Approximate Point Research Articles

Predictive power of transverse-momentum-dependent distributions

We investigate the predictive power of transverse-momentum-dependent (TMD) distributions as a function of the light-cone momentum fraction $x$ and the hard scale $Q$ defined by the process. We apply the saddle point approximation to the unpolarized quark and gluon transverse momentum distributions and evaluate the position of the saddle point as a function of the kinematics. We determine quantitatively that the predictive power for an unpolarized transverse momentum distribution is maximal in the large-$Q$ and small-$x$ region. For cross sections the predictive power of the TMD factorization formalism is generally enhanced by considering the convolution of two distributions, and we explicitly consider the case of $Z$ and $H^0$ boson production. In the kinematic regions where the predictive power is not maximal, the distributions are sensitive to the non-perturbative hadron structure. Thus, these regions are critical for investigating hadron tomography in a three-dimensional momentum space.

Uncertainty-Based Design and Optimization Using First Order Saddle Point Approximation Method for Multidisciplinary Engineering Systems

AbstractThe widely used First Order Reliability Method (FORM) is efficient for uncertainty quantification and safety assessment in Uncertainty-based Multidisciplinary Design Optimization (UBMDO). H...

Entanglement entropy and its quench dynamics for pure states of the Sachdev-Ye-Kitaev model

Sachdev-Ye-Kitaev (SYK) is a concrete solvable model with non-Fermi liquid behavior and maximal chaos. In this work, we study the entanglement Rényi entropy for the subsystems of the SYK model in the Kourkoulou-Maldacena states. We use the path-integral approach and take the saddle point approximation in the large-N limit. We find a first-order transition exist when tuning the subsystem size for the q = 4 case, while it is absent for the q = 2 case. We further study the entanglement dynamics for such states under the real-time evolution for noninteracting, weakly interacting and strongly interacting SYK(-like) models.

Configurational mean-field reduced transfer matrix method for Ising systems

A mean-field method for the hypercubic nearest-neighbor Ising system is introduced and applications to the method are demonstrated. The main idea of this work is to combine the Kadanoff’s mean-field approach with the model presented by one of us previously. The mean-field approximation is introduced with the replacement of the central spin in Ising Hamiltonian with an average value of particular spin configuration, i.e, the approximation is taken into account within each configuration. This approximation is used in two different mean-field-type approaches. The first consideration is a pure-mean-field-type treatment in which all the neighboring spins are replaced with the assumed configurational average. The second consideration is introduced by the reduced transfer matrix method. The estimations of critical coupling values of the systems are evaluated both numerically and also analytically by the using of saddle point approximation. The analytic estimation of critical values in the first and second considerations are [Formula: see text] and [Formula: see text] respectively. Obviously, both of the considerations have some significant deviation from the exact treatment. In this work, we conclude that the method introduced here is more appropriate physical picture than self-consistent mean-field-type models, because the method introduced here does not presume the presence of the phase transition from the outset. Consequently, the introduced approach potentially makes our research very valuable mean-field-type picture for phase transition treatment.

A general non-adiabatic quantum instanton approximation.

We present a general quantum instanton approach to calculating reaction rates for systems with two electronic states and arbitrary values of the electronic coupling. This new approach, which we call the non-adiabatic quantum instanton (NAQI) approximation, reduces to Wolynes theory in the golden rule limit and to a recently proposed projected quantum instanton method in the adiabatic limit. As in both of these earlier theories, the NAQI approach is based on making a saddle point approximation to the time integral of a reactive flux autocorrelation function, although with a generalized definition of the projection operator onto the product states. We illustrate the accuracy of the approach by comparison with exact rates for one dimensional scattering problems and discuss its applicability to more complex reactions.

A transference inequality for rational approximation to points in geometric progression

We establish a transference inequality conjectured by Badziahin and Bugeaud relating exponents of rational approximation of points in geometric progression.

