minimum mass of $\sim$0.085 M$_{sun}$. We demonstrate that the disk around L1448 IRS3B appears susceptible to disk fragmentation at radii between 150~AU and 320~AU, overlapping with the location of the tertiary protostar. This is consistent with models for a protostellar disk that has recently undergone gravitational instability, spawning one or two companion stars.Comment: Published in Nature on Oct. 27th. 24 pages, 8 figure

Holographic Vitrification

We establish the existence of stable and metastable stationary black hole bound states at finite temperature and chemical potentials in global and planar four-dimensional asymptotically anti-de Sitter space. We determine a number of features of their holographic duals and argue they represent structural glasses. We map out their thermodynamic landscape in the probe approximation, and show their relaxation dynamics exhibits logarithmic aging, with aging rates determined by the distribution of barriers.Comment: 100 pages, 25 figure

Ernst Freund as Precursor of the Rational Study of Corporate Law

Gindis, David, Ernst Freund as Precursor of the Rational Study of Corporate Law (October 27, 2017). Journal of Institutional Economics, Forthcoming. Available at SSRN: https://ssrn.com/abstract=2905547, doi: https://dx.doi.org/10.2139/ssrn.2905547The rise of large business corporations in the late 19th century compelled many American observers to admit that the nature of the corporation had yet to be understood. Published in this context, Ernst Freund's little-known The Legal Nature of Corporations (1897) was an original attempt to come to terms with a new legal and economic reality. But it can also be described, to paraphrase Oliver Wendell Holmes, as the earliest example of the rational study of corporate law. The paper shows that Freund had the intuitions of an institutional economist, and engaged in what today would be called comparative institutional analysis. Remarkably, his argument that the corporate form secures property against insider defection and against outsiders anticipated recent work on entity shielding and capital lock-in, and can be read as an early contribution to what today would be called the theory of the firm.Peer reviewe

Female social and sexual interest across the menstrual cycle: the roles of pain, sleep and hormones

<p>Abstract</p> <p>Background</p> <p>Although research suggests that socio-sexual behavior changes in conjunction with the menstrual cycle, several potential factors are rarely taken into consideration. We investigated the role of changing hormone concentrations on self-reported physical discomfort, sleep, exercise and socio-sexual interest in young, healthy women.</p> <p>Methods</p> <p>Salivary hormones (dehydroepiandrosterone sulfate-DHEAS, progesterone, cortisol, testosterone, estradiol and estriol) and socio-sexual variables were measured in 20 women taking oral contraceptives (OC group) and 20 not using OCs (control group). Outcome measures were adapted from questionnaires of menstrual cycle-related symptoms, physical activity, and interpersonal relations. Testing occurred during menstruation (T1), mid-cycle (T2), and during the luteal phase (T3). Changes in behavior were assessed across time points and between groups. Additionally, correlations between hormones and socio-behavioral characteristics were determined.</p> <p>Results</p> <p>Physical discomfort and sleep disturbances peaked at T1 for both groups. Exercise levels and overall socio-sexual interest did not change across the menstrual cycle for both groups combined. However, slight mid-cycle increases in general and physical attraction were noted among the control group, whereas the OC group experienced significantly greater socio-sexual interest across all phases compared to the control group. Associations with hormones differed by group and cycle phase. The estrogens were correlated with socio-sexual and physical variables at T1 and T3 in the control group; whereas progesterone, cortisol, and DHEAS were more closely associated with these variables in the OC group across test times. The direction of influence further varies by behavior, group, and time point. Among naturally cycling women, higher concentrations of estradiol and estriol are associated with lower attraction scores at T1 but higher scores at T3. Among OC users, DHEAS and progesterone exhibit opposing relationships with attraction scores at T1 and invert at T3.</p> <p>Conclusions</p> <p>Data from this study show no change across the cycle in socio-sexual interest among healthy, reproductive age women but higher social and physical attraction among OC users. Furthermore, a broader range of hormones may be associated with attraction than previously thought. Such relationships differ by use of oral contraceptives, and may either reflect endogenous hormone modulation by OCs and/or self-selection of sexually active women to practice contraceptive techniques.</p

The state of the Martian climate

60°N was +2.0°C, relative to the 1981–2010 average value (Fig. 5.1). This marks a new high for the record. The average annual surface air temperature (SAT) anomaly for 2016 for land stations north of starting in 1900, and is a significant increase over the previous highest value of +1.2°C, which was observed in 2007, 2011, and 2015. Average global annual temperatures also showed record values in 2015 and 2016. Currently, the Arctic is warming at more than twice the rate of lower latitudes

Sharing vocabularies: towards horizontal alignment of values-driven business functions

This paper highlights the emergence of different ‘vocabularies’ that describe various values-driven business functions within large organisations and argues for improved horizontal alignment between them. We investigate two established functions that have long-standing organisational histories: Ethics and Compliance (E&C) and Corporate Social Responsibility (CSR). By drawing upon research on organisational alignment, we explain both the need for and the potential benefit of greater alignment between these values-driven functions. We then examine the structural and socio-cultural dimensions of organisational systems through which E&C and CSR horizontal alignment can be coordinated to improve synergies, address tensions, and generate insight to inform future research and practice in the field of Business and Society. The paper concludes with research questions that can inform future scholarly research and a practical model to guide organizations’ efforts towards inter-functional, horizontal alignment of values-driven organizational practice