Multiplier Equations Research Articles

Unique continuation for water waves and dispersive multiplier equations

Unique continuation for water waves and dispersive multiplier equations

Numerical simulation for fractional optimal control problems via euler wavelets

Abstract This paper suggests employing Euler wavelets to provide a precise and effective computational approach for certain fractional optimal control problems. The primary objective of the approach is to transform the fractional optimal control problem defined by the dynamical system and performance index into systems of algebraic equations, which then can be readily solved using matrix techniques. Since Euler polynomials are employed to build Euler wavelets and since Euler polynomials have fewer terms than most of commonly used other polynomials to build wavelets, employing them for the numerical approach results in sparser operational matrices. The speed of the recommended numerical method is improved due to the fewer terms in Euler wavelet operational matrices. We obtain the corresponding systems of algebraic equations for state variable, control variable, and Lagrange multipliers (used to determine the essential conditions of optimality) by incorporating operational matrices of Euler wavelets. Subsequently, those systems of algebraic equations are solved to obtain the numerical solution. The suggested method’s efficiency is demonstrated using a few typical examples. The suggested procedure is accurate and efficient, according to the results.

Climate change and air pollution impacts on cultural heritage building materials in Europe and Mexico

Climate and air pollution have adverse effects on cultural heritage building materials. However, the quantified damage due to modeled changes in climate and air pollution is still poorly studied. Here, we review first the damage affecting these materials and the associated damage equations in the literature. Across all relevant studies (n = 87), we found only nine independent equations to estimate different damage categories, mainly limited to limestones. Then, by using current meteorological data and future bias-corrected CMIP6 climate and air pollution data at high resolution (1 km; historical and business-as-usual scenario) and applying these equations, we quantified the relative contributions of climate and air pollution changes on the building materials of eight cultural heritage sites of the European project Sustainable COnservation and REstoration of built cultural heritage (SCORE) from 2020 to 2100. On average across the sites, a significant decrease in damage is projected in surface recession (−10 % ± 10 %), biomass accumulation (−20 % ± 18 %), and wind–rain erosion (−7 % ± 6 %) in response to future climate and air pollution changes, except in the regions where precipitation substantially increases (Northern Europe). A large uncertainty in the relative magnitude of the damage to built cultural heritage materials was found for the same site, changes in surface recession vary up to a 40 % difference across the equations. Moreover, thermal expansion and lifetime multiplier equations project an increase in the related damage while all the other types of damage are significantly reduced. Finally, in general, but not systematically, climate-induced damage was found to be predominant over the pollution-induced one. Our results allow prioritizing cultural heritage maintenance decisions in regions where damage will further increase. Beyond simulated damages which are still limited use, we urge more campaign studies to determine real in situ damage in different climate locations to validate or build the best equations.

Lagrange multiplier and variational equations in mechanics

Lagrange multiplier and variational equations in mechanics

A Generalized (3+1)-Dimensional Nonlinear Wave Equation in Liquid with Gas Bubbles: Symmetry Reductions; Exact Solutions; Conservation Laws

The analysis of a generalised (3+1)-dimensional nonlinear wave equation that simulates a variety of nonlinear processes that occur in liquids including gas bubbles will be performed. After some cosmetic adjustments to the underlying equation, this generalised (3+1)-dimensional nonlinear wave equation naturally degenerates into the (3+1)-dimensional Kadomtsev-Petviashvili equation, the (3+1)-dimensional nonlinear wave equation, and the Korteweg-de Vries equation. To completely investigate this fundamental equation, a clear and rigorous technique is used. In order to obtain innovative symmetry reductions, group invariant solutions, conservation laws, and eventually kink wave solutions, the Lie symmetry, multiplier, and simplest equation methods are used. Complex waves and their dealing dynamics in fluids can be well imitated by the verdicts.

Euclidean distance degree of zero-set of two polynomials

In this note, we recall the study of the Euclidean distance degree of an algebraic set X which is the zero-point set of a polynomial (see [BSW]). Specifically, consider a hypersurface defined by a general polynomial f with its support and contains the origin i.e 0 support of f. In the paper [BSW], the authors study about the Euclidean distance degree (EDD) and found that the EDD of this hypersurface is approximately by the mixed volume (MV) of some Newton polytopes. The main purpose of this note is to study the case that the manifold is defined by two polynomials . We show that the Euclidean distance degree is equal to the solution of the Lagrange multiplier equation. Furthermore, we also find out that the EDD of this variety is not greater than the mixed volume of Newton polytopes of the associated Lagrange multiplier equations.

