Modular retractions of numerical semigroups

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote ${\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}$ (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that ${\rm R}(S,a,m)$ is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n ₁,…,n _p 〉 to computing the genus of a “smaller” numerical semigroup 〈n ₁?an ₁modm,…,n _p ?an _p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.     

On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

A note on extreme points in dual spaces

Given a normed space X it can be easily proven that every extreme point in B _X *, the unit ball of X*, is the restriction of an extreme point in B _X ***. Our purpose is to study when the restrictions of extreme points in B _X *** are extreme points in B _X *. Namely, we characterize L ₁-preduals satisfying the aforementioned property.     

Weak Allee effect in a predator–prey model involving memory with a hump

In this paper we consider a predator–prey system which has a factor that allows for a reduction in fitness due to declining population sizes, often termed an Allee effect. We study the influence of the weak Allee effect which is included in the prey equation and we determine conditions for the occurrence of Hopf bifurcation. The prey population is limited by the carrying capacity of the environment, and the predator growth rate depends on past quantities of the prey which is represented by a weight function that specifies a moment in the past when the quantity of food is the most important from the point of view of the present growth of the predator. The stability properties of the system and the biological issues of the memory and Allee effect on the coexistence of the two species are studied. Finally we present some simulations to verify the veracity of the analytical conclusions.     

Robust Certified Numerical Homotopy Tracking

We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.     

Using multi-objective evolutionary algorithms for single-objective optimization

In recent decades, several multi-objective evolutionary algorithms have been successfully applied to a wide variety of multi-objective optimization problems. Along the way, several new concepts, paradigms and methods have emerged. Additionally, some authors have claimed that the application of multi-objective approaches might be useful even in single-objective optimization. Thus, several guidelines for solving single-objective optimization problems using multi-objective methods have been proposed. This paper offers a survey of the main methods that allow the use of multi-objective schemes for single-objective optimization. In addition, several open topics and some possible paths of future work in this area are identified.     

Classical versus Bayesian risks in acceptance sampling: a sensitivity analysis

Assuming a beta prior distribution on the fraction defective, $p$ , failure-censored sampling plans for Weibull lifetime models using classical (or average) and Bayesian (or posterior) producer’s and consumer’s risks are designed to determine the acceptability of lots of a given product. The average risk criterion provides a certain assurance that good (bad) lots will be accepted (rejected), whereas the posterior risk criterion provides a determined confidence that an accepted (rejected) lot is indeed good (bad). The performance of classical and Bayesian risks are analyzed in developing sampling plans when the lifetime variable follows the Weibull distribution. Several figures and tables illustrate the sensitivity of the risks and optimal sample sizes for selected censoring levels and specifications according to the available prior information on $p$ . The analysis clarifies the distinction among the different risks for a given sampling plan, and the effect of the prior knowledge on the required sample size. The study shows that, under uncertainty in the prior variance of $p$ , the designs using Bayesian risks are more appropriate.     

Optimization methodology assessment for the inlet velocity profile of a hydraulic turbine draft tube: part II—performance evaluation of draft tube model

Computational Fluids Dynamics (CFD) tools guide engineers and designers to estimate the performance of new designs. However, a CFD analysis can be very time-consuming depending mainly on the grid size and domain complexity. Thus, this paper aims to describe the tools used to evaluate and compare the performance of different 3D draft tube models for reducing the time-effort needed in an optimization procedure. The results presented here, are the second part of an overall research to establish a global optimization methodology to improve the performance of an hydraulic draft tube through the inlet velocity profile. Previously, three steps of optimization methodology to minimize the energy losses were studied: the inlet velocity profile parameterization, the numerical optimization set-up and the objective function validation. In the latter step, a global optimization method called Multi Island Genetic Algorithm (MIGA) was considered, which requires a large number of iterations before producing a reliable result. This step is able to identify an efficient inlet velocity profile to minimize the energy losses through the draft tube model. However, each iteration is expensive in terms of computational time due to the need for 3D Navier–Stokes (NS) computations to evaluate each profile’s fitness. Thus, in this work the methodology attempts to accelerate the optimization process with accurate results. In order to achieve the goal, the grid size of the 3D draft tube model was minimized, resulting in a much lower computational cost. Specifically, the draft tube calculations were performed on a sequence of five different grids each having approximately twice the number of elements compared to the previous. The measurements of the sensitivity of the draft tube performance quantities to the change of the inlet velocity parameters during the process showed that, in spite of the numerical difference between its performance, the results have the same tendency. Consequently, the 3D draft tube numerical model with a minimal grid size, is reliable and left record of its capabilities for being integrated in the optimization process.