Construction of blow-up sequences for the prescribed scalar curvature equation on S n . II. Annular domains

Using the Lyapunov–Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on S ⁿ (n ≥ 6), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have C ^{n - 1, β} smoothness.     

Implied Volatility of Leveraged ETF Options

Abstract

This paper studies the problem of understanding implied volatilities from options written on leveraged exchanged-traded funds (LETFs), with an emphasis on the relations between LETF options with different leverage ratios. We first examine from empirical data the implied volatility skews for LETF options based on the S&P 500. In order to enhance their comparison with non-leveraged ETFs, we introduce the concept of moneyness scaling and provide a new formula that links option implied volatilities between leveraged and unleveraged ETFs. Under a multiscale stochastic volatility framework, we apply asymptotic techniques to derive an approximation for both the LETF option price and implied volatility. The approximation formula reflects the role of the leverage ratio, and thus allows us to link implied volatilities of options on an ETF and its leveraged counterparts. We apply our result to quantify matches and mismatches in the level and slope of the implied volatility skews for various LETF options using data from the underlying ETF option prices. This reveals some apparent biases in the leverage implied by the market prices of different products, long and short with leverage ratios two times and three times.     

A generalized geometric-programming solution to “An economic production quantity model with flexibility and reliability considerations”

The classical economic production quantity (EPQ) model assumes that items are produced by a perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be perfected cost-free, the set-up cost can be reduced by investment in flexibility improvement. In this paper, we propose an EPQ model with a flexible and imperfect production process. We formulate this inventory decision problem using geometric programming (GP), establish more general results using the arithmetic-geometric mean inequality, and solve the problem to obtain a closed-form optimal solution. Following the theoretical treatment, we provide a numerical example to demonstrate that GP has potential as a valuable analytical tool for studying a certain class of inventory control problems. Finally we discuss some aspects of sensitivity analysis of the optimal solution based on the GP approach.     

Probing the interior of neutron stars with gravitational waves

We show here how the internal structure of a neutron star can be inferred from its gravitational wave spectrum. Under the premise that the frequencies and damping rates of a few w-mode oscillations are found, we apply an inversion scheme to determine its mass, radius, and density distribution. In addition, an accurate equation of state of nuclear matter can also be determined.     

Empirical study of the scaling behavior of the amplitude–frequency distribution of the Hilbert–Huang transform and its application in sunspot time series analysis

Investigating long-range correlation by the Hurst exponent, H$H$, is crucial in the study of time series. Recently, empirical-mode-decomposition-based arbitrary-order Hilbert spectral analysis (EMD-HSA) has been proposed to numerically obtain without proof a scaling relationship, generated from the amplitude–frequency distribution, related to H$H$. We propose a formalism to empirically study EMD-HSA, to deduce its scaling exponent ξ(q)$\xi \left(q\right)$ from the perspective of EMD-based arbitrary-order Hilbert marginal spectrum (EMD-HMS), and to numerically compare the results with the expected H$H$. EMD-HSA and EMD-HMS experiments show that, by incompletely removing (quasi-)periodic trends, the sunspot series should have an H$H$ value around 0.12.     

Quantum key distribution based on arbitrarily weak distillable entangled states

States with private correlations but little or no distillable entanglement were recently reported. Here, we consider the secure distribution of such states, i.e., the situation when an adversary gives two parties such states and they have to verify privacy. We present a protocol which enables the parties to extract from such untrusted states an arbitrarily long and secure key, even though the amount of distillable entanglement of the untrusted states can be arbitrarily small.     

Spiral cracks in drying precipitates

We investigate the formation of spiral crack patterns during the desiccation of thin layers of precipitates in contact with a substrate. This symmetry-breaking fracturing mode is found to arise naturally not from torsion forces but from a propagating stress front induced by the foldup of the fragments. We model their formation mechanism using a coarse-grain model for fragmentation and successfully reproduce the spiral cracks. Fittings of experimental and simulation data show that the spirals are logarithmic. Theoretical aspects of the logarithmic spirals are discussed. In particular we show that this occurs generally when the crack speed is proportional to the propagating speed of stress front.     

Monotone scheme for finite difference equations concerning steady-state prey-predator interactions

In this article a system of semilinear elliptic partial differential equations is studied. This system determines the equilibria of the Volterra-Lotka equations describing prey-predator interactions with diffusion. To analyze the system, a new monotone scheme is presented. A rigorous foundation is given for numerical calculations by adapting a suitable finite difference method to the new monotone scheme. Earlier theories in finite differences are not successful in solving the system without this scheme.