Discontinuous Step Function Research Articles

Weak Approximation for a Black-Scholes Type Regime Switching Model

ABSTRACT We provide a Monte Carlo simulation method for a European type option under a regime switching type Black-Scholes model where the volatility is a discontinuous step function which takes n + 1 different values depending on the value of the underlying within n + 1 disjoint partition intervals. Our simulation method is based on a combination of the Euler scheme which is used in the case that the process is away from the boundary points of the partition intervals and the skew Brownian motion in the case that it the process in near the boundary points of the partition intervals. We show that the weak error of approximation is exponentially small for the proposed method and show some simulation results.

Beyond superoscillation: general theory of approximation with bandlimited functions

We give a general strategy to construct superoscillating/supergrowing functions using an orthogonal polynomial expansion of a bandlimited function. The degree of superoscillation/supergrowth is controlled by an anomalous expectation value of a pseudodistribution that exceeds the band limit. The function is specified via the rest of its cumulants of the pseudodistribution. We give an explicit construction using Legendre polynomials in the Fourier space, which leads to an expansion in terms of spherical Bessel functions in the real space. The other expansion coefficients may be chosen to optimize other desirable features, such as the range of super behavior. We provide a prescription to generate bandlimited functions that mimic an arbitrary behavior in a finite interval. As target behaviors, we give examples of a superoscillating function, a supergrowing function, and even a discontinuous step function. We also look at the energy content in a superoscillating/supergrowing region and provide a bound that depends on the minimum value of the logarithmic derivative in that interval. Our work offers a new approach to analyzing superoscillations/supergrowth and is relevant to the optical field spot generation endeavors for far-field superresolution imaging.

Is Detecting Discontinuity Difficult? Evidence from the Visual Trend Classification of Scatterplots

Abstract Visualization of data comes with the promise that even unexpected aspects of the data can be detected. For instance, viewers might discover that patterns in a scatterplot follow a non-linear trend. Using two experiments, we examined whether and when scatterplots depicting different types of mathematical functions are classified correctly and confidently. In Experiment 1, 237 participants categorized scatterplots that depicted more or less noisy linear, negative exponential, or discontinuous step functions. Results indicated a classification advantage for the continuous linear and negative exponential functions over the discontinuous step function. Generalizing from these findings, in Experiment 2, 231 participants categorized scatterplots that depicted more or less noisy versions of a broader range of functions, including rising continuous linear, quadratic (half parabola), and discontinuous linear (kinked) functions and rising and falling continuous quadratic (full parabola), sine wave, and discontinuous linear (inverted-v) functions. Generally, results suggested that scatterplots based on discontinuous functions are difficult to classify. Overall, the results indicated a classification advantage for the continuous functions over the discontinuous functions and underscore the need for scientists and practitioners to be extremely careful when developing theory and making decisions based on exploration of raw data visualizations.

Volume‐of‐fluid based two‐phase flow methods on structured multiblock and overset grids

AbstractA geometric volume‐of‐fluid (VOF) method is implemented on the static/dynamic overset grids with a new VOF interpolation algorithm developed. The VOF value cannot be directly interpolated through the overset grid boundaries since the VOF function is a discontinuous step function. In the present study, the interface is reconstructed in the fringe cells of the overset grids using the interpolated distance function and normal vector. The VOF value in the fringe cells is calculated geometrically based on the reconstructed interface on the physical domain. The gravity and surface tension forces are implemented using a VOF based ghost fluid method, where a ghost pressure interpolation scheme is developed to handle the pressure jump across the interface due to the density changes and surface tension effect. Numerical tests and application examples have been performed for the verification and validation of the numerical methods developed in the present study. The advection tests show that the effect of the VOF overset interpolation on the accuracy of the VOF method is small. The numerical test for the static pressure shows that the present VOF based ghost fluid method can greatly reduce the spurious velocity observed in the body force method. The sharp pressure profile due to the surface tension effects can be preserved across the interface. Application examples demonstrate the capability and superiority of the VOF based numerical methods for the prediction of the violent ship flows involving strong air water interface interactions and complex geometries.

