Loop Equations Research Articles

Effect of salinity-clay variation on the transient magnetic field in the Quaternary aquifer, theoretically and practically

This study focuses on understanding the physical foundations of the transient electromagnetic method, such as the transition from a simple basic principle to a more complex principle, starting from the Biot-Savart law until the current concept. It calculates the steady and transient magnetic fields around any electrical wire system. Accordingly, this law will be used to determine the magnetic field around the square loop configuration, which matches the magnetic field arising from the circular loop configuration. The square loop equation was derived and extended to include changes in the half-space, which allows us to understand how the electromagnetic waves travel in the subsurface and the farthest point, that can record the TEM response from the TEM loop, according to the loop size and loop current. Accordingly, the deduced equations were applied to predict the transient magnetic field curves of the Quaternary aquifer in three different water zones along the Nile delta, depending on the values of resistivity and chargeability caused by the variations in salinity and clay contents. The calculated transient magnetic curves were then compared with other measured field curves to confirm the validity of the derived equations. Therefore, these curves are expected to be used as guide curves for the transient magnetic field response in this aquifer. Also, these curves are important in estimating the hydro-geophysical characteristics of the main groundwater aquifer in the selected area and in other areas with the same geologic and hydrogeologic settings. Also, a common model is presented for any delta environment in the world in measured and calculated data.

Solutions of the Loop Equations of a Class of Generalized Frobenius Manifolds

Solutions of the Loop Equations of a Class of Generalized Frobenius Manifolds

Dynamical loop equation

Dynamical loop equation

Bootstrapping the Abelian lattice gauge theories

We study the ℤ2 and U(1) Abelian lattice gauge theories using a bootstrap method, in which the loop equations and positivity conditions are employed for Wilson loops with lengths L ⩽ Lmax to derive two-sided bounds on the Wilson loop averages. We address a fundamental question that whether the constraints from loop equations and positivity are strong enough to solve lattice gauge theories. We answer this question by bootstrapping the 2D U(1) lattice gauge theory. We show that with sufficiently large Lmax = 60, the two-sided bounds provide estimates for the plaquette averages with precision near 10−8 or even higher, suggesting the bootstrap constraints are sufficient to numerically pin down this theory. We compute the bootstrap bounds on the plaquette averages in the 3D ℤ2 and U(1) lattice gauge theories with Lmax = 16. In the regions with weak or strong coupling, the two-sided bootstrap bounds converge quickly and coincide with the perturbative results to high precision. The bootstrap bounds are well consistent with the Monte Carlo results in the nonperturbative region. We observe interesting connections between the bounds generated by the bootstrap computations and the Griffiths’ inequalities. We present results towards bootstrapping the string tension and glueball mass in Abelian lattice gauge theories.

Unorientable topological gravity and orthogonal random matrix universality

The duality of Jackiw-Teitelboim (JT) gravity and a double scaled matrix integral has led to studies of the canonical spectral form factor (SFF) in the so called τ−scaled limit of large times, t → ∞, and fixed temperature, in order to demonstrate agreement with universal random matrix theory (RMT). Though this has been established for the unitary case, extensions to other symmetry classes requires the inclusion of unorientable manifolds in the sum over geometries, necessary to address time reversal invariance, and regularization of the corresponding prime geometrical objects, the Weil-Petersson (WP) volumes. We report here how universal signatures of quantum chaos, witnessed by the fidelity to the Gaussian orthogonal ensemble, emerge for the low-energy limit of unorientable JT gravity, i.e. the unorientable Airy model/topological gravity. To this end, we implement the loop equations for the corresponding dual (double-scaled) matrix model and find the generic form of the unorientable Airy WP volumes, supported by calculations using unorientable Kontsevich graphs. In an apparent violation of the gravity/chaos duality, the τ−scaled SFF on the gravity side acquires both logarithmic and power law contributions in t, not manifestly present on the RMT side. We show the expressions can be made to agree by means of bootstrapping-like relations hidden in the asymptotic expansions of generalized hypergeometric functions. Thus, we are able to establish strong evidence of the quantum chaotic nature of unorientable topological gravity.

