Closed-form Formula Research Articles

A general Monte-Carlo approach to consider a maximum admissible risk in decision-making procedures based on measurement results

According to the standards, decision-making procedures generally consider both a threshold that should not be exceeded and the measurement uncertainty that is associated to the measurement result. However, the general indications given in the Standards, in their examples, refer to the particular case when the measurand distributes according to a normal PDF. But a generalization to other cases is not considered and is not straightforward. In a previous paper, the Authors proposed a decision-making procedure which not only considers the measurement uncertainty and the threshold, but also considers a Maximum Admissible Risk. The proposed procedure leads to decisions taken with a risk of a wrong decision lower than the given Maximum Admissible Risk. In particular, closed-form formulas were derived under specific assumptions for the distributions of the measured values. Hence, the aim of this paper is to generalize the proposed decision rule and method for setting acceptance and rejection limits, by applying the Monte-Carlo method. In this way, it can be generally applied, even when the distribution associated to the measurement result is not a priori known in closed form.

Valuing European Option Under Double 3/2-Volatility Jump-Diffusion Model With Stochastic Interest Rate and Stochastic Intensity Under Approximative Fractional Brownian Motion

In this study, we propose a more comprehensive and realistic option pricing model based on approximative fractional Brownian motion, building upon recent advancements in this area. Specifically, we utilize the double 3/2-volatility Jump-Diffusion model, which incorporates approximative fractional Brownian motion with 3/2-volatility, stochastic interest rate, and stochastic intensity. To account for the stochastic interest rate, we employ a two-factor Vasicek model. Notably, our model accommodates negative interest rates. Consequently, we develop a multi-factor model with a stochastic interest rate structure for pricing European options and derive a closed-form pricing formula with an analytical solution by applying some algebraic calculations and Lie symmetries. In order to demonstrate the superiority of our proposed model over other classical approaches, we present numerical results that showcase the value of a European call option. This comparative analysis underscores the advantages of our model in comparison to traditional models.

Stability analysis of unsaturated soil slopes with cracks under rainfall infiltration conditions

Pre-existing (open) cracks at slope crests, especially tension cracks, are a significant factor in the transition from shallow to deep-seated landslides in unsaturated soil slopes induced by rainfall. This paper presents an analytical framework to assess the stability of unsaturated soil slopes for deep-seated rotational failure considering arbitrary crack depths and unsteady infiltration, and its effectiveness is verified by previous literature and finite element analysis. A closed-form formula is proposed for the maximum depth of cracks in unsaturated soils by the stability of the vertical crack boundary. A parametric study is performed to investigate the impact of some key parameters on the stability of unsaturated soil slopes with cracks. Dimensionless stability charts are developed, which can calculate the factor of safety for the slopes with cracksof any depth and can reflect that the potential slip surfaces through or without passing through the crack tip. In addition, given the existence of a critical crack depth and maximum crack depth, with a ratio ranging from 0.5 to 0.6 based on the stability charts, deep-seated rotational failure is more likely to occur than failure caused by a crack boundary.

Nonlinear symbolic regression for bit error rate prediction of NOMA systems in 5G cellular communications

Machine learning (ML) has been suggested as a promising tool in the design of telecommunication systems to overcome the challenges of traditional methods and meet the requirements regarding future cellular networks. As an enabler of 5G communications, Non-orthogonal multiple access (NOMA) has emerged to adaptively manage the existing resources and received huge attention in terms of performance analysis to confirm its efficiency in resource management. This work addresses the applications of nonlinear symbolic regression (NLSR) techniques to evolve mapping models for bit error rate (BER) prediction in the downlink NOMA systems with perfect and imperfect successive interference cancellation (SIC) receivers. In this regard, a multi-stage structure is proposed to evaluate the BER using the related inputs as well as the output of the previous stages for imperfect SIC signal decoding. Each step consists of an NLSR block with different inputs and an output, which inputs include the effective parameters and the output of the previous stages. Experimental results confirm that the NLSR techniques are able to establish modeling expressions with good prediction accuracy compared to other benchmark methods. In particular, these ML-based techniques overcome the limitation of the classical approaches in modeling a closed-form BER formula, especially for the NOMA systems with more than two users. In addition, this BER prediction tool can be utilized by the NOMA base station as new side information for more optimal resource management.