Quantum parallel multi-layer Monte Carlo optimization algorithm for controller parameters optimization of doubly-fed induction generator-based wind turbines

With larger searching space, intelligent algorithms could have insufficient global search capability and accuracy for optimization problems. This paper proposes a novel quantum parallel multi-layer Monte Carlo optimization algorithm (QPMMCOA) to optimize the rotor-side controller (RSC) parameters based on proportional–integral of a doubly-fed induction generator (DFIG) for achieving maximum power and improving generation efficiency. The QPMMCOA is proposed to find optimal solutions in an accurate small searching space. The QPMMCOA combines qubit probability amplitude with Monte Carlo random numbers to generate a diverse population and expand accurate search space. The optimization process of the QPMMCOA with strong global search ability and local development ability is divided into the rough search, precise search, and re-precise search. These three search processes are searched in the ever-shrinking feasible region. The QPMMCOA mainly seeks the optimized solution by continuously changing and narrowing the feasible region. The QPMMCOA is utilized to optimize a discontinuous step function and a multimodal function for confirming the efficacy and feasibility. With broader exploration and deeper development capabilities, the QPMMCOA can achieve the optimization result of the fitness function of the RSC is at least 0.51% lower than other algorithms. Compared with other algorithms, the QPMMCOA-based RSC can enhance the average power coefficient of the DFIG-based wind power system by at least 0.0028%.

Proof Discontinuities and Civil Settlements

Abstract This Article explores settlement incentives under three different burden of proof rules. The conventional burden of proof is a discontinuous step-function, jumping from no damages to full damages at the 0.5 jury confidence level. Continuous burdens of proof, by contrast, would permit sanctions to steadily increase as juror confidence rises from 0 to 1, with no discontinuity. Linear burdens, which have received extensive attention in prior literature, escalate sanctions steadily across the whole range of confidence levels, while the logistic burden takes a nonlinear form. Using a data simulation approach guided by the empirical realities of American civil litigation, I consider the incentives that each of these rules creates for parties contemplating settlement, using a model in which parties make divergent forecasts of their expected outcomes at trial due to optimism bias. Based on this analysis, I conclude that a linear burden would likely raise our settlement rate by a modest amount, except in very large cases and in “easy” cases, in which an unbiased person would predict that a trial factfinder would have a level of confidence in liability quite close to either zero or one. I also compare the expected error rate of the settlements that each rule produces, and find that the linear rule modestly lowers the expected error rate of settlement overall, although this benefit does not hold for easy cases or those with very high damages. Lastly, I conduct a similar analysis for the logistic burden, finding that it induces a similar quality and quantity of settlements as we currently achieve using conventional burdens.

Существование и устойчивость периодических решений уравнения нейронного поля

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.

A hybrid bat algorithm with natural-inspired algorithms for continuous optimization problem

This paper proposes a hybrid bat algorithm with natural-inspired algorithms for continuous optimization problem. In this study, the proposed algorithm combines the reproduction step from weed algorithm and genetic algorithm. The reproduction step is applied to clone each bat population by fitness values and the genetic algorithm is applied in order to expand the population. The algorithm is evaluated on eighteen benchmark problems. The computational results of the proposed algorithm are compared with the methods in the literature which are self-adaptive differential evolution (DE), traditional DE algorithm, intersection mutation differential evolution (IMDE) algorithm, and the JDE self-adaptive algorithm. Findings show that the algorithm produces several solutions obtained by the previously published methods especially for the continuous unimodal function, the quartic function, the multimodal function and the discontinuous step function. In addition, the finding shows that the proposed algorithm can produce optimal solutions efficiently on benchmark instances within short computational time.

Quantifying the Likelihood of Co-existence for Communities with Asymmetric Competition

Trade-offs in performance of different ecological functions within a species are commonly offered as an explanation for co-existence in natural communities. Single trade-offs between competitive ability and other life history traits have been shown to support a large number of species, as a result of strong competitive asymmetry. We consider a single competition-fecundity trade-off in a homogeneous environment, and examine the effect of the form of asymmetry on the likelihood of species co-existing. We find conditions that allow co-existence of two species for a general competition function, and show that (1) two species can only co-exist if the competition function is sufficiently steep when the species are similar; (2) when competition is determined by a linear function, no more than two species can co-exist; (3) when the competition between two individuals is determined by a discontinuous step function, this single trade-off can support an arbitrarily large number of species. Further, we show analytically that as the degree of asymmetry in competition increases, the probability of a given number of species co-existing also increases, but note that even in the most favourable conditions, large numbers of species co-existing along a single trade-off is highly unlikely. On this basis, we suggest it is unlikely that single trade-offs are able to support high levels of bio-diversity without interacting other processes.