A New Two-Phase Multiplier for Flow Pressure Drop in Multiport Minichannel Condensers and Evaporators

Due to flow maldistribution, the condensation and evaporation flow patterns in multiport minichannels are considerably different from single minichannels or macrochannels. This article compiled the friction pressure drop data from experimental phase-change investigations in multiport minichannel condensers and evaporators over the past two decades. The data was reduced to friction pressure gradients, and a new two-phase multiplier was proposed for estimating the phase-change pressure drop in multiport condensers and evaporators. Thirteen hundred and forty-four condensation pressure drop and six hundred and twenty-three evaporation pressure drop data from twenty-nine studies were correlated to yield four predictive two-phase multiplier equations in the laminar and turbulent flow regimes. The predictive correlations fit 67% of the laminar condensation and 80% of the turbulent pressure drop data within ±50%. Further, 57% of the laminar evaporation and 100% of the turbulent evaporation pressure drop data were fit within ±50%. The correlations were compared with widely published correlations and were a significant improvement. Meta-analysis revealed that multiport minichannels are most effective for reducing turbulent flow condensation pressure drop and laminar flow evaporation pressure drop. The compiled data and presented correlations and analysis should be helpful to the process, electronics packaging, aviation, and aerospace industries designing compact, lightweight, and high-efficiency condensers, and evaporators.

Cluster perturbation theory. VII. The convergence of cluster perturbation expansions.

The convergence of the recently developed cluster perturbation (CP) expansions [Pawlowski et al., J. Chem. Phys. 150, 134108 (2019)] is analyzed with the double purpose of developing the mathematical tools and concepts needed to describe these expansions at general order and to identify the factors that define the rate of convergence of CP series. To this end, the CP energy, amplitude, and Lagrangian multiplier equations as a function of the perturbation strength are developed. By determining the critical points, defined as the perturbation strengths for which the Jacobian becomes singular, the rate of convergence and the intruder and critical states are determined for five small molecules: BH, CO, H2O, NH3, and HF. To describe the patterns of convergence for these expansions at orders lower than the high-order asymptotic limit, a model is developed where the perturbation corrections arise from two critical points. It is shown that this model allows for rationalization of the behavior of the perturbation corrections at much lower order than required for the onset of the asymptotic convergence. For the H2O, CO, and HF molecules, the pattern and rate of convergence are defined by critical states where the Fock-operator underestimates the excitation energies, whereas the pattern and rate of convergence for BH are defined by critical states where the Fock-operator overestimates the excitation energy. For the NH3 molecule, both forms of critical points are required to describe the convergence behavior up to at least order 25.

How are Banks and the Fed Linked? Teaching Key Concepts Today

ABSTRACT This article walks through the roles of banks and the Federal Reserve (the Fed) in the financial system and describes the key linkages between the two, using current concepts. Because many introductory economics textbooks are not up-to-date on key concepts, we highlight two crucial mistakes often taught in the classroom: (1) using an outdated description of Fed operations that rely on open market operations instead of focusing on the Fed’s administered interest rates, and (2) relying on the money multiplier equation, which broke down over time and is now no longer definable. We provide recommendations for the key themes that should be part of a current introductory economics curriculum and note many resources that are available for instructors seeking to update their materials.

An effective operator splitting scheme for two-dimensional conservative nonlocal Allen–Cahn equation

An effective operator splitting scheme for solving the nonlocal Allen–Cahn equation with a Lagrange multiplier is studied. Firstly, based on the operator splitting method, the original equation is discretized into a nonlinear equation, a nonlocal equation and a Lagrange multiplier equation, respectively. Then, the nonlinear equation is solved analytically, the nonlocal equation is discretized using Crank–Nicolson format, and the Lagrange multiplier equation is solved explicitly. The mass conservation and convergence analysis of the numerical algorithm are analyzed theoretically. Numerical experiments are presented to confirm the validity of the proposed method, including the convergence, energy decline and mass conservation.

Laplacian-perturbed Lagrange-multiplier formulation for fluid-structure interaction

This paper presents a stabilized Lagrange-multiplier formulation for Euler-Lagrange type fluid-structure interaction (FSI) computations. The formulation adds a Laplacian perturbation term to the multiplier equation to relax over-constraint condition caused by the multimer formulation. The over-constraint occurs when the Euler mesh for fluid computation is finer than Lagrange meshes for structures around the FSI interfaces. We show a two-dimensional computation that demonstrates relaxation effect of the proposed method.

Euclidean Distance Degree and Mixed Volume

We initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface f=0 defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of f=0 equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.

Utilizing dynamic parallelism in CUDA to accelerate a 3D red-black successive over relaxation wind-field solver

QES-Winds is a fast-response wind modeling platform for simulating high-resolution mean wind fields for optimization and prediction. The code uses a variational analysis technique to solve the Poisson equation for Lagrange multipliers to obtain a mean wind field and GPU parallelization to accelerate the numerical solution of the Poisson equation. QES-Winds benefits from CUDA dynamic parallelism (launching the kernel from the GPU) to speed up calculations by a factor of 128 compared to the serial solver for a domain with 145 million cells. The dynamic parallelism enables QES-Winds to calculate mean velocity fields for domains with sizes of 10km2 and horizontal resolutions of 1-3m in under 1 min. As a result, QES-Winds is a numerical code suitable for computing high-resolution wind fields on large domains in real time, which can be used to model a wide range of real-world problems including wildfires and urban air quality.