Loop equations for generalised eigenvalue models

Loop equations for generalised eigenvalue models

Modeling, design and validation of DC-DC landsman converter for low-power electric vehicle battery charging applications

As the world moves towards sustainable development, the reduction of air pollution is considered an important factor. Consequently, the transportation sector is moving from conventional internal combustion engine vehicles to electric vehicles (EVs). To encourage EV usage, governments are working towards installing more public charging systems. These systems invariably use a DC-DC converter to ensure the supply of an appropriate voltage to charge the battery. In developing countries, the majority of users prefer electric bikes and electric rickshaws, which require low power. Therefore, it becomes necessary to use low-power converters suitable to charge these batteries. In such systems, non-isolated DC-DC converters play an important role. This work is intended to explore the use of the Landsman converter for EV battery charging. The motivation behind the work is explained, and the state of the art of EV battery charging research is reviewed to show the research gap. The operation of the proposed converter is explained in detail with its dynamic loop and nodal equations. The gain equation and the memory elements design are detailed with the corresponding equations. The model of the proposed converter is provided with the loss calculation. The system is simulated on MATLAB R2022b software, and the simulation results are provided to show the behavior of voltage and current in each component. To validate the design, a 100 W hardware prototype is implemented using discrete components to supply a resistive load, and to charge a 48 V Li-ion battery. The corresponding results are provided to validate the design and operation. A comparison of the proposed converters’ performance parameters with other non-isolated converters is provided. The topology is seen to provide an instantaneous efficiency of 88% during the prototype testing for resistive load, and an efficiency of 89.6% while charging a 48 V Li-ion battery, wherein, the efficiency magnitudes are the experimental values obtained through real-time measurements on Keysight IntegraVision Power Analyzer PA2203A, during the testing of the hardware prototype, as presented in section . Consequently, Landsman converter is seen to be an attractive converter for EV battery charging applications.

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

AbstractWe study the ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.

A replaceable-component method to construct single-degree-of-freedom multi-mode planar mechanisms with up to eight links

Abstract. The multi-mode planar mechanisms (MMPMs) are excellent-performance reconfigurable mechanisms, which not only inherit structural characteristics of planar mechanisms but also have the multi-task, multi-working-condition application advantages of multi-mode mechanisms. However, lacking common bifurcation analysis and construction methods, their industrial application and development are seriously hindered. This paper presents a replaceable-component method to construct a set of single-degree-of-freedom (single-DOF) MMPMs based on the branch graphs of the corresponding planar mechanisms and the proposed multi-mode modules (MMMs). First, according to the established loop equations, all the kinematic information of the original planar mechanism is obtained by the branch graphs and singularity points using Maple. Then, compared to the relationship between the concepts of the branch and motion mode, the number and continuity of branches are taken as the index to identify the potential bifurcation and mode conversion ability for the corresponding planar mechanisms. Subsequently, the MMM is presented to help the planar mechanisms break the singularity positions to form the corresponding MMPMs, and the steps of constructing single-DOF MMPMs are summarized. Finally, a single-DOF Stephenson six-bar three-mode planar mechanism, a Watt six-bar three-mode planar mechanism, and an eight-bar four-mode planar mechanism are constructed for the first time, and the corresponding multi-mode motion analyses are made. The results can give the available configuration for the design of corresponding MMPMs. The proposed method will provide strong guidance for the configuration design of MMPMs.

Construction and Multi-Mode Motion Analysis of Single-Degree-of-Freedom Four-Bar Multi-Mode Planar Mechanisms Based on Singular Configuration

Abstract The traditional four-bar mechanism is renowned for its simple structure, dependable performance, and wide range of applications. The single-degree-of-freedom (DOF) four-bar multi-mode planar mechanism (MMPM) is a type of four-bar mechanism that not only has the structural characteristics of the traditional four-bar mechanisms but also can achieve multiple motion modes by changing its structure. It has the advantage of performing diverse functions while conserving resources, which opens up new possibilities for research and application of the four-bar mechanism. However, due to the lack of a systematic configuration construction method, the design and application of single-DOF four-bar MMPMs are seriously limited. This paper presents a systematic method to construct a set of single-DOF four-bar MMPMs based on the loop equations and the proposed multi-mode modules (MMMs). First, depending on the loop equations, the four-bar planar mechanism containing two branches is identified by the corresponding branch graphs. Then, three kinds of MMMs are systematically proposed for the first time, helping the identified mechanism realize multiple motion modes. Subsequently, single-DOF four-bar MMPMs are constructed by replacing the specific component of the planar mechanism with the MMMs. Furthermore, the replacement rules of MMMs and the corresponding construction steps are summarized. Finally, 14 kinds of single-DOF four-bar MMPMs are listed, and the corresponding multi-mode motion analysis is discussed at the end of this paper. The proposed method is a straightforward one, which will provide great convenience for the configuration design of single-DOF four-bar MMPMs and promote the development and application of MMPMs.

A new derivation of the finite N master loop equation for lattice Yang-Mills

A new derivation of the finite N master loop equation for lattice Yang-Mills

Expanding the Fourier Transform of the Scaled Circular Jacobi beta Ensemble Density

The family of circular Jacobi beta ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding N rightarrow infty bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobibeta ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for beta = 2 and beta = 4, and the loop equation hierarchy. The polynomials in the variable u=2/beta which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular beta ensemble, specifically in relation to the zeros lying on the unit circle |u|=1 and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.

To the Theory of Decaying Turbulence

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d>2. This solution family is equivalent to a fractal curve in complex space Cd with random steps parametrized by N Ising variables σi=±1, in addition to a rational number pq and an integer winding number r, related by ∑σi=qr. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in Rd space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N→∞ or its chemical potential μ→0. The system with fixed N has different asymptotics at odd and even N→∞, but the limit μ→0 is well defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν/μ2 in the continuum limit μ→0, leading to anomalous dissipation at μ∝ν→0. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as t−λ, with a continuous spectrum of indexes λ in the local limit μ→0. The spectrum is determined by a resolvent, which is represented as an infinite product of 3⊗3 matrices depending of the element of the Euler ensemble.

Loop equations and a proof of Zvonkine's $qr$-ELSV formula

Loop equations and a proof of Zvonkine's $qr$-ELSV formula

Energy Balance Control of Energy Storage System Based on Improved Virtual DC Motor

Energy storage units have a big role in microgrids. To enhance the inertia of the DC microgrid while achieving energy balancing of each energy storage system, an energy balancing control of the energy storage system with virtual DC motor characteristics is proposed. By adding the VDCM technique to the traditional constant voltage control and adding the SoC information of the respective energy storage system to the virtual armature induction potential of the armature loop equation in the VDCM control, compared to existing controls, this control method can enhance DC microgrid inertia while balancing the SoC of the energy storage system, as well as charging and discharging mode switching without changing the control strategy. Finally, a multi-storage DC microgrid model is built in Matlab/Simulink, and the feasibility of the method is verified by simulation.

Degree zero Gromov–Witten invariants for smooth curves

AbstractFor a smooth projective curve, we derive a closed formula for the generating series of its Gromov–Witten invariants in genus and degree zero. It is known that the calculation of these invariants can be reduced to that of the and integrals on the moduli space of stable algebraic curves. The closed formula for the integrals is given by the conjecture, proved by Faber and Pandharipande. We compute in this paper the integrals via solving the degree zero limit of the loop equation associated to the complex projective line.

The Kinematic Investigation of the Stephenson-III Spherical Mechanism with a Spherical Slider Containing a Spherical Prismatic Pair

Multi-loop spherical mechanisms are extremely beneficial for creating versatile mechanical devices, including robotic joints and surgical tools, since multi-loop spherical mechanisms possess unique capabilities to operate in spatial situations with relatively simple movement. Nevertheless, the research on multi-loop spherical mechanisms with spherical sliders containing spherical prismatic pairs is lacking. Therefore, the main innovation of this paper is to propose the Stephenson-III two-loop spherical mechanism that possesses a spherical slider containing a spherical prismatic pair and to analyze the proposed spherical mechanism’s motion characteristics. An algebraic approach was employed to obtain the branch graphs of the proposed spherical mechanism with a spherical slider. The branch graphs were categorized into two types, according to whether branch points existed. With the algebraic approach, loop equations of the two spherical kinematic chains inside the proposed spherical mechanism were established to identify the input–output curves and singularity curves, with which the branch graphs were obtained. With the branch graphs, the joint rotation spaces (JRSs) of the proposed mechanism were recognized and so were the dead center positions, branches, sub-branches, and branch points. The results from the mathematical analysis were simulated and verified by three-dimensional (3D) models of the proposed spherical mechanism. The analytical results demonstrate that the spherical prismatic pair diversifies the motion of the proposed spherical mechanism by producing rotational sliding movement, which can cover the entire circumference of a specific greater circle on the proposed mechanism’s sphere.

Comprehensive fault reporting for three-terminal transmission line using adaptive estimation of line parameters

In this study, the post-fault synchronized phasors acquired from the local phasor measurement units (PMUs) are used to provide comprehensive fault reporting in Three terminal transmission line system (TTTL). The reporting data incorporates the classification, section identification, and localization of the fault based on the post-fault measurements of positive, negative, and zero sequence quantities. For accurate reporting, a proposed algorithm computes transmission line parameters, on hourly basis, to account for fluctuations in characteristics caused by temperature influences arising from either the loading factor or the climatic vagaries. Upon fault occurrence, the algorithm firstly determines fault classification and faulty section. Thereafter, localization of symmetrical faults is achieved via the positive sequence quantities to solve the positive sequence fault loop equation while that of asymmetrical faults is obtained through the negative sequence quantities to solve the negative sequence fault loop equation. The fault localization speed of the reporting proposed algorithm is attributed to the fact that synchronized data are obtained after one and half cycles from the fault onset. PSCAD/EMTDC platform is dedicated for time-domain investigations, while the proposed algorithm is carried out in MATLAB. The proposed algorithm was tested against 1537 fault characteristics at different locations along the TTTL, including varying fault resistances, loading conditions, and network configurations, in addition to assessing its sensitivity to changes in the TTTL parameters. These tests confirmed the algorithm’s reliability to be used with confidence in a variety of scenarios as the only 13 tests were observed with an error of estimation greater than 1%.

Parasitic Motions of 3-PRS Parallel Mechanisms with Two Different Branch Chain Arrangements

The kinematics of 3-PRS parallel mechanisms (PMs), an important category of 2R1T PMs, is very important for the motion control and error compensation of PMs. In order to produce a spindle head that could meet the requirements of various machining accuracy levels, a definite kinematics analysis of the PMs par should be taken with caution firstly. Here, the mathematical models of two 3-PRS PMs with different branch chain arrangements were established using the Euler angle. The kinematic equations of two 3-PRS PMs were obtained by establishing the closed-vector loop equation according to structural conditions. Parasitic motions that would cause a great effect were obtained, and their variations with structural and kinematic parameters were also obtained through MATLAB. Since the relationship between parasitic motions, and the given structural and kinematic parameters, is inextricable, an analysis is performed for the purpose of reducing or eliminating parasitic motion to ensure the accuracy of end-effector movement. The caused effect of parasitic motion can be improved with scale optimization and precision compensation. Lastly, two improved 3-PRS PMs without the parasitic motion of different branch chain arrangements used as the spindle head were obtained. Parasitic motion was related to the mechanism configuration and could be avoided in the configuration design stage with the comprehensive optimization of the structural and kinematic parameters, which provides a reference to obtain a sufficient workspace and improve the precision of traditional five-axis CNC machine tools.

Complete Solution of the LSZ Model via Topological Recursion

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials $\omega_{g,n}$ of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fieldss) and their consequences for the resulting particular type of topological recursion that governs the models.