A Maximum Power Point Tracking in Wind Energy Conversion Systems using Machine Learning

In this paper, an efficient and feasible algorithm to extract the maximum power point (MPP) in wind energy conversion systems (WECS) by implementing machine learning (ML) into perturb and observe (P&O) algorithm is presented. The proposed algorithm is simulated on a separately-excited DC generator. This model uses instantaneous measurements of wind speed, humidity, temperature, pressure and generator speed to estimate a MPP by using ML at the end of each iteration. From this estimated power point, the controller follows quick perturbation to calculate the accurate MPP and is used as training data for further predictions in the next iteration. The controller learns from this training set and estimates the MPP closer to the maximum achievable power (MAP) which is corrected again through perturbation and is recorded. With the progress of time, the approximation of the maximum power point becomes more accurate whilst the time in further perturbation required for modification decreases. This model adapts to the versatile climatic conditions and yields an efficiency of 99.95% in predicting the MAP at the end of 1000 iterations corresponding to 2 hours 30 minutes.

Indirect Predicates for Geometric Constructions

Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art predicates are guaranteed to be always exact while being only slightly slower than corresponding (inexact) floating point implementations. Unfortunately, if the input to these predicates is an intermediate construction of an algorithm, its floating point representation may be affected by an approximation error, and correctness is no longer guaranteed.This paper introduces the concept of indirect geometric predicate: instead of taking the intermediate construction as an explicit input, an indirect predicate considers the primitive geometric elements which are combined to produce such a construction. This makes it possible to keep track of the floating point approximation, and thus to exploit efficient filters and expansion arithmetic to exactly resolve the predicate with minimal overhead with respect to a naive floating point implementation. As a representative example, we show how to extend standard predicates to the case of points of intersection of linear elements (i.e. lines and planes) and show that, on classical problems, this approach outperforms state-of-the-art solutions based on lazy exact intermediate representations.

Novel probabilistic model for searching most probable point in structural reliability analysis

The Most Probable Point (MPP) is typically searched by transforming the basic random variables from original to standard normal space in the traditional first order reliability method (FORM). However, the FORM formulation may provide inaccurate MPP for those reliability problems with several local MPPs. The FORM based probabilistic model is a computational method and it may provide inaccuracy and unstable results for complex problems. In this paper, a probabilistic model is proposed to evaluate the MPP using cumulative distribution function (CDF) of basic random variables. The MPP is determined by solving an optimization problem in the original space which maximizes a multiplier function in terms of the CDF of the random variables. The performance of the proposed probabilistic model, usual FORM and first order saddle point approximation method (FOSAM) using SQP optimization solver is demonstrated by several numerical and engineering problems with normal and non-normal variables. The results of the numerical study illustrate that the proposed model provides an efficient approach to obtain the MPP which is simpler and more accurate than the usual FORM and FOSAM; particularly for reliability problems with non-normal random variables.

Finite-approximate controllability of fractional stochastic evolution equations with nonlocal conditions

This paper deals with the finite-approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We establish sufficient conditions for the finite-approximate controllability of the control system when the compactness conditions or Lipschitz conditions for the nonlocal term and uniform boundedness conditions for the nonlinear term are not required. The discussion is based on the fixed point theorem, approximation techniques and diagonal argument. In the end, an example is presented to illustrate the abstract theory. Our result improves and extends some relevant results in this area.

Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums

Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums

Universal 3d Cardy block and black hole entropy

We discuss the Cardy limit of 3d supersymmetric partition functions which allow the factorization into the hemisphere indices: the generalized superconformal index, the refined topologically twisted index and the squashed sphere partition function. In the Cardy limit, the hemisphere index can be evaluated by the saddle point approximation where there exists a dominant saddle point contribution, which we call the Cardy block. The Cardy block turns out to be a simple but powerful object as it is a building block of other partition functions in the Cardy limit. The factorization to the Cardy block allows us to find universal relations among the partition functions, which we formulate as index theorems. Furthermore, if we consider a holographic 3d SCFT and its large N limit, those partition functions relate to various entropic quantities of the dual gravity theory in AdS4. As a result, our result provides the microscopic derivation of the universal relations among those entropic quantities of the gravity theory. We also discuss explicit examples, which confirm our general index theorems.