Queue-aware computation offloading for UAV-assisted edge computing in wind farm routine inspection

Integration of unmanned aerial vehicles (UAVs) and edge computing into the wind farm routine inspection provides a promising approach to enhancing inspection effectiveness and decreasing operation maintenance costs. In light of the finite battery power and computational capacity of UAVs, a dynamic queue-aware UAV-assisted edge computing inspection wind farm framework is investigated with the goal of minimizing the long-term energy consumption of UAVs. The Lyapunov optimization theory is utilized to decouple the long-term stochastic optimization problem into four short-term deterministic subproblems, including the task splitting, the UAV-side computing resource allocation, the task offloading, and the edge server-side computing resource allocation. Furthermore, a Lyapunov optimization-based dynamic queue-aware computation offloading algorithm (LODQCO) is presented to optimize task offloading and resource allocation jointly. The optimal UAV-side computing resource is determined by a closed form formula, and then the optimal task offloading decision is tackled by applying the classical interior point method. Finally, the edge server-side computing resource is addressed via a linear optimization CPLEX solver. Based on simulation results, LODQCO is superior to the benchmark algorithms with respect to the energy consumption, queue backlogs, and queuing delays.

N-Bridge braids and the braid index

In this work, we find a closed form formula for the braid index of an [Formula: see text]-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman–Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree.

The Curious Case of Integrator Reach Sets, Part I: Basic Theory

This is the first of a two part paper investigating the geometry of the integrator reach sets, and the applications thereof. In this Part I, assuming box-valued input uncertainties, we establish that this compact convex reach set is semialgebraic, translated zonoid, and not a spectrahedron. We derive the parametric as well as the implicit representation of the boundary of this reach set. We also deduce the closed form formula for the volume and diameter of this set, and discuss their scaling with state dimension and time. We point out that these results may be utilized in benchmarking the performance of the reach set over-approximation algorithms.

The Schur decomposition of discrete Sine and Cosine transformations of type IV

Eigendecomposition of discrete sine and cosine transformations of type IV is the main objective of this study. A closed form formula for eigenvectors is derived using the property that a square of the transform matrix is proportional to the identity matrix. Odd and even transform sizes are considered separately. In both cases, it has been demonstrated that the eigenvectors can be obtained by a straightforward modification of the columns of the transformation matrix. Comparing this approach with the related works, it provides a simple explanation of the eigendecomposition with a reliable numerical stability. Additionally, these eigenvectors can be used to obtain the eigendecomposition of the counterpart offset discrete Fourier transform.

Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform

In this paper, we derive closed-form formulas of first-order approximation for down-and-out barrier and floating strike lookback put option prices under a stochastic volatility model using an asymptotic approach. To find the explicit closed-form formulas for the zero-order term and the first-order correction term, we use Mellin transform. We also conduct a sensitivity analysis on these formulas, and compare the option prices calculated by them with those generated by Monte-Carlo simulation.

Derivative of Reduced Cumulative Distribution Function and Applications

The reduced cumulative distribution function (rCDF) is the maximal lower bound for the cumulative distribution function (CDF). It is equivalent to the inverse of the conditional value at risk (CVaR), or one minus the buffered probability of exceedance (bPOE). This paper introduces the reduced probability density function (rPDF), the derivative of rCDF. We first explore the relation between rCDF and other risk measures. Then we describe three means of calculating rPDF for a distribution, depending on what is known about the distribution. For functions with a closed-form formula for bPOE, we derive closed-form formulae for rPDF. Further, we describe formulae for rPDF based on a numerical bPOE when there is a closed-form formula for CVaR but no closed-form formula for bPOE. Finally, we give a method for numerically calculating rPDF for an empirical distribution, and compare the results with other methods for known distributions. We conducted a case study and used rPDF for sensitivity analysis and parameter estimation with a method similar to the maximum likelihood method.

Computing indefinite integrals by difference equations

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5).

Uncertainty laws of experimental modal analysis with known broadband input

‘Uncertainty law’ aims at closed-form asymptotic formulas for the relationship between the identification uncertainties of modal properties (e.g., natural frequency, damping ratio) and test configuration (e.g., noise level, number and location of sensors, data duration). Existing developments focused on the case of unknown-input (ambient), where it has been found that identification uncertainty does not vanish even for noiseless instruments, essentially because the input is unknown. A natural question is then on how the uncertainty depends on test configuration when the input is known, not to mention how the configuration should be quantified. Motivated by these and related questions, this paper develops the uncertainty laws of modal parameters for well-separated modes with known single broadband input, e.g., vibration test with a single shaker as in experimental modal analysis. Asymptotic expressions for the posterior coefficient of variation of modal parameters are derived via the Fisher Information Matrix for long data and small damping scenarios. Assumptions and theory are validated using synthetic and field test data. Governing factors motivated by the theory are investigated, including the equivalent modal signal-to-noise ratio (for known input), the number of measured degrees of freedom, shaker location, and data duration. By virtue of the Cramér-Rao bound in classical statistics, the developed uncertainty laws represent the lower bound of identification uncertainty with known broadband input that can be achieved by any unbiased estimator. They provide a scientific basis for planning and managing identification uncertainties in vibration tests with known input.

Lagrangian Voronoï meshes and particle dynamics with shocks

We present a new first order particle method on lagrangian and moving Voronoï meshes for the numerical simulation of compressible flows with shocks and internal interfaces between different gas. The method is based on the closed form formula of the partial derivative of the volume of Voronoï cells with respect to the generators. The mathematical proof of the formula seems original with respect to the literature. A corollary is that the volume of Voronoï cells is generically of class C1 with respect to the generators. The final scheme is conservative in local mass, total momentum and total energy, and it is endowed with an entropy inequality which insures the correctness of shocks calculations. Numerical illustrations in dimension d=2 are displayed for basic problems on coarse meshes. The implementation developed to obtain the numerical illustrations uses a freely available library for the generation of the Voronoï cells at all time steps.

Vibration-Theoretic Approach to Vulnerability Analysis of Nonlinear Vehicle Platoons

This research explores the inherent vulnerability of nonlinear vehicle platoons characterized by the oscillatory behavior triggered by external perturbations. The perturbation exerted on the vehicle platoon is regarded as an external force on an object. Following the mechanical vibration analysis in mechanics, this research proposes a vibration-theoretic approach that advances our understanding of platoon vulnerability from two aspects. First, the proposed approach introduces damping intensity to characterize vehicular platoon vulnerability, which divides platoon oscillations into two types, i.e., underdamped and overdamped. The damping intensity measures the platoon’s recovery strength in responding to perturbations. Second, the proposed approach can obtain the resonance frequency of a nonlinear vehicle platoon, where resonance amplifies platoon oscillation magnitude when the external perturbation frequency equals the platoon’s damping oscillation frequency. The main contribution of this research lies in the analytical derivation of the closed-form formulas of damping intensity and resonance frequency. In particular, the proposed approach formulates platoon dynamics under perturbation as a second-order non-homogeneous ordinary differential equation, enabling rigorous derivations and analyses for platoons with complicated nonlinear car-following behaviors. Through simulations built on real-world data, this paper demonstrates that an overdamped vehicle platoon is more robust against perturbations, and an underdamped platoon can be destabilized easily by exerting a perturbation at the platoon’s resonance frequency. The theoretical derivations and simulation results shed light on the design of reliable platooning control, either for human-driven or automated vehicles, to suppress the adverse effects of oscillations.

A simple formula for coherent specular reflection loss of underwater sound from rough ocean surfaces

The interaction of underwater sound with rough sea surfaces affects its propagation. The interaction may involve several coupled mechanisms: (1) a rough air-sea interface scatters and spreads the incident sound energy over a range of angles, (2) entrapped sub-surface air bubbles absorb and scatter sound energy, and (3) medium of bubble clouds refract sound energy upwards. In this paper, we ignore the effect of bubbles and consider the effect of rough air-sea interface only. We further limit our consideration to the coherent component in the specular direction. Two simple closed-form formulas are often used to model the coherent reflection loss from rough ocean surfaces. One is derived from perturbation approximation and suitable for smaller grazing angles. The other is from Kirchhoff approximation and suitable for larger grazing angles. More accurate approximations over a wide-range of grazing angles require numerical evaluation of integrals or complex special functions. In this work, we simplify and adapt earlier results by other authors to obtain a simple expression that transits from the perturbation to the Kirchhoff formulas as grazing angle increases. The new simple analytical formula is easy to use and compares well with results from complex numerical methods.

Power Analysis of Exposure Mixture Studies via Monte Carlo Simulations.

Estimating sample size and statistical power is an essential part of a good epidemiological study design. Closed-form formulas exist for simple hypothesis tests but not for advanced statistical methods designed for exposure mixture studies. Estimating power with Monte Carlo simulations is flexible and applicable to these methods. However, it is not straightforward to code a simulation for non-experienced programmers and is often hard for a researcher to manually specify multivariate associations among exposure mixtures to set up a simulation. To simplify this process, we present the R package mpower for power analysis of observational studies of environmental exposure mixtures involving recently-developed mixtures analysis methods. The components within mpower are also versatile enough to accommodate any mixtures methods that will developed in the future. The package allows users to simulate realistic exposure data and mixed-typed covariates based on public data set such as the National Health and Nutrition Examination Survey or other existing data set from prior studies. Users can generate power curves to assess the trade-offs between sample size, effect size, and power of a design. This paper presents tutorials and examples of power analysis using mpower.

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

An analytical closed-form solution for the in-plane elastic modulus of plain-woven textile composites

Engineering application of textile composites needs simple relations for the estimation of their mechanical properties. Many homogenization and simplified models have been already proposed for the analytical characterization of the elastic moduli of this class of materials. Along this route, the current study introduces an analytical solution for the longitudinal elastic modulus of plain-woven textile composites. A hybrid beam-spring model of the representative unit cell (RUC) is postulated consisting of a horizontal beam connected to an inclined beam, two supporting springs beneath the free end of the horizontal beam and the common end of the horizontal and inclined beams, and a torsion spring at the free end of the inclined beam. The mechanical and geometrical properties of the beams and springs are calculated from the weave structure and the symmetry conditions. A reduction technique from the matrix analysis of structures is borrowed to consider the effect of internal degrees of freedom on the in-plane stiffness coefficient and a closed-form formula is extracted for the elastic modulus. The proposed analytical formula is finally verified against available experimental data and a wide range of FE predictions for the GFRP and CFRP plain-woven composites.

On the Witten index of 3d $\mathcal{N}=2$ unitary SQCD with general CS levels

We consider unitary SQCD, a three-dimensional \mathcal{N}=2𝒩=2 supersymmetric Chern-Simons-matter theory consisting of one U(N_c)_{k, k+l N_c}U(Nc)k,k+lNc vector multiplet coupled to n_fnf fundamental and n_ana antifundamental chiral multiplets, where kk and ll parameterise generic CS levels for U(N_c)=(SU(N_c)\times U(1))/\mathbb{Z}_{N_c}U(Nc)=(SU(Nc)×U(1))/ℤNc. We study the moduli space of vacua of this theory with n_a=0na=0, for generic values of the parameters N_c, k, l, n_fNc,k,l,nf and with a non-zero Fayet-Ilopoulos parameter turned on. We uncover a rich pattern of vacua including Higgs, topological and hybrid phases. This allows us to derive a closed-form formula for the flavoured Witten index of unitary SQCD for any n_f\neq n_anf≠na, generalising previously known results for either l=0 or n_f=n_anf=na. Finally, we analyse the vacuum structure of recently proposed infrared-dual gauge theories and we match vacua across the dualities, thus providing intricate new checks of those dualities. Incidentally, we also discuss a seemingly new level/rank duality for pure CS theories with U(N) x U(N’) gauge group.

Funding startups using contingent option of value appreciation: theory and formula

PurposeThis paper provides a structural model to value startup companies and determine the optimal level of research and development (R&D) spending by these companies.Design/methodology/approachThis paper describes a new variant of float-the-money options, which can act as a financial instrument for financing R&D expenses for a specific time horizon or development stage, allowing the investor to share in the startup's value appreciation over that duration. Another innovation of this paper is that it develops a structural model for evaluating optimal level of R&D spending over a given time horizon. The paper deploys the Gompertz-Cox model for the R&D project outcomes, which facilitates investigation of how increased level of R&D input can enhance the company's value growth.FindingsThe author first introduces a time-varying drift term into standard Black-Scholes model to account for the varying growth rates of the startup at different stages, and the author interprets venture capital's investment in the startup as a “float-the-money” option. The author then incorporates the probabilities of startup failures at multiple stages into their financial valuation. The author gets a closed-form pricing formula for the contingent option of value appreciation. Finally, the author utilizes Cox proportional hazards model to analyze the optimal level of R&D input that maximizes the return on investment.Research limitations/implicationsThe integrated contingent claims model links the change in the financial valuation of startups with the incremental R&D spending. The Gompertz-Cox contingency model for R&D success rate is used to quantify the optimal level of R&D input. This model assumption may be simplistic, but nevertheless illustrative.Practical implicationsOnce supplemented with actual transaction data, the model can serve as a reference benchmark valuation of new project deals and previously invested projects seeking exit.Social implicationsThe integrated structural model can potentially have much wider applications beyond valuation of startup companies. For instance, in valuing a company's risk management, the level of R&D spending in the model can be replaced by the company's budget for risk management. As another promising application, in evaluating a country's economic growth rate in the face of rising climate risks, the level of R&D spending in this paper can be replaced by a country's investment in addressing climate risks.Originality/valueThis paper is the first to develop an integrated valuation model for startups by combining the real-world R&D project contingencies with risk-neutral valuation of the potential payoffs.