Could any black holes be produced at the LHC?

We introduce analytical quantum gravity modifications of the production cross section for terascale black holes by employing an effective ultraviolet cut off $l$. We find the new cross sections approach the usual "black disk" form at high energy, while they differ significantly near the fundamental scale from the standard increase with respect to $s$. We show that the heretofore discontinuous step function used to represent the cross section threshold can realistically be modeled by two functions representing the incoming and final parton states in a high energy collision. The growth of the cross section with collision energy is thus a unique signature of $l$ and number of spatial dimensions $d$. Contrary to the classical black disk result, our cross section is able to explain why black holes might not be observable in LHC experiments while they could be still at the reach of ultra-high energy cosmic ray events.

Fractals and ultrasmooth microeffects

In this paper, a portion of the nonstandard methods first introduced by Nottale and Schneider [J. Math. Phys. 25, 1296 (1984)] for the investigation of fractal behavior are replaced by methods employing polysaturated enlargements. This process yields internal functions that are ultrasmooth and have a standard part equal to the original fractal. These ultrasmooth nonstandard functions also possess a well-behaved * integral length concept. Additionally, a method is presented that shows that if certain behavior is modeled after a simple finitely discontinuous step function, then this function is also the standard part of an internal hypersmooth nonstandard approximation.

The hard-sphere and square-well models for the surface properties of liquid metals

Abstract The statistical mechanical deduction of Fowler for surface tension from the assumption of a discontinuous step function profile of the surface density is used in conjunction with Evan's pseudo-atom theory to calculate surface properties such as surface entropy and tension for hard-sphere and square-well types of interactions in liquid metals. The resulting values of surface tension are in reasonable agreement.

Wave concepts in the theory of heat

The known fundamental solution of the heat-conduction equation in the form of Poisson's integral is obtained by summing-up particular similarity solutions derived with the help of a similarity group. As these similarity solutions do not satisfy the initial data, then the integral condition is obtained from which the arbitrary constant of a general solution is determined. Then the solution in the form of Poisson's integral is shown to satisfy approximately the initial condition in the form of the discontinuous step function approximating the initial function. This is one of the drawbacks of the fundamental solution of the classical heat-conduction equation. The paradox on an infinite “heat propagation velocity” is discussed. Analysing the isotherm behaviour and using Green's theorem, the general non-linear wave equation is derived here in which the speed of isotherm propagation along the normal is used as an experimental parameter. The relationship between this speed and thermal diffusivity is shown. The different particular cases of this wave equation are studied. Some wave equation solutions are shown to correspond to those of the non-linear parabolic equation. Then, the derivation is given of the wave heat conduction equation from the point of view of molecular kinetic considerations. The conception on time relaxation is generalized using the Maxwell method and taking into account the correlation between components of a heat velocity of atoms of molecules.

TEMPERATURE AND TURBULENCE IN THE LOWER ATMOSPHERE

Abstract The equation of turbulent transfer is derived from first principles, following Taylor, to focus attention on the physical nature of eddy diffusivity in the atmosphere. The equation is here extended to “polytropic” atmospheric processes. It is shown that physical considerations imply that the eddy diffusivity is a discontinuous function of elevation through the atmosphere. The fundamental assumption is made that eddy diffusivity is a discontinuous step-function of elevation, and the equation of transfer is solved for the general case where the surface temperature is known as a function of time. The theory is tested by comparing computations with the Lindenberg observations of diurnal variations of temperature aloft. Computations and observations show good agreement at all levels, and in particular the computations agree well with the large diurnal variations observed at and above 1000 meters. Qualitative comparison is also made between theory and observations made in the lowest 100 feet of the atm...