Probabilistic Load Flow Based on Holomorphic Embedding, Kernel Density Estimator and Saddle Point Approximation Including Correlated Uncertainty Variables

Probabilistic Load Flow (PLF) methods have been extensively regarded due to the growth of uncertainty and renewable energies penetration in power systems. Given that, the current study aims to propose two novel probabilistic load flow methods based on holomorphic embedding method. Holomorphic method, unlike other iterative methods, introduces a nonlinear equation solver in an independent class, recursively. To that end, Kernel Density Estimator (KDE) and Saddle Point Approximation (SPA) methods are used to efficiently estimate probabilistic characteristics of load flow outputs. The correlations between random input variables with non-normal/normal probability distributions have been considered. The proposed algorithms have been examined on the modified IEEE 14- and 118-bus systems. Finally, to consider the uncorrelated and correlated conditions, the researcher compared the simulation results of the two proposed methods with some well-known and published probabilistic iterative load flow methods such as Monte Carlo Simulation (MCS), Parzen Window (PW), Diffusion method, and 2n+1 Point Estimate Method (PEM). MCS is always a dependable solution but time-consuming that make it useless for large power systems. In the present study, the accurate results obtained from MCS are regarded as a reference. The comparison of results shows high efficiency and low computational burden of proposed methods, and also the accuracy of these methods in density function estimation of output random variables.

A two-stage robust optimisation for terminal traffic flow problem

Airport congestion witnesses potential conflicts: insufficient terminal airspace and delay propagation within scrambled the competition in the terminal manoeuvring area. Re-scheduling of flights is needed in numerous situations, heavy traffic in air segments, holding patterns, runway schedules and airport surface operations. Robust optimisation for terminal traffic flow problem, providing a practical point of view in hedging uncertainty, can leverage the adverse effect of uncertainty and schedule intervention. To avoid delay propagation throughout the air traffic flow network and reduce the vulnerability to disruption, this research adopts a two-stage robust optimisation approach in terminal traffic flow. It further enhances the quality of Pareto-optimality Benders-dual cutting plane based on core point approximation in the second stage recourse decision. The efficiency of the cutting plane algorithm is evaluated by a set of medium sized real-life scenarios. The numerical results show that the proposed scheme outperforms the well-known Pareto-optimal cuts in Benders-dual method from the literature.

Optimal heat treatment furnace based on new robust point approximation strategy

In this work, a new model of air to air heat pump (AAHP) using industrial continuous heat treatment furnace (ICHTF) that runs by wind generation and solar Stirling engine are presented. The suggested approach for making fire includes an internal and an external airflow fans, extension valve, compressor, condenser, and evaporator to provide energy with temperature higher than 1000 K for thermal demands. While the combustion reactions is happening, AAHP does not provide thermal energy; it is moved from the environment to furnace section. Accordingly, application of wind, solar and AAHP not only decreases the emission and ozone depleting substances, but also significantly rises the saving rate in electricity and petroleum products consumptions. The paper presents a mathematical approach, that utilizes energy and exergy, which is used for modeling the suggested ICHTF in presence of green energy, e.g. solar, wind and air. The proposed method is compared with different optimization algorithms. In addition, a robust periodic optimization technique for economic issues is presented for minimization of the overall energy supplement costs utilizing the predicted wind velocity as the problem input and considering the higher and lower bounds of wind energy for simulating wind volatility. Obtained results proof the validity of proposed model.

APPROXIMATING SMOOTH, MULTIVARIATE FUNCTIONS ON IRREGULAR DOMAINS

In this paper, we introduce a method known aspolynomial frame approximationfor approximating smooth, multivariate functions defined on irregular domains in$d$dimensions, where$d$can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the correspondingNikolskii inequalityfor the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

UK Biobank Whole-Exome Sequence Binary Phenome Analysis with Robust Region-Based Rare-Variant Test

In biobank data analysis, most binary phenotypes have unbalanced case-control ratios, and this can cause inflation of type I error rates. Recently, a saddle point approximation (SPA) based single-variant test has been developed to provide an accurate and scalable method to test for associations of such phenotypes. For gene- or region-based multiple-variant tests, a few methods exist that can adjust for unbalanced case-control ratios; however, these methods are either less accurate when case-control ratios are extremely unbalanced or not scalable for large data analyses. To address these problems, we propose SKAT- and SKAT-O- type region-based tests; in these tests, the single-variant score statistic is calibrated based on SPA and efficient resampling (ER). Through simulation studies, we show that the proposed method provides well-calibrated p values. In contrast, when the case-control ratio is 1:99, the unadjusted approach has greatly inflated type I error rates (90 times that of exome-wide sequencing α = 2.5× 10-6). Additionally, the proposed method has similar computation time to the unadjusted approaches and is scalable for large sample data. In our application, the UK Biobank whole-exome sequence data analysis of 45,596 unrelated European samples and 791 PheCode phenotypes identified 10 rare-variant associations with p value < 10-7, including the associations between JAK2 and myeloproliferative disease, HOXB13 and cancer of prostate, and F11 and congenital coagulation defects. All analysis summary results are publicly available through a web-based visual server, and this availability can help facilitate the identification of the genetic basis of complex diseases.

Enhanced Positioning Algorithm Using a Single Image in an LCD-Camera System by Mesh Elements’ Recalculation and Angle Error Orientation

In this article, we present a method to position the tool in a micromachine system based on a camera-LCD screen positioning system that also provides information about angular deviations of the tool axis during its running. Both position and angular deviations are obtained by reducing a matrix of LEDs in the image to a single rectangle in the conical perspective that is treated by a photogrammetry method. This method computes the coordinates and orientation of the camera with respect to the fixed screen coordinate system. The used image consists of 5 × 5 lit LEDs, which are analyzed by the algorithm to determine a rectangle with known dimensions. The coordinates of the vertices of the rectangle in space are obtained by an inverse perspective computation from the image. The method presents a good approximation of the central point of the rectangle and provides the inclination of the workpiece with respect to the LCD screen reference system of coordinates. A test of the method is designed with the assistance of a Coordinate Measurement Machine (CMM) to check the accuracy of the positioning method. The performed test delivers a good accuracy in the position measurement of the designed method. A high dispersion in the angular deviation is detected, although the orientation of the inclination is appropriate in almost every case. This is due to the small values of the angles that makes the trigonometric function approximations very erratic. This method is a good starting point for the compensation of angular deviation in vision based micromachine tools, which is the principal source of errors in these operations and represents the main volume in the cost of machine elements’ parts.

Method of reducing the level of underground waters

A method is proposed to calculate water irrigation rates for crops that considers hysteresis in the volumetric water content (θ)-water pressure head (ψ) relations, including scanning paths, by the method of point approximation. Three θ(ψ) models are described and computational experiments were carried out using data from literary sources on the hydrophysical properties of soils with different textures. The error analysis of the point approximation of the main branches and the predictive calculation of the scanning branches of the hysteresis loop were carried out. The practical significance of the research lies in the possibility to calculate the precise irrigation rates for crops. The use of such standards minimizes unproductive spending of irrigation water, fertilizers, ameliorants and plant protection products, as well as prevents the pollution of natural waters with agrochemicals and reduces the risk of eutrophication of water sources.

Estimation of precise irrigation rates taking into account the hysteresis of soil water-retention capacity

A method is proposed to calculate water irrigation rates for crops that considers hysteresis in the volumetric water content (θ)-water pressure head (ψ) relations, including scanning paths, by the method of point approximation. Three θ(ψ) models are described and computational experiments were carried out using data from literary sources on the hydrophysical properties of soils with different textures. The error analysis of the point approximation of the main branches and the predictive calculation of the scanning branches of the hysteresis loop were carried out. The practical significance of the research lies in the possibility to calculate the precise irrigation rates for crops. The use of such standards minimizes unproductive spending of irrigation water, fertilizers, ameliorants and plant protection products, as well as prevents the pollution of natural waters with agrochemicals and reduces the risk of eutrophication of water sources.