Generalized symmetries and higher-order Conservation laws of the Camassa–Holm equation

In the present paper, we derive generalized symmetries of order three of the Camassa–Holm equation by infinite prolongation of a generalized vector field and applying infinitesimal symmetry criterion. In addition, one-dimensional optimal system of Lie subalgebras investigated by applying the adjoint representation. Furthermore, determining equation for multipliers and the 2- dimensional homotopy formula employed to construct higher–order conservation laws for the Camassa–Holm equation.

Phillips model with exponentially distributed lag and power-law memory

In this paper, we propose two generalizations of the Phillips model of multiplier–accelerator by taking into account memory with power-law fading. In the first generalization we consider the model, where we replace the exponential weighting function by the power-law memory function. In this model we consider two power-law fading memories, one on the side of the accelerator (induced investment responding to changes in output with memory-fading parameter α) and the other on the supply side (output responding to demand with memory-fading parameter β). To describe power-law memory we use the fractional derivatives in the accelerator equation and the fractional integral in multiplier equation. The solution of the model fractional differential equation is suggested. In the second generalization of the Phillips model of multiplier–accelerator we consider the power-law memory in addition to the continuously (exponentially) distributed lag. Equation, which describes generalized Phillips model of multiplier–accelerator with distributed lag and power-law memory, is solved using Laplace method.

Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.

Influential factors responsible for the effect of tax reduction on GDP

The factors responsible for the effect of a tax reduction on GDP are analyzed using both agent-based modeling, based on the authors’ previous study, and a derived set of equations for tax reduction multipliers, based on Morishima’s economic linkage table. The findings are that, under the condition of balanced government finance, the tax reduction multiplier is determined by the difference between the increase in demand by consumers or firms as a result of the tax reduction and the decrease in demand by the government. To increase the effect of a tax reduction, it is necessary that the increased disposable income of consumers or firms, as a result of the tax reduction, is more likely to be used for consumption and investment. The analyzed factors that are proved to be indispensable to reproduce positive influence of a tax reduction in ABM are consistent with this mechanism.

Pressure Drop Through Orifices for Single- and Two-Phase Vertically Upward Flow—Implication for Metering

Pressure drop has been measured for upward single- and two-phase gas–liquid flow across an orifice in a vertical pipe. A conductance probe provided average void fraction upstream of the orifice. Six orifices with different apertures/thickness were mounted in turn in a 34 mm diameter transparent acrylic resin pipe. Gas and liquid superficial velocities of 0–4 m/s and 0.3–0.91 m/s, respectively, were studied. For single-phase flow, pressure drop, expressed as an Euler number, was seen to be independent of Reynolds number in turbulent region. The Euler number increased with decreasing the open area ratio/orifice thickness and increasing velocity. The pressure drop was well predicted by the correlation of Idel'chik et al. (1994, Handbook of Hydraulic Resistances, 3rd ed., CRC Press, Boca, Raton, FL.), which uses a form of Euler number. The corresponding two-phase flow pressure drop depends on the flow pattern. Decreasing open area ratio/orifice thickness increased the pressure drop. For a given liquid superficial velocity, the pressure drop increases with gas superficial velocity except for low open area ratio where this increase is followed by a decrease beyond a critical superficial gas velocity for the high liquid superficial velocities. Relevant correlations were assessed using the present data via a systematic statistical approach. The two-phase multiplier equations of Morris (1985, “Two-Phase Pressure Drop Across Valves and Orifice Plates,” European Two Phase Flow Group Meeting, Marchwood Engineering Laboratories, Southampton, UK.) and Simpson et al. (1983, “Two-Phase Flow Through Gate Valves and Orifice Plates,” International Conference on Physical Modelling of Multiphase Flow, Coventry, UK.) are the most reliable ones.

Hydrodynamics of suspensions of passive and active rigid particles: a rigid multiblob approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally-resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. For unbounded suspensions and suspensions sedimented against a single no-slip boundary, we rely on existing analytical expressions for the Rotne-Prager tensor combined with a fast multipole method or a direct summation on a Graphical Processing Unit to obtain an simple yet efficient and scalable implementation. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently-developed rigid-body immersed boundary method to suspensions of freely-moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid body equations converges in a bounded number of iterations regardless of the system size. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.

Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion

Location estimation plays a crucial role in Location-Based Services (LBSs) with satisfactory user experience. The Wireless Local Area Network (WLAN) localization approach is preferred as a cost-efficient solution to indoor localization on account of the widely-deployed WLAN infrastructures. In this paper, we propose a new WLAN Received Signal Strength (RSS)-based indoor localization approach using the semi-supervised manifold alignment with dimension expansion. In concrete terms, we first construct an innovative objective function based on the augmented physical coordinates and the corresponding WLAN RSS measurements. Second, the closed-form solution to the objective function is derived out according to the Lagrange multiplier equation, which results in the manifold in physical coordinate space. Third, the target location is estimated by matching the transformed newly-collected RSS against the manifold. The localization performance with noise perturbation is analyzed upon the constructed objective function, and meanwhile, the closed-form solution to the objective function with respect to multiple types of measurements is also derived out for the sake of leveraging all of the potential measurements for indoor localization. The extensive testing results show that the proposed approach performs well in localization accuracy even at low calibration load, and its performance can be further improved by using multiple types of measurements